Subgroups — C₆₇×#25908 #30705159

Order 17152.

1320 conjugacy classes of subgroups.

The subgroup lattice has 486 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, complemented
2	C₂×#489×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2850	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2024	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2023	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1993	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1874	C₂	1	2	yes	maximal
8	C₆₇×#1872	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₆₇	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	8	no	—
12	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#489×C₆₇	—	2	8	no	—
23	C₂×#103×C₆₇	C₂²	1	4	yes	—
24	C₂×#102×C₆₇	C₂²	1	4	yes	—
25	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×#97×C₆₇	C₂²	1	2	yes	—
27	#391×C₆₇	C₂²	1	1	yes	characteristic
28	#390×C₆₇	C₂²	1	1	yes	characteristic
29	#388×C₆₇	—	2	1	no	—
30	#388×C₆₇	—	2	1	no	—
31	#386×C₆₇	—	2	1	no	—
32	#386×C₆₇	—	2	1	no	—
33	#382×C₆₇	C₂²	1	1	yes	characteristic
34	#340×C₆₇	C₂²	1	4	yes	—
35	#340×C₆₇	C₂²	1	4	yes	—
36	#336×C₆₇	C₂²	1	2	yes	—
37	C₂²×Q₈×C₆₇	—	2	8	no	—
38	C₂²×Q₈×C₆₇	—	2	2	no	—
39	C₂²×Q₈×C₆₇	—	2	2	no	—
40	C₂²×Q₈×C₆₇	—	4	4	no	—
41	C₂²×Q₈×C₆₇	—	4	4	no	—
42	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₆₇	—	2	1	no	—
44	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	—	2	2	no	—
46	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
50	C₄⋊Q₈×C₆₇	—	2	8	no	—
51	C₄⋊Q₈×C₆₇	—	2	8	no	—
52	C₄⋊Q₈×C₆₇	—	2	8	no	—
53	C₄⋊Q₈×C₆₇	—	2	4	no	—
54	C₄⋊Q₈×C₆₇	D₄	1	8	yes	complemented
55	#124×C₆₇	—	2	4	no	—
56	C₂²⋊Q₈×C₆₇	—	2	4	no	—
57	#121×C₆₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄²×C₆₇	—	2	4	no	—
74	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	#103×C₆₇	—	2	8	no	—
79	#102×C₆₇	—	2	8	no	—
80	C₂²⋊C₈×C₆₇	—	2	4	no	—
81	#97×C₆₇	—	2	4	no	—
82	#95×C₆₇	—	2	1	no	—
83	#95×C₆₇	—	2	1	no	—
84	#95×C₆₇	—	2	1	no	—
85	#95×C₆₇	—	2	1	no	—
86	C₂⁴×C₆₇	—	2	1	no	—
87	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂×Q₈×C₆₇	—	2	8	no	—
89	C₂×Q₈×C₆₇	—	2	8	no	—
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	4	32	no	—
94	C₂×Q₈×C₆₇	—	4	4	no	—
95	C₂×Q₈×C₆₇	—	4	4	no	—
96	C₂×Q₈×C₆₇	—	4	4	no	—
97	C₂×Q₈×C₆₇	—	4	4	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	—	4	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	8	no	—
113	C₂²×C₄×C₆₇	—	2	4	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	—	4	4	no	—
118	C₂²×C₄×C₆₇	—	4	4	no	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	#41	1	4	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
126	C₂×C₈×C₆₇	—	2	4	no	—
127	C₂×C₈×C₆₇	—	2	4	no	—
128	C₂×C₈×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	8	no	—
130	C₄⋊C₄×C₆₇	—	2	8	no	—
131	C₄⋊C₄×C₆₇	—	2	8	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	4	2	no	—
136	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
137	C₂²⋊C₄×C₆₇	—	4	2	no	—
138	C₂²⋊C₄×C₆₇	—	4	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₂²⋊C₄×C₆₇	—	4	2	no	—
141	C₂²⋊C₄×C₆₇	—	4	2	no	—
142	C₂²⋊C₄×C₆₇	—	4	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	2	2	no	—
