Subgroups — C₆₇×#17872 #30697123

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#137×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1587	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1585	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1584	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1582	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1477	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	C₄	1	1	yes	characteristic
13	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#446×C₆₇	—	2	1	no	—
16	#446×C₆₇	—	2	1	no	—
17	#446×C₆₇	C₂²	1	2	yes	—
18	#446×C₆₇	C₂²	1	2	yes	—
19	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	#441×C₆₇	C₄	1	2	yes	—
22	#440×C₆₇	C₄	1	2	yes	—
23	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
24	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
25	C₈×D₄×C₆₇	—	2	1	no	—
26	C₈×D₄×C₆₇	—	2	1	no	—
27	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	#431×C₆₇	—	2	1	no	—
30	#431×C₆₇	C₂²	1	1	yes	characteristic
31	#431×C₆₇	C₂²	1	1	yes	characteristic
32	#428×C₆₇	C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#419×C₆₇	C₄	1	1	yes	characteristic
36	#417×C₆₇	C₄	1	1	yes	characteristic
37	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#97×C₆₇	—	2	1	no	—
39	C₂×C₄×C₈×C₆₇	—	2	1	no	—
40	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	#335×C₆₇	C₂²	1	2	yes	—
42	#334×C₆₇	C₂²	1	2	yes	—
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
45	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
46	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
47	#322×C₆₇	C₂²	1	2	yes	—
48	#322×C₆₇	C₂²	1	2	yes	—
49	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	#137×C₆₇	—	2	4	no	—
52	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₂²×C₈×C₆₇	—	2	1	no	—
58	C₂²×C₈×C₆₇	—	2	1	no	—
59	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₄×Q₈×C₆₇	—	2	1	no	—
62	C₄×Q₈×C₆₇	—	2	1	no	—
63	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
64	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
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	C₂×C₄	1	1	yes	characteristic
70	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
71	#117×C₆₇	—	2	1	no	—
72	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	—	2	1	no	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	C₂×C₄	1	2	yes	—
79	#106×C₆₇	C₂×C₄	1	2	yes	—
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	C₄⋊C₈×C₆₇	—	2	2	no	—
82	C₄⋊C₈×C₆₇	—	2	1	no	—
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	C₂³	1	2	yes	—
87	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
89	#103×C₆₇	C₂×C₄	1	2	yes	—
90	#103×C₆₇	C₂×C₄	1	2	yes	—
91	#103×C₆₇	C₂×C₄	1	2	yes	—
92	#102×C₆₇	C₂×C₄	1	2	yes	—
93	C₂²⋊C₈×C₆₇	—	2	1	no	—
94	C₂²⋊C₈×C₆₇	—	2	1	no	—
95	C₂²⋊C₈×C₆₇	—	2	1	no	—
96	C₂²⋊C₈×C₆₇	—	2	1	no	—
97	#97×C₆₇	—	2	2	no	—
98	#97×C₆₇	—	2	2	no	—
99	#97×C₆₇	C₂×C₄	1	2	yes	—
100	C₄×C₈×C₆₇	—	2	1	no	—
101	C₄×C₈×C₆₇	—	2	1	no	—
102	C₄×C₈×C₆₇	—	2	1	no	—
103	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
104	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
105	C₄×C₈×C₆₇	C₂³	1	2	yes	—
106	C₄×C₈×C₆₇	C₂³	1	2	yes	—
107	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
109	#41×C₆₇	—	2	2	no	—
110	#41×C₆₇	M₁₆	1	4	yes	—
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	C₂×Q₈×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	—	2	2	no	—
114	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
115	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×D₄×C₆₇	—	2	1	no	—
