Subgroups — #31897212

Order 18176.

526 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589302	C₂	1	2	yes	maximal
2	#22589143	C₂	1	2	yes	maximal
3	#22588500	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556062	—	2	2	no	—
7	C₂×#14556052	—	2	4	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	#17985869	C₂²	1	2	yes	—
11	#17985761	—	2	4	no	—
12	C₈×C₇₁⋊C₈	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
14	#500×C₇₁	—	2	2	no	—
15	#500×C₇₁	C₄	1	4	yes	complemented
16	C₂×#107×C₇₁	—	2	2	no	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
19	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	2	2	no	—
22	C₂×C₄⋊Dic₇₁	—	2	2	no	—
23	#14556062	—	4	4	no	—
24	#14556052	—	4	8	no	—
25	C₄×C₇₁⋊C₈	—	2	1	no	—
26	C₄×C₇₁⋊C₈	—	2	1	no	—
27	C₄×C₇₁⋊C₈	—	2	1	no	—
28	C₄×C₇₁⋊C₈	—	2	1	no	—
29	C₂²×Q₈×C₇₁	—	2	2	no	—
30	C₂²×C₈×C₇₁	D₄	1	2	yes	—
31	C₄⋊Q₈×C₇₁	—	2	2	no	—
32	C₄⋊Q₈×C₇₁	—	2	2	no	—
33	C₄⋊Q₈×C₇₁	—	2	2	no	—
34	C₄⋊Q₈×C₇₁	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
37	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
38	#107×C₇₁	—	2	8	no	—
39	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₇₁	D₄	1	2	yes	—
41	C₂²×Dic₇₁	—	4	2	no	—
42	C₂×C₇₁⋊C₈	—	2	2	no	—
43	C₂×C₇₁⋊C₈	—	2	2	no	—
44	C₂×C₇₁⋊C₈	—	2	2	no	—
45	C₂×C₇₁⋊C₈	—	2	2	no	—
46	C₂×C₇₁⋊C₈	D₈	1	4	yes	—
47	C₂×C₇₁⋊C₈	Q₁₆	1	4	yes	—
48	C₄⋊Dic₇₁	—	4	4	no	—
49	C₄⋊Dic₇₁	—	4	4	no	—
50	C₄⋊Dic₇₁	—	4