Subgroups — #30772292

Order 17152.

462 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₆₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₈×(C₂×D₆₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#22558286	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₆₇	—	2	2	no	—
12	C₂×C₁₆×D₆₇	—	2	2	no	—
13	C₂×C₁₆×D₆₇	—	2	2	no	—
14	C₂×(C₂×D₆₇)⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₄×(C₂×D₆₇)⋊C₄	C₄	1	1	yes	characteristic
17	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
18	Dic₆₇⋊C₁₆	—	2	2	no	—
19	(C₂×D₆₇)⋊C₁₆	—	2	2	no	—
20	#16863151	C₄	1	1	yes	characteristic
21	#16863150	C₄	1	1	yes	characteristic
22	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₆₇	—	2	1	no	—
29	C₂×C₈×D₆₇	—	2	1	no	—
30	C₂×C₈×D₆₇	—	2	1	no	—
31	C₂×C₈×D₆₇	—	2	1	no	—
32	C₂×C₈×D₆₇	D₄	1	2	yes	—
33	C₂×C₈×D₆₇	Q₈	1	2	yes	—
34	C₂×(C₂×D₆₇)⋊C₄	C₈	1	1	yes	characteristic
35	C₂×(C₂×D₆₇)⋊C₄	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₆₇⋊C₁₆	—	2	1	no	—
38	C₂×C₆₇⋊C₁₆	—	2	1	no	—
39	C₂×C₆₇⋊C₁₆	—	2	1	no	—
40	C₂×C₆₇⋊C₁₆	—	2	1	no	—
41	C₂×C₆₇⋊C₁₆	—	2	1	no	—
42	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
43	C₁₆×D₆₇	—	2	8	no	—
44	(C₂×D₆₇)⋊C₈	—	2	2	no	—
45	C₈×Dic₆₇	—	2	1	no	—
46	C₈×Dic₆₇	—	2	1	no	—
47	C₆₇⋊#95	C₈	1	1	yes	characteristic
48	C₆₇⋊#95	C₈	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₆₇	—	2	1	no	—
53	C₂×C₁₆×C₆₇	—	2	1	no	—
54	C₂×C₁₆×C₆₇	—	2	1	no	—
55	C₂×C₁₆×C₆₇	—	2	1	no	—
56	C₂×C₁₆×C₆₇	—	2	1	no	—
57	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
58	C₄×C₈×C₆₇	—	2	1	no	—
59	C₄×C₈×C₆₇	—	2	1	no	—
60	C₂³×D₆₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄×D₆₇	—	2	1	no	—
65	C₂×C₄×D₆₇	—	2	1	no	—
66	C₂×C₄×D₆₇	—	2	1	no	—
67	C₂×C₄×D₆₇	—	2	1	no	—
68	C₂×C₄×D₆₇	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
70	C₂×C₆₇⋊C₈	—	2	1	no	—
71	C₂×C₆₇⋊C₈	—	2	1	no	—
72	C₂×C₆₇⋊C₈	—	2	1	no	—
73	C₂×C₆₇⋊C₈	—	2	1	no	—
74	C₂×C₆₇⋊C₈	—	2	1	no	—
75	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
76	C₈×D₆₇	—	2	2	no	—
77	C₈×D₆₇	—	2	2	no	—
78	C₈×D₆₇	—	2	2	no	—
79	C₈×D₆₇	—	2	2	no	—
80	(C₂×D₆₇)⋊C₄	—	2	2	no	—
81	(C₂×D₆₇)⋊C₄	C₁₆	1	4	yes	complemented
82	C₄×Dic₆₇	—	2	1	no	—
83	C₄×Dic₆₇	—	2	1	no	—
84	C₆₇⋊C₁₆	—	2	2	no	—
85	C₆₇⋊C₁₆	—	4	1	no	—
86	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	—	2	1	no	—
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
97	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
98	C₄²×C₆₇	—	2	1	no	—
99	C₄²×C₆₇	—	2	1	no	—
100	C₁₆×C₆₇	—	2	2	no	—
101	C₁₆×C₆₇	—	4	1	no	—
