Subgroups — #31980694

Order 18176.

606 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22599288	C₂	1	2	yes	maximal
2	#22597274	C₂	1	1	yes	maximal, characteristic
3	#22587855	C₂	1	4	yes	maximal, complemented
4	#22587731	C₂	1	4	yes	maximal, complemented
5	#22587088	C₂	1	1	yes	maximal, characteristic
6	#22587025	C₂	1	2	yes	maximal
7	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#17986556	C₂²	1	1	yes	characteristic
9	#17986378	—	2	2	no	—
10	#17986349	—	2	2	no	—
11	#17986206	C₂²	1	8	yes	—
12	#17986155	—	2	4	no	—
13	#17986129	—	2	1	no	—
14	#17986129	C₂²	1	2	yes	—
15	C₈×Dic₁₄₂	—	2	1	no	—
16	C₈×Dic₁₄₂	C₂²	1	2	yes	—
17	#17986052	C₂²	1	2	yes	—
18	#17986025	—	2	1	no	—
19	#17986023	C₂²	1	1	yes	characteristic
20	#17985954	—	2	1	no	—
21	#17985752	C₂²	1	2	yes	—
22	#17985750	C₂²	1	2	yes	—
23	#17985745	C₂²	1	8	yes	—
24	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
25	#500×C₇₁	—	2	1	no	—
26	#500×C₇₁	—	2	1	no	—
27	#500×C₇₁	C₂²	1	2	yes	—
28	#500×C₇₁	C₂²	1	2	yes	—
29	C₂×#107×C₇₁	—	2	1	no	—
30	C₂×#107×C₇₁	—	2	1	no	—
31	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
32	C₂×#11765855	—	2	8	no	—
33	C₂×#11765849	—	2	1	no	—
34	#14556128	D₄	1	1	yes	characteristic
35	#14556128	D₄	1	1	yes	characteristic
36	#14556120	—	2	1	no	—
37	#14556107	—	2	1	no	—
38	#14556107	D₄	1	2	yes	—
39	C₄×Dic₁₄₂	—	2	1	no	—
40	C₄×Dic₁₄₂	C₂³	1	2	yes	—
41	#14556087	—	2	1	no	—
42	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
43	C₄⋊(C₇₁⋊C₈)	C₂³	1	4	yes	—
44	C₄⋊(C₇₁⋊C₈)	C₂³	1	2	yes	—
45	Dic₇₁⋊C₈	—	2	2	no	—
46	Dic₇₁⋊C₈	—	2	2	no	—
47	#14556052	—	2	8	no	—
48	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
49	#14556038	—	2	2	no	—
50	C₄×C₇₁⋊C₈	—	2	2	no	—
51	C₈×Dic₇₁	—	2	2	no	—
52	C₂²×Q₈×C₇₁	—	2	2	no	—
53	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
55	C₄⋊Q₈×C₇₁	—	2	2	no	—
56	C₄⋊Q₈×C₇₁	—	2	2	no	—
57	C₄⋊Q₈×C₇₁	—	2	2	no	—
58	C₄⋊Q₈×C₇₁	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
61	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
62	#107×C₇₁	—	2	2	no	—
63	#107×C₇₁	—	2	2	no	—
64	#107×C₇₁	—	2	2	no	—
65	#107×C₇₁	—	2	2	no	—
66	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
70	C₂×Dic₁₄₂	—	2	1	no	—
71	C₂×Dic₁₄₂	C₂×D₄	1	2	yes	—
72	#11765855	—	8	8	no	—
73	#11765849	—	4	2	no	—
74	C₂×C₇₁⋊C₈	—	2	2	no	—
75	C₂×C₇₁⋊C₈	—	2	4	no	—
76	C₄⋊Dic₇₁	—	2	1	no	—
77	C₄⋊Dic₇₁	—	2	1	no	—
78	C₄⋊Dic₇₁	—	2	1	no	—
79	C₄⋊Dic₇₁	C₂×D₄	1	2	yes	—
80	Dic₇₁⋊C₄	—	2	2	no	—
81	Dic₇₁⋊C₄	—	2	2	no	—
82	C₂²⋊Dic₇₁	—	2	2	no	—
83	C₄×Dic₇₁	—	2	2	no	—
84	C₂×Q₈×C₇₁	—	2	4	no	—
85	C₂×Q₈×C₇₁	—	4	2	no	—
86	C₂×Q₈×C₇₁	—	8	4	no	—
87	C₂²×C₄×C₇₁	—	4	2	no	—
88	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
91	C₂×C₈×C₇₁	—	2	2	no	—
92	C₂×C₈×C₇₁	Q₁₆	1	4	yes	complemented
93	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₇₁	—	2	4	no	—
98	C₄⋊C₄×C₇₁	—	2	4	no	—
99	C₄⋊C₄×C₇₁	—	4	2	no	—
100	C₄⋊C₄×C₇₁	—	4	2	no	—
