Subgroups — C₆₇×#26043 #30705294

Order 17152.

758 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#479×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2552	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2088	C₂	1	2	yes	maximal
5	C₆₇×#2059	C₂	1	2	yes	maximal
6	C₆₇×#1947	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1865	C₂	1	2	yes	maximal
9	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1803	C₂	1	1	yes	maximal, characteristic
11	C₄×#103×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#125×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#479×C₆₇	—	2	8	no	—
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×#106×C₆₇	C₂²	1	2	yes	—
18	C₂×#106×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#103×C₆₇	C₂²	1	2	yes	—
22	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#398×C₆₇	—	2	2	no	—
26	#389×C₆₇	—	2	2	no	—
27	#389×C₆₇	—	2	1	no	—
28	#389×C₆₇	C₂²	1	2	yes	—
29	#389×C₆₇	C₂²	1	2	yes	—
30	#384×C₆₇	—	2	1	no	—
31	#384×C₆₇	—	2	1	no	—
32	#384×C₆₇	C₂²	1	1	yes	characteristic
33	#384×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	#340×C₆₇	C₂²	1	1	yes	characteristic
38	#340×C₆₇	C₂²	1	1	yes	characteristic
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	#125×C₆₇	—	2	4	no	—
47	#125×C₆₇	—	4	2	no	—
48	C₄×Q₈×C₆₇	—	2	2	no	—
49	C₄×Q₈×C₆₇	—	2	4	no	—
50	C₄×Q₈×C₆₇	—	4	2	no	—
51	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	—	2	1	no	—
70	C₂×C₄²×C₆₇	—	2	2	no	—
71	C₂×C₄²×C₆₇	—	2	1	no	—
72	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	#107×C₆₇	—	2	4	no	—
75	#106×C₆₇	—	2	4	no	—
76	#106×C₆₇	—	4	2	no	—
77	C₄⋊C₈×C₆₇	—	2	4	no	—
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	#103×C₆₇	—	2	4	no	—
80	#103×C₆₇	D₄	1	4	yes	—
81	#103×C₆₇	Q₈	1	4	yes	—
82	C₄×C₈×C₆₇	—	2	2	no	—
83	#95×C₆₇	—	2	2	no	—
84	#95×C₆₇	—	2	1	no	—
85	#95×C₆₇	—	2	2	no	—
86	#95×C₆₇	D₄	1	2	yes	—
87	C₂×Q₈×C₆₇	—	2	1	no	—
88	C₂×Q₈×C₆₇	—	2	1	no	—
89	C₂×Q₈×C₆₇	—	2	1	no	—
90	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
91	C₂×Q₈×C₆₇	C₂×Q₈	1	2	yes	—
92	C₂×Q₈×C₆₇	C₂×Q₈	1	2	yes	—
93	C₂²×C₄×C₆₇	—	2	1	no	—
94	C₂²×C₄×C₆₇	—	2	1	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
104	C₂×C₈×C₆₇	—	2	1	no	—
105	C₂×C₈×C₆₇	—	2	1	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	—	2	1	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	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	1	no	—
117	C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	4	4	no	—
124	C₄⋊C₄×C₆₇	—	4	4	no	—
125	C₄⋊C₄×C₆₇	—	4	4	no	—
126	C₄⋊C₄×C₆₇	—	4	2	no	—
127	C₄⋊C₄×C₆₇	—	4	2	no	—
128	C₄⋊C₄×C₆₇	—	8	1	no	—
129	C₄⋊C₄×C₆₇	—	8	1	no	—
130	C₄⋊C₄×C₆₇	—	8	2	no	—
131	C₄⋊C₄×C₆₇	—	8	2	no	—
132	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
133	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
134	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
135	C₄²×C₆₇	—	2	4	no	—
136	C₄²×C₆₇	—	2	2	no	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	—	4	4	no	—
139	C₄²×C₆₇	—	4	2	no	—
140	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
141	C₄²×C₆₇	#41	1	2	yes	—
142	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	Q₈×C₆₇	—	2	2	no	—
144	Q₈×C₆₇	—	2	2	no	—
145	Q₈×C₆₇	—	2	2	no	—
146	Q₈×C₆₇	C₄⋊Q₈	1	4	yes	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	4	1	no	—
162	C₂×C₄×C₆₇	—	4	1	no	—
163	C₂×C₄×C₆₇	—	4	1	no	—
164	C₂×C₄×C₆₇	—	4	1	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
174	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