102	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₆₇	—	2	1	no	—
104	C₂²×D₆₇	—	2	1	no	—
105	C₂²×D₆₇	—	2	1	no	—
106	C₂²×D₆₇	—	2	1	no	—
107	C₂²×D₆₇	C₄⋊C₈	1	2	yes	—
108	C₂²×D₆₇	C₂×C₁₆	1	2	yes	—
109	C₂²×D₆₇	M₃₂	1	2	yes	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	complemented
116	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
122	C₂×Dic₆₇	—	2	1	no	—
123	C₂×Dic₆₇	—	2	1	no	—
124	C₂×Dic₆₇	—	2	1	no	—
125	C₂×Dic₆₇	—	2	1	no	—
126	C₂×Dic₆₇	—	2	1	no	—
127	C₂×Dic₆₇	—	2	1	no	—
128	C₂×Dic₆₇	—	2	1	no	—
129	C₂×Dic₆₇	C₄⋊C₈	1	2	yes	—
130	C₂×Dic₆₇	C₂×C₁₆	1	2	yes	complemented
131	C₄×D₆₇	—	2	2	no	—
132	C₄×D₆₇	—	2	2	no	—
133	C₄×D₆₇	—	2	2	no	—
134	C₄×D₆₇	—	2	2	no	—
135	C₈×C₆₇	—	4	1	no	—
136	C₈×C₆₇	#120	1	1	yes	characteristic
137	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
138	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
139	C₈×C₆₇	#123	1	1	yes	characteristic
140	C₆₇⋊C₈	—	2	2	no	—
141	C₆₇⋊C₈	—	4	1	no	—
142	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
143	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
144	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
145	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
147	C₂²×C₆₇	#504	1	1	yes	characteristic
148	C₂²×C₆₇	#504	1	1	yes	characteristic
149	C₂×D₆₇	—	2	2	no	—
150	C₂×D₆₇	—	2	2	no	—
151	C₂×D₆₇	—	2	2	no	—
152	C₂×D₆₇	—	2	2	no	—
153	C₂×D₆₇	—	2	2	no	—
154	C₂×D₆₇	—	2	2	no	—
155	C₂×D₆₇	C₄⋊C₁₆	1	4	yes	complemented
156	C₄×C₆₇	—	2	2	no	—
157	C₄×C₆₇	—	4	1	no	—
158	C₄×C₆₇	#385	1	1	yes	characteristic
159	C₄×C₆₇	#424	1	1	yes	characteristic
160	C₄×C₆₇	#435	1	1	yes	characteristic
161	C₄×C₆₇	#435	1	1	yes	characteristic
162	Dic₆₇	—	2	2	no	—
163	Dic₆₇	—	2	2	no	—
164	Dic₆₇	—	4	1	no	—
165	#12912	—	67	1	no	maximal, Sylow
166	C₂×C₆₇	#1850	1	1	yes	characteristic
167	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
168	C₂×C₆₇	#2151	1	1	yes	characteristic
169	C₂×C₆₇	D₄×C₁₆	1	1	yes	characteristic
170	C₂×C₆₇	D₄×C₁₆	1	1	yes	characteristic
171	C₂×C₆₇	#2168	1	1	yes	characteristic
172	C₂×C₆₇	#2168	1	1	yes	characteristic
173	D₆₇	—	2	4	no	—
174	C₂³×C₁₆	—	67	1	no	—
175	C₂×C₄⋊C₁₆	—	67	1	no	—
176	C₂×C₄⋊C₁₆	—	67	1	no	—
177	C₂×C₂²⋊C₁₆	—	67	1	no	—
178	C₈×C₂²⋊C₄	—	67	1	no	—
179	#1374	—	67	1	no	—
180	#1374	D₆₇	1	1	yes	characteristic, complemented
181	C₆₇	#12912	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₈	—	67	1	no	—
183	C₂²×C₁₆	—	67	1	no	—
184	C₂²×C₁₆	—	67	1	no	—
185	C₂²×C₁₆	—	134	2	no	—
186	C₂²×C₁₆	—	134	2	no	—
187	C₂²×C₁₆	—	134	2	no	—
