Subgroups — C₆₇×#17852 #30697103

Order 17152.

754 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2988	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1580	C₂	1	2	yes	maximal, complemented
5	C₆₇×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₈×SD₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₈×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1476	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄⋊C₈×C₆₇	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	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
16	#442×C₆₇	C₄	1	2	yes	—
17	#440×C₆₇	C₄	1	2	yes	—
18	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
19	C₄×D₈×C₆₇	C₂²	1	2	yes	—
20	C₈×D₄×C₆₇	—	2	2	no	—
21	C₈×D₄×C₆₇	—	2	1	no	—
22	C₈×D₄×C₆₇	—	2	1	no	—
23	C₈×D₄×C₆₇	C₂²	1	2	yes	—
24	C₈×D₄×C₆₇	C₂²	1	2	yes	—
25	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
26	#431×C₆₇	—	2	1	no	—
27	#431×C₆₇	C₂²	1	1	yes	characteristic
28	#431×C₆₇	C₂²	1	1	yes	characteristic
29	#428×C₆₇	C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
31	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#418×C₆₇	C₄	1	1	yes	characteristic
33	#417×C₆₇	C₄	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
35	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#335×C₆₇	C₂²	1	2	yes	complemented
38	#334×C₆₇	C₂²	1	2	yes	complemented
39	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
42	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
43	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	#322×C₆₇	C₂²	1	2	yes	complemented
46	C₈²×C₆₇	C₂²	1	2	yes	complemented
47	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	—	2	1	no	—
50	AGL(1,ℤ/8)×C₆₇	C₈	1	8	yes	complemented
51	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₂²×C₈×C₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	—	2	2	no	—
60	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₄×D₄×C₆₇	—	2	1	no	—
63	C₄×D₄×C₆₇	—	2	1	no	—
64	C₄×D₄×C₆₇	—	2	1	no	—
65	C₄×D₄×C₆₇	—	2	1	no	—
66	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₄×D₄×C₆₇	C₂³	1	2	yes	—
69	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
70	#117×C₆₇	—	2	1	no	—
71	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄²×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	C₂×C₄	1	2	yes	—
77	#106×C₆₇	C₂×C₄	1	2	yes	—
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	—	2	1	no	—
80	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
82	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
83	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
85	#103×C₆₇	C₂×C₄	1	2	yes	—
86	#102×C₆₇	C₂×C₄	1	2	yes	—
87	#102×C₆₇	C₂×C₄	1	2	yes	—
88	#102×C₆₇	C₂×C₄	1	2	yes	—
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	C₂²⋊C₈×C₆₇	—	2	2	no	—
91	C₂²⋊C₈×C₆₇	—	2	2	no	—
92	C₂²⋊C₈×C₆₇	—	2	2	no	—
93	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
94	#97×C₆₇	C₂×C₄	1	2	yes	—
95	C₄×C₈×C₆₇	—	2	2	no	—
96	C₄×C₈×C₆₇	—	2	1	no	—
97	C₄×C₈×C₆₇	—	2	1	no	—
98	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₆₇	C₂³	1	2	yes	—
101	C₄×C₈×C₆₇	C₂³	1	2	yes	—
102	C₂⁴×C₆₇	—	2	1	no	—
103	#41×C₆₇	—	2	2	no	—
104	#41×C₆₇	C₂×C₈	1	4	yes	complemented
