Subgroups — #31011334

Order 17152.

474 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570742	C₂	1	2	yes	maximal
2	#22562493	C₂	1	2	yes	maximal, complemented
3	#22562492	C₂	1	2	yes	maximal, complemented
4	#22562488	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562486	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562483	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561994	C₂	1	2	yes	maximal
8	#22561990	C₂	1	1	yes	maximal, characteristic
9	#366×D₆₇	C₂	1	1	yes	maximal, characteristic
10	#22561804	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864023	—	2	2	no	—
13	C₂×#14523982	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
15	#16863804	—	2	2	no	—
16	#16863802	—	2	2	no	—
17	#16863800	C₂²	1	2	yes	—
18	#16863798	C₂²	1	2	yes	—
19	#16863797	C₂²	1	2	yes	—
20	#16863437	—	2	2	no	—
21	#16863433	C₂²	1	2	yes	—
22	#107×D₆₇	C₂²	1	2	yes	—
23	#16863350	C₂²	1	1	yes	characteristic
24	#16863224	C₂²	1	2	yes	—
25	#16863223	C₂²	1	2	yes	—
26	#16863223	C₂²	1	2	yes	—
27	#16863222	C₂²	1	1	yes	characteristic
28	#16863222	C₂²	1	1	yes	characteristic
29	#16863221	C₂²	1	2	yes	—
30	#16863221	C₂²	1	2	yes	—
31	#16863172	C₂²	1	1	yes	characteristic
32	#16863171	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
34	#500×C₆₇	C₂²	1	2	yes	—
35	#366×C₆₇	C₂²	1	1	yes	characteristic
36	#366×C₆₇	C₂²	1	1	yes	characteristic
37	#366×C₆₇	C₂²	1	2	yes	—
38	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#11761611	D₄	1	1	yes	characteristic
40	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
41	#14524042	—	2	2	no	—
42	#14524041	—	4	2	no	—
43	#14524035	—	2	2	no	—
44	#14524034	—	2	2	no	—
45	#14524033	—	4	2	no	—
46	#14524021	—	4	2	no	—
47	#14524019	—	4	2	no	—
48	#14524001	—	2	2	no	—
49	#14524000	D₄	1	1	yes	characteristic
50	C₄⋊C₄×D₆₇	—	2	2	no	—
51	#14523982	Q₈	1	4	yes	complemented
52	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
54	C₁₆×D₆₇	Q₈	1	4	yes	complemented
55	#14523977	C₂³	1	2	yes	—
56	#14523977	C₂³	1	2	yes	—
57	#14523976	—	2	2	no	—
58	#14523976	—	2	2	no	—
59	#14523975	—	2	2	no	—
60	#14523974	—	2	2	no	—
61	#14523974	—	2	2	no	—
62	#14523954	D₄	1	1	yes	characteristic
63	#14523953	D₄	1	1	yes	characteristic
64	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₆₇	—	2	2	no	—
66	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
68	#107×C₆₇	—	2	2	no	—
69	#107×C₆₇	C₂³	1	2	yes	—
70	#107×C₆₇	C₂³	1	2	yes	—
71	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₃₄	—	4	2	no	—
73	C₂×C₄×D₆₇	—	4	2	no	—
74	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
75	#11761611	Q₁₆	1	4	yes	—
76	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
78	C₈×D₆₇	C₂×Q₈	1	2	yes	—
79	C₈×D₆₇	C₂×Q₈	1	2	yes	—
80	C₄⋊Dic₆₇	—	2	2	no	—
81	C₄⋊Dic₆₇	—	2	2	no	—
82	C₄⋊Dic₆₇	—	4	2	no	—
83	Dic₆₇⋊C₄	—	4	2	no	—
84	Dic₆₇⋊C₄	—	4	2	no	—
85	Dic₆₇⋊C₄	—	4	2	no	—
86	Dic₆₇⋊C₄	—	4	2	no	—
87	Dic₆₇⋊C₄	—	4	2	no	—
88	Dic₆₇⋊C₄	D₈	1	1	yes	characteristic
89	Dic₆₇⋊C₄	D₈	1	1	yes	characteristic
90	(C₂×D₆₇)⋊C₄	—	4	2	no	—
91	(C₂×D₆₇)⋊C₄	D₈	1	1	yes	characteristic