101	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₇₁	#41	1	1	yes	characteristic
104	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
105	C₂³×C₇₁	#121	1	1	yes	characteristic
106	Q₈×C₇₁	—	4	8	no	—
107	Q₈×C₇₁	—	8	4	no	—
108	C₂×C₄×C₇₁	—	2	2	no	—
109	C₂×C₄×C₇₁	—	2	2	no	—
110	C₂×C₄×C₇₁	—	4	4	no	—
111	C₂×C₄×C₇₁	—	8	4	no	—
112	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
113	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
114	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
116	C₂×C₄×C₇₁	#137	1	2	yes	—
117	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₇₁	—	2	2	no	—
121	C₂×Dic₇₁	—	2	2	no	—
122	Dic₁₄₂	—	2	4	no	—
123	Dic₁₄₂	—	4	2	no	—
124	C₈×C₇₁	—	2	1	no	—
125	C₈×C₇₁	—	2	1	no	—
126	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
127	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
128	C₇₁⋊C₈	—	4	2	no	—
129	C₇₁⋊C₈	—	4	4	no	—
130	C₂²×C₇₁	—	2	1	no	—
131	C₂²×C₇₁	—	2	1	no	—
132	C₂²×C₇₁	#462	1	2	yes	—
133	C₂²×C₇₁	C₂×#121	1	1	yes	characteristic
134	C₄×C₇₁	—	2	1	no	—
135	C₄×C₇₁	—	2	1	no	—
136	C₄×C₇₁	—	8	4	no	—
137	C₄×C₇₁	#462	1	2	yes	—
138	C₄×C₇₁	#473	1	2	yes	—
139	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
140	C₄×C₇₁	C₂×#137	1	1	yes	characteristic
141	C₄×C₇₁	#576	1	2	yes	—
142	Dic₇₁	—	4	2	no	—
143	Dic₇₁	—	4	2	no	—
144	#19208	—	71	1	no	maximal, Sylow
145	C₂×C₇₁	—	2	2	no	—
146	C₂×C₇₁	C₂×#462	1	1	yes	characteristic
147	C₂×C₇₁	#3058	1	1	yes	characteristic
148	C₂×C₇₁	C₂×#473	1	1	yes	characteristic
149	#3209	—	71	2	no	—
150	C₂×#500	D₇₁	1	1	yes	characteristic
151	#2960	—	71	1	no	—
152	#1694	—	71	4	no	—
153	#1674	—	71	4	no	—
154	#1564	—	71	1	no	—
155	#1549	—	71	2	no	—
156	C₇₁	#19208	1	1	yes	characteristic, complemented, Sylow
157	#533	—	71	1	no	—
158	C₂×C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
159	#500	—	2	1	no	—
160	#500	—	2	1	no	—
161	#500	C₂×D₇₁	1	2	yes	—
162	#500	C₂×D₇₁	1	2	yes	—
163	#494	—	142	2	no	—
164	#487	—	142	2	no	—
165	#462	—	71	8	no	—
166	#451	—	142	4	no	—
167	#446	—	71	2	no	—
168	#446	—	142	1	no	—
169	C₈×Q₈	—	71	2	no	—
170	C₈×Q₈	—	142	1	no	—
171	#431	—	71	2	no	—
172	C₂×#107	—	2	1	no	—
173	C₂×#107	—	2	1	no	—
174	#424	—	142	1	no	—
175	#423	—	71	1	no	—
176	#405	—	142	1	no	—
177	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
178	#335	—	71	2	no	—
179	#334	—	71	2	no	—
180	#332	—	71	8	no	—
181	C₂²×Q₈	—	2	2	no	—
182	C₂×Q₁₆	—	142	8	no	—
183	C₂×M₁₆	—	142	1	no	—
184	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
185	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	—	2	2	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	71	1	no	—
190	C₄⋊Q₈	—	71	1	no	—
191	C₄⋊Q₈	C₂²×D₇₁	1	4	yes	—
192	#125	—	142	1	no	—
193	C₂²⋊Q₈	—	71	2	no	—
194	C₂²⋊Q₈	—	142	1	no	—
195	C₄×Q₈	—	71	2	no	—
196	C₄×Q₈	—	142	1	no	—
197	#117	—	142	1	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
201	#107	—	2	2	no	—
