Subgroups — #31011357

Order 17152.

482 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	#22570722	C₂	1	1	yes	maximal, characteristic
2	#499×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22562494	C₂	1	2	yes	maximal, complemented
4	#22562493	C₂	1	2	yes	maximal, complemented
5	#22562488	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562485	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562485	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561996	C₂	1	2	yes	maximal
9	#22561994	C₂	1	2	yes	maximal
10	C₄×#14523982	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863952	—	2	1	no	—
13	#125×D₆₇	—	2	1	no	—
14	C₂×#14523982	C₂²	1	1	yes	characteristic
15	C₂×#14523982	C₂²	1	1	yes	characteristic
16	#16863800	C₂²	1	2	yes	—
17	#16863798	C₂²	1	1	yes	characteristic
18	#16863798	C₂²	1	1	yes	characteristic
19	#16863797	C₂²	1	2	yes	—
20	#16863796	—	2	1	no	—
21	#16863795	—	2	1	no	—
22	#16863795	—	2	1	no	—
23	#16863793	—	2	1	no	—
24	#16863433	C₂²	1	2	yes	—
25	#16863431	—	2	1	no	—
26	#107×D₆₇	C₂²	1	2	yes	—
27	#106×D₆₇	—	2	1	no	—
28	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
29	#16863224	C₂²	1	2	yes	—
30	#16863223	C₂²	1	2	yes	—
31	#16863223	C₂²	1	2	yes	—
32	#16863222	C₂²	1	2	yes	—
33	#16863221	C₂²	1	2	yes	—
34	#16863221	C₂²	1	2	yes	—
35	#16863171	C₂²	1	1	yes	characteristic
36	#16863171	C₂²	1	1	yes	characteristic
37	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
38	#499×C₆₇	C₂²	1	1	yes	characteristic
39	#499×C₆₇	C₂²	1	1	yes	characteristic
40	#367×C₆₇	C₂²	1	2	yes	—
41	#366×C₆₇	C₂²	1	2	yes	—
42	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
44	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
45	#14524042	—	2	2	no	—
46	#14524035	—	2	1	no	—
47	#14524035	—	2	1	no	—
48	#14524034	—	2	2	no	—
49	#14524033	—	4	2	no	—
50	#14524021	—	4	2	no	—
51	#14524008	—	4	2	no	—
52	#14524001	—	2	2	no	—
53	C₄⋊C₄×D₆₇	—	2	2	no	—
54	C₄⋊C₄×D₆₇	—	4	2	no	—
55	C₄²×D₆₇	D₄	1	1	yes	characteristic
56	#14523982	Q₈	1	4	yes	complemented
57	#14523982	Q₈	1	4	yes	complemented
58	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	#14523977	C₂³	1	2	yes	—
61	#14523977	C₂³	1	2	yes	—
62	#14523976	—	2	2	no	—
63	#14523976	—	2	2	no	—
64	#14523975	—	2	1	no	—
65	#14523975	—	2	1	no	—
66	#14523974	—	2	2	no	—
67	#14523974	—	2	2	no	—
68	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
69	C₈×Dic₆₇	D₄	1	1	yes	characteristic
70	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
71	#125×C₆₇	—	2	1	no	—
72	#125×C₆₇	—	2	1	no	—
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	#107×C₆₇	C₂³	1	2	yes	—
77	#106×C₆₇	—	2	1	no	—
78	#106×C₆₇	—	2	1	no	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×Dic₁₃₄	—	4	2	no	—
81	C₂×C₄×D₆₇	—	4	2	no	—
82	C₂×C₄×D₆₇	D₈	1	2	yes	—
83	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
86	C₈×D₆₇	Q₁₆	1	4	yes	—
87	C₈×D₆₇	C₂×Q₈	1	2	yes	—
88	C₈×D₆₇	C₂×Q₈	1	2	yes	—
89	C₄⋊Dic₆₇	—	2	2	no	—
90	C₄⋊Dic₆₇	—	2	2	no	—
91	C₄⋊Dic₆₇	—	4	2	no	—