188	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
189	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
190	C₂×C₂²⋊C₈	—	67	1	no	—
191	C₂×C₄×C₈	—	67	1	no	—
192	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
193	C₄×C₂²⋊C₄	—	67	1	no	—
194	C₄⋊C₁₆	—	134	2	no	—
195	C₄⋊C₁₆	—	134	2	no	—
196	C₂²⋊C₁₆	—	134	2	no	—
197	#336	—	67	1	no	—
198	#336	—	67	1	no	—
199	C₂³×C₄	—	67	1	no	—
200	C₂²×C₈	—	67	1	no	—
201	C₂²×C₈	—	67	2	no	—
202	C₂²×C₈	—	67	2	no	—
203	C₂²×C₈	—	67	1	no	—
204	C₂²×C₈	—	134	1	no	—
205	C₂²×C₈	—	134	1	no	—
206	C₂²×C₈	—	134	1	no	—
207	C₂²×C₈	—	134	1	no	—
208	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	67	1	no	—
211	C₂×C₂²⋊C₄	—	67	1	no	—
212	C₂×C₄²	—	67	1	no	—
213	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	2	1	no	—
219	C₂×C₁₆	—	67	2	no	—
220	C₂×C₁₆	—	134	8	no	—
221	C₂×C₁₆	—	134	1	no	—
222	C₂×C₁₆	—	134	1	no	—
223	C₂×C₁₆	—	134	1	no	—
224	C₂×C₁₆	—	134	1	no	—
225	C₂×C₁₆	—	134	1	no	—
226	C₂×C₁₆	D₂₆₈	1	2	yes	—
227	C₂²⋊C₈	—	134	2	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	134	1	no	—
231	C₄×C₈	—	134	1	no	—
232	#95	—	67	1	no	—
233	#95	—	67	1	no	—
234	C₂⁴	—	67	1	no	—
235	C₂²×C₄	—	67	1	no	—
236	C₂²×C₄	—	67	1	no	—
237	C₂²×C₄	—	67	2	no	—
238	C₂²×C₄	—	67	2	no	—
239	C₂²×C₄	—	67	1	no	—
240	C₂²×C₄	—	134	1	no	—
241	C₂²×C₄	—	134	1	no	—
242	C₂²×C₄	—	134	1	no	—
243	C₂²×C₄	—	134	1	no	—
244	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	67	2	no	—
251	C₂×C₈	—	134	1	no	—
252	C₂×C₈	—	134	1	no	—
253	C₂×C₈	—	134	1	no	—
254	C₂×C₈	—	134	1	no	—
255	C₂×C₈	—	134	1	no	—
256	C₂×C₈	—	134	2	no	—
257	C₂×C₈	—	134	2	no	—
258	C₂×C₈	—	134	2	no	—
259	C₂×C₈	—	134	2	no	—
260	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
261	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
262	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
263	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
264	C₂×C₈	#11761627	1	1	yes	characteristic
265	C₂×C₈	#11761628	1	1	yes	characteristic
266	C₂²⋊C₄	—	67	4	no	—
267	C₂²⋊C₄	—	134	2	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	134	1	no	—
271	C₄²	—	134	1	no	—
272	C₁₆	—	2	2	no	—
273	C₁₆	—	4	1	no	—
274	C₁₆	—	268	1	no	—
275	C₁₆	—	134	2	no	—
276	C₂³	—	67	2	no	—
277	C₂³	—	67	2	no	—
278	C₂³	—	67	2	no	—
279	C₂³	—	134	1	no	—
280	C₂³	—	134	1	no	—
281	C₂³	—	134	1	no	—
282	C₂³	—	134	1	no	—
283	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	67	2	no	—
290	C₂×C₄	—	67	2	no	—
291	C₂×C₄	—	134	1	no	—
292	C₂×C₄	—	134	1	no	—
293	C₂×C₄	—	134	1	no	—