105	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×D₄×C₆₇	—	2	1	no	—
107	C₂×D₄×C₆₇	—	2	1	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	complemented
110	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	2	1	no	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	8	1	no	—
119	C₂²×C₄×C₆₇	—	8	1	no	—
120	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
123	SD₁₆×C₆₇	C₂×C₈	1	8	yes	complemented
124	D₈×C₆₇	C₂×C₈	1	8	yes	complemented
125	M₁₆×C₆₇	—	2	2	no	—
126	M₁₆×C₆₇	C₂×C₈	1	4	yes	complemented
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	—	2	1	no	—
131	C₂×C₈×C₆₇	—	2	1	no	—
132	C₂×C₈×C₆₇	—	4	2	no	—
133	C₂×C₈×C₆₇	—	8	2	no	—
134	C₂×C₈×C₆₇	—	8	2	no	—
135	C₂×C₈×C₆₇	—	8	2	no	—
136	C₂×C₈×C₆₇	—	8	1	no	—
137	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
141	C₄⋊C₄×C₆₇	—	2	1	no	—
142	C₄⋊C₄×C₆₇	—	2	1	no	—
143	C₄⋊C₄×C₆₇	—	2	1	no	—
144	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
147	C₂²⋊C₄×C₆₇	—	2	1	no	—
148	C₂²⋊C₄×C₆₇	—	2	2	no	—
149	C₂²⋊C₄×C₆₇	—	2	1	no	—
150	C₂²⋊C₄×C₆₇	—	2	2	no	—
151	C₂²⋊C₄×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	—	2	1	no	—
153	C₄²×C₆₇	—	2	1	no	—
154	C₄²×C₆₇	—	2	1	no	—
155	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₆₇	#41	1	1	yes	characteristic
158	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	—	2	2	no	—
161	C₂³×C₆₇	—	2	4	no	—
162	C₂³×C₆₇	—	8	1	no	—
163	C₂³×C₆₇	—	8	1	no	—
164	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
165	Q₈×C₆₇	—	2	1	no	—
166	Q₈×C₆₇	C₂²×C₈	1	2	yes	—
167	D₄×C₆₇	—	2	4	no	—
168	D₄×C₆₇	—	2	2	no	—
169	D₄×C₆₇	—	2	1	no	—
170	D₄×C₆₇	—	2	4	no	—
171	D₄×C₆₇	C₂²×C₈	1	2	yes	—
172	D₄×C₆₇	C₂²×C₈	1	4	yes	—
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	4	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	4	1	no	—
185	C₂×C₄×C₆₇	—	8	1	no	—
186	C₂×C₄×C₆₇	—	8	2	no	—
187	C₂×C₄×C₆₇	—	8	1	no	—
188	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
192	C₂×C₄×C₆₇	#131	1	2	yes	—
193	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
196	C₈×C₆₇	—	2	2	no	—
197	C₈×C₆₇	—	2	2	no	—
198	C₈×C₆₇	—	2	2	no	—
199	C₈×C₆₇	—	2	2	no	—
200	C₈×C₆₇	—	4	1	no	—
201	C₈×C₆₇	—	4	2	no	—
202	C₈×C₆₇	—	4	1	no	—
203	C₈×C₆₇	—	4	1	no	—
204	C₈×C₆₇	C₂²×C₈	1	4	yes	—
205	C₂²×C₆₇	—	2	1	no	—
206	C₂²×C₆₇	—	2	1	no	—
207	C₂²×C₆₇	—	2	2	no	—
208	C₂²×C₆₇	—	2	4	no	—
209	C₂²×C₆₇	—	4	2	no	—
210	C₂²×C₆₇	—	4	1	no	—
211	C₂²×C₆₇	—	4	1	no	—
212	C₂²×C₆₇	—	4	2	no	—
213	C₂²×C₆₇	—	8	4	no	—
214	C₂²×C₆₇	C₈×D₄	1	2	yes	—
215	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
216	C₄×C₆₇	—	2	2	no	—
217	C₄×C₆₇	—	2	1	no	—
218	C₄×C₆₇	—	2	1	no	—
219	C₄×C₆₇	—	2	1	no	—
220	C₄×C₆₇	—	4	1	no	—
221	C₄×C₆₇	—	4	2	no	—
222	C₄×C₆₇	—	4	1	no	—
223	C₄×C₆₇	C₈×D₄	1	2	yes	—
224	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
225	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
226	C₄×C₆₇	#576	1	1	yes	characteristic
227	C₄×C₆₇	#577	1	1	yes	characteristic
228	#17852	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂×C₆₇	—	2	2	no	—
230	C₂×C₆₇	—	4	2	no	—
231	C₂×C₆₇	—	4	4	no	—
232	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
233	C₂×C₆₇	#2946	1	1	yes	characteristic
