Subgroups — C₆₇×#14053 #30693304

Order 17152.

1838 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#391×C₆₇	C₂	1	2	yes	maximal
3	C₂×#389×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1446	C₂	1	2	yes	maximal
6	C₆₇×#1441	C₂	1	2	yes	maximal
7	C₆₇×#1436	C₂	1	2	yes	maximal
8	C₆₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
9	C₆₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
10	C₆₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
11	C₄×#95×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₆₇	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₆₇	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₆₇	C₄	1	8	yes	complemented
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
26	#391×C₆₇	—	2	8	no	—
27	#389×C₆₇	—	2	4	no	—
28	C₄×C₄⋊C₄×C₆₇	—	2	4	no	—
29	C₂×#95×C₆₇	C₂²	1	2	yes	—
30	C₂×#95×C₆₇	C₂²	1	2	yes	—
31	C₂×#95×C₆₇	C₂²	1	2	yes	—
32	C₂×#95×C₆₇	C₂²	1	2	yes	—
33	C₂×#95×C₆₇	C₂²	1	2	yes	—
34	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#95×C₆₇	C₂²	1	2	yes	—
37	C₂×#95×C₆₇	C₂²	1	2	yes	—
38	C₂×#95×C₆₇	C₂²	1	2	yes	—
39	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₆₇	—	2	4	no	—
42	C₂²×Q₈×C₆₇	—	2	4	no	—
43	C₂²×Q₈×C₆₇	—	2	4	no	—
44	C₂²×Q₈×C₆₇	—	2	4	no	—
45	C₂²×Q₈×C₆₇	—	4	2	no	—
46	C₂²×Q₈×C₆₇	—	4	2	no	—
47	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₆₇	—	2	32	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
72	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
75	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
77	C₂×C₄⋊C₄×C₆₇	Q₈	1	4	yes	complemented
78	C₂×C₄²×C₆₇	—	2	2	no	—
79	C₂×C₄²×C₆₇	—	2	2	no	—
80	C₂×C₄²×C₆₇	—	2	2	no	—
81	C₂×C₄²×C₆₇	—	2	2	no	—
82	C₂×C₄²×C₆₇	—	2	4	no	—
83	C₂×C₄²×C₆₇	—	2	2	no	—
84	C₂×C₄²×C₆₇	—	2	4	no	—
85	C₂×C₄²×C₆₇	—	2	2	no	—
86	C₂×C₄²×C₆₇	—	2	2	no	—
87	C₂×C₄²×C₆₇	—	4	2	no	—
88	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
89	C₂×C₄²×C₆₇	Q₈	1	4	yes	complemented
90	#95×C₆₇	—	2	4	no	—
91	#95×C₆₇	—	2	4	no	—
92	#95×C₆₇	—	2	4	no	—
93	#95×C₆₇	—	2	4	no	—
94	#95×C₆₇	—	2	4	no	—
95	#95×C₆₇	—	2	2	no	—
96	#95×C₆₇	—	2	2	no	—
97	#95×C₆₇	—	2	4	no	—
98	#95×C₆₇	—	4	2	no	—
99	#95×C₆₇	—	4	2	no	—
100	#95×C₆₇	D₄	1	4	yes	—
101	#95×C₆₇	Q₈	1	4	yes	complemented
102	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
103	C₂×Q₈×C₆₇	—	2	16	no	—
104	C₂×Q₈×C₆₇	—	2	16	no	—
105	C₂×Q₈×C₆₇	—	4	8	no	—
106	C₂×Q₈×C₆₇	—	4	8	no	—
107	C₂×Q₈×C₆₇	—	4	8	no	—
108	C₂×Q₈×C₆₇	—	4	8	no	—
109	C₂×Q₈×C₆₇	—	4	8	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	1	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	—	2	1	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	1	no	—
126	C₂²×C₄×C₆₇	—	2	1	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	1	no	—
151	C₂²×C₄×C₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	—	2	1	no	—
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	2	1	no	—
156	C₂²×C₄×C₆₇	—	2	2	no	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	2	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	2	2	no	—
163	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
164	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
165	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
170	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
177	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
178	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
183	C₂²×C₄×C₆₇	#41	1	2	yes	—
184	C₂²×C₄×C₆₇	#41	1	2	yes	—
185	C₂²×C₄×C₆₇	#41	1	2	yes	—
186	C₂²×C₄×C₆₇	#41	1	2	yes	—
187	C₄⋊C₄×C₆₇	—	2	16	no	—
188	C₄⋊C₄×C₆₇	—	2	8	no	—
189	C₄⋊C₄×C₆₇	—	2	8	no	—
190	C₄⋊C₄×C₆₇	—	4	8	no	—
191	C₄⋊C₄×C₆₇	—	4	4	no	—
192	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	16	yes	—
193	C₄²×C₆₇	—	2	8	no	—
194	C₄²×C₆₇	—	2	8	no	—
195	C₄²×C₆₇	—	4	8	no	—
196	C₄²×C₆₇	—	4	4	no	—
197	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
198	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
199	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
200	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
201	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
