Subgroups — C₆₇×#17891 #30697142

Order 17152.

724 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2965	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1594	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1592	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1590	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1589	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1478	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1477	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	—	2	1	no	—
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	#446×C₆₇	C₂²	1	1	yes	characteristic, complemented
17	#446×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	#442×C₆₇	C₄	1	2	yes	—
19	#440×C₆₇	C₄	1	2	yes	—
20	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
21	C₄×D₈×C₆₇	C₂²	1	2	yes	—
22	#436×C₆₇	—	2	1	no	—
23	#436×C₆₇	—	2	1	no	—
24	#436×C₆₇	C₂²	1	2	yes	—
25	#436×C₆₇	C₂²	1	2	yes	—
26	#436×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	#436×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	#435×C₆₇	—	2	1	no	—
29	#435×C₆₇	—	2	1	no	—
30	#431×C₆₇	—	2	1	no	—
31	#431×C₆₇	C₂²	1	1	yes	characteristic
32	#431×C₆₇	C₂²	1	1	yes	characteristic
33	#428×C₆₇	C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#418×C₆₇	C₄	1	1	yes	characteristic
36	#417×C₆₇	C₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
38	C₄×M₁₆×C₆₇	—	2	1	no	—
39	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	#335×C₆₇	C₂²	1	2	yes	complemented
42	#334×C₆₇	C₂²	1	2	yes	complemented
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	complemented
44	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
45	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
46	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
47	#322×C₆₇	C₂²	1	2	yes	complemented
48	C₈²×C₆₇	C₂²	1	2	yes	complemented
49	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	—	2	1	no	—
52	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
53	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂×M₁₆×C₆₇	—	2	2	no	—
57	C₂×M₁₆×C₆₇	—	2	1	no	—
58	C₂×M₁₆×C₆₇	—	2	1	no	—
59	C₂×M₁₆×C₆₇	—	4	4	no	—
60	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	—	2	1	no	—
63	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
65	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₆₇	—	2	1	no	—
68	C₄×D₄×C₆₇	—	2	1	no	—
69	C₄×D₄×C₆₇	—	2	1	no	—
70	C₄×D₄×C₆₇	—	2	1	no	—
71	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
72	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×D₄×C₆₇	C₂³	1	2	yes	—
74	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
75	#117×C₆₇	—	2	1	no	—
76	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₄²×C₆₇	—	2	1	no	—
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	#107×C₆₇	C₂×C₄	1	2	yes	—
82	#106×C₆₇	C₂×C₄	1	2	yes	—
83	C₄⋊C₈×C₆₇	—	2	1	no	—
84	C₄⋊C₈×C₆₇	—	2	1	no	—
85	C₄⋊C₈×C₆₇	—	2	1	no	—
86	C₄⋊C₈×C₆₇	—	2	1	no	—
87	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
90	#103×C₆₇	C₂×C₄	1	2	yes	—
91	#102×C₆₇	C₂×C₄	1	2	yes	—
92	#102×C₆₇	C₂×C₄	1	2	yes	—
93	#102×C₆₇	C₂×C₄	1	2	yes	—
94	C₂²⋊C₈×C₆₇	—	2	1	no	—
95	C₂²⋊C₈×C₆₇	—	2	1	no	—
96	C₂²⋊C₈×C₆₇	—	2	2	no	—
97	C₂²⋊C₈×C₆₇	—	2	2	no	—
98	C₂²⋊C₈×C₆₇	—	2	1	no	—
99	C₂²⋊C₈×C₆₇	—	2	1	no	—
100	#97×C₆₇	—	2	1	no	—
101	#97×C₆₇	—	2	1	no	—
102	#97×C₆₇	C₂×C₄	1	2	yes	—
103	C₄×C₈×C₆₇	—	2	1	no	—
104	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
105	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
106	C₄×C₈×C₆₇	C₂³	1	2	yes	—
107	C₄×C₈×C₆₇	C₂³	1	2	yes	—
108	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
110	C₂⁴×C₆₇	—	2	1	no	—
111	#41×C₆₇	—	2	2	no	—
112	#41×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₄×C₆₇	—	2	1	no	—
115	C₂×D₄×C₆₇	—	2	1	no	—
116	C₂×D₄×C₆₇	—	2	2	no	—
117	C₂×D₄×C₆₇	—	2	2	no	—
118	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	4	2	no	—
127	C₂²×C₄×C₆₇	—	4	2	no	—
128	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
131	SD₁₆×C₆₇	—	2	4	no	—
132	D₈×C₆₇	—	2	4	no	—