234	C₂×C₆₇	#2956	1	1	yes	characteristic
235	#2988	C₂×C₆₇	1	1	yes	characteristic, complemented
236	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
237	C₄×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic
238	#1580	C₂×C₆₇	1	2	yes	complemented
239	SD₁₆⋊C₈	C₂×C₆₇	1	2	yes	complemented
240	C₈×SD₁₆	C₂×C₆₇	1	2	yes	complemented
241	C₈×D₈	C₂×C₆₇	1	2	yes	complemented
242	#1565	C₂×C₆₇	1	1	yes	characteristic, complemented
243	#1476	C₂×C₆₇	1	1	yes	characteristic, complemented
244	C₂×D₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
245	C₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
246	C₆₇	#17852	1	1	yes	characteristic, direct product factor, Sylow
247	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	1	yes	characteristic
248	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
249	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
250	C₈×Q₈	C₂²×C₆₇	1	2	yes	complemented
251	#442	C₄×C₆₇	1	2	yes	—
252	#440	C₄×C₆₇	1	2	yes	—
253	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
254	C₄×D₈	C₂²×C₆₇	1	2	yes	—
255	C₈×D₄	—	2	1	no	—
256	C₈×D₄	—	2	1	no	—
257	C₈×D₄	—	2	2	no	—
258	C₈×D₄	C₂²×C₆₇	1	2	yes	—
259	C₈×D₄	C₂²×C₆₇	1	2	yes	—
260	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
261	#431	—	2	1	no	—
262	#431	C₂²×C₆₇	1	1	yes	characteristic
263	#431	C₂²×C₆₇	1	1	yes	characteristic
264	#428	C₄×C₆₇	1	1	yes	characteristic
265	C₂×C₄⋊C₈	—	2	1	no	—
266	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
267	#418	C₄×C₆₇	1	1	yes	characteristic
268	#417	C₄×C₆₇	1	1	yes	characteristic
269	C₂×C₂²⋊C₈	—	2	2	no	—
270	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
272	#335	C₂²×C₆₇	1	2	yes	complemented
273	#334	C₂²×C₆₇	1	2	yes	complemented
274	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
275	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
276	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
277	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
278	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
279	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
280	#322	C₂²×C₆₇	1	2	yes	complemented
281	C₈²	C₂²×C₆₇	1	2	yes	complemented
282	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
283	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
284	C₂³×C₄	—	2	1	no	—
285	AGL(1,ℤ/8)	C₈×C₆₇	1	8	yes	complemented
286	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
287	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
288	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
289	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
290	C₂²×C₈	—	2	1	no	—
291	C₂²×C₈	—	2	1	no	—
292	C₂²×C₈	—	2	2	no	—
293	C₂²×C₈	—	2	2	no	—
294	C₂²×C₈	—	2	2	no	—
295	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
296	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₄×D₄	—	2	1	no	—
298	C₄×D₄	—	2	1	no	—
299	C₄×D₄	—	2	1	no	—
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
302	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
303	C₄×D₄	C₂³×C₆₇	1	2	yes	—
304	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
305	#117	—	2	1	no	—
306	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
311	#107	C₂×C₄×C₆₇	1	2	yes	—