202	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
203	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
204	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
205	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
206	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
207	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
208	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
209	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
211	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
212	Q₈×C₆₇	—	4	32	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	4	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	2	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	4	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	4	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	4	1	no	—
253	C₂×C₄×C₆₇	—	4	1	no	—
254	C₂×C₄×C₆₇	—	4	2	no	—
255	C₂×C₄×C₆₇	—	4	2	no	—
256	C₂×C₄×C₆₇	—	4	1	no	—
257	C₂×C₄×C₆₇	—	4	1	no	—
258	C₂×C₄×C₆₇	—	4	2	no	—
259	C₂×C₄×C₆₇	—	4	2	no	—
260	C₂×C₄×C₆₇	—	4	2	no	—
261	C₂×C₄×C₆₇	—	4	1	no	—
262	C₂×C₄×C₆₇	—	4	1	no	—
263	C₂×C₄×C₆₇	—	4	2	no	—
264	C₂×C₄×C₆₇	—	4	2	no	—
265	C₂×C₄×C₆₇	—	4	1	no	—
266	C₂×C₄×C₆₇	—	4	2	no	—
267	C₂×C₄×C₆₇	—	4	1	no	—
268	C₂×C₄×C₆₇	—	4	1	no	—
269	C₂×C₄×C₆₇	—	4	1	no	—
270	C₂×C₄×C₆₇	—	4	1	no	—
271	C₂×C₄×C₆₇	—	4	1	no	—
272	C₂×C₄×C₆₇	—	4	2	no	—
273	C₂×C₄×C₆₇	—	4	2	no	—
274	C₂×C₄×C₆₇	—	4	2	no	—
275	C₂×C₄×C₆₇	—	4	2	no	—
276	C₂×C₄×C₆₇	—	4	2	no	—
277	C₂×C₄×C₆₇	—	4	2	no	—
278	C₂×C₄×C₆₇	—	4	2	no	—
279	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
280	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
281	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
282	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
283	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
284	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
285	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
286	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
287	C₂×C₄×C₆₇	#124	1	4	yes	—
288	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
289	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	complemented
290	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
291	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
292	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
293	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
294	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
295	C₂²×C₆₇	#518	1	1	yes	characteristic
296	C₂²×C₆₇	#519	1	1	yes	characteristic
297	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
298	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
299	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
300	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
301	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
302	C₂²×C₆₇	#532	1	1	yes	characteristic
303	C₂²×C₆₇	#533	1	1	yes	characteristic
304	C₂²×C₆₇	#533	1	1	yes	characteristic
305	C₂²×C₆₇	#542	1	1	yes	characteristic
306	C₂²×C₆₇	#542	1	1	yes	characteristic
307	C₂²×C₆₇	#544	1	1	yes	characteristic
308	C₂²×C₆₇	#544	1	1	yes	characteristic
309	C₂²×C₆₇	D₄²	1	1	yes	characteristic
310	C₂²×C₆₇	D₄²	1	1	yes	characteristic
311	C₂²×C₆₇	#546	1	1	yes	characteristic
312	C₂²×C₆₇	#546	1	1	yes	characteristic
313	C₂²×C₆₇	#547	1	1	yes	characteristic
314	C₂²×C₆₇	#548	1	1	yes	characteristic
315	C₂²×C₆₇	#548	1	1	yes	characteristic
316	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
317	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
318	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
319	C₂²×C₆₇	#554	1	1	yes	characteristic
320	C₂²×C₆₇	#554	1	1	yes	characteristic
321	C₂²×C₆₇	#555	1	1	yes	characteristic
322	C₂²×C₆₇	#556	1	1	yes	characteristic
323	C₂²×C₆₇	#557	1	1	yes	characteristic
324	C₂²×C₆₇	Q₈²	1	1	yes	characteristic
325	C₄×C₆₇	—	2	8	no	—
326	C₄×C₆₇	—	2	4	no	—
327	C₄×C₆₇	—	2	4	no	—
328	C₄×C₆₇	—	2	4	no	—
329	C₄×C₆₇	—	4	4	no	—
330	C₄×C₆₇	—	4	2	no	—
331	C₄×C₆₇	—	4	4	no	—
332	C₄×C₆₇	—	4	2	no	—
333	C₄×C₆₇	—	4	2	no	—
334	C₄×C₆₇	—	8	2	no	—
335	C₄×C₆₇	#391	1	8	yes	complemented