92	Dic₆₇⋊C₄	—	4	2	no	—
93	Dic₆₇⋊C₄	—	4	2	no	—
94	Dic₆₇⋊C₄	—	4	2	no	—
95	Dic₆₇⋊C₄	—	4	2	no	—
96	Dic₆₇⋊C₄	—	4	2	no	—
97	(C₂×D₆₇)⋊C₄	—	4	2	no	—
98	C₄×Dic₆₇	—	4	2	no	—
99	C₄×Dic₆₇	D₈	1	2	yes	—
100	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
101	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
102	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
103	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	4	2	no	—
108	C₄⋊C₄×C₆₇	—	4	2	no	—
109	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
111	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
112	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₆₇	—	4	2	no	—
118	C₂×Dic₆₇	—	4	2	no	—
119	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
120	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
121	C₄×D₆₇	—	4	2	no	—
122	C₄×D₆₇	—	8	4	no	—
123	C₄×D₆₇	C₄⋊Q₈	1	2	yes	—
124	C₄×D₆₇	C₂×Q₁₆	1	2	yes	—
125	Dic₁₃₄	—	8	4	no	—
126	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
127	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
128	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
129	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
130	C₆₇⋊C₈	C₂×Q₁₆	1	2	yes	—
131	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
132	C₂×D₆₇	—	2	2	no	—
133	C₂×D₆₇	#500	1	2	yes	—
134	C₄×C₆₇	—	2	2	no	—
135	C₄×C₆₇	—	8	2	no	—
136	C₄×C₆₇	—	8	2	no	—
137	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
139	Dic₆₇	—	2	2	no	—
140	Dic₆₇	—	8	2	no	—
141	Dic₆₇	—	8	2	no	—
142	Dic₆₇	#500	1	2	yes	—
143	#33732	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
145	C₂×C₆₇	#3414	1	2	yes	—
146	D₆₇	—	2	2	no	—
147	#3160	—	67	1	no	—
148	C₂×#499	—	67	1	no	—
149	#2255	—	67	2	no	—
150	#2255	—	67	2	no	—
151	#2253	—	67	1	no	—
152	#2253	—	67	1	no	—
153	#2253	—	67	1	no	—
154	#2253	D₆₇	1	1	yes	characteristic, complemented
155	#2159	—	67	2	no	—
156	#2159	—	67	2	no	—
157	C₄×M₃₂	—	67	1	no	—
158	C₆₇	#33732	1	1	yes	characteristic, complemented, Sylow
159	#533	—	134	1	no	—
160	C₂×#125	—	134	1	no	—
161	C₂×M₃₂	—	67	1	no	—
162	C₂×M₃₂	—	67	1	no	—
163	#500	—	67	2	no	—
164	#499	—	67	2	no	—
165	#499	—	67	1	no	—
166	#499	—	67	1	no	—
167	#499	—	134	1	no	—
168	#499	—	134	1	no	—
169	#499	—	134	1	no	—
170	#499	C₂×D₆₇	1	1	yes	characteristic
171	#499	C₂×D₆₇	1	1	yes	characteristic
172	#498	—	134	1	no	—
173	#427	—	67	2	no	—
174	#427	—	134	1	no	—
175	C₂×#107	—	67	2	no	—
176	C₂×#106	—	134	1	no	—
177	C₂×C₄×C₈	—	67	1	no	—
178	#367	—	67	2	no	—
179	#367	—	67	2	no	—
180	#367	—	67	2	no	—
181	#367	C₂×D₆₇	1	2	yes	—
182	#366	—	67	2	no	—
183	#366	—	67	2	no	—
184	#366	—	67	2	no	—
185	#366	C₂×D₆₇	1	2	yes	—
186	#346	—	67	1	no	—
187	#346	—	67	1	no	—
188	C₄×C₁₆	—	67	1	no	—
189	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
190	C₂²×C₈	—	67	1	no	—
191	C₂²×C₈	—	67	1	no	—
192	C₄⋊Q₈	—	134	2	no	—
193	#125	—	2	1	no	—
194	#125	—	2	1	no	—
195	#125	—	268	2	no	—
196	#125	—	134	2	no	—
197	#125	—	134	1	no	—
198	#125	—	134	1	no	—
199	C₂²⋊Q₈	—	268	2	no	—