312	#106	C₂×C₄×C₆₇	1	2	yes	—
313	C₄⋊C₈	—	2	2	no	—
314	C₄⋊C₈	—	2	1	no	—
315	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
316	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
317	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
318	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
319	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
320	#103	C₂×C₄×C₆₇	1	2	yes	—
321	#102	C₂×C₄×C₆₇	1	2	yes	—
322	#102	C₂×C₄×C₆₇	1	2	yes	—
323	#102	C₂×C₄×C₆₇	1	2	yes	—
324	C₂²⋊C₈	—	2	1	no	—
325	C₂²⋊C₈	—	2	2	no	—
326	C₂²⋊C₈	—	2	2	no	—
327	C₂²⋊C₈	—	2	2	no	—
328	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
329	#97	C₂×C₄×C₆₇	1	2	yes	—
330	C₄×C₈	—	2	2	no	—
331	C₄×C₈	—	2	1	no	—
332	C₄×C₈	—	2	1	no	—
333	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
334	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
335	C₄×C₈	C₂³×C₆₇	1	2	yes	—
336	C₄×C₈	C₂³×C₆₇	1	2	yes	—
337	C₂⁴	—	2	1	no	—
338	#41	—	2	2	no	—
339	#41	C₂×C₈×C₆₇	1	4	yes	complemented
340	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	C₂×C₈×C₆₇	1	4	yes	complemented
345	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
346	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	8	1	no	—
354	C₂²×C₄	—	8	1	no	—
355	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
357	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
358	SD₁₆	C₂×C₈×C₆₇	1	8	yes	complemented
359	D₈	C₂×C₈×C₆₇	1	8	yes	complemented
360	M₁₆	—	2	2	no	—
361	M₁₆	C₂×C₈×C₆₇	1	4	yes	complemented
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	4	2	no	—
368	C₂×C₈	—	8	2	no	—
369	C₂×C₈	—	8	2	no	—
370	C₂×C₈	—	8	2	no	—
371	C₂×C₈	—	8	1	no	—
372	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
373	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
374	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
375	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
380	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
381	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
382	C₂²⋊C₄	—	2	1	no	—
383	C₂²⋊C₄	—	2	2	no	—
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	2	2	no	—
386	C₂²⋊C₄	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
391	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
392	C₄²	#41×C₆₇	1	1	yes	characteristic
393	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
394	C₂³	—	2	1	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	4	no	—
397	C₂³	—	8	1	no	—
398	C₂³	—	8	1	no	—
399	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
400	Q₈	—	2	1	no	—
401	Q₈	C₂²×C₈×C₆₇	1	2	yes	—
402	D₄	—	2	4	no	—
403	D₄	—	2	2	no	—
404	D₄	—	2	1	no	—
405	D₄	—	2	4	no	—
406	D₄	C₂²×C₈×C₆₇	1	2	yes	—
407	D₄	C₂²×C₈×C₆₇	1	4	yes	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	8	2	no	—
421	C₂×C₄	—	8	1	no	—
422	C₂×C₄	—	8	1	no	—
423	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
424	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
425	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
426	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
427	C₂×C₄	#131×C₆₇	1	2	yes	—
428	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
