Subgroups — C₆₇×#22317 #30701568

Order 17152.

758 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1984	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1923	C₂	1	2	yes	maximal
4	C₆₇×#1923	C₂	1	2	yes	maximal
5	C₆₇×#1916	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1916	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1817	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	#391×C₆₇	C₄	1	1	yes	characteristic
23	#389×C₆₇	C₄	1	1	yes	characteristic
24	#387×C₆₇	C₄	1	2	yes	—
25	#385×C₆₇	C₄	1	2	yes	—
26	#380×C₆₇	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	#336×C₆₇	C₂²	1	2	yes	—
31	#336×C₆₇	C₂²	1	2	yes	—
32	#336×C₆₇	C₂²	1	2	yes	—
33	#336×C₆₇	C₂²	1	2	yes	—
34	#336×C₆₇	C₂²	1	2	yes	—
35	#336×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	2	yes	—
37	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂²⋊Q₈×C₆₇	C₈	1	8	yes	complemented
46	C₄×Q₈×C₆₇	—	4	1	no	—
47	C₄×Q₈×C₆₇	—	4	1	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₇	—	4	1	no	—
57	C₂×C₄²×C₆₇	—	4	1	no	—
58	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
64	C₄⋊C₈×C₆₇	—	2	4	no	—
65	C₄⋊C₈×C₆₇	—	4	2	no	—
66	C₂²⋊C₈×C₆₇	—	2	4	no	—
67	C₂²⋊C₈×C₆₇	—	4	1	no	—
68	C₂²⋊C₈×C₆₇	—	4	1	no	—
69	C₄×C₈×C₆₇	—	2	4	no	—
70	C₄×C₈×C₆₇	—	4	1	no	—
71	C₄×C₈×C₆₇	—	4	1	no	—
72	C₄×C₈×C₆₇	—	4	2	no	—
73	#95×C₆₇	C₂×C₄	1	2	yes	—
74	#95×C₆₇	C₂×C₄	1	2	yes	—
75	#95×C₆₇	C₂×C₄	1	2	yes	—
76	#95×C₆₇	C₂×C₄	1	2	yes	—
77	#95×C₆₇	C₂×C₄	1	2	yes	—
78	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Q₈×C₆₇	—	4	1	no	—
80	C₂×Q₈×C₆₇	—	4	1	no	—
81	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	2	1	no	—
85	C₂²×C₄×C₆₇	—	2	1	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	4	1	no	—
88	C₂²×C₄×C₆₇	—	4	1	no	—
89	C₂²×C₄×C₆₇	—	4	1	no	—
90	C₂²×C₄×C₆₇	—	4	1	no	—
91	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
92	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	#41	1	4	yes	—
114	C₄⋊C₄×C₆₇	—	2	4	no	—
115	C₄⋊C₄×C₆₇	—	4	1	no	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₄⋊C₄×C₆₇	—	4	2	no	—
118	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
119	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
120	C₂²⋊C₄×C₆₇	—	4	2	no	—
121	C₂²⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
122	C₄²×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	1	no	—
126	C₄²×C₆₇	—	2	1	no	—
127	C₄²×C₆₇	—	2	1	no	—
128	C₄²×C₆₇	—	2	1	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	4	1	no	—
131	C₄²×C₆₇	—	4	1	no	—
132	C₂³×C₆₇	—	2	1	no	—
133	C₂³×C₆₇	—	2	1	no	—
134	C₂³×C₆₇	—	2	1	no	—
135	C₂³×C₆₇	—	2	1	no	—
136	C₂³×C₆₇	—	2	1	no	—
137	C₂³×C₆₇	—	2	1	no	—
138	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
139	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
140	Q₈×C₆₇	—	4	4	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	4	4	no	—
164	C₂×C₄×C₆₇	—	4	4	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
168	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
169	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
170	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
171	C₂×C₄×C₆₇	#131	1	2	yes	—
172	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
173	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
174	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
176	C₈×C₆₇	—	2	4	no	—
177	C₈×C₆₇	—	4	2	no	—
178	C₈×C₆₇	—	4	2	no	—
179	C₈×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	2	2	no	—
181	C₂²×C₆₇	—	2	2	no	—
182	C₂²×C₆₇	—	2	2	no	—
183	C₂²×C₆₇	—	4	1	no	—
184	C₂²×C₆₇	—	4	1	no	—
185	C₂²×C₆₇	—	4	1	no	—
186	C₂²×C₆₇	—	4	1	no	—
187	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
188	C₂²×C₆₇	#518	1	1	yes	characteristic
189	C₂²×C₆₇	#519	1	1	yes	characteristic
190	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