336	#14053	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
337	C₂×C₆₇	C₂×#391	1	1	yes	characteristic
338	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
339	C₂×C₆₇	#2331	1	1	yes	characteristic
340	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
341	C₂×C₆₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
342	C₂×C₆₇	#2365	1	1	yes	characteristic
343	C₂×C₆₇	#2369	1	1	yes	characteristic
344	C₂×C₆₇	#2439	1	1	yes	characteristic
345	C₂×C₆₇	#2440	1	1	yes	characteristic
346	C₂×C₆₇	#2451	1	1	yes	characteristic
347	C₂×C₆₇	#2452	1	1	yes	characteristic
348	C₂×C₆₇	#2491	1	1	yes	characteristic
349	C₂×C₆₇	#2492	1	1	yes	characteristic
350	C₂×C₆₇	#2550	1	1	yes	characteristic
351	C₂×C₆₇	#2551	1	1	yes	characteristic
352	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
353	C₂×#391	C₂×C₆₇	1	2	yes	—
354	C₂×#389	C₂×C₆₇	1	1	yes	characteristic
355	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
356	#1446	C₂×C₆₇	1	2	yes	—
357	#1441	C₂×C₆₇	1	2	yes	—
358	#1436	C₂×C₆₇	1	2	yes	—
359	C₄⋊#95	C₂×C₆₇	1	1	yes	characteristic
360	C₄⋊#95	C₂×C₆₇	1	1	yes	characteristic
361	C₄⋊#95	C₂×C₆₇	1	1	yes	characteristic
362	C₄×#95	C₂×C₆₇	1	1	yes	characteristic
363	C₆₇	#14053	1	1	yes	characteristic, direct product factor, Sylow
364	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
365	C₂×C₄⋊Q₈	—	2	4	no	—
366	C₂×C₄⋊Q₈	—	2	4	no	—
367	C₂×C₄⋊Q₈	C₄×C₆₇	1	8	yes	complemented
368	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
369	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
370	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
371	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
372	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
373	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
374	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
375	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
376	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
377	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
378	#391	—	2	8	no	—
379	#389	—	2	4	no	—
380	C₄×C₄⋊C₄	—	2	4	no	—
381	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
382	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
383	C₂×#95	C₂²×C₆₇	1	2	yes	—
384	C₂×#95	C₂²×C₆₇	1	2	yes	—
385	C₂×#95	C₂²×C₆₇	1	2	yes	—
386	C₂×#95	C₂²×C₆₇	1	2	yes	—
387	C₂×#95	C₂²×C₆₇	1	2	yes	—
388	C₂×#95	C₂²×C₆₇	1	2	yes	—
389	C₂×#95	C₂²×C₆₇	1	2	yes	—
390	C₂×#95	C₂²×C₆₇	1	2	yes	—
391	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
392	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
393	C₂²×Q₈	—	2	4	no	—
394	C₂²×Q₈	—	2	4	no	—
395	C₂²×Q₈	—	2	4	no	—
396	C₂²×Q₈	—	2	4	no	—
397	C₂²×Q₈	—	4	2	no	—
398	C₂²×Q₈	—	4	2	no	—
399	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	—
400	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
401	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
402	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
403	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
404	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
405	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
406	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
407	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
408	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
409	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
410	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
411	C₄⋊Q₈	—	2	32	no	—
412	C₂×C₄⋊C₄	—	2	4	no	—
413	C₂×C₄⋊C₄	—	2	4	no	—
414	C₂×C₄⋊C₄	—	2	2	no	—
415	C₂×C₄⋊C₄	—	2	2	no	—
416	C₂×C₄⋊C₄	—	2	2	no	—
417	C₂×C₄⋊C₄	—	2	2	no	—
418	C₂×C₄⋊C₄	—	2	4	no	—
419	C₂×C₄⋊C₄	—	2	2	no	—
420	C₂×C₄⋊C₄	—	2	4	no	—
421	C₂×C₄⋊C₄	—	2	2	no	—
422	C₂×C₄⋊C₄	—	2	2	no	—
423	C₂×C₄⋊C₄	—	4	2	no	—
424	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
425	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
426	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
427	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
428	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
429	C₂×C₄⋊C₄	Q₈×C₆₇	1	4	yes	complemented
430	C₂×C₄²	—	2	2	no	—
431	C₂×C₄²	—	2	2	no	—
