Subgroups — #31890313

Order 18176.

462 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22587088	C₂	1	2	yes	maximal, complemented
2	#22586654	C₂	1	2	yes	maximal
3	#22586171	C₂	1	2	yes	maximal, complemented
4	C₂×#405×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986052	C₂²	1	4	yes	—
6	#17985752	—	2	2	no	—
7	#17985750	—	2	2	no	—
8	#17985725	—	2	2	no	—
9	#17985723	—	2	2	no	—
10	C₂²×M₁₆×C₇₁	C₄	1	1	yes	characteristic
11	C₂³×C₈×C₇₁	C₄	1	1	yes	characteristic
12	C₂×#117×C₇₁	C₂²	1	1	yes	characteristic
13	#405×C₇₁	—	2	2	no	—
14	#405×C₇₁	—	2	2	no	—
15	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
16	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
21	C₄×C₇₁⋊C₈	—	2	4	no	—
22	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₇₁	—	2	2	no	—
24	C₂×M₁₆×C₇₁	—	2	2	no	—
25	C₂×M₁₆×C₇₁	—	2	2	no	—
26	C₂²×C₈×C₇₁	—	2	1	no	—
27	C₂²×C₈×C₇₁	—	2	1	no	—
28	C₂²×C₈×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	C₈	1	4	yes	complemented
30	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	#117×C₇₁	—	2	2	no	—
35	#117×C₇₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₇₁	Q₈	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₇₁	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₇₁	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	#97×C₇₁	—	2	2	no	—
43	#97×C₇₁	—	2	2	no	—
44	C₄×C₈×C₇₁	—	2	1	no	—
45	C₄×C₈×C₇₁	—	2	1	no	—
46	C₄×C₈×C₇₁	D₄	1	2	yes	—
47	C₄×C₈×C₇₁	D₄	1	2	yes	—
48	C₂×C₇₁⋊C₈	—	4	4	no	—
49	C₂⁴×C₇₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	—	2	1	no	—
51	C₂²×C₄×C₇₁	—	2	1	no	—
52	C₂²×C₄×C₇₁	—	2	1	no	—
53	C₂²×C₄×C₇₁	—	2	1	no	—
54	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
59	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
62	M₁₆×C₇₁	—	2	8	no	—
63	C₂×C₈×C₇₁	—	2	1	no	—
64	C₂×C₈×C₇₁	—	2	1	no	—
65	C₂×C₈×C₇₁	—	2	1	no	—
66	C₂×C₈×C₇₁	—	2	1	no	—
67	C₂×C₈×C₇₁	—	2	1	no	—
68	C₂×C₈×C₇₁	—	2	1	no	—
69	C₂×C₈×C₇₁	—	2	4	no	—
70	C₂×C₈×C₇₁	—	2	4	no	—
71	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
77	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
78	C₄⋊C₄×C₇₁	—	2	2	no	—
79	C₄⋊C₄×C₇₁	—	2	2	no	—
80	C₂²⋊C₄×C₇₁	—	2	2	no	—
81	C₂²⋊C₄×C₇₁	—	2	2	no	—
82	C₄²×C₇₁	—	2	1	no	—
83	C₄²×C₇₁	—	2	1	no	—
84	C₄²×C₇₁	C₂×D₄	1	2	yes	—
85	C₄²×C₇₁	#41	1	2	yes	—
86	C₂³×C₇₁	—	2	1	no	—
87	C₂³×C₇₁	—	2	1	no	—
88	C₂³×C₇₁	—	4	1	no	—
89	C₂³×C₇₁	—	4	1	no	—
90	C₂³×C₇₁	C₄⋊C₈	1	2	yes	—
91	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₁	—	2	1	no	—
93	C₂×C₄×C₇₁	—	2	1	no	—
94	C₂×C₄×C₇₁	—	2	2	no	—
95	C₂×C₄×C₇₁	—	2	2	no	—
96	C₂×C₄×C₇₁	—	2	2	no	—
97	C₂×C₄×C₇₁	—	2	2	no	—
98	C₂×C₄×C₇₁	—	2	1	no	—
99	C₂×C₄×C₇₁	—	2	1	no	—
100	C₂×C₄×C₇₁	—	4	1	no	—
101	C₂×C₄×C₇₁	—	4	1	no	—
102	C₂×C₄×C₇₁	—	4	1	no	—
103	C₂×C₄×C₇₁	—	4	1	no	—
104	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
105	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
107	C₂×C₄×C₇₁	#121	1	2	yes	—
108	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
113	C₈×C₇₁	—	2	2	no	—
114	C₈×C₇₁	—	2	2	no	—
115	C₈×C₇₁	—	2	2	no	—
116	C₈×C₇₁	C₂²⋊C₈	1	2	yes	—
117	C₈×C₇₁	C₂²⋊C₈	1	2	yes	—
118	C₇₁⋊C₈	—	8	4	no	—
119	C₂²×C₇₁	—	2	1	no	—
120	C₂²×C₇₁	—	2	1	no	—
121	C₂²×C₇₁	—	2	2	no	—
122	C₂²×C₇₁	—	4	2	no	—
123	C₂²×C₇₁	—	4	2	no	—
124	C₂²×C₇₁	—	4	1	no	—
125	C₂²×C₇₁	—	4	1	no	—
126	C₂²×C₇₁	#380	1	1	yes	characteristic
127	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₂²×C₇₁	#423	1	1	yes	characteristic
129	C₄×C₇₁	—	2	2	no	—
130	C₄×C₇₁	—	2	2	no	—
131	C₄×C₇₁	—	2	2	no	—
132	C₄×C₇₁	—	4	1	no	—
133	C₄×C₇₁	—	4	1	no	—
134	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₄×C₇₁	#407	1	1	yes	characteristic
136	C₄×C₇₁	#431	1	1	yes	characteristic
137	C₄×C₇₁	#432	1	1	yes	characteristic
138	#8601	—	71	1	no	maximal, Sylow
139	C₂×C₇₁	—	2	1	no	—
140	C₂×C₇₁	—	2	1	no	—
141	C₂×C₇₁	—	4	2	no	—
142	C₂×C₇₁	#1797	1	1	yes	characteristic
143	C₂×C₇₁	#1812	1	1	yes	characteristic
144	C₂×C₇₁	#1813	1	1	yes	characteristic