133	M₁₆×C₆₇	—	2	2	no	—
134	M₁₆×C₆₇	—	4	2	no	—
135	M₁₆×C₆₇	—	4	4	no	—
136	M₁₆×C₆₇	—	4	2	no	—
137	M₁₆×C₆₇	—	4	4	no	—
138	M₁₆×C₆₇	M₁₆	1	4	yes	complemented
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	2	1	no	—
142	C₂×C₈×C₆₇	—	2	1	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₂×C₈×C₆₇	—	4	1	no	—
145	C₂×C₈×C₆₇	—	4	2	no	—
146	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
147	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
148	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₄⋊C₄×C₆₇	—	2	1	no	—
153	C₄⋊C₄×C₆₇	—	2	1	no	—
154	C₄⋊C₄×C₆₇	—	2	1	no	—
155	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
158	C₂²⋊C₄×C₆₇	—	2	1	no	—
159	C₂²⋊C₄×C₆₇	—	2	2	no	—
160	C₂²⋊C₄×C₆₇	—	2	1	no	—
161	C₂²⋊C₄×C₆₇	—	2	2	no	—
162	C₂²⋊C₄×C₆₇	—	2	1	no	—
163	C₄²×C₆₇	—	2	1	no	—
164	C₄²×C₆₇	—	2	1	no	—
165	C₄²×C₆₇	—	2	1	no	—
166	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₆₇	#41	1	1	yes	characteristic
169	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₂³×C₆₇	—	2	1	no	—
171	C₂³×C₆₇	—	2	2	no	—
172	C₂³×C₆₇	—	4	2	no	—
173	C₂³×C₆₇	—	4	2	no	—
174	C₂³×C₆₇	—	4	2	no	—
175	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
176	Q₈×C₆₇	—	2	1	no	—
177	Q₈×C₆₇	—	2	1	no	—
178	D₄×C₆₇	—	2	1	no	—
179	D₄×C₆₇	—	2	2	no	—
180	D₄×C₆₇	—	2	2	no	—
181	D₄×C₆₇	—	2	1	no	—
182	D₄×C₆₇	—	4	2	no	—
183	D₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	4	4	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	2	no	—
195	C₂×C₄×C₆₇	—	4	1	no	—
196	C₂×C₄×C₆₇	—	4	1	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	2	no	—
199	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
203	C₂×C₄×C₆₇	#131	1	2	yes	—
204	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
207	C₈×C₆₇	—	2	2	no	—
208	C₈×C₆₇	—	2	1	no	—
209	C₈×C₆₇	—	2	1	no	—
210	C₈×C₆₇	—	2	1	no	—
211	C₈×C₆₇	—	4	2	no	—
212	C₈×C₆₇	—	4	2	no	—
213	C₈×C₆₇	—	4	1	no	—
214	C₈×C₆₇	—	4	1	no	—
215	C₈×C₆₇	C₂×M₁₆	1	4	yes	complemented
216	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	complemented
217	C₂²×C₆₇	—	2	1	no	—
218	C₂²×C₆₇	—	2	1	no	—
219	C₂²×C₆₇	—	4	1	no	—
220	C₂²×C₆₇	—	4	2	no	—
221	C₂²×C₆₇	—	4	2	no	—
222	C₂²×C₆₇	—	4	1	no	—
223	C₂²×C₆₇	—	4	1	no	—
224	C₂²×C₆₇	—	4	2	no	—
225	C₂²×C₆₇	—	8	4	no	—
226	C₂²×C₆₇	#436	1	2	yes	—
227	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
228	C₄×C₆₇	—	2	1	no	—
229	C₄×C₆₇	—	2	1	no	—
230	C₄×C₆₇	—	2	1	no	—
231	C₄×C₆₇	—	4	1	no	—
232	C₄×C₆₇	—	4	1	no	—
233	C₄×C₆₇	—	4	2	no	—
234	C₄×C₆₇	—	4	1	no	—
235	C₄×C₆₇	#436	1	2	yes	—
236	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
237	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
238	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₄×C₆₇	#575	1	1	yes	characteristic
240	#17891	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂×C₆₇	—	2	2	no	—
242	C₂×C₆₇	—	8	1	no	—
243	C₂×C₆₇	—	8	2	no	—
244	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
245	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
246	C₂×C₆₇	#2955	1	1	yes	characteristic
247	#2965	C₂×C₆₇	1	1	yes	characteristic, complemented
248	C₄×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic
249	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#1594	C₂×C₆₇	1	2	yes	complemented
251	#1592	C₂×C₆₇	1	2	yes	complemented
252	#1590	C₂×C₆₇	1	2	yes	complemented
253	#1589	C₂×C₆₇	1	2	yes	complemented
254	#1565	C₂×C₆₇	1	1	yes	characteristic, complemented
255	#1478	C₂×C₆₇	1	1	yes	characteristic, complemented
256	#1477	C₂×C₆₇	1	1	yes	characteristic, complemented
257	C₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₆₇	#17891	1	1	yes	characteristic, direct product factor, Sylow
259	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	1	yes	characteristic
260	C₂²×M₁₆	—	2	1	no	—
261	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
263	#446	C₂²×C₆₇	1	1	yes	characteristic, complemented
264	#446	C₂²×C₆₇	1	1	yes	characteristic, complemented
265	#442	C₄×C₆₇	1	2	yes	—
266	#440	C₄×C₆₇	1	2	yes	—
267	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