191	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
192	C₂²×C₆₇	#568	1	1	yes	characteristic
193	C₂²×C₆₇	#568	1	1	yes	characteristic
194	C₄×C₆₇	—	2	4	no	—
195	C₄×C₆₇	—	2	2	no	—
196	C₄×C₆₇	—	2	2	no	—
197	C₄×C₆₇	—	2	4	no	—
198	C₄×C₆₇	—	2	2	no	—
199	C₄×C₆₇	—	4	1	no	—
200	C₄×C₆₇	—	4	2	no	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	—	4	1	no	—
203	C₄×C₆₇	#539	1	2	yes	—
204	C₄×C₆₇	#542	1	2	yes	—
205	#22317	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂×C₆₇	—	4	2	no	—
207	C₂×C₆₇	#2332	1	1	yes	characteristic
208	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
209	C₂×C₆₇	#2967	1	1	yes	characteristic
210	C₂×C₆₇	#2972	1	1	yes	characteristic
211	C₂×C₆₇	#2976	1	1	yes	characteristic
212	C₂×C₆₇	#2982	1	1	yes	characteristic
213	C₂×C₆₇	#2985	1	1	yes	characteristic
214	C₄×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
215	#1984	C₂×C₆₇	1	2	yes	complemented
216	#1923	C₂×C₆₇	1	2	yes	—
217	#1923	C₂×C₆₇	1	2	yes	—
218	#1916	C₂×C₆₇	1	2	yes	complemented
219	#1916	C₂×C₆₇	1	2	yes	complemented
220	#1817	C₂×C₆₇	1	1	yes	characteristic
221	C₈×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
222	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
223	C₆₇	#22317	1	1	yes	characteristic, direct product factor, Sylow
224	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
225	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
226	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
228	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
229	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
230	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
232	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
233	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
235	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
236	#391	C₄×C₆₇	1	1	yes	characteristic
237	#389	C₄×C₆₇	1	1	yes	characteristic
238	#387	C₄×C₆₇	1	2	yes	—
239	#385	C₄×C₆₇	1	2	yes	—
240	#380	C₄×C₆₇	1	1	yes	characteristic
241	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
242	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
243	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
244	#336	C₂²×C₆₇	1	2	yes	—
245	#336	C₂²×C₆₇	1	2	yes	—
246	#336	C₂²×C₆₇	1	2	yes	—
247	#336	C₂²×C₆₇	1	2	yes	—
248	#336	C₂²×C₆₇	1	2	yes	—
249	#336	C₂²×C₆₇	1	2	yes	—
250	#336	C₂²×C₆₇	1	2	yes	—
251	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
252	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
253	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
254	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
255	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
256	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
257	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
258	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
259	C₂²⋊Q₈	C₈×C₆₇	1	8	yes	complemented
260	C₄×Q₈	—	4	1	no	—
261	C₄×Q₈	—	4	1	no	—
262	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
263	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
265	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
267	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
268	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
269	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
270	C₂×C₄²	—	4	1	no	—
271	C₂×C₄²	—	4	1	no	—
272	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
273	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
276	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
277	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
278	C₄⋊C₈	—	2	4	no	—
279	C₄⋊C₈	—	4	2	no	—
280	C₂²⋊C₈	—	2	4	no	—
281	C₂²⋊C₈	—	4	1	no	—
282	C₂²⋊C₈	—	4	1	no	—
283	C₄×C₈	—	2	4	no	—
284	C₄×C₈	—	4	1	no	—
285	C₄×C₈	—	4	1	no	—
