Subgroups — #31009206

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₆₇	C₂	1	1	yes	maximal, characteristic
2	#22569072	C₂	1	2	yes	maximal
3	#22562081	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562080	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562079	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14523989	C₂	1	2	yes	maximal, complemented
7	#22562065	C₂	1	2	yes	maximal, complemented
8	#22561992	C₂	1	1	yes	maximal, characteristic
9	#22561927	C₂	1	2	yes	maximal
10	#22561807	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2176	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₇	C₂²	1	2	yes	—
13	#16863872	—	2	2	no	—
14	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic
15	C₂×#14523989	C₂²	1	2	yes	—
16	Q₃₂×D₆₇	C₄	1	8	yes	complemented
17	C₂×#14523982	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
19	#16863501	—	2	2	no	—
20	#16863500	—	2	2	no	—
21	C₄×Dic₂₆₈	C₂²	1	2	yes	—
22	Q₁₆×Dic₆₇	C₂²	1	2	yes	—
23	#16863490	C₂²	1	2	yes	—
24	#16863433	C₂²	1	2	yes	—
25	#16863403	—	2	2	no	—
26	#16863350	C₂²	1	1	yes	characteristic
27	#16863224	C₂²	1	1	yes	characteristic
28	#16863221	C₂²	1	2	yes	—
29	#16863201	C₂²	1	2	yes	—
30	#16863200	C₂²	1	2	yes	—
31	#16863199	C₂²	1	2	yes	—
32	#16863171	C₂²	1	1	yes	characteristic
33	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
37	#367×C₆₇	C₂²	1	1	yes	characteristic
38	#358×C₆₇	C₂²	1	2	yes	—
39	#346×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₆₇	—	2	2	no	—
41	C₂×Dic₂₆₈	C₂³	1	2	yes	—
42	C₂×#11761618	—	2	2	no	—
43	Q₁₆×D₆₇	—	2	4	no	—
44	Q₁₆×D₆₇	C₂×C₄	1	8	yes	—
45	C₂×#11761611	D₄	1	1	yes	characteristic
46	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
47	Q₈×Dic₆₇	—	2	2	no	—
48	C₄×Dic₁₃₄	—	2	2	no	—
49	C₄×Dic₁₃₄	—	4	2	no	—
50	#14524008	—	2	2	no	—
51	#14524008	—	4	2	no	—
52	#14524001	—	2	2	no	—
53	#14524001	—	4	2	no	—
54	#14524000	—	4	2	no	—
55	#14524000	D₄	1	1	yes	characteristic
56	Dic₅₃₆	C₂×C₄	1	4	yes	—
57	#14523989	C₂×C₄	1	8	yes	—
58	#14523982	—	2	2	no	—
59	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₁₆×D₆₇	C₂×C₄	1	4	yes	—
62	#14523977	C₂³	1	2	yes	—
63	#14523974	—	2	2	no	—
64	#14523968	—	2	2	no	—
65	#14523967	—	2	2	no	—
66	#14523966	—	2	2	no	—
67	#14523954	D₄	1	1	yes	characteristic
68	#14523953	D₄	1	1	yes	characteristic
69	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
71	C₄×Q₈×C₆₇	—	2	2	no	—
72	Q₃₂×C₆₇	C₂×C₄	1	4	yes	—
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	C₂³	1	2	yes	—
76	#103×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₃₄	—	2	2	no	—
79	C₂×Dic₁₃₄	—	4	2	no	—
80	Q₈×D₆₇	—	2	8	no	—
81	Q₈×D₆₇	—	4	4	no	—
82	C₂×C₄×D₆₇	—	4	2	no	—
83	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
84	Dic₂₆₈	—	2	2	no	—
85	Dic₂₆₈	C₂²×C₄	1	4	yes	—
86	#11761618	—	2	8	no	—
87	#11761611	—	2	2	no	—
88	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
90	C₈×D₆₇	C₂²×C₄	1	2	yes	—
91	C₈×D₆₇	#41	1	2	yes	—
92	C₄⋊Dic₆₇	—	2	2	no	—
93	C₄⋊Dic₆₇	—	4	2	no	—
94	Dic₆₇⋊C₄	—	4	2	no	—
95	Dic₆₇⋊C₄	—	4	2	no	—
96	Dic₆₇⋊C₄	D₈	1	2	yes	—
97	(C₂×D₆₇)⋊C₄	—	4	2	no	—
98	(C₂×D₆₇)⋊C₄	D₈	1	2	yes	—
99	C₄×Dic₆₇	—	4	2	no	—
100	C₄×Dic₆₇	—	4	2	no	—
