Subgroups — C₆₇×#17921 #30697172

Order 17152.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3107	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2988	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2985	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1587	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1585	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1584	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1582	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1579	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#1573	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#1488	C₂	1	1	yes	maximal, characteristic
14	C₆₇×#1481	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1454	C₂	1	1	yes	maximal, characteristic, complemented
16	#533×C₆₇	C₂²	1	1	yes	characteristic
17	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
18	#485×C₆₇	C₄	1	1	yes	characteristic
19	#485×C₆₇	C₄	1	1	yes	characteristic
20	#475×C₆₇	C₄	1	1	yes	characteristic
21	#474×C₆₇	C₄	1	1	yes	characteristic
22	#462×C₆₇	C₄	1	1	yes	characteristic
23	#461×C₆₇	C₄	1	1	yes	characteristic
24	#452×C₆₇	C₄	1	1	yes	characteristic
25	#452×C₆₇	C₄	1	1	yes	characteristic
26	#446×C₆₇	—	2	1	no	—
27	#446×C₆₇	—	2	1	no	—
28	#446×C₆₇	C₂²	1	1	yes	characteristic
29	#446×C₆₇	C₂²	1	1	yes	characteristic
30	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	C₈×D₄×C₆₇	—	2	1	no	—
39	C₈×D₄×C₆₇	—	2	1	no	—
40	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
42	#433×C₆₇	—	2	1	no	—
43	#433×C₆₇	—	2	1	no	—
44	#432×C₆₇	C₂²	1	1	yes	characteristic
45	#431×C₆₇	—	2	1	no	—
46	#431×C₆₇	C₂²	1	1	yes	characteristic
47	#431×C₆₇	C₂²	1	1	yes	characteristic
48	#431×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
50	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
51	#335×C₆₇	C₂²	1	1	yes	characteristic
52	#335×C₆₇	C₂²	1	1	yes	characteristic
53	#334×C₆₇	C₂²	1	1	yes	characteristic
54	#334×C₆₇	C₂²	1	1	yes	characteristic
55	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
56	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
57	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
58	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
59	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
60	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
61	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
62	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
63	#322×C₆₇	C₂²	1	1	yes	characteristic
64	#322×C₆₇	C₂²	1	1	yes	characteristic
65	#322×C₆₇	C₂²	1	1	yes	characteristic
66	C₈²×C₆₇	C₂²	1	1	yes	characteristic
67	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	—	2	1	no	—
73	C₂²×C₈×C₆₇	—	2	1	no	—
74	C₂²×C₈×C₆₇	—	2	1	no	—
75	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
77	#125×C₆₇	—	2	1	no	—
78	C₂²⋊Q₈×C₆₇	—	2	1	no	—
79	C₂²⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₄×Q₈×C₆₇	—	2	1	no	—
82	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄×D₄×C₆₇	—	2	1	no	—
87	C₄×D₄×C₆₇	—	2	1	no	—
88	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
90	#117×C₆₇	—	2	1	no	—
91	#117×C₆₇	—	2	1	no	—
92	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	—	2	1	no	—
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
96	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
97	#106×C₆₇	C₂×C₄	1	1	yes	characteristic
98	#106×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₄⋊C₈×C₆₇	—	2	1	no	—
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	—	2	1	no	—
104	C₄⋊C₈×C₆₇	—	2	1	no	—
105	C₄⋊C₈×C₆₇	—	2	1	no	—
106	C₄⋊C₈×C₆₇	—	2	1	no	—
107	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
108	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
110	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
111	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