117	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	—	2	1	no	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
126	Q₁₆×C₆₇	—	2	4	no	—
127	SD₁₆×C₆₇	—	2	4	no	—
128	M₁₆×C₆₇	—	2	2	no	—
129	M₁₆×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	4	1	no	—
136	C₂×C₈×C₆₇	—	4	2	no	—
137	C₂×C₈×C₆₇	—	4	2	no	—
138	C₂×C₈×C₆₇	—	4	2	no	—
139	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
141	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₄⋊C₄×C₆₇	—	2	2	no	—
146	C₄⋊C₄×C₆₇	—	2	1	no	—
147	C₄⋊C₄×C₆₇	—	2	1	no	—
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	1	no	—
150	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
151	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²⋊C₄×C₆₇	—	2	1	no	—
154	C₂²⋊C₄×C₆₇	—	2	1	no	—
155	C₂²⋊C₄×C₆₇	—	2	1	no	—
156	C₄²×C₆₇	—	2	1	no	—
157	C₄²×C₆₇	—	2	1	no	—
158	C₄²×C₆₇	—	2	2	no	—
159	C₄²×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	—	2	2	no	—
161	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₆₇	#41	1	1	yes	characteristic
164	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
165	C₂³×C₆₇	—	2	1	no	—
166	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
167	Q₈×C₆₇	—	2	1	no	—
168	Q₈×C₆₇	—	2	2	no	—
169	Q₈×C₆₇	—	2	2	no	—
170	Q₈×C₆₇	—	4	2	no	—
171	Q₈×C₆₇	C₂×M₁₆	1	2	yes	—
172	D₄×C₆₇	—	2	4	no	—
173	D₄×C₆₇	—	2	1	no	—
174	D₄×C₆₇	C₂×M₁₆	1	2	yes	—
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	4	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	1	no	—
188	C₂×C₄×C₆₇	—	4	1	no	—
189	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
193	C₂×C₄×C₆₇	#131	1	2	yes	—
194	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
197	C₈×C₆₇	—	2	2	no	—
198	C₈×C₆₇	—	2	2	no	—
199	C₈×C₆₇	—	2	1	no	—
200	C₈×C₆₇	—	2	1	no	—
201	C₈×C₆₇	—	2	1	no	—
202	C₈×C₆₇	—	4	2	no	—
203	C₈×C₆₇	—	4	1	no	—
204	C₈×C₆₇	—	4	2	no	—
205	C₈×C₆₇	—	4	1	no	—
206	C₈×C₆₇	#137	1	2	yes	complemented
207	C₂²×C₆₇	—	2	1	no	—
208	C₂²×C₆₇	—	2	1	no	—
209	C₂²×C₆₇	—	2	2	no	—
210	C₂²×C₆₇	—	4	1	no	—
211	C₂²×C₆₇	—	4	1	no	—
212	C₂²×C₆₇	#435	1	2	yes	—
213	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
214	C₄×C₆₇	—	2	2	no	—
215	C₄×C₆₇	—	2	1	no	—
216	C₄×C₆₇	—	2	1	no	—
217	C₄×C₆₇	—	2	1	no	—
218	C₄×C₆₇	—	4	2	no	—
219	C₄×C₆₇	—	4	1	no	—
220	C₄×C₆₇	—	4	1	no	—
221	C₄×C₆₇	—	4	2	no	—
222	C₄×C₆₇	#435	1	2	yes	—
223	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
224	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
225	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
226	C₄×C₆₇	#575	1	1	yes	characteristic
227	#17872	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂×C₆₇	—	2	2	no	—
229	C₂×C₆₇	—	4	2	no	—
230	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
231	C₂×C₆₇	C₄×#137	1	1	yes	characteristic
232	C₂×C₆₇	#2955	1	1	yes	characteristic
233	C₈×#41	C₂×C₆₇	1	1	yes	characteristic
234	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
235	C₄×#137	C₂×C₆₇	1	1	yes	characteristic
236	#1587	C₂×C₆₇	1	2	yes	complemented
237	#1585	C₂×C₆₇	1	2	yes	complemented
238	#1584	C₂×C₆₇	1	2	yes	complemented
239	#1582	C₂×C₆₇	1	2	yes	complemented
240	#1565	C₂×C₆₇	1	1	yes	characteristic, complemented
241	#1479	C₂×C₆₇	1	1	yes	characteristic, complemented
242	#1477	C₂×C₆₇	1	1	yes	characteristic
243	#1451	C₂×C₆₇	1	1	yes	characteristic, complemented
244	C₆₇	#17872	1	1	yes	characteristic, direct product factor, Sylow
245	C₂×#137	C₄×C₆₇	1	1	yes	characteristic
246	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
248	#446	—	2	1	no	—
249	#446	—	2	1	no	—
250	#446	C₂²×C₆₇	1	2	yes	—
251	#446	C₂²×C₆₇	1	2	yes	—
252	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