286	C₄×C₈	—	4	2	no	—
287	#95	C₂×C₄×C₆₇	1	2	yes	—
288	#95	C₂×C₄×C₆₇	1	2	yes	—
289	#95	C₂×C₄×C₆₇	1	2	yes	—
290	#95	C₂×C₄×C₆₇	1	2	yes	—
291	#95	C₂×C₄×C₆₇	1	2	yes	—
292	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
293	C₂×Q₈	—	4	1	no	—
294	C₂×Q₈	—	4	1	no	—
295	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	4	1	no	—
302	C₂²×C₄	—	4	1	no	—
303	C₂²×C₄	—	4	1	no	—
304	C₂²×C₄	—	4	1	no	—
305	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
306	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
308	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
310	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
312	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
313	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	#41×C₆₇	1	4	yes	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	—	4	2	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
333	C₄⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₂²⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	1	no	—
339	C₄²	—	2	1	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	1	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	4	1	no	—
345	C₄²	—	4	1	no	—
346	C₂³	—	2	1	no	—
347	C₂³	—	2	1	no	—
348	C₂³	—	2	1	no	—
349	C₂³	—	2	1	no	—
350	C₂³	—	2	1	no	—
351	C₂³	—	2	1	no	—
352	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
353	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
354	Q₈	—	4	4	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	4	4	no	—
378	C₂×C₄	—	4	4	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
382	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
383	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
384	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
385	C₂×C₄	#131×C₆₇	1	2	yes	—
386	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
387	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
388	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
389	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
390	C₈	—	2	4	no	—
391	C₈	—	4	2	no	—
392	C₈	—	4	2	no	—
393	C₈	—	4	2	no	—
394	C₂²	—	2	2	no	—
395	C₂²	—	2	2	no	—
396	C₂²	—	2	2	no	—
397	C₂²	—	4	1	no	—
398	C₂²	—	4	1	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	1	no	—
401	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
402	C₂²	#518×C₆₇	1	1	yes	characteristic
403	C₂²	#519×C₆₇	1	1	yes	characteristic
404	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
405	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
406	C₂²	#568×C₆₇	1	1	yes	characteristic
407	C₂²	#568×C₆₇	1	1	yes	characteristic
408	C₄	—	2	4	no	—
409	C₄	—	2	2	no	—
410	C₄	—	2	2	no	—
411	C₄	—	2	4	no	—
412	C₄	—	2	2	no	—
413	C₄	—	4	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	1	no	—
416	C₄	—	4	2	no	—
417	C₄	#539×C₆₇	1	2	yes	—
418	C₄	#542×C₆₇	1	2	yes	—
419	C₂	—	4	2	no	—
420	C₂	C₆₇×#2332	1	1	yes	characteristic
421	C₂	C₈×#41×C₆₇	1	1	yes	characteristic
422	C₂	C₆₇×#2967	1	1	yes	characteristic
423	C₂	C₆₇×#2972	1	1	yes	characteristic
424	C₂	C₆₇×#2976	1	1	yes	characteristic
425	C₂	C₆₇×#2982	1	1	yes	characteristic
426	C₂	C₆₇×#2985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1984	C₂	semidirect
5. C₆₇×#1916	C₂	semidirect
6. C₆₇×#1916	C₂	semidirect
8. C₈×C₄⋊C₄×C₆₇	C₂	semidirect
45. C₂²⋊Q₈×C₆₇	C₈	semidirect
205. #22317	C₆₇	direct (N × H)
215. #1984	C₂×C₆₇	semidirect
218. #1916	C₂×C₆₇	semidirect
219. #1916	C₂×C₆₇	semidirect
221. C₈×C₄⋊C₄	C₂×C₆₇	semidirect
223. C₆₇	#22317	direct (N × H)
259. C₂²⋊Q₈	C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊Q₈×C₆₇	2
C₆₇×#1984	2
C₆₇×#1923	2
C₆₇×#1916	2
C₆₇×#1817	2
C₈×C₄⋊C₄×C₆₇	2
C₄×C₂²⋊C₈×C₆₇	2
#22317	67

Subgroups — C67×#22317 #30701568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22317 #30701568