Subgroups — C₆₇×#34276 #30713527

Order 17152.

2930 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#519×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2362	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2360	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2359	C₂	1	8	yes	maximal, complemented
7	C₆₇×#2327	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2326	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2317	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊Q₈×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₄×#124×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2297	C₂	1	1	yes	maximal, characteristic
13	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	#536×C₆₇	C₄	1	16	yes	complemented
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#124×C₆₇	C₂²	1	2	yes	—
20	C₂×#123×C₆₇	C₂²	1	4	yes	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
22	#520×C₆₇	—	2	8	no	—
23	#519×C₆₇	—	2	8	no	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
31	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#117×C₆₇	C₂²	1	4	yes	—
33	C₂×#117×C₆₇	C₂²	1	2	yes	—
34	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
38	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
39	#391×C₆₇	C₂²	1	2	yes	—
40	#391×C₆₇	C₂²	1	2	yes	—
41	#391×C₆₇	C₂²	1	4	yes	—
42	#391×C₆₇	C₂²	1	1	yes	characteristic
43	#391×C₆₇	C₂²	1	1	yes	characteristic
44	#390×C₆₇	C₂²	1	1	yes	characteristic
45	#390×C₆₇	C₂²	1	1	yes	characteristic
46	#389×C₆₇	C₂²	1	8	yes	—
47	#389×C₆₇	C₂²	1	4	yes	—
48	#388×C₆₇	C₂²	1	8	yes	—
49	#388×C₆₇	C₂²	1	8	yes	—
50	#387×C₆₇	C₂²	1	8	yes	—
51	#387×C₆₇	C₂²	1	8	yes	—
52	#386×C₆₇	C₂²	1	4	yes	—
53	#385×C₆₇	C₂²	1	4	yes	—
54	#385×C₆₇	C₂²	1	8	yes	—
55	#384×C₆₇	C₂²	1	2	yes	—
56	#382×C₆₇	C₂²	1	2	yes	—
57	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
58	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
59	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
60	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
61	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
62	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
63	#376×C₆₇	C₂²	1	2	yes	—
64	C₂×#95×C₆₇	C₂²	1	4	yes	—
65	C₄³×C₆₇	C₂²	1	2	yes	—
66	C₂×#41×C₆₇	—	2	4	no	—
67	C₂×#41×C₆₇	—	2	4	no	—
68	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
69	C₂²×Q₈×C₆₇	—	2	4	no	—
70	C₂²×Q₈×C₆₇	—	2	4	no	—
71	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
72	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
74	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
76	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
78	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₄⋊Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
84	#124×C₆₇	C₂×C₄	1	16	yes	complemented
85	#123×C₆₇	C₂×C₄	1	32	yes	complemented
86	C₂²⋊Q₈×C₆₇	C₂×C₄	1	32	yes	complemented
87	C₄×Q₈×C₆₇	—	2	8	no	—
88	C₄×Q₈×C₆₇	—	2	8	no	—
89	C₄×Q₈×C₆₇	—	2	8	no	—
90	C₄×Q₈×C₆₇	—	2	8	no	—
91	C₄×Q₈×C₆₇	—	2	8	no	—
92	C₄×D₄×C₆₇	—	2	16	no	—
93	C₄×D₄×C₆₇	—	2	8	no	—
94	#117×C₆₇	—	2	4	no	—
95	#117×C₆₇	—	2	8	no	—
96	#117×C₆₇	—	2	16	no	—
97	#117×C₆₇	C₂×C₄	1	8	yes	complemented
98	#117×C₆₇	D₄	1	8	yes	—
99	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
100	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
120	C₂×C₄²×C₆₇	—	2	4	no	—
121	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
124	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
127	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
128	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
129	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
130	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
132	#95×C₆₇	C₂³	1	8	yes	—
133	#95×C₆₇	C₂³	1	8	yes	—
134	#95×C₆₇	C₂³	1	4	yes	—
135	#95×C₆₇	C₂³	1	4	yes	—
136	#95×C₆₇	C₂³	1	4	yes	—
137	#95×C₆₇	C₂³	1	8	yes	—
138	#95×C₆₇	C₂³	1	4	yes	—
139	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
140	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
141	#41×C₆₇	—	2	32	no	—
142	C₂×Q₈×C₆₇	—	2	32	no	—
143	C₂×Q₈×C₆₇	—	2	4	no	—