429	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	2	2	no	—
433	C₈	—	2	2	no	—
434	C₈	—	2	2	no	—
435	C₈	—	4	1	no	—
436	C₈	—	4	1	no	—
437	C₈	—	4	2	no	—
438	C₈	—	4	1	no	—
439	C₈	C₂²×C₈×C₆₇	1	4	yes	—
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	2	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	1	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	8	4	no	—
449	C₂²	C₈×D₄×C₆₇	1	2	yes	—
450	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
451	C₄	—	2	2	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	2	1	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	1	no	—
457	C₄	—	4	1	no	—
458	C₄	C₈×D₄×C₆₇	1	2	yes	—
459	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
460	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
461	C₄	#576×C₆₇	1	1	yes	characteristic
462	C₄	#577×C₆₇	1	1	yes	characteristic
463	C₂	—	2	2	no	—
464	C₂	—	4	4	no	—
465	C₂	—	4	2	no	—
466	C₂	C₂×C₈×D₄×C₆₇	1	1	yes	characteristic
467	C₂	C₆₇×#2946	1	1	yes	characteristic
468	C₂	C₆₇×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₄	C₂³×C₈×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×AGL(1,ℤ/8)×C₆₇	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ #431×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2988	C₂	semidirect
2. C₆₇×#2972	C₂	semidirect
4. C₆₇×#1580	C₂	semidirect
5. C₆₇×SD₁₆⋊C₈	C₂	semidirect
6. C₈×SD₁₆×C₆₇	C₂	semidirect
7. C₈×D₈×C₆₇	C₂	semidirect
8. C₆₇×#1565	C₂	semidirect
9. C₆₇×#1476	C₂	semidirect
10. C₂×D₄⋊C₈×C₆₇	C₂	semidirect
11. C₈×M₁₆×C₆₇	C₂	semidirect
15. C₈×Q₈×C₆₇	C₂²	semidirect
25. C₈×D₄×C₆₇	C₂²	semidirect
37. #335×C₆₇	C₂²	semidirect
38. #334×C₆₇	C₂²	semidirect
39. Q₈⋊C₈×C₆₇	C₂²	semidirect
40. Q₈⋊C₈×C₆₇	C₂²	semidirect
43. D₄⋊C₈×C₆₇	C₂²	semidirect
44. D₄⋊C₈×C₆₇	C₂²	semidirect
45. #322×C₆₇	C₂²	semidirect
46. C₈²×C₆₇	C₂²	semidirect
50. AGL(1,ℤ/8)×C₆₇	C₈	semidirect
104. #41×C₆₇	C₂×C₈	semidirect
109. C₂×D₄×C₆₇	C₂×C₈	semidirect
123. SD₁₆×C₆₇	C₂×C₈	semidirect
124. D₈×C₆₇	C₂×C₈	semidirect
126. M₁₆×C₆₇	C₂×C₈	semidirect
228. #17852	C₆₇	direct (N × H)
235. #2988	C₂×C₆₇	semidirect
236. #2972	C₂×C₆₇	semidirect
238. #1580	C₂×C₆₇	semidirect
239. SD₁₆⋊C₈	C₂×C₆₇	semidirect
240. C₈×SD₁₆	C₂×C₆₇	semidirect
241. C₈×D₈	C₂×C₆₇	semidirect
242. #1565	C₂×C₆₇	semidirect
243. #1476	C₂×C₆₇	semidirect
244. C₂×D₄⋊C₈	C₂×C₆₇	semidirect
245. C₈×M₁₆	C₂×C₆₇	semidirect
246. C₆₇	#17852	direct (N × H)
250. C₈×Q₈	C₂²×C₆₇	semidirect
260. C₈×D₄	C₂²×C₆₇	semidirect
272. #335	C₂²×C₆₇	semidirect
273. #334	C₂²×C₆₇	semidirect
274. Q₈⋊C₈	C₂²×C₆₇	semidirect
275. Q₈⋊C₈	C₂²×C₆₇	semidirect
278. D₄⋊C₈	C₂²×C₆₇	semidirect
279. D₄⋊C₈	C₂²×C₆₇	semidirect
280. #322	C₂²×C₆₇	semidirect
281. C₈²	C₂²×C₆₇	semidirect
285. AGL(1,ℤ/8)	C₈×C₆₇	semidirect
339. #41	C₂×C₈×C₆₇	semidirect
344. C₂×D₄	C₂×C₈×C₆₇	semidirect
358. SD₁₆	C₂×C₈×C₆₇	semidirect
359. D₈	C₂×C₈×C₆₇	semidirect
361. M₁₆	C₂×C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2988	2
C₆₇×#2972	2
C₄×AGL(1,ℤ/8)×C₆₇	2
C₆₇×#1580	2
C₆₇×SD₁₆⋊C₈	2
C₈×SD₁₆×C₆₇	2
C₈×D₈×C₆₇	2
C₆₇×#1565	2
C₆₇×#1476	2
C₂×D₄⋊C₈×C₆₇	2
C₈×M₁₆×C₆₇	2
#17852	67

Subgroups — C67×#17852 #30697103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17852 #30697103