432	C₂×C₄²	—	2	2	no	—
433	C₂×C₄²	—	2	2	no	—
434	C₂×C₄²	—	2	4	no	—
435	C₂×C₄²	—	2	2	no	—
436	C₂×C₄²	—	2	4	no	—
437	C₂×C₄²	—	2	2	no	—
438	C₂×C₄²	—	2	2	no	—
439	C₂×C₄²	—	4	2	no	—
440	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
441	C₂×C₄²	Q₈×C₆₇	1	4	yes	complemented
442	#95	—	2	4	no	—
443	#95	—	2	4	no	—
444	#95	—	2	4	no	—
445	#95	—	2	4	no	—
446	#95	—	2	4	no	—
447	#95	—	2	2	no	—
448	#95	—	2	2	no	—
449	#95	—	2	4	no	—
450	#95	—	4	2	no	—
451	#95	—	4	2	no	—
452	#95	D₄×C₆₇	1	4	yes	—
453	#95	Q₈×C₆₇	1	4	yes	complemented
454	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
455	C₂×Q₈	—	2	16	no	—
456	C₂×Q₈	—	2	16	no	—
457	C₂×Q₈	—	4	8	no	—
458	C₂×Q₈	—	4	8	no	—
459	C₂×Q₈	—	4	8	no	—
460	C₂×Q₈	—	4	8	no	—
461	C₂×Q₈	—	4	8	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	2	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	2	2	no	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	2	2	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	2	no	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	2	2	no	—
511	C₂²×C₄	—	2	2	no	—
512	C₂²×C₄	—	2	2	no	—
513	C₂²×C₄	—	2	2	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
516	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
517	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
518	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
519	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
520	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
521	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
522	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
523	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
524	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
525	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
526	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
527	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
528	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
529	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
530	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
531	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
532	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
533	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
534	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
535	C₂²×C₄	#41×C₆₇	1	2	yes	—
536	C₂²×C₄	#41×C₆₇	1	2	yes	—
537	C₂²×C₄	#41×C₆₇	1	2	yes	—
538	C₂²×C₄	#41×C₆₇	1	2	yes	—
539	C₄⋊C₄	—	2	16	no	—
540	C₄⋊C₄	—	2	8	no	—
541	C₄⋊C₄	—	2	8	no	—
542	C₄⋊C₄	—	4	8	no	—
543	C₄⋊C₄	—	4	4	no	—
544	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	16	yes	—
545	C₄²	—	2	8	no	—
546	C₄²	—	2	8	no	—
547	C₄²	—	4	8	no	—
548	C₄²	—	4	4	no	—
549	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
550	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
551	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
552	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
553	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
554	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
555	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
556	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
557	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
558	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
559	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
560	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
561	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
562	C₂³	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
563	C₂³	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
564	Q₈	—	4	32	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	4	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	4	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	2	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	2	no	—
612	C₂×C₄	—	4	2	no	—
613	C₂×C₄	—	4	2	no	—
614	C₂×C₄	—	4	2	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	2	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	2	no	—
625	C₂×C₄	—	4	2	no	—
626	C₂×C₄	—	4	2	no	—
627	C₂×C₄	—	4	2	no	—
628	C₂×C₄	—	4	2	no	—