112	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
113	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
114	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
115	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
116	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
117	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
118	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
119	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
120	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
121	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
122	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
123	C₂²⋊C₈×C₆₇	—	2	1	no	—
124	C₂²⋊C₈×C₆₇	—	2	1	no	—
125	C₂²⋊C₈×C₆₇	—	2	1	no	—
126	C₂²⋊C₈×C₆₇	—	2	1	no	—
127	C₂²⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
128	C₂²⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
129	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
130	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
131	#97×C₆₇	—	2	1	no	—
132	#97×C₆₇	—	2	1	no	—
133	#97×C₆₇	—	2	1	no	—
134	C₄×C₈×C₆₇	—	2	1	no	—
135	C₄×C₈×C₆₇	—	2	1	no	—
136	C₄×C₈×C₆₇	—	2	1	no	—
137	C₄×C₈×C₆₇	—	2	1	no	—
138	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
144	#41×C₆₇	—	2	2	no	—
145	#41×C₆₇	—	2	2	no	—
146	C₂×Q₈×C₆₇	—	2	1	no	—
147	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₂×D₄×C₆₇	—	2	1	no	—
151	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	—	2	1	no	—
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
158	Q₁₆×C₆₇	—	2	2	no	—
159	Q₁₆×C₆₇	—	2	2	no	—
160	SD₁₆×C₆₇	—	2	2	no	—
161	SD₁₆×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	1	no	—
163	C₂×C₈×C₆₇	—	2	1	no	—
164	C₂×C₈×C₆₇	—	2	1	no	—
165	C₂×C₈×C₆₇	—	2	1	no	—
166	C₂×C₈×C₆₇	—	2	1	no	—
167	C₂×C₈×C₆₇	—	2	1	no	—
168	C₂×C₈×C₆₇	—	8	1	no	—
169	C₂×C₈×C₆₇	—	8	1	no	—
170	C₂×C₈×C₆₇	—	8	1	no	—
171	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₄⋊C₄×C₆₇	—	2	1	no	—
180	C₄⋊C₄×C₆₇	—	2	1	no	—
181	C₄⋊C₄×C₆₇	—	2	1	no	—
182	C₄⋊C₄×C₆₇	—	2	1	no	—
183	C₄⋊C₄×C₆₇	—	2	1	no	—
184	C₄⋊C₄×C₆₇	—	2	1	no	—
185	C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₄⋊C₄×C₆₇	—	2	1	no	—
187	C₄⋊C₄×C₆₇	—	2	1	no	—
188	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
189	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
192	C₂²⋊C₄×C₆₇	—	2	1	no	—
193	C₂²⋊C₄×C₆₇	—	2	1	no	—
194	C₂²⋊C₄×C₆₇	—	2	1	no	—
195	C₂²⋊C₄×C₆₇	—	2	1	no	—
196	C₂²⋊C₄×C₆₇	—	2	1	no	—
197	C₄²×C₆₇	—	2	1	no	—
198	C₄²×C₆₇	—	2	1	no	—
199	C₄²×C₆₇	—	2	1	no	—
200	C₄²×C₆₇	—	2	1	no	—
201	C₄²×C₆₇	—	2	1	no	—
202	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
203	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
204	C₄²×C₆₇	#41	1	1	yes	characteristic
205	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
206	C₂³×C₆₇	—	2	1	no	—
207	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
208	Q₈×C₆₇	—	2	1	no	—
209	Q₈×C₆₇	—	2	1	no	—
210	Q₈×C₆₇	—	2	1	no	—
211	Q₈×C₆₇	—	2	1	no	—
212	Q₈×C₆₇	—	2	1	no	—
213	Q₈×C₆₇	—	8	1	no	—
214	Q₈×C₆₇	#131	1	2	yes	—
215	D₄×C₆₇	—	2	4	no	—
216	D₄×C₆₇	—	2	1	no	—
217	D₄×C₆₇	#131	1	2	yes	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	1	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	1	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	4	1	no	—
231	C₂×C₄×C₆₇	—	4	1	no	—
232	C₂×C₄×C₆₇	—	4	1	no	—
233	C₂×C₄×C₆₇	—	8	1	no	—
234	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
240	C₈×C₆₇	—	2	1	no	—
241	C₈×C₆₇	—	2	1	no	—
242	C₈×C₆₇	—	2	1	no	—
243	C₈×C₆₇	—	2	1	no	—
244	C₈×C₆₇	—	2	1	no	—
245	C₈×C₆₇	—	2	1	no	—
246	C₈×C₆₇	—	2	1	no	—
247	C₈×C₆₇	—	2	1	no	—
248	C₈×C₆₇	—	4	1	no	—
249	C₈×C₆₇	—	4	1	no	—
250	C₈×C₆₇	—	4	1	no	—
251	C₈×C₆₇	—	4	1	no	—
252	C₈×C₆₇	—	4	1	no	—
253	C₈×C₆₇	—	4	1	no	—
254	C₂²×C₆₇	—	2	1	no	—
255	C₂²×C₆₇	—	2	2	no	—
256	C₂²×C₆₇	—	2	1	no	—
257	C₂²×C₆₇	—	2	1	no	—