144	C₂×Q₈×C₆₇	—	2	4	no	—
145	C₂×Q₈×C₆₇	—	2	2	no	—
146	C₂×Q₈×C₆₇	—	2	4	no	—
147	C₂×Q₈×C₆₇	—	2	4	no	—
148	C₂×Q₈×C₆₇	—	2	2	no	—
149	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
150	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
151	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	4	no	—
154	C₂×D₄×C₆₇	—	2	2	no	—
155	C₂×D₄×C₆₇	—	2	2	no	—
156	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
157	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	4	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	2	8	no	—
166	C₂²×C₄×C₆₇	—	4	2	no	—
167	C₂²×C₄×C₆₇	—	4	8	no	—
168	C₂²×C₄×C₆₇	—	4	4	no	—
169	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
170	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
171	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
172	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₆₇	#41	1	4	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
186	C₄⋊C₄×C₆₇	—	2	4	no	—
187	C₄⋊C₄×C₆₇	—	2	8	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	8	no	—
190	C₄⋊C₄×C₆₇	—	2	8	no	—
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	—	2	2	no	—
193	C₄⋊C₄×C₆₇	—	4	8	no	—
194	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
195	C₄⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
196	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
197	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
198	C₂²⋊C₄×C₆₇	—	2	4	no	—
199	C₂²⋊C₄×C₆₇	—	2	8	no	—
200	C₂²⋊C₄×C₆₇	—	2	8	no	—
201	C₂²⋊C₄×C₆₇	—	4	4	no	—
202	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
203	C₂²⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
204	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
205	C₄²×C₆₇	—	2	8	no	—
206	C₄²×C₆₇	—	2	8	no	—
207	C₄²×C₆₇	—	2	8	no	—
208	C₄²×C₆₇	—	2	4	no	—
209	C₄²×C₆₇	—	2	2	no	—
210	C₄²×C₆₇	—	2	4	no	—
211	C₄²×C₆₇	—	2	2	no	—
212	C₄²×C₆₇	—	2	8	no	—
213	C₄²×C₆₇	—	4	4	no	—
214	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
215	C₄²×C₆₇	C₂×D₄	1	8	yes	—
216	C₂³×C₆₇	—	2	2	no	—
217	C₂³×C₆₇	—	2	2	no	—
218	C₂³×C₆₇	—	2	2	no	—
219	C₂³×C₆₇	—	2	2	no	—
220	C₂³×C₆₇	—	2	4	no	—
221	C₂³×C₆₇	—	2	4	no	—
222	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
223	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
224	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
225	C₂³×C₆₇	C₂×#41	1	2	yes	—
226	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
227	Q₈×C₆₇	—	2	16	no	—
228	Q₈×C₆₇	—	2	16	no	—
229	Q₈×C₆₇	—	2	8	no	—
230	D₄×C₆₇	—	2	16	no	—
231	D₄×C₆₇	—	2	8	no	—
232	C₂×C₄×C₆₇	—	2	16	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	4	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	4	no	—
243	C₂×C₄×C₆₇	—	2	16	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	4	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	2	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	8	no	—
253	C₂×C₄×C₆₇	—	4	8	no	—
254	C₂×C₄×C₆₇	—	8	8	no	—
255	C₂×C₄×C₆₇	—	8	4	no	—
256	C₂×C₄×C₆₇	C₄×D₄	1	16	yes	—
257	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
258	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
259	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
260	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
261	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
262	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
267	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
268	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
269	C₂²×C₆₇	—	2	4	no	—
270	C₂²×C₆₇	—	2	4	no	—
271	C₂²×C₆₇	—	2	2	no	—
272	C₂²×C₆₇	—	2	2	no	—
273	C₂²×C₆₇	—	2	4	no	—
274	C₂²×C₆₇	—	2	8	no	—
275	C₂²×C₆₇	—	4	4	no	—
276	C₂²×C₆₇	—	4	2	no	—
277	C₂²×C₆₇	—	4	2	no	—
278	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
279	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
280	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
281	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
282	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
283	C₂²×C₆₇	#585	1	2	yes	—
284	C₄×C₆₇	—	2	4	no	—
285	C₄×C₆₇	—	2	8	no	—
286	C₄×C₆₇	—	2	8	no	—