294	C₂×C₄	—	134	1	no	—
295	C₂×C₄	—	134	1	no	—
296	C₂×C₄	—	134	1	no	—
297	C₂×C₄	—	134	1	no	—
298	C₂×C₄	—	134	2	no	—
299	C₂×C₄	—	134	2	no	—
300	C₂×C₄	—	134	2	no	—
301	C₂×C₄	—	134	2	no	—
302	C₂×C₄	C₁₆×D₆₇	1	2	yes	complemented
303	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
304	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
306	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
307	C₂×C₄	#14524049	1	1	yes	characteristic
308	C₂×C₄	#14524051	1	1	yes	characteristic
309	C₈	—	4	1	no	—
310	C₈	—	268	1	no	—
311	C₈	—	134	2	no	—
312	C₈	#14524013	1	1	yes	characteristic
313	C₈	#14524021	1	1	yes	characteristic
314	C₈	#14524023	1	1	yes	characteristic
315	C₈	#14524025	1	1	yes	characteristic
316	C₂²	—	67	4	no	—
317	C₂²	—	134	2	no	—
318	C₂²	—	134	2	no	—
319	C₂²	—	134	2	no	—
320	C₂²	—	134	2	no	—
321	C₂²	—	134	2	no	—
322	C₂²	—	134	2	no	—
323	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
324	C₂²	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
325	C₂²	C₈×D₂₆₈	1	1	yes	characteristic
326	C₂²	C₂×C₁₆×D₆₇	1	1	yes	characteristic
327	C₂²	M₃₂×D₆₇	1	1	yes	characteristic
328	C₂²	#16863812	1	1	yes	characteristic
329	C₂²	#16863814	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	4	1	no	—
332	C₄	—	268	1	no	—
333	C₄	—	134	2	no	—
334	C₄	—	134	2	no	—
335	C₄	#16863270	1	1	yes	characteristic
336	C₄	#16863421	1	1	yes	characteristic
337	C₄	#16863469	1	1	yes	characteristic
338	C₄	#16863470	1	1	yes	characteristic
339	C₂	—	134	4	no	—
340	C₂	#22560388	1	1	yes	characteristic
341	C₂	C₄⋊C₁₆×D₆₇	1	1	yes	characteristic
342	C₂	#22561966	1	1	yes	characteristic
343	C₂	(C₂²⋊D₆₇)⋊C₁₆	1	1	yes	characteristic
344	C₂	C₁₆×D₂₆₈	1	1	yes	characteristic
345	C₂	#22562036	1	1	yes	characteristic
346	C₂	#22562037	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₁₆
Fitting	C₆₇×#1374	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂³×D₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1374	D₆₇
hypocenter	C₆₇	#12912
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
3. C₂×Dic₆₇⋊C₁₆	C₂	semidirect
6. #22558286	C₂	semidirect
7. C₆₇×#1374	C₂	semidirect
81. (C₂×D₆₇)⋊C₄	C₁₆	semidirect
115. C₂×C₄×C₆₇	C₂×C₁₆	semidirect
130. C₂×Dic₆₇	C₂×C₁₆	semidirect
155. C₂×D₆₇	C₄⋊C₁₆	semidirect
180. #1374	D₆₇	semidirect
181. C₆₇	#12912	semidirect
302. C₂×C₄	C₁₆×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₆₇	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×Dic₆₇⋊C₁₆	2
C₂×(C₂×D₆₇)⋊C₁₆	2
C₈×(C₂×D₆₇)⋊C₄	2
#22558286	2
C₆₇×#1374	2
#12912	67