258	C₂²×C₆₇	—	4	1	no	—
259	C₂²×C₆₇	—	4	1	no	—
260	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
261	C₄×C₆₇	—	2	1	no	—
262	C₄×C₆₇	—	2	2	no	—
263	C₄×C₆₇	—	2	1	no	—
264	C₄×C₆₇	—	2	1	no	—
265	C₄×C₆₇	—	2	1	no	—
266	C₄×C₆₇	—	4	1	no	—
267	C₄×C₆₇	—	4	1	no	—
268	C₄×C₆₇	—	4	1	no	—
269	C₄×C₆₇	—	4	1	no	—
270	C₄×C₆₇	—	4	1	no	—
271	C₄×C₆₇	—	4	1	no	—
272	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
273	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
274	C₄×C₆₇	#577	1	1	yes	characteristic
275	C₄×C₆₇	#578	1	1	yes	characteristic
276	#17921	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
277	C₂×C₆₇	—	4	1	no	—
278	C₂×C₆₇	—	4	2	no	—
279	C₂×C₆₇	#2929	1	1	yes	characteristic
280	C₂×C₆₇	#2947	1	1	yes	characteristic
281	C₂×C₆₇	#2957	1	1	yes	characteristic
282	#3107	C₂×C₆₇	1	1	yes	characteristic
283	#2988	C₂×C₆₇	1	1	yes	characteristic
284	#2985	C₂×C₆₇	1	1	yes	characteristic, complemented
285	#1587	C₂×C₆₇	1	1	yes	characteristic, complemented
286	#1585	C₂×C₆₇	1	1	yes	characteristic, complemented
287	#1584	C₂×C₆₇	1	1	yes	characteristic, complemented
288	#1582	C₂×C₆₇	1	1	yes	characteristic, complemented
289	#1579	C₂×C₆₇	1	1	yes	characteristic, complemented
290	SD₁₆⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
291	C₈×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
292	C₈×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
293	#1573	C₂×C₆₇	1	1	yes	characteristic, complemented
294	#1488	C₂×C₆₇	1	1	yes	characteristic
295	#1481	C₂×C₆₇	1	1	yes	characteristic, complemented
296	#1454	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₆₇	#17921	1	1	yes	characteristic, direct product factor, Sylow
298	#533	C₂²×C₆₇	1	1	yes	characteristic
299	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
300	#485	C₄×C₆₇	1	1	yes	characteristic
301	#485	C₄×C₆₇	1	1	yes	characteristic
302	#475	C₄×C₆₇	1	1	yes	characteristic
303	#474	C₄×C₆₇	1	1	yes	characteristic
304	#462	C₄×C₆₇	1	1	yes	characteristic
305	#461	C₄×C₆₇	1	1	yes	characteristic
306	#452	C₄×C₆₇	1	1	yes	characteristic
307	#452	C₄×C₆₇	1	1	yes	characteristic
308	#446	—	2	1	no	—
309	#446	—	2	1	no	—
310	#446	C₂²×C₆₇	1	1	yes	characteristic
311	#446	C₂²×C₆₇	1	1	yes	characteristic
312	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
313	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
314	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
315	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
316	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
317	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
318	C₄×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
319	C₄×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
320	C₈×D₄	—	2	1	no	—
321	C₈×D₄	—	2	1	no	—
322	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
323	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
324	#433	—	2	1	no	—
325	#433	—	2	1	no	—
326	#432	C₂²×C₆₇	1	1	yes	characteristic
327	#431	—	2	1	no	—
328	#431	C₂²×C₆₇	1	1	yes	characteristic
329	#431	C₂²×C₆₇	1	1	yes	characteristic
330	#431	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×C₄⋊C₈	—	2	1	no	—
332	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
333	#335	C₂²×C₆₇	1	1	yes	characteristic
334	#335	C₂²×C₆₇	1	1	yes	characteristic
335	#334	C₂²×C₆₇	1	1	yes	characteristic
336	#334	C₂²×C₆₇	1	1	yes	characteristic
337	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
338	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
339	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
340	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
341	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
342	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
343	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
344	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