253	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
254	#441	C₄×C₆₇	1	2	yes	—
255	#440	C₄×C₆₇	1	2	yes	—
256	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
257	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
258	C₈×D₄	—	2	1	no	—
259	C₈×D₄	—	2	1	no	—
260	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
261	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
262	#431	—	2	1	no	—
263	#431	C₂²×C₆₇	1	1	yes	characteristic
264	#431	C₂²×C₆₇	1	1	yes	characteristic
265	#428	C₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄⋊C₈	—	2	1	no	—
267	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
268	#419	C₄×C₆₇	1	1	yes	characteristic
269	#417	C₄×C₆₇	1	1	yes	characteristic
270	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×#97	—	2	1	no	—
272	C₂×C₄×C₈	—	2	1	no	—
273	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
274	#335	C₂²×C₆₇	1	2	yes	—
275	#334	C₂²×C₆₇	1	2	yes	—
276	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
277	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
278	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
279	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
280	#322	C₂²×C₆₇	1	2	yes	—
281	#322	C₂²×C₆₇	1	2	yes	—
282	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
283	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
284	#137	—	2	4	no	—
285	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
286	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
287	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
288	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
289	C₂²×C₈	—	2	1	no	—
290	C₂²×C₈	—	2	1	no	—
291	C₂²×C₈	—	2	1	no	—
292	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
293	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
294	C₄×Q₈	—	2	1	no	—
295	C₄×Q₈	—	2	1	no	—
296	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
297	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
298	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
299	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	—	2	1	no	—
302	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
303	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
304	#117	—	2	1	no	—
305	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
308	C₂×C₄²	—	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	2	no	—
315	C₄⋊C₈	—	2	1	no	—
316	C₄⋊C₈	—	2	1	no	—
317	C₄⋊C₈	—	2	1	no	—
318	C₄⋊C₈	—	2	1	no	—
319	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
320	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
321	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
322	#103	C₂×C₄×C₆₇	1	2	yes	—
323	#103	C₂×C₄×C₆₇	1	2	yes	—
324	#103	C₂×C₄×C₆₇	1	2	yes	—
325	#102	C₂×C₄×C₆₇	1	2	yes	—
326	C₂²⋊C₈	—	2	1	no	—
327	C₂²⋊C₈	—	2	1	no	—
328	C₂²⋊C₈	—	2	1	no	—
329	C₂²⋊C₈	—	2	1	no	—
330	#97	—	2	2	no	—
331	#97	—	2	2	no	—
332	#97	C₂×C₄×C₆₇	1	2	yes	—
333	C₄×C₈	—	2	1	no	—
334	C₄×C₈	—	2	1	no	—
335	C₄×C₈	—	2	1	no	—
336	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
337	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
338	C₄×C₈	C₂³×C₆₇	1	2	yes	—
339	C₄×C₈	C₂³×C₆₇	1	2	yes	—
340	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
341	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
342	#41	—	2	2	no	—
343	#41	M₁₆×C₆₇	1	4	yes	—
344	C₂×Q₈	—	2	1	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
348	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
359	Q₁₆	—	2	4	no	—
360	SD₁₆	—	2	4	no	—
361	M₁₆	—	2	2	no	—
362	M₁₆	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	4	1	no	—