202	#107	—	2	2	no	—
203	#107	—	2	2	no	—
204	#107	—	2	2	no	—
205	C₄⋊C₈	—	71	2	no	—
206	C₄⋊C₈	—	71	4	no	—
207	C₄⋊C₈	—	142	2	no	—
208	C₄⋊C₈	—	142	2	no	—
209	C₄⋊C₈	—	142	1	no	—
210	#103	—	142	8	no	—
211	C₂²⋊C₈	—	142	2	no	—
212	#97	—	142	2	no	—
213	C₄×C₈	—	142	2	no	—
214	C₄×C₈	—	142	2	no	—
215	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	—	8	4	no	—
222	C₂×Q₈	—	71	2	no	—
223	C₂×Q₈	—	142	1	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
227	C₂²×C₄	#11765864	1	1	yes	characteristic
228	Q₁₆	—	568	8	no	—
229	M₁₆	—	284	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	142	4	no	—
232	C₂×C₈	—	142	2	no	—
233	C₂×C₈	#11765855	1	4	yes	complemented
234	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	71	2	no	—
243	C₄⋊C₄	—	142	2	no	—
244	C₄⋊C₄	—	142	2	no	—
245	C₄⋊C₄	—	142	1	no	—
246	C₄⋊C₄	—	142	1	no	—
247	C₄⋊C₄	—	142	1	no	—
248	C₂²⋊C₄	—	142	2	no	—
249	C₄²	—	142	2	no	—
250	C₄²	D₄×D₇₁	1	1	yes	characteristic
251	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
252	C₄²	#11765864	1	1	yes	characteristic
253	C₄²	C₂³×D₇₁	1	1	yes	characteristic
254	C₂³	#14556099	1	1	yes	characteristic
255	Q₈	—	4	8	no	—
256	Q₈	—	8	4	no	—
257	Q₈	—	284	2	no	—
258	Q₈	—	142	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	—	142	2	no	—
264	C₂×C₄	—	142	2	no	—
265	C₂×C₄	#14556099	1	1	yes	characteristic
266	C₂×C₄	#14556099	1	1	yes	characteristic
267	C₂×C₄	#14556099	1	1	yes	characteristic
268	C₂×C₄	C₂×#11765855	1	2	yes	—
269	C₂×C₄	#14556159	1	2	yes	—
270	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
272	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	284	2	no	—
276	C₈	—	284	4	no	—
277	C₈	C₂×#11765855	1	2	yes	—
278	C₈	C₂×#11765855	1	2	yes	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	#17986206	1	2	yes	—
282	C₂²	C₂×#14556099	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	8	4	no	—
286	C₄	—	284	2	no	—
287	C₄	—	284	2	no	—
288	C₄	#17986206	1	2	yes	—
289	C₄	#17986271	1	2	yes	—
290	C₄	C₂²×#11765855	1	1	yes	characteristic
291	C₄	C₂×#14556159	1	1	yes	characteristic
292	C₄	#17986800	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	C₂×#17986206	1	1	yes	characteristic
295	C₂	#22597984	1	1	yes	characteristic
296	C₂	C₂×#17986271	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556099
commutator, cosocle	C₄²×C₇₁	C₂⁴
Frattini	C₄²	C₂³×D₇₁
Fitting	C₂×#500×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#121
prime radical	C₂³×C₇₁	#121
hypercenter	C₂×#500	D₇₁
hypocenter	C₇₁	#19208
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #22587855	C₂	semidirect
4. #22587731	C₂	semidirect
92. C₂×C₈×C₇₁	Q₁₆	semidirect
156. C₇₁	#19208	semidirect
233. C₂×C₈	#11765855	semidirect

Maximal subgroups

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

subgroup	index
#22599288	2
#22597274	2
#22587855	2
#22587731	2
#22587088	2
#22587025	2
C₂×#500×C₇₁	2
#19208	71