145	C₄²×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	C₂×D₄	1	4	yes	—
147	C₂³×C₆₇	—	2	1	no	—
148	C₂³×C₆₇	—	2	1	no	—
149	C₂³×C₆₇	—	2	1	no	—
150	C₂³×C₆₇	—	2	1	no	—
151	C₂³×C₆₇	—	4	1	no	—
152	C₂³×C₆₇	—	4	1	no	—
153	C₂³×C₆₇	—	4	1	no	—
154	C₂³×C₆₇	—	4	1	no	—
155	C₂³×C₆₇	—	4	1	no	—
156	C₂³×C₆₇	—	4	1	no	—
157	C₂³×C₆₇	—	4	1	no	—
158	C₂³×C₆₇	#124	1	2	yes	—
159	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
160	C₂³×C₆₇	#137	1	2	yes	—
161	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
162	Q₈×C₆₇	—	4	16	no	—
163	Q₈×C₆₇	—	4	16	no	—
164	Q₈×C₆₇	—	4	4	no	—
165	D₄×C₆₇	—	4	4	no	—
166	D₄×C₆₇	—	8	4	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	8	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	8	no	—
180	C₂×C₄×C₆₇	—	2	8	no	—
181	C₂×C₄×C₆₇	—	4	16	no	—
182	C₂×C₄×C₆₇	—	4	4	no	—
183	C₂×C₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	—	4	1	no	—
185	C₂×C₄×C₆₇	—	4	1	no	—
186	C₂×C₄×C₆₇	—	4	1	no	—
187	C₂×C₄×C₆₇	—	4	1	no	—
188	C₂×C₄×C₆₇	—	4	1	no	—
189	C₂×C₄×C₆₇	—	4	1	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	4	1	no	—
193	C₂×C₄×C₆₇	#120	1	2	yes	—
194	C₂×C₄×C₆₇	#120	1	2	yes	—
195	C₂×C₄×C₆₇	#124	1	2	yes	—
196	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
197	C₂×C₄×C₆₇	#137	1	2	yes	—
198	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
199	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
200	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
201	C₈×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	2	2	no	—
203	C₂²×C₆₇	—	2	2	no	—
204	C₂²×C₆₇	—	2	2	no	—
205	C₂²×C₆₇	—	2	2	no	—
206	C₂²×C₆₇	—	2	2	no	—
207	C₂²×C₆₇	—	2	2	no	—
208	C₂²×C₆₇	—	8	1	no	—
209	C₂²×C₆₇	—	8	1	no	—
210	C₂²×C₆₇	—	8	1	no	—
211	C₂²×C₆₇	—	8	1	no	—
212	C₂²×C₆₇	—	8	1	no	—
213	C₂²×C₆₇	—	8	1	no	—
214	C₂²×C₆₇	—	8	1	no	—
215	C₂²×C₆₇	#489	1	4	yes	complemented
216	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
217	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
218	C₂²×C₆₇	#536	1	1	yes	characteristic
219	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
222	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
223	C₄×C₆₇	—	2	2	no	—
224	C₄×C₆₇	—	2	2	no	—
225	C₄×C₆₇	—	2	2	no	—
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	2	2	no	—
228	C₄×C₆₇	—	4	8	no	—
229	C₄×C₆₇	—	8	1	no	—
230	C₄×C₆₇	—	8	1	no	—
231	C₄×C₆₇	—	8	1	no	—
232	C₄×C₆₇	#548	1	4	yes	—
233	#25908	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂×C₆₇	—	2	4	no	—
235	C₂×C₆₇	—	16	1	no	—
236	C₂×C₆₇	#2436	1	1	yes	characteristic
237	C₂×C₆₇	#3003	1	1	yes	characteristic
238	C₂×C₆₇	#3004	1	1	yes	characteristic
239	C₂×C₆₇	C₂×#489	1	1	yes	characteristic
240	C₂×C₆₇	#3141	1	1	yes	characteristic
241	C₂×C₆₇	#3238	1	1	yes	characteristic
242	C₂×C₆₇	#3239	1	1	yes	characteristic
243	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
244	C₂×#489	C₂×C₆₇	1	2	yes	complemented
245	#2850	C₂×C₆₇	1	1	yes	characteristic
246	#2024	C₂×C₆₇	1	2	yes	complemented
247	#2023	C₂×C₆₇	1	2	yes	complemented
248	#1993	C₂×C₆₇	1	2	yes	complemented
249	#1874	C₂×C₆₇	1	2	yes	—
250	#1872	C₂×C₆₇	1	2	yes	complemented
251	#1749	C₂×C₆₇	1	1	yes	characteristic, complemented