92	(C₂×D₆₇)⋊C₄	D₈	1	1	yes	characteristic
93	C₄×Dic₆₇	—	4	2	no	—
94	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
95	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
96	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
97	C₂×Q₈×C₆₇	—	4	2	no	—
98	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	4	2	no	—
103	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
105	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
106	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
107	Q₈×C₆₇	—	16	2	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×Dic₆₇	—	4	2	no	—
114	C₂×Dic₆₇	—	4	2	no	—
115	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
116	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
117	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
118	C₄×D₆₇	—	16	2	no	—
119	C₄×D₆₇	C₄⋊Q₈	1	2	yes	—
120	C₄×D₆₇	C₂×Q₁₆	1	2	yes	—
121	Dic₁₃₄	—	16	2	no	—
122	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
123	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
125	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
126	C₆₇⋊C₈	C₂×Q₁₆	1	2	yes	—
127	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
128	C₂×D₆₇	—	2	1	no	—
129	C₂×D₆₇	—	2	1	no	—
130	C₂×D₆₇	#500	1	2	yes	—
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	—	8	2	no	—
134	C₄×C₆₇	—	8	2	no	—
135	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
137	Dic₆₇	—	2	1	no	—
138	Dic₆₇	—	2	1	no	—
139	Dic₆₇	—	8	2	no	—
140	Dic₆₇	—	8	2	no	—
141	Dic₆₇	#500	1	2	yes	—
142	#33729	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
144	C₂×C₆₇	#3415	1	1	yes	characteristic
145	C₂×C₆₇	#3417	1	1	yes	characteristic
146	D₆₇	—	2	2	no	—
147	#3164	—	67	2	no	—
148	#2255	—	67	2	no	—
149	#2255	—	67	2	no	—
150	#2253	—	67	1	no	—
151	#2253	—	67	1	no	—
152	#2252	—	67	1	no	—
153	#2252	D₆₇	1	1	yes	characteristic, complemented
154	#2159	—	67	2	no	—
155	#2158	—	67	1	no	—
156	C₂×#366	—	67	1	no	—
157	#2108	—	67	1	no	—
158	C₆₇	#33729	1	1	yes	characteristic, complemented, Sylow
159	#544	—	134	2	no	—
160	C₂×M₃₂	—	67	1	no	—
161	C₂²×C₁₆	—	67	1	no	—
162	#501	—	134	2	no	—
163	#501	—	134	2	no	—
164	#500	—	67	2	no	—
165	#500	C₂×D₆₇	1	2	yes	—
166	#499	—	67	2	no	—
167	#499	—	67	2	no	—
168	#428	—	134	2	no	—
169	#427	—	67	2	no	—
170	C₂×#107	—	67	2	no	—
171	#405	—	67	1	no	—
172	#367	—	67	2	no	—
173	#367	—	67	2	no	—
174	#367	—	67	2	no	—
175	#366	—	67	1	no	—
176	#366	—	67	1	no	—
177	#366	—	67	2	no	—
178	#366	—	67	2	no	—
179	#366	C₂×D₆₇	1	1	yes	characteristic
180	#366	C₂×D₆₇	1	1	yes	characteristic
181	#366	C₂×D₆₇	1	2	yes	—
182	#346	—	67	1	no	—
183	#346	—	67	1	no	—
184	C₄×C₁₆	—	67	1	no	—
185	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
186	C₂×M₁₆	—	67	1	no	—
187	C₂²×C₈	—	67	1	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	268	2	no	—
190	C₄⋊Q₈	—	134	2	no	—
191	#125	—	268	2	no	—
192	#125	—	134	2	no	—
193	#125	—	134	2	no	—
194	C₂²⋊Q₈	—	268	2	no	—
195	C₂²⋊Q₈	—	268	2	no	—
196	#117	—	67	1	no	—
197	#117	—	134	2	no	—
198	C₂×C₄⋊C₄	—	134	2	no	—