287	C₄×C₆₇	—	2	4	no	—
288	C₄×C₆₇	—	2	4	no	—
289	C₄×C₆₇	—	2	4	no	—
290	C₄×C₆₇	—	4	2	no	—
291	C₄×C₆₇	—	4	2	no	—
292	C₄×C₆₇	—	4	2	no	—
293	C₄×C₆₇	—	4	4	no	—
294	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
295	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
296	#34276	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
297	C₂×C₆₇	—	2	4	no	—
298	C₂×C₆₇	—	4	4	no	—
299	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
300	C₂×C₆₇	C₄×2₋¹⁺⁴	1	2	yes	—
301	C₂×C₆₇	#3495	1	2	yes	—
302	C₂×C₆₇	#3499	1	1	yes	characteristic
303	C₂×C₆₇	#3500	1	1	yes	characteristic
304	C₂×#536	C₂×C₆₇	1	1	yes	characteristic
305	C₂×#520	C₂×C₆₇	1	1	yes	characteristic
306	C₂×#519	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#2362	C₂×C₆₇	1	4	yes	complemented
308	#2360	C₂×C₆₇	1	4	yes	complemented
309	#2359	C₂×C₆₇	1	8	yes	complemented
310	#2327	C₂×C₆₇	1	2	yes	complemented
311	#2326	C₂×C₆₇	1	2	yes	complemented
312	#2317	C₂×C₆₇	1	1	yes	characteristic, complemented
313	C₄×C₄⋊Q₈	C₂×C₆₇	1	2	yes	complemented
314	C₄×#124	C₂×C₆₇	1	2	yes	complemented
315	#2297	C₂×C₆₇	1	1	yes	characteristic
316	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
317	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
318	C₆₇	#34276	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
320	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
321	#536	C₄×C₆₇	1	16	yes	complemented
322	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
323	C₂×#124	C₂²×C₆₇	1	2	yes	—
324	C₂×#123	C₂²×C₆₇	1	4	yes	—
325	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
326	#520	—	2	8	no	—
327	#519	—	2	8	no	—
328	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
329	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
330	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
331	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
332	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
333	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
334	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
335	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
336	C₂×#117	C₂²×C₆₇	1	4	yes	—
337	C₂×#117	C₂²×C₆₇	1	2	yes	—
338	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
340	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
341	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
342	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
343	#391	C₂²×C₆₇	1	1	yes	characteristic
344	#391	C₂²×C₆₇	1	2	yes	—
345	#391	C₂²×C₆₇	1	2	yes	—
346	#391	C₂²×C₆₇	1	1	yes	characteristic
347	#391	C₂²×C₆₇	1	4	yes	—
348	#390	C₂²×C₆₇	1	1	yes	characteristic
349	#390	C₂²×C₆₇	1	1	yes	characteristic
350	#389	C₂²×C₆₇	1	8	yes	—
351	#389	C₂²×C₆₇	1	4	yes	—
352	#388	C₂²×C₆₇	1	8	yes	—
353	#388	C₂²×C₆₇	1	8	yes	—
354	#387	C₂²×C₆₇	1	8	yes	—
355	#387	C₂²×C₆₇	1	8	yes	—
356	#386	C₂²×C₆₇	1	4	yes	—
357	#385	C₂²×C₆₇	1	4	yes	—
358	#385	C₂²×C₆₇	1	8	yes	—
359	#384	C₂²×C₆₇	1	2	yes	—
360	#382	C₂²×C₆₇	1	2	yes	—
361	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
362	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
363	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
364	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
365	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
366	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
367	#376	C₂²×C₆₇	1	2	yes	—
368	C₂×#95	C₂²×C₆₇	1	4	yes	—
369	C₄³	C₂²×C₆₇	1	2	yes	—
370	C₂×#41	—	2	4	no	—
371	C₂×#41	—	2	4	no	—
372	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
373	C₂²×Q₈	—	2	4	no	—
374	C₂²×Q₈	—	2	4	no	—
375	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
376	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
377	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
378	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
379	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
380	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
381	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
382	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
383	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