175	C₂×C₄×C₆₇	#135	1	2	yes	—
176	C₂×C₄×C₆₇	#137	1	2	yes	—
177	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
182	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
183	C₈×C₆₇	—	2	2	no	—
184	C₈×C₆₇	—	2	2	no	—
185	C₈×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
187	C₂²×C₆₇	D₄²	1	1	yes	characteristic
188	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
189	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
190	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
191	C₂²×C₆₇	#577	1	1	yes	characteristic
192	C₂²×C₆₇	#577	1	1	yes	characteristic
193	C₄×C₆₇	—	2	2	no	—
194	C₄×C₆₇	—	2	4	no	—
195	C₄×C₆₇	—	4	2	no	—
196	C₄×C₆₇	—	4	2	no	—
197	C₄×C₆₇	—	4	2	no	—
198	C₄×C₆₇	—	4	2	no	—
199	C₄×C₆₇	—	4	2	no	—
200	C₄×C₆₇	—	8	2	no	—
201	C₄×C₆₇	—	8	2	no	—
202	C₄×C₆₇	#479	1	4	yes	complemented
203	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
204	C₄×C₆₇	#544	1	1	yes	characteristic
205	C₄×C₆₇	#547	1	1	yes	characteristic
206	C₄×C₆₇	#554	1	1	yes	characteristic
207	#26043	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂×C₆₇	#2440	1	1	yes	characteristic
209	C₂×C₆₇	C₂×#479	1	1	yes	characteristic
210	C₂×C₆₇	#3084	1	1	yes	characteristic
211	C₂×C₆₇	#3278	1	1	yes	characteristic
212	C₂×C₆₇	#3284	1	1	yes	characteristic
213	C₂×C₆₇	#3381	1	1	yes	characteristic
214	C₂×C₆₇	#3387	1	1	yes	characteristic
215	C₂×#479	C₂×C₆₇	1	2	yes	—
216	#2552	C₂×C₆₇	1	1	yes	characteristic
217	Q₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
218	#2088	C₂×C₆₇	1	2	yes	—
219	#2059	C₂×C₆₇	1	2	yes	—
220	#1947	C₂×C₆₇	1	1	yes	characteristic
221	#1939	C₂×C₆₇	1	1	yes	characteristic
222	#1865	C₂×C₆₇	1	2	yes	—
223	#1844	C₂×C₆₇	1	1	yes	characteristic
224	#1803	C₂×C₆₇	1	1	yes	characteristic
225	C₄×#103	C₂×C₆₇	1	1	yes	characteristic
226	C₆₇	#26043	1	1	yes	characteristic, direct product factor, Sylow
227	C₂×#125	C₂²×C₆₇	1	2	yes	—
228	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
229	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
230	#479	—	2	8	no	—
231	C₂×#107	C₂²×C₆₇	1	2	yes	—
232	C₂×#106	C₂²×C₆₇	1	2	yes	—
233	C₂×#106	C₂²×C₆₇	1	2	yes	—
234	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
235	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×#103	C₂²×C₆₇	1	2	yes	—
237	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
240	#398	—	2	2	no	—
241	#389	—	2	1	no	—
242	#389	—	2	2	no	—
243	#389	C₂²×C₆₇	1	2	yes	—
244	#389	C₂²×C₆₇	1	2	yes	—
245	#384	—	2	1	no	—
246	#384	—	2	1	no	—
247	#384	C₂²×C₆₇	1	1	yes	characteristic
248	#384	C₂²×C₆₇	1	1	yes	characteristic
249	C₄×C₄⋊C₄	—	2	1	no	—
250	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
251	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
252	#340	C₂²×C₆₇	1	2	yes	—
253	#340	C₂²×C₆₇	1	1	yes	characteristic
254	#340	C₂²×C₆₇	1	1	yes	characteristic
255	#340	C₂²×C₆₇	1	2	yes	—
256	#336	C₂²×C₆₇	1	2	yes	—
257	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
259	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
261	#125	—	2	4	no	—
262	#125	—	4	2	no	—
263	C₄×Q₈	—	2	2	no	—
264	C₄×Q₈	—	2	4	no	—
265	C₄×Q₈	—	4	2	no	—
266	C₄×Q₈	Q₈×C₆₇	1	4	yes	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
278	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
284	C₂×C₄²	—	2	1	no	—
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	—	2	1	no	—
287	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
288	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
289	#107	—	2	4	no	—
290	#106	—	2	4	no	—
291	#106	—	4	2	no	—