268	C₄×D₈	C₂²×C₆₇	1	2	yes	—
269	#436	—	2	1	no	—
270	#436	—	2	1	no	—
271	#436	C₂²×C₆₇	1	2	yes	—
272	#436	C₂²×C₆₇	1	2	yes	—
273	#436	C₂²×C₆₇	1	1	yes	characteristic, complemented
274	#436	C₂²×C₆₇	1	1	yes	characteristic, complemented
275	#435	—	2	1	no	—
276	#435	—	2	1	no	—
277	#431	—	2	1	no	—
278	#431	C₂²×C₆₇	1	1	yes	characteristic
279	#431	C₂²×C₆₇	1	1	yes	characteristic
280	#428	C₄×C₆₇	1	1	yes	characteristic
281	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
282	#418	C₄×C₆₇	1	1	yes	characteristic
283	#417	C₄×C₆₇	1	1	yes	characteristic
284	C₂×C₂²⋊C₈	—	2	1	no	—
285	C₄×M₁₆	—	2	1	no	—
286	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
287	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
288	#335	C₂²×C₆₇	1	2	yes	complemented
289	#334	C₂²×C₆₇	1	2	yes	complemented
290	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	complemented
291	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
292	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
293	D₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
294	#322	C₂²×C₆₇	1	2	yes	complemented
295	C₈²	C₂²×C₆₇	1	2	yes	complemented
296	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
297	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
298	C₂³×C₄	—	2	1	no	—
299	AGL(1,ℤ/8)	—	2	4	no	—
300	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
301	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
302	C₂×M₁₆	—	2	2	no	—
303	C₂×M₁₆	—	2	2	no	—
304	C₂×M₁₆	—	2	1	no	—
305	C₂×M₁₆	—	2	1	no	—
306	C₂×M₁₆	—	4	4	no	—
307	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
308	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₈	—	2	1	no	—
310	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
312	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
313	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
319	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
320	C₄×D₄	C₂³×C₆₇	1	2	yes	—
321	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
322	#117	—	2	1	no	—
323	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
325	C₂×C₂²⋊C₄	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
328	#107	C₂×C₄×C₆₇	1	2	yes	—
329	#106	C₂×C₄×C₆₇	1	2	yes	—
330	C₄⋊C₈	—	2	1	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	—	2	1	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
335	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
336	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
337	#103	C₂×C₄×C₆₇	1	2	yes	—
338	#102	C₂×C₄×C₆₇	1	2	yes	—
339	#102	C₂×C₄×C₆₇	1	2	yes	—
340	#102	C₂×C₄×C₆₇	1	2	yes	—
341	C₂²⋊C₈	—	2	1	no	—
342	C₂²⋊C₈	—	2	1	no	—
343	C₂²⋊C₈	—	2	2	no	—
344	C₂²⋊C₈	—	2	2	no	—
345	C₂²⋊C₈	—	2	1	no	—
346	C₂²⋊C₈	—	2	1	no	—
347	#97	—	2	1	no	—
348	#97	—	2	1	no	—
349	#97	C₂×C₄×C₆₇	1	2	yes	—
350	C₄×C₈	—	2	1	no	—
351	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
352	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
353	C₄×C₈	C₂³×C₆₇	1	2	yes	—
354	C₄×C₈	C₂³×C₆₇	1	2	yes	—
355	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
356	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
357	C₂⁴	—	2	1	no	—
358	#41	—	2	2	no	—
359	#41	—	2	2	no	—
360	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	2	1	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
366	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
377	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
378	SD₁₆	—	2	4	no	—
379	D₈	—	2	4	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	4	4	no	—
382	M₁₆	—	4	2	no	—
383	M₁₆	—	4	4	no	—
384	M₁₆	—	4	2	no	—
385	M₁₆	M₁₆×C₆₇	1	4	yes	complemented
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	4	1	no	—
392	C₂×C₈	—	4	2	no	—
393	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
394	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
395	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
396	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
397	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
398	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