345	#322	C₂²×C₆₇	1	1	yes	characteristic
346	#322	C₂²×C₆₇	1	1	yes	characteristic
347	#322	C₂²×C₆₇	1	1	yes	characteristic
348	C₈²	C₂²×C₆₇	1	1	yes	characteristic
349	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
350	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
351	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
352	C₂×SD₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
353	C₂×SD₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
354	C₂²×C₈	—	2	1	no	—
355	C₂²×C₈	—	2	1	no	—
356	C₂²×C₈	—	2	1	no	—
357	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
358	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
359	#125	—	2	1	no	—
360	C₂²⋊Q₈	—	2	1	no	—
361	C₂²⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
362	C₂²⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
363	C₄×Q₈	—	2	1	no	—
364	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
365	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
366	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
367	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
368	C₄×D₄	—	2	1	no	—
369	C₄×D₄	—	2	1	no	—
370	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
371	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
372	#117	—	2	1	no	—
373	#117	—	2	1	no	—
374	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
375	C₂×C₄²	—	2	1	no	—
376	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
377	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
378	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
379	#106	C₂×C₄×C₆₇	1	1	yes	characteristic
380	#106	C₂×C₄×C₆₇	1	1	yes	characteristic
381	C₄⋊C₈	—	2	1	no	—
382	C₄⋊C₈	—	2	1	no	—
383	C₄⋊C₈	—	2	1	no	—
384	C₄⋊C₈	—	2	1	no	—
385	C₄⋊C₈	—	2	1	no	—
386	C₄⋊C₈	—	2	1	no	—
387	C₄⋊C₈	—	2	1	no	—
388	C₄⋊C₈	—	2	1	no	—
389	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
390	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
391	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
392	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
393	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
394	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
395	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
396	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
397	#103	C₂×C₄×C₆₇	1	1	yes	characteristic
398	#103	C₂×C₄×C₆₇	1	1	yes	characteristic
399	#103	C₂×C₄×C₆₇	1	1	yes	characteristic
400	#103	C₂×C₄×C₆₇	1	1	yes	characteristic
401	#103	C₂×C₄×C₆₇	1	1	yes	characteristic
402	#103	C₂×C₄×C₆₇	1	1	yes	characteristic
403	#102	C₂×C₄×C₆₇	1	1	yes	characteristic
404	#102	C₂×C₄×C₆₇	1	1	yes	characteristic
405	C₂²⋊C₈	—	2	1	no	—
406	C₂²⋊C₈	—	2	1	no	—
407	C₂²⋊C₈	—	2	1	no	—
408	C₂²⋊C₈	—	2	1	no	—
409	C₂²⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
410	C₂²⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
411	C₂²⋊C₈	D₄×C₆₇	1	1	yes	characteristic
412	C₂²⋊C₈	D₄×C₆₇	1	1	yes	characteristic
413	#97	—	2	1	no	—
414	#97	—	2	1	no	—
415	#97	—	2	1	no	—
416	C₄×C₈	—	2	1	no	—
417	C₄×C₈	—	2	1	no	—
418	C₄×C₈	—	2	1	no	—
419	C₄×C₈	—	2	1	no	—
420	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
421	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
422	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
423	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
424	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
425	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
426	#41	—	2	2	no	—
427	#41	—	2	2	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
430	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
431	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
439	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
440	Q₁₆	—	2	2	no	—
441	Q₁₆	—	2	2	no	—
442	SD₁₆	—	2	2	no	—
443	SD₁₆	—	2	2	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	8	1	no	—