384	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
385	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
386	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
387	C₄⋊Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
388	#124	C₂×C₄×C₆₇	1	16	yes	complemented
389	#123	C₂×C₄×C₆₇	1	32	yes	complemented
390	C₂²⋊Q₈	C₂×C₄×C₆₇	1	32	yes	complemented
391	C₄×Q₈	—	2	8	no	—
392	C₄×Q₈	—	2	8	no	—
393	C₄×Q₈	—	2	8	no	—
394	C₄×Q₈	—	2	8	no	—
395	C₄×Q₈	—	2	8	no	—
396	C₄×D₄	—	2	16	no	—
397	C₄×D₄	—	2	8	no	—
398	#117	—	2	4	no	—
399	#117	—	2	8	no	—
400	#117	—	2	16	no	—
401	#117	C₂×C₄×C₆₇	1	8	yes	complemented
402	#117	D₄×C₆₇	1	8	yes	—
403	C₂×C₄⋊C₄	—	2	4	no	—
404	C₂×C₄⋊C₄	—	2	8	no	—
405	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
415	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
417	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
424	C₂×C₄²	—	2	4	no	—
425	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
426	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
427	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
428	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
429	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
430	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
431	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
433	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
434	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
435	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
436	#95	C₂³×C₆₇	1	8	yes	—
437	#95	C₂³×C₆₇	1	8	yes	—
438	#95	C₂³×C₆₇	1	4	yes	—
439	#95	C₂³×C₆₇	1	4	yes	—
440	#95	C₂³×C₆₇	1	4	yes	—
441	#95	C₂³×C₆₇	1	8	yes	—
442	#95	C₂³×C₆₇	1	4	yes	—
443	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
444	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
445	#41	—	2	32	no	—
446	C₂×Q₈	—	2	32	no	—
447	C₂×Q₈	—	2	4	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	—	2	4	no	—
451	C₂×Q₈	—	2	4	no	—
452	C₂×Q₈	—	2	2	no	—
453	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
454	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
455	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
461	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	8	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	—	4	8	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
474	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
475	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
476	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
477	C₂²×C₄	#41×C₆₇	1	4	yes	—
478	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
480	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
483	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
484	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
485	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
487	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
488	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
489	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	8	no	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	8	no	—
494	C₄⋊C₄	—	2	8	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	4	8	no	—
498	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
499	C₄⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
500	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
501	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	8	no	—
504	C₂²⋊C₄	—	2	8	no	—
505	C₂²⋊C₄	—	4	4	no	—
506	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
507	C₂²⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
508	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
509	C₄²	—	2	8	no	—
510	C₄²	—	2	8	no	—
511	C₄²	—	2	8	no	—
512	C₄²	—	2	4	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	4	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	8	no	—
517	C₄²	—	4	4	no	—
518	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
519	C₄²	C₂×D₄×C₆₇	1	8	yes	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	2	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	4	no	—