629	C₂×C₄	—	4	2	no	—
630	C₂×C₄	—	4	2	no	—
631	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
632	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
633	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
634	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
635	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
636	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
637	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
638	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
639	C₂×C₄	#124×C₆₇	1	4	yes	—
640	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
641	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	complemented
642	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
643	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
644	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
645	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
646	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
647	C₂²	#518×C₆₇	1	1	yes	characteristic
648	C₂²	#519×C₆₇	1	1	yes	characteristic
649	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
650	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
651	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
652	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
653	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
654	C₂²	#532×C₆₇	1	1	yes	characteristic
655	C₂²	#533×C₆₇	1	1	yes	characteristic
656	C₂²	#533×C₆₇	1	1	yes	characteristic
657	C₂²	#542×C₆₇	1	1	yes	characteristic
658	C₂²	#542×C₆₇	1	1	yes	characteristic
659	C₂²	#544×C₆₇	1	1	yes	characteristic
660	C₂²	#544×C₆₇	1	1	yes	characteristic
661	C₂²	D₄²×C₆₇	1	1	yes	characteristic
662	C₂²	D₄²×C₆₇	1	1	yes	characteristic
663	C₂²	#546×C₆₇	1	1	yes	characteristic
664	C₂²	#546×C₆₇	1	1	yes	characteristic
665	C₂²	#547×C₆₇	1	1	yes	characteristic
666	C₂²	#548×C₆₇	1	1	yes	characteristic
667	C₂²	#548×C₆₇	1	1	yes	characteristic
668	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
669	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
670	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
671	C₂²	#554×C₆₇	1	1	yes	characteristic
672	C₂²	#554×C₆₇	1	1	yes	characteristic
673	C₂²	#555×C₆₇	1	1	yes	characteristic
674	C₂²	#556×C₆₇	1	1	yes	characteristic
675	C₂²	#557×C₆₇	1	1	yes	characteristic
676	C₂²	Q₈²×C₆₇	1	1	yes	characteristic
677	C₄	—	2	8	no	—
678	C₄	—	2	4	no	—
679	C₄	—	2	4	no	—
680	C₄	—	2	4	no	—
681	C₄	—	4	4	no	—
682	C₄	—	4	2	no	—
683	C₄	—	4	4	no	—
684	C₄	—	4	2	no	—
685	C₄	—	4	2	no	—
686	C₄	—	8	2	no	—
687	C₄	#391×C₆₇	1	8	yes	complemented
688	C₂	C₂×#391×C₆₇	1	1	yes	characteristic
689	C₂	C₄×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
690	C₂	C₆₇×#2331	1	1	yes	characteristic
691	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
692	C₂	Q₈×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
693	C₂	C₆₇×#2365	1	1	yes	characteristic
694	C₂	C₆₇×#2369	1	1	yes	characteristic
695	C₂	C₆₇×#2439	1	1	yes	characteristic
696	C₂	C₆₇×#2440	1	1	yes	characteristic
697	C₂	C₆₇×#2451	1	1	yes	characteristic
698	C₂	C₆₇×#2452	1	1	yes	characteristic
699	C₂	C₆₇×#2491	1	1	yes	characteristic
700	C₂	C₆₇×#2492	1	1	yes	characteristic
701	C₂	C₆₇×#2550	1	1	yes	characteristic
702	C₂	C₆₇×#2551	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂⁴	C₂⁴×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
15. C₂×C₄⋊Q₈×C₆₇	C₄	semidirect
77. C₂×C₄⋊C₄×C₆₇	Q₈	semidirect
89. C₂×C₄²×C₆₇	Q₈	semidirect
101. #95×C₆₇	Q₈	semidirect
289. C₂×C₄×C₆₇	C₄⋊Q₈	semidirect
335. C₄×C₆₇	#391	semidirect
336. #14053	C₆₇	direct (N × H)
363. C₆₇	#14053	direct (N × H)
367. C₂×C₄⋊Q₈	C₄×C₆₇	semidirect
429. C₂×C₄⋊C₄	Q₈×C₆₇	semidirect
441. C₂×C₄²	Q₈×C₆₇	semidirect
453. #95	Q₈×C₆₇	semidirect
641. C₂×C₄	C₄⋊Q₈×C₆₇	semidirect
687. C₄	#391×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄⋊Q₈×C₆₇	2
C₂×#391×C₆₇	2
C₂×#389×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1446	2
C₆₇×#1441	2
C₆₇×#1436	2
C₆₇×C₄⋊#95	2
C₄×#95×C₆₇	2
#14053	67

Subgroups — C67×#14053 #30693304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#14053 #30693304