199	M₃₂	—	67	4	no	—
200	C₂×C₁₆	—	67	1	no	—
201	C₂×C₁₆	—	67	1	no	—
202	C₂×C₁₆	—	67	4	no	—
203	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
204	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
205	#107	—	2	2	no	—
206	#107	—	67	2	no	—
207	#107	—	67	2	no	—
208	#107	—	134	2	no	—
209	#107	—	134	2	no	—
210	#107	C₂²×D₆₇	1	2	yes	—
211	#107	C₂²×D₆₇	1	2	yes	—
212	#106	—	134	2	no	—
213	#106	—	134	2	no	—
214	#106	—	134	2	no	—
215	#97	—	67	1	no	—
216	#97	—	67	1	no	—
217	C₄×C₈	—	67	1	no	—
218	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	268	2	no	—
221	C₂²×C₄	—	268	2	no	—
222	C₂²×C₄	—	67	1	no	—
223	M₁₆	—	67	4	no	—
224	C₂×C₈	—	67	1	no	—
225	C₂×C₈	—	67	1	no	—
226	C₂×C₈	—	67	2	no	—
227	C₂×C₈	—	67	2	no	—
228	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	268	2	no	—
234	C₄⋊C₄	—	268	2	no	—
235	C₄⋊C₄	—	268	2	no	—
236	C₄⋊C₄	—	268	2	no	—
237	C₄⋊C₄	—	268	2	no	—
238	C₄⋊C₄	—	268	2	no	—
239	C₄⋊C₄	—	67	1	no	—
240	C₄⋊C₄	—	67	1	no	—
241	C₄⋊C₄	—	134	2	no	—
242	C₄⋊C₄	—	134	2	no	—
243	C₂²⋊C₄	—	268	2	no	—
244	C₂²⋊C₄	—	67	1	no	—
245	C₂²⋊C₄	—	67	1	no	—
246	C₄²	—	268	2	no	—
247	C₄²	—	67	1	no	—
248	C₄²	D₄×D₆₇	1	1	yes	characteristic
249	C₁₆	—	67	2	no	—
250	C₁₆	—	67	2	no	—
251	C₁₆	Q₈×D₆₇	1	2	yes	—
252	C₁₆	Q₈×D₆₇	1	2	yes	—
253	C₂³	—	67	1	no	—
254	Q₈	—	16	2	no	—
255	Q₈	—	1072	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	268	2	no	—
259	C₂×C₄	—	268	2	no	—
260	C₂×C₄	—	1072	2	no	—
261	C₂×C₄	—	67	1	no	—
262	C₂×C₄	—	67	2	no	—
263	C₂×C₄	—	67	2	no	—
264	C₂×C₄	—	67	1	no	—
265	C₂×C₄	—	67	1	no	—
266	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
267	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
269	C₈	—	67	2	no	—
270	C₈	—	67	2	no	—
271	C₈	Q₁₆×D₆₇	1	2	yes	—
272	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
273	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
274	C₂²	—	67	2	no	—
275	C₂²	—	134	1	no	—
276	C₂²	—	134	1	no	—
277	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	536	2	no	—
283	C₄	—	536	2	no	—
284	C₄	—	67	2	no	—
285	C₄	—	134	1	no	—
286	C₄	—	134	1	no	—
287	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
288	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
289	C₂	—	134	2	no	—
290	C₂	#500×D₆₇	1	1	yes	characteristic
291	C₂	#22573346	1	1	yes	characteristic
292	C₂	#22573358	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2252	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₆₇	C₂×D₈
hypercenter	#2252	D₆₇
hypocenter	C₆₇	#33729
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. #22562493	C₂	semidirect
3. #22562492	C₂	semidirect
4. #22562488	C₂	semidirect
5. #22562486	C₂	semidirect
6. #22562483	C₂	semidirect
11. C₆₇×#2252	C₂	semidirect
51. #14523982	Q₈	semidirect
54. C₁₆×D₆₇	Q₈	semidirect
153. #2252	D₆₇	semidirect
158. C₆₇	#33729	semidirect

Maximal subgroups

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

subgroup	index
#22570742	2
#22562493	2
#22562492	2
#22562488	2
#22562486	2
#22562483	2
#22561994	2
#22561990	2
#366×D₆₇	2
#22561804	2
C₆₇×#2252	2
#33729	67