200	C₄×Q₈	—	268	2	no	—
201	#117	—	134	2	no	—
202	C₂×C₄⋊C₄	—	268	2	no	—
203	C₂×C₄⋊C₄	—	134	2	no	—
204	C₂×C₄²	—	67	1	no	—
205	M₃₂	—	67	4	no	—
206	M₃₂	—	67	4	no	—
207	C₂×C₁₆	—	67	1	no	—
208	C₂×C₁₆	—	67	1	no	—
209	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
210	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
211	#107	—	67	2	no	—
212	#107	—	67	2	no	—
213	#107	—	134	2	no	—
214	#107	—	134	2	no	—
215	#107	C₂²×D₆₇	1	2	yes	—
216	#107	C₂²×D₆₇	1	2	yes	—
217	#106	—	2	1	no	—
218	#106	—	2	1	no	—
219	#106	—	134	2	no	—
220	#106	—	134	2	no	—
221	#106	—	134	1	no	—
222	#106	—	134	1	no	—
223	C₄×C₈	—	67	1	no	—
224	C₄×C₈	—	67	1	no	—
225	C₄×C₈	—	67	1	no	—
226	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
227	C₂×Q₈	—	268	2	no	—
228	C₂²×C₄	—	268	2	no	—
229	C₂²×C₄	—	67	2	no	—
230	C₂²×C₄	—	67	1	no	—
231	C₂×C₈	—	67	1	no	—
232	C₂×C₈	—	67	1	no	—
233	C₂×C₈	—	67	4	no	—
234	C₂×C₈	—	67	2	no	—
235	C₂×C₈	—	67	2	no	—
236	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	268	2	no	—
243	C₄⋊C₄	—	268	2	no	—
244	C₄⋊C₄	—	268	2	no	—
245	C₄⋊C₄	—	268	2	no	—
246	C₄⋊C₄	—	268	2	no	—
247	C₄⋊C₄	—	268	2	no	—
248	C₄⋊C₄	—	134	2	no	—
249	C₄⋊C₄	—	134	2	no	—
250	C₂²⋊C₄	—	268	2	no	—
251	C₄²	—	268	2	no	—
252	C₄²	—	67	2	no	—
253	C₄²	—	67	1	no	—
254	C₄²	D₄×D₆₇	1	1	yes	characteristic
255	C₁₆	—	67	2	no	—
256	C₁₆	—	67	2	no	—
257	C₁₆	Q₈×D₆₇	1	2	yes	—
258	C₁₆	Q₈×D₆₇	1	2	yes	—
259	C₂³	—	67	1	no	—
260	Q₈	—	536	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	268	2	no	—
264	C₂×C₄	—	268	2	no	—
265	C₂×C₄	—	268	2	no	—
266	C₂×C₄	—	536	4	no	—
267	C₂×C₄	—	67	1	no	—
268	C₂×C₄	—	67	2	no	—
269	C₂×C₄	—	67	2	no	—
270	C₂×C₄	—	67	2	no	—
271	C₂×C₄	D₈×D₆₇	1	2	yes	—
272	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
273	C₈	—	67	2	no	—
274	C₈	—	67	2	no	—
275	C₈	Q₁₆×D₆₇	1	2	yes	—
276	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
277	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
278	C₂²	—	67	2	no	—
279	C₂²	—	134	2	no	—
280	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	536	2	no	—
285	C₄	—	536	2	no	—
286	C₄	—	67	2	no	—
287	C₄	—	134	2	no	—
288	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
290	C₂	—	134	2	no	—
291	C₂	#500×D₆₇	1	1	yes	characteristic
292	C₂	#22573342	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2253	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₆₇	C₂×D₈
hypercenter	#2253	D₆₇
hypocenter	C₆₇	#33732
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
3. #22562494	C₂	semidirect
4. #22562493	C₂	semidirect
5. #22562488	C₂	semidirect
6. #22562485	C₂	semidirect
7. #22562485	C₂	semidirect
11. C₆₇×#2253	C₂	semidirect
56. #14523982	Q₈	semidirect
57. #14523982	Q₈	semidirect
154. #2253	D₆₇	semidirect
158. C₆₇	#33732	semidirect

Maximal subgroups

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

subgroup	index
#22570722	2
#499×D₆₇	2
#22562494	2
#22562493	2
#22562488	2
#22562485	2
#22561996	2
#22561994	2
C₄×#14523982	2
C₆₇×#2253	2
#33732	67