252	C₆₇	#25908	1	1	yes	characteristic, direct product factor, Sylow
253	C₂³×Q₈	—	2	1	no	—
254	C₂×C₄⋊Q₈	—	2	8	no	—
255	C₂×C₄⋊Q₈	—	2	1	no	—
256	C₂×C₄⋊Q₈	—	2	1	no	—
257	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
258	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
259	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
262	C₂²×C₄⋊C₄	—	2	1	no	—
263	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
264	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
265	#489	—	2	8	no	—
266	C₂×#103	C₂²×C₆₇	1	4	yes	—
267	C₂×#102	C₂²×C₆₇	1	4	yes	—
268	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
269	C₂×#97	C₂²×C₆₇	1	2	yes	—
270	#391	C₂²×C₆₇	1	1	yes	characteristic
271	#390	C₂²×C₆₇	1	1	yes	characteristic
272	#388	—	2	1	no	—
273	#388	—	2	1	no	—
274	#386	—	2	1	no	—
275	#386	—	2	1	no	—
276	#382	C₂²×C₆₇	1	1	yes	characteristic
277	#340	C₂²×C₆₇	1	4	yes	—
278	#340	C₂²×C₆₇	1	4	yes	—
279	#336	C₂²×C₆₇	1	2	yes	—
280	C₂²×Q₈	—	2	8	no	—
281	C₂²×Q₈	—	2	2	no	—
282	C₂²×Q₈	—	2	2	no	—
283	C₂²×Q₈	—	4	4	no	—
284	C₂²×Q₈	—	4	4	no	—
285	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
286	C₂²×D₄	—	2	1	no	—
287	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
288	C₂³×C₄	—	2	2	no	—
289	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
290	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
291	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
293	C₄⋊Q₈	—	2	8	no	—
294	C₄⋊Q₈	—	2	8	no	—
295	C₄⋊Q₈	—	2	8	no	—
296	C₄⋊Q₈	—	2	4	no	—
297	C₄⋊Q₈	D₄×C₆₇	1	8	yes	complemented
298	#124	—	2	4	no	—
299	C₂²⋊Q₈	—	2	4	no	—
300	#121	—	2	4	no	—
301	C₂×C₄⋊C₄	—	2	8	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	—	2	2	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	8	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
308	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
310	C₂×C₂²⋊C₄	—	2	1	no	—
311	C₂×C₂²⋊C₄	—	2	1	no	—
312	C₂×C₂²⋊C₄	—	2	1	no	—
313	C₂×C₂²⋊C₄	—	2	1	no	—
314	C₂×C₂²⋊C₄	—	2	1	no	—
315	C₂×C₂²⋊C₄	—	2	1	no	—
316	C₂×C₄²	—	2	4	no	—
317	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
318	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
321	#103	—	2	8	no	—
322	#102	—	2	8	no	—
323	C₂²⋊C₈	—	2	4	no	—
324	#97	—	2	4	no	—
325	#95	—	2	1	no	—
326	#95	—	2	1	no	—
327	#95	—	2	1	no	—
328	#95	—	2	1	no	—
329	C₂⁴	—	2	1	no	—
330	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₂×Q₈	—	2	8	no	—
332	C₂×Q₈	—	2	8	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	4	32	no	—
337	C₂×Q₈	—	4	4	no	—
338	C₂×Q₈	—	4	4	no	—
339	C₂×Q₈	—	4	4	no	—
340	C₂×Q₈	—	4	4	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	4	2	no	—
345	C₂×D₄	—	4	2	no	—
346	C₂×D₄	—	4	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	8	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	4	4	no	—
361	C₂²×C₄	—	4	4	no	—
362	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	#41×C₆₇	1	4	yes	—
368	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
369	C₂×C₈	—	2	4	no	—
370	C₂×C₈	—	2	4	no	—
371	C₂×C₈	—	2	4	no	—
372	C₄⋊C₄	—	2	8	no	—
373	C₄⋊C₄	—	2	8	no	—
374	C₄⋊C₄	—	2	8	no	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
380	C₂²⋊C₄	—	4	2	no	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₂²⋊C₄	—	4	2	no	—