526	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
527	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
528	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
529	C₂³	C₂×#41×C₆₇	1	2	yes	—
530	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
531	Q₈	—	2	16	no	—
532	Q₈	—	2	16	no	—
533	Q₈	—	2	8	no	—
534	D₄	—	2	16	no	—
535	D₄	—	2	8	no	—
536	C₂×C₄	—	2	16	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	16	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	8	no	—
557	C₂×C₄	—	4	8	no	—
558	C₂×C₄	—	8	8	no	—
559	C₂×C₄	—	8	4	no	—
560	C₂×C₄	C₄×D₄×C₆₇	1	16	yes	—
561	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
562	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
563	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
564	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
565	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
566	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
567	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
568	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
569	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
570	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
571	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
572	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
573	C₂²	—	2	4	no	—
574	C₂²	—	2	4	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	4	no	—
578	C₂²	—	2	8	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
583	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
584	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
585	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
586	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
587	C₂²	#585×C₆₇	1	2	yes	—
588	C₄	—	2	4	no	—
589	C₄	—	2	8	no	—
590	C₄	—	2	8	no	—
591	C₄	—	2	4	no	—
592	C₄	—	2	4	no	—
593	C₄	—	2	4	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	2	no	—
598	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
599	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
600	C₂	—	2	4	no	—
601	C₂	—	4	4	no	—
602	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
603	C₂	C₄×2₋¹⁺⁴×C₆₇	1	2	yes	—
604	C₂	C₆₇×#3495	1	2	yes	—
605	C₂	C₆₇×#3499	1	1	yes	characteristic
606	C₂	C₆₇×#3500	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#519×C₆₇	C₂	semidirect
4. C₆₇×#2362	C₂	semidirect
5. C₆₇×#2360	C₂	semidirect
6. C₆₇×#2359	C₂	semidirect
7. C₆₇×#2327	C₂	semidirect
8. C₆₇×#2326	C₂	semidirect
9. C₆₇×#2317	C₂	semidirect
10. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
11. C₄×#124×C₆₇	C₂	semidirect
13. C₂×#391×C₆₇	C₂	semidirect
14. C₄×#117×C₆₇	C₂	semidirect
17. #536×C₆₇	C₄	semidirect
71. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
83. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
84. #124×C₆₇	C₂×C₄	semidirect
85. #123×C₆₇	C₂×C₄	semidirect
86. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
97. #117×C₆₇	C₂×C₄	semidirect
296. #34276	C₆₇	direct (N × H)
306. C₂×#519	C₂×C₆₇	semidirect
307. #2362	C₂×C₆₇	semidirect
308. #2360	C₂×C₆₇	semidirect
309. #2359	C₂×C₆₇	semidirect
310. #2327	C₂×C₆₇	semidirect
311. #2326	C₂×C₆₇	semidirect
312. #2317	C₂×C₆₇	semidirect
313. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
314. C₄×#124	C₂×C₆₇	semidirect
316. C₂×#391	C₂×C₆₇	semidirect
317. C₄×#117	C₂×C₆₇	semidirect
318. C₆₇	#34276	direct (N × H)
321. #536	C₄×C₆₇	semidirect
375. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
387. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
388. #124	C₂×C₄×C₆₇	semidirect
389. #123	C₂×C₄×C₆₇	semidirect
390. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
401. #117	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#536×C₆₇	2
C₂×#520×C₆₇	2
C₂×#519×C₆₇	2
C₆₇×#2362	2
C₆₇×#2360	2
C₆₇×#2359	2
C₆₇×#2327	2
C₆₇×#2326	2
C₆₇×#2317	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#124×C₆₇	2
C₆₇×#2297	2
C₂×#391×C₆₇	2
C₄×#117×C₆₇	2
#34276	67

Subgroups — C67×#34276 #30713527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34276 #30713527