101	C₄×Dic₆₇	—	4	2	no	—
102	C₄×Dic₆₇	—	4	2	no	—
103	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
104	C₆₇⋊C₁₆	—	2	1	no	—
105	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
106	C₂×Q₈×C₆₇	—	2	2	no	—
107	Q₁₆×C₆₇	—	2	2	no	—
108	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	4	2	no	—
113	C₄²×C₆₇	—	4	2	no	—
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₆₇	—	2	1	no	—
116	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
117	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₆₇	—	2	4	no	—
119	Q₈×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₆₇	—	4	2	no	—
125	C₂×Dic₆₇	—	4	2	no	—
126	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
127	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
128	C₄×D₆₇	—	4	8	no	—
129	C₄×D₆₇	C₄×D₄	1	2	yes	—
130	C₄×D₆₇	#135	1	2	yes	—
131	Dic₁₃₄	—	2	4	no	—
132	Dic₁₃₄	—	4	2	no	—
133	Dic₁₃₄	—	4	8	no	—
134	C₈×C₆₇	—	2	1	no	—
135	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
136	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₆₇⋊C₈	—	2	1	no	—
138	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
139	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₆₇	—	2	2	no	—
141	C₂×D₆₇	C₄×D₈	1	2	yes	—
142	C₄×C₆₇	—	2	2	no	—
143	C₄×C₆₇	—	4	4	no	—
144	C₄×C₆₇	—	8	2	no	—
145	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
147	Dic₆₇	—	2	2	no	—
148	Dic₆₇	—	4	4	no	—
149	Dic₆₇	—	8	2	no	—
150	Dic₆₇	C₄×D₈	1	2	yes	—
151	#33458	—	67	1	no	maximal, Sylow
152	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
153	C₂×C₆₇	#3417	1	2	yes	—
154	D₆₇	—	2	2	no	—
155	C₂²×Q₃₂	—	67	1	no	—
156	#2950	—	67	2	no	—
157	#2176	—	67	1	no	—
158	#2176	—	67	1	no	—
159	#2176	—	67	1	no	—
160	#2176	D₆₇	1	1	yes	characteristic, complemented
161	C₄×Q₃₂	—	67	2	no	—
162	C₄×Q₃₂	—	67	2	no	—
163	#2158	—	67	1	no	—
164	#2141	—	67	2	no	—
165	#2108	—	67	1	no	—
166	C₆₇	#33458	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₁₆	—	67	2	no	—
168	#519	—	134	2	no	—
169	C₂×Q₃₂	—	67	2	no	—
170	C₂×Q₃₂	—	67	8	no	—
171	C₂×Q₃₂	—	67	1	no	—
172	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic
173	C₂×M₃₂	—	67	1	no	—
174	C₂²×C₁₆	—	67	1	no	—
175	#441	—	134	2	no	—
176	#441	—	134	2	no	—
177	C₄×Q₁₆	—	67	2	no	—
178	C₄×Q₁₆	—	67	2	no	—
179	C₄×Q₁₆	—	67	2	no	—
180	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
181	#427	—	67	2	no	—
182	#419	—	134	2	no	—
183	#405	—	67	1	no	—
184	#367	—	67	1	no	—
185	#367	C₂×D₆₇	1	1	yes	characteristic
186	#366	—	67	2	no	—
187	#358	—	67	2	no	—
188	#358	—	67	2	no	—
189	#358	—	67	2	no	—
190	#358	C₂×D₆₇	1	2	yes	—
191	#346	—	67	1	no	—
192	#346	C₂×D₆₇	1	1	yes	characteristic
193	C₄×C₁₆	—	67	1	no	—
194	C₄×C₁₆	—	67	1	no	—
195	C₂²×Q₈	—	134	2	no	—
196	C₂×Q₁₆	—	67	8	no	—
197	C₂×Q₁₆	—	67	2	no	—
198	C₂×Q₁₆	—	134	4	no	—
199	C₂×Q₁₆	—	134	2	no	—
200	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
201	C₂×M₁₆	—	67	1	no	—
202	C₂²×C₈	—	67	1	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	—	268	2	no	—
205	C₄×Q₈	—	268	2	no	—
206	C₄×Q₈	—	134	2	no	—
207	C₄×Q₈	—	134	2	no	—
208	C₄×Q₈	—	134	2	no	—
209	#117	—	268	2	no	—
210	#117	—	268	2	no	—
211	#117	—	67	1	no	—
212	#117	—	134	2	no	—