451	C₂×C₈	—	8	1	no	—
452	C₂×C₈	—	8	1	no	—
453	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
454	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
455	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
456	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
457	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
458	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
459	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
460	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
471	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
472	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
473	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
474	C₂²⋊C₄	—	2	1	no	—
475	C₂²⋊C₄	—	2	1	no	—
476	C₂²⋊C₄	—	2	1	no	—
477	C₂²⋊C₄	—	2	1	no	—
478	C₂²⋊C₄	—	2	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
485	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
486	C₄²	#41×C₆₇	1	1	yes	characteristic
487	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
488	C₂³	—	2	1	no	—
489	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
490	Q₈	—	2	1	no	—
491	Q₈	—	2	1	no	—
492	Q₈	—	2	1	no	—
493	Q₈	—	2	1	no	—
494	Q₈	—	2	1	no	—
495	Q₈	—	8	1	no	—
496	Q₈	#131×C₆₇	1	2	yes	—
497	D₄	—	2	4	no	—
498	D₄	—	2	1	no	—
499	D₄	#131×C₆₇	1	2	yes	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	8	1	no	—
516	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
517	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
518	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
519	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
521	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
522	C₈	—	2	1	no	—
523	C₈	—	2	1	no	—
524	C₈	—	2	1	no	—
525	C₈	—	2	1	no	—
526	C₈	—	2	1	no	—
527	C₈	—	2	1	no	—
528	C₈	—	2	1	no	—
529	C₈	—	2	1	no	—
530	C₈	—	4	1	no	—
531	C₈	—	4	1	no	—
532	C₈	—	4	1	no	—
533	C₈	—	4	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₂²	—	2	1	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	4	1	no	—
541	C₂²	—	4	1	no	—
542	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
543	C₄	—	2	1	no	—
544	C₄	—	2	2	no	—
545	C₄	—	2	1	no	—
546	C₄	—	2	1	no	—
547	C₄	—	2	1	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	1	no	—
550	C₄	—	4	1	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
555	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
556	C₄	#577×C₆₇	1	1	yes	characteristic
557	C₄	#578×C₆₇	1	1	yes	characteristic
558	C₂	—	4	1	no	—
559	C₂	—	4	2	no	—
560	C₂	C₆₇×#2929	1	1	yes	characteristic
561	C₂	C₆₇×#2947	1	1	yes	characteristic
562	C₂	C₆₇×#2957	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 ◁ #431×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2985	C₂	semidirect
4. C₆₇×#1587	C₂	semidirect
5. C₆₇×#1585	C₂	semidirect
6. C₆₇×#1584	C₂	semidirect
7. C₆₇×#1582	C₂	semidirect
8. C₆₇×#1579	C₂	semidirect
9. C₆₇×SD₁₆⋊C₈	C₂	semidirect
10. C₈×Q₁₆×C₆₇	C₂	semidirect
11. C₈×SD₁₆×C₆₇	C₂	semidirect
12. C₆₇×#1573	C₂	semidirect
14. C₆₇×#1481	C₂	semidirect
15. C₆₇×#1454	C₂	semidirect
276. #17921	C₆₇	direct (N × H)
284. #2985	C₂×C₆₇	semidirect
285. #1587	C₂×C₆₇	semidirect
286. #1585	C₂×C₆₇	semidirect
287. #1584	C₂×C₆₇	semidirect
288. #1582	C₂×C₆₇	semidirect
289. #1579	C₂×C₆₇	semidirect
290. SD₁₆⋊C₈	C₂×C₆₇	semidirect
291. C₈×Q₁₆	C₂×C₆₇	semidirect
292. C₈×SD₁₆	C₂×C₆₇	semidirect
293. #1573	C₂×C₆₇	semidirect
295. #1481	C₂×C₆₇	semidirect
296. #1454	C₂×C₆₇	semidirect
297. C₆₇	#17921	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3107	2
C₆₇×#2988	2
C₆₇×#2985	2
C₆₇×#1587	2
C₆₇×#1585	2
C₆₇×#1584	2
C₆₇×#1582	2
C₆₇×#1579	2
C₆₇×SD₁₆⋊C₈	2
C₈×Q₁₆×C₆₇	2
C₈×SD₁₆×C₆₇	2
C₆₇×#1573	2
C₆₇×#1488	2
C₆₇×#1481	2
C₆₇×#1454	2
#17921	67

Subgroups — C67×#17921 #30697172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17921 #30697172