369	C₂×C₈	—	4	2	no	—
370	C₂×C₈	—	4	2	no	—
371	C₂×C₈	—	4	2	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	1	yes	characteristic
375	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
376	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
377	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
384	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
385	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
386	C₂²⋊C₄	—	2	1	no	—
387	C₂²⋊C₄	—	2	1	no	—
388	C₂²⋊C₄	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	—	2	1	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	1	no	—
393	C₄²	—	2	2	no	—
394	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
395	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
396	C₄²	#41×C₆₇	1	1	yes	characteristic
397	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
398	C₂³	—	2	1	no	—
399	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
400	Q₈	—	2	1	no	—
401	Q₈	—	2	2	no	—
402	Q₈	—	2	2	no	—
403	Q₈	—	4	2	no	—
404	Q₈	C₂×M₁₆×C₆₇	1	2	yes	—
405	D₄	—	2	4	no	—
406	D₄	—	2	1	no	—
407	D₄	C₂×M₁₆×C₆₇	1	2	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	—	2	2	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
423	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
424	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
425	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
426	C₂×C₄	#131×C₆₇	1	2	yes	—
427	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
428	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
429	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
430	C₈	—	2	2	no	—
431	C₈	—	2	2	no	—
432	C₈	—	2	1	no	—
433	C₈	—	2	1	no	—
434	C₈	—	2	1	no	—
435	C₈	—	4	2	no	—
436	C₈	—	4	1	no	—
437	C₈	—	4	1	no	—
438	C₈	—	4	2	no	—
439	C₈	#137×C₆₇	1	2	yes	complemented
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	2	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	4	1	no	—
445	C₂²	#435×C₆₇	1	2	yes	—
446	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
447	C₄	—	2	2	no	—
448	C₄	—	2	1	no	—
449	C₄	—	2	1	no	—
450	C₄	—	2	1	no	—
451	C₄	—	4	2	no	—
452	C₄	—	4	1	no	—
453	C₄	—	4	1	no	—
454	C₄	—	4	2	no	—
455	C₄	#435×C₆₇	1	2	yes	—
456	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
457	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
458	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
459	C₄	#575×C₆₇	1	1	yes	characteristic
460	C₂	—	2	2	no	—
461	C₂	—	4	2	no	—
462	C₂	C₂×#435×C₆₇	1	1	yes	characteristic
463	C₂	C₄×#137×C₆₇	1	1	yes	characteristic
464	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×#41×C₆₇	C₂×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
2. C₂×#446×C₆₇	C₂	semidirect
4. C₆₇×#1587	C₂	semidirect
5. C₆₇×#1585	C₂	semidirect
6. C₆₇×#1584	C₂	semidirect
7. C₆₇×#1582	C₂	semidirect
8. C₆₇×#1565	C₂	semidirect
9. C₆₇×#1479	C₂	semidirect
11. C₆₇×#1451	C₂	semidirect
206. C₈×C₆₇	#137	semidirect
227. #17872	C₆₇	direct (N × H)
234. C₂×#446	C₂×C₆₇	semidirect
236. #1587	C₂×C₆₇	semidirect
237. #1585	C₂×C₆₇	semidirect
238. #1584	C₂×C₆₇	semidirect
239. #1582	C₂×C₆₇	semidirect
240. #1565	C₂×C₆₇	semidirect
241. #1479	C₂×C₆₇	semidirect
243. #1451	C₂×C₆₇	semidirect
244. C₆₇	#17872	direct (N × H)
439. C₈	#137×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41×C₆₇	2
C₂×#446×C₆₇	2
C₄×#137×C₆₇	2
C₆₇×#1587	2
C₆₇×#1585	2
C₆₇×#1584	2
C₆₇×#1582	2
C₆₇×#1565	2
C₆₇×#1479	2
C₆₇×#1477	2
C₆₇×#1451	2
#17872	67

Subgroups — C67×#17872 #30697123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17872 #30697123