292	C₄⋊C₈	—	2	4	no	—
293	C₄⋊C₈	—	2	2	no	—
294	#103	—	2	4	no	—
295	#103	D₄×C₆₇	1	4	yes	—
296	#103	Q₈×C₆₇	1	4	yes	—
297	C₄×C₈	—	2	2	no	—
298	#95	—	2	2	no	—
299	#95	—	2	1	no	—
300	#95	—	2	2	no	—
301	#95	D₄×C₆₇	1	2	yes	—
302	C₂×Q₈	—	2	1	no	—
303	C₂×Q₈	—	2	1	no	—
304	C₂×Q₈	—	2	1	no	—
305	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
306	C₂×Q₈	C₂×Q₈×C₆₇	1	2	yes	—
307	C₂×Q₈	C₂×Q₈×C₆₇	1	2	yes	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
318	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
327	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	1	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	4	4	no	—
339	C₄⋊C₄	—	4	4	no	—
340	C₄⋊C₄	—	4	4	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	—	8	1	no	—
344	C₄⋊C₄	—	8	1	no	—
345	C₄⋊C₄	—	8	2	no	—
346	C₄⋊C₄	—	8	2	no	—
347	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
348	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
349	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
350	C₄²	—	2	4	no	—
351	C₄²	—	2	2	no	—
352	C₄²	—	2	2	no	—
353	C₄²	—	4	4	no	—
354	C₄²	—	4	2	no	—
355	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
356	C₄²	#41×C₆₇	1	2	yes	—
357	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
358	Q₈	—	2	2	no	—
359	Q₈	—	2	2	no	—
360	Q₈	—	2	2	no	—
361	Q₈	C₄⋊Q₈×C₆₇	1	4	yes	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
389	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
390	C₂×C₄	#135×C₆₇	1	2	yes	—
391	C₂×C₄	#137×C₆₇	1	2	yes	—
392	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
393	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
394	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
395	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
396	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
397	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
398	C₈	—	2	2	no	—
399	C₈	—	2	2	no	—
400	C₈	—	4	2	no	—
401	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
402	C₂²	D₄²×C₆₇	1	1	yes	characteristic
403	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
404	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
405	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
406	C₂²	#577×C₆₇	1	1	yes	characteristic
407	C₂²	#577×C₆₇	1	1	yes	characteristic
408	C₄	—	2	2	no	—
409	C₄	—	2	4	no	—
410	C₄	—	4	2	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	2	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	2	no	—
415	C₄	—	8	2	no	—
416	C₄	—	8	2	no	—
417	C₄	#479×C₆₇	1	4	yes	complemented
418	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
419	C₄	#544×C₆₇	1	1	yes	characteristic
420	C₄	#547×C₆₇	1	1	yes	characteristic
421	C₄	#554×C₆₇	1	1	yes	characteristic
422	C₂	C₆₇×#2440	1	1	yes	characteristic
423	C₂	C₂×#479×C₆₇	1	1	yes	characteristic
424	C₂	C₆₇×#3084	1	1	yes	characteristic
425	C₂	C₆₇×#3278	1	1	yes	characteristic
426	C₂	C₆₇×#3284	1	1	yes	characteristic
427	C₂	C₆₇×#3381	1	1	yes	characteristic
428	C₂	C₆₇×#3387	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₄⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
202. C₄×C₆₇	#479	semidirect
207. #26043	C₆₇	direct (N × H)
226. C₆₇	#26043	direct (N × H)
417. C₄	#479×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#479×C₆₇	2
C₆₇×#2552	2
Q₈×C₄⋊C₄×C₆₇	2
C₆₇×#2088	2
C₆₇×#2059	2
C₆₇×#1947	2
C₆₇×#1939	2
C₆₇×#1865	2
C₆₇×#1844	2
C₆₇×#1803	2
C₄×#103×C₆₇	2
#26043	67

Subgroups — C67×#26043 #30705294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#26043 #30705294