213	Q₃₂	—	67	8	no	—
214	Q₃₂	—	67	4	no	—
215	Q₃₂	C₄×D₆₇	1	4	yes	—
216	M₃₂	—	134	2	no	—
217	C₂×C₁₆	—	67	1	no	—
218	C₂×C₁₆	—	67	4	no	—
219	C₂×C₁₆	—	67	1	no	—
220	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
221	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
222	#107	—	67	2	no	—
223	#107	C₂²×D₆₇	1	2	yes	—
224	#106	—	134	2	no	—
225	#103	—	2	2	no	—
226	#103	—	134	2	no	—
227	#103	—	134	2	no	—
228	#103	—	134	2	no	—
229	#97	—	67	1	no	—
230	#97	—	67	1	no	—
231	C₄×C₈	—	67	1	no	—
232	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	268	2	no	—
235	C₂×Q₈	—	268	4	no	—
236	C₂×Q₈	—	134	8	no	—
237	C₂×Q₈	—	134	2	no	—
238	C₂²×C₄	—	268	2	no	—
239	C₂²×C₄	—	67	1	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	67	4	no	—
242	Q₁₆	—	134	8	no	—
243	Q₁₆	—	134	2	no	—
244	Q₁₆	C₂×C₄×D₆₇	1	4	yes	—
245	M₁₆	—	134	2	no	—
246	C₂×C₈	—	67	1	no	—
247	C₂×C₈	—	67	2	no	—
248	C₂×C₈	—	67	2	no	—
249	C₂×C₈	—	67	1	no	—
250	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	268	2	no	—
255	C₄⋊C₄	—	268	2	no	—
256	C₄⋊C₄	—	268	2	no	—
257	C₄⋊C₄	—	67	2	no	—
258	C₄⋊C₄	—	134	2	no	—
259	C₂²⋊C₄	—	268	2	no	—
260	C₂²⋊C₄	—	67	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	268	2	no	—
263	C₄²	—	268	2	no	—
264	C₄²	—	268	2	no	—
265	C₄²	—	268	2	no	—
266	C₄²	—	67	1	no	—
267	C₄²	D₄×D₆₇	1	1	yes	characteristic
268	C₁₆	—	2	1	no	—
269	C₁₆	—	67	2	no	—
270	C₁₆	—	134	1	no	—
271	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
272	C₂³	—	67	1	no	—
273	Q₈	—	2	4	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	268	8	no	—
276	Q₈	—	268	2	no	—
277	Q₈	—	134	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	268	8	no	—
281	C₂×C₄	—	268	2	no	—
282	C₂×C₄	—	268	2	no	—
283	C₂×C₄	—	67	1	no	—
284	C₂×C₄	—	67	2	no	—
285	C₂×C₄	—	67	2	no	—
286	C₂×C₄	—	67	2	no	—
287	C₂×C₄	D₈×D₆₇	1	2	yes	—
288	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	67	2	no	—
291	C₈	—	134	1	no	—
292	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
293	C₈	#41×D₆₇	1	1	yes	characteristic
294	C₂²	—	67	2	no	—
295	C₂²	—	134	2	no	—
296	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	8	2	no	—
300	C₄	—	268	4	no	—
301	C₄	—	536	2	no	—
302	C₄	—	67	2	no	—
303	C₄	—	134	2	no	—
304	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
305	C₄	#135×D₆₇	1	1	yes	characteristic
306	C₂	—	134	2	no	—
307	C₂	C₄×D₈×D₆₇	1	1	yes	characteristic
308	C₂	#22573361	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2176	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₆₇	C₂×D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2176	D₆₇
hypocenter	C₆₇	#33458
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

Complements

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

N	H ≅ G/N	type
3. #22562081	C₂	semidirect
4. #22562080	C₂	semidirect
5. #22562079	C₂	semidirect
6. C₄×#14523989	C₂	semidirect
7. #22562065	C₂	semidirect
11. C₆₇×#2176	C₂	semidirect
16. Q₃₂×D₆₇	C₄	semidirect
160. #2176	D₆₇	semidirect
166. C₆₇	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₆₇	2
#22569072	2
#22562081	2
#22562080	2
#22562079	2
C₄×#14523989	2
#22562065	2
#22561992	2
#22561927	2
#22561807	2
C₆₇×#2176	2
#33458	67