403	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
404	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
405	C₂²⋊C₄	—	2	1	no	—
406	C₂²⋊C₄	—	2	2	no	—
407	C₂²⋊C₄	—	2	1	no	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	1	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	1	no	—
413	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
414	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
415	C₄²	#41×C₆₇	1	1	yes	characteristic
416	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
417	C₂³	—	2	1	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
423	Q₈	—	2	1	no	—
424	Q₈	—	2	1	no	—
425	D₄	—	2	1	no	—
426	D₄	—	2	2	no	—
427	D₄	—	2	2	no	—
428	D₄	—	2	1	no	—
429	D₄	—	4	2	no	—
430	D₄	—	4	2	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
447	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
448	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
449	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
450	C₂×C₄	#131×C₆₇	1	2	yes	—
451	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
452	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
453	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	2	1	no	—
458	C₈	—	4	2	no	—
459	C₈	—	4	2	no	—
460	C₈	—	4	1	no	—
461	C₈	—	4	1	no	—
462	C₈	C₂×M₁₆×C₆₇	1	4	yes	complemented
463	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	complemented
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	4	1	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	1	no	—
470	C₂²	—	4	1	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	8	4	no	—
473	C₂²	#436×C₆₇	1	2	yes	—
474	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	4	1	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	1	no	—
481	C₄	—	4	1	no	—
482	C₄	#436×C₆₇	1	2	yes	—
483	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
484	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
485	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
486	C₄	#575×C₆₇	1	1	yes	characteristic
487	C₂	—	2	2	no	—
488	C₂	—	8	2	no	—
489	C₂	—	8	1	no	—
490	C₂	C₂×#436×C₆₇	1	1	yes	characteristic
491	C₂	C₄×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
492	C₂	C₆₇×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×AGL(1,ℤ/8)×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2965	C₂	semidirect
3. C₂×#436×C₆₇	C₂	semidirect
4. C₆₇×#1594	C₂	semidirect
5. C₆₇×#1592	C₂	semidirect
6. C₆₇×#1590	C₂	semidirect
7. C₆₇×#1589	C₂	semidirect
8. C₆₇×#1565	C₂	semidirect
9. C₆₇×#1478	C₂	semidirect
10. C₆₇×#1477	C₂	semidirect
11. C₈×M₁₆×C₆₇	C₂	semidirect
16. #446×C₆₇	C₂²	semidirect
17. #446×C₆₇	C₂²	semidirect
26. #436×C₆₇	C₂²	semidirect
27. #436×C₆₇	C₂²	semidirect
41. #335×C₆₇	C₂²	semidirect
42. #334×C₆₇	C₂²	semidirect
43. Q₈⋊C₈×C₆₇	C₂²	semidirect
46. D₄⋊C₈×C₆₇	C₂²	semidirect
47. #322×C₆₇	C₂²	semidirect
48. C₈²×C₆₇	C₂²	semidirect
138. M₁₆×C₆₇	M₁₆	semidirect
215. C₈×C₆₇	C₂×M₁₆	semidirect
216. C₈×C₆₇	AGL(1,ℤ/8)	semidirect
240. #17891	C₆₇	direct (N × H)
247. #2965	C₂×C₆₇	semidirect
249. C₂×#436	C₂×C₆₇	semidirect
250. #1594	C₂×C₆₇	semidirect
251. #1592	C₂×C₆₇	semidirect
252. #1590	C₂×C₆₇	semidirect
253. #1589	C₂×C₆₇	semidirect
254. #1565	C₂×C₆₇	semidirect
255. #1478	C₂×C₆₇	semidirect
256. #1477	C₂×C₆₇	semidirect
257. C₈×M₁₆	C₂×C₆₇	semidirect
258. C₆₇	#17891	direct (N × H)
263. #446	C₂²×C₆₇	semidirect
264. #446	C₂²×C₆₇	semidirect
273. #436	C₂²×C₆₇	semidirect
274. #436	C₂²×C₆₇	semidirect
288. #335	C₂²×C₆₇	semidirect
289. #334	C₂²×C₆₇	semidirect
290. Q₈⋊C₈	C₂²×C₆₇	semidirect
293. D₄⋊C₈	C₂²×C₆₇	semidirect
294. #322	C₂²×C₆₇	semidirect
295. C₈²	C₂²×C₆₇	semidirect
385. M₁₆	M₁₆×C₆₇	semidirect
462. C₈	C₂×M₁₆×C₆₇	semidirect
463. C₈	AGL(1,ℤ/8)×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2965	2
C₄×AGL(1,ℤ/8)×C₆₇	2
C₂×#436×C₆₇	2
C₆₇×#1594	2
C₆₇×#1592	2
C₆₇×#1590	2
C₆₇×#1589	2
C₆₇×#1565	2
C₆₇×#1478	2
C₆₇×#1477	2
C₈×M₁₆×C₆₇	2
#17891	67

Subgroups — C67×#17891 #30697142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17891 #30697142