384	C₂²⋊C₄	—	4	2	no	—
385	C₂²⋊C₄	—	4	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	2	no	—
389	C₄²	C₂×D₄×C₆₇	1	4	yes	—
390	C₂³	—	2	1	no	—
391	C₂³	—	2	1	no	—
392	C₂³	—	2	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	4	1	no	—
395	C₂³	—	4	1	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	1	no	—
398	C₂³	—	4	1	no	—
399	C₂³	—	4	1	no	—
400	C₂³	—	4	1	no	—
401	C₂³	#124×C₆₇	1	2	yes	—
402	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
403	C₂³	#137×C₆₇	1	2	yes	—
404	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
405	Q₈	—	4	16	no	—
406	Q₈	—	4	16	no	—
407	Q₈	—	4	4	no	—
408	D₄	—	4	4	no	—
409	D₄	—	8	4	no	—
410	C₂×C₄	—	2	8	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	8	no	—
423	C₂×C₄	—	2	8	no	—
424	C₂×C₄	—	4	16	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	#120×C₆₇	1	2	yes	—
437	C₂×C₄	#120×C₆₇	1	2	yes	—
438	C₂×C₄	#124×C₆₇	1	2	yes	—
439	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
440	C₂×C₄	#137×C₆₇	1	2	yes	—
441	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
442	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
443	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
444	C₈	—	4	4	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	8	1	no	—
453	C₂²	—	8	1	no	—
454	C₂²	—	8	1	no	—
455	C₂²	—	8	1	no	—
456	C₂²	—	8	1	no	—
457	C₂²	—	8	1	no	—
458	C₂²	#489×C₆₇	1	4	yes	complemented
459	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
460	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
461	C₂²	#536×C₆₇	1	1	yes	characteristic
462	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
463	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
464	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
465	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
466	C₄	—	2	2	no	—
467	C₄	—	2	2	no	—
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	4	8	no	—
472	C₄	—	8	1	no	—
473	C₄	—	8	1	no	—
474	C₄	—	8	1	no	—
475	C₄	#548×C₆₇	1	4	yes	—
476	C₂	—	2	4	no	—
477	C₂	—	16	1	no	—
478	C₂	C₆₇×#2436	1	1	yes	characteristic
479	C₂	C₆₇×#3003	1	1	yes	characteristic
480	C₂	C₆₇×#3004	1	1	yes	characteristic
481	C₂	C₂×#489×C₆₇	1	1	yes	characteristic
482	C₂	C₆₇×#3141	1	1	yes	characteristic
483	C₂	C₆₇×#3238	1	1	yes	characteristic
484	C₂	C₆₇×#3239	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₆₇	C₂	semidirect
2. C₂×#489×C₆₇	C₂	semidirect
4. C₆₇×#2024	C₂	semidirect
5. C₆₇×#2023	C₂	semidirect
6. C₆₇×#1993	C₂	semidirect
8. C₆₇×#1872	C₂	semidirect
9. C₆₇×#1749	C₂	semidirect
54. C₄⋊Q₈×C₆₇	D₄	semidirect
215. C₂²×C₆₇	#489	semidirect
233. #25908	C₆₇	direct (N × H)
243. C₂²×C₄⋊Q₈	C₂×C₆₇	semidirect
244. C₂×#489	C₂×C₆₇	semidirect
246. #2024	C₂×C₆₇	semidirect
247. #2023	C₂×C₆₇	semidirect
248. #1993	C₂×C₆₇	semidirect
250. #1872	C₂×C₆₇	semidirect
251. #1749	C₂×C₆₇	semidirect
252. C₆₇	#25908	direct (N × H)
297. C₄⋊Q₈	D₄×C₆₇	semidirect
458. C₂²	#489×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×#489×C₆₇	2
C₆₇×#2850	2
C₆₇×#2024	2
C₆₇×#2023	2
C₆₇×#1993	2
C₆₇×#1874	2
C₆₇×#1872	2
C₆₇×#1749	2
#25908	67

Subgroups — C67×#25908 #30705159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25908 #30705159