145	C₂×#405	D₇₁	1	1	yes	characteristic
146	#1564	—	71	2	no	—
147	#1458	—	71	2	no	—
148	#1275	—	71	2	no	—
149	C₇₁	#8601	1	1	yes	characteristic, complemented, Sylow
150	C₂²×M₁₆	Dic₇₁	1	1	yes	characteristic
151	C₂³×C₈	Dic₇₁	1	1	yes	characteristic
152	C₂×#117	C₂×D₇₁	1	1	yes	characteristic
153	#431	—	71	4	no	—
154	#405	—	2	2	no	—
155	#405	—	2	2	no	—
156	C₂×#97	Dic₇₁	1	1	yes	characteristic
157	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
158	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
160	#335	—	142	2	no	—
161	#334	—	142	2	no	—
162	#324	—	142	2	no	—
163	#323	—	142	2	no	—
164	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	C₇₁⋊C₈	1	4	yes	complemented
172	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
173	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
174	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
175	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	C₂×C₄⋊C₄	Dic₁₄₂	1	1	yes	characteristic
179	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	Dic₁₄₂	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	D₂₈₄	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
184	C₄⋊C₈	—	142	4	no	—
185	C₂²⋊C₈	—	142	4	no	—
186	#97	—	2	2	no	—
187	#97	—	2	2	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	142	4	no	—
191	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
192	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
193	C₂⁴	C₄⋊Dic₇₁	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
199	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
202	C₂²×C₄	#11765849	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
205	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
206	M₁₆	—	2	8	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	284	4	no	—
216	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
217	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
219	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
220	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
221	C₂×C₈	#11765849	1	2	yes	—
222	C₂×C₈	#11765849	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
230	C₄²	#11765865	1	2	yes	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	4	1	no	—
234	C₂³	—	4	1	no	—
235	C₂³	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
236	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
250	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
251	C₂×C₄	#14556087	1	1	yes	characteristic
252	C₂×C₄	#14556102	1	2	yes	—
253	C₂×C₄	#14556107	1	2	yes	—
254	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
255	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
256	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
257	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	—	568	4	no	—
262	C₈	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
263	C₈	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	#17985859	1	1	yes	characteristic
272	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
273	C₂²	#17986023	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
280	C₄	#17985963	1	1	yes	characteristic
281	C₄	#17986052	1	1	yes	characteristic
282	C₄	#17986056	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	4	2	no	—
286	C₂	#22588320	1	1	yes	characteristic
287	C₂	#22588386	1	1	yes	characteristic
288	C₂	#22588390	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556087
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#405×C₇₁	C₂
socle	C₂²×C₇₁	#380
prime radical	C₂⁴×C₇₁	C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#405	D₇₁
hypocenter	C₇₁	#8601
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. #22587088	C₂	semidirect
3. #22586171	C₂	semidirect
29. C₂²×C₈×C₇₁	C₈	semidirect
149. C₇₁	#8601	semidirect
171. C₂²×C₈	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22587088	2
#22586654	2
#22586171	2
C₂×#405×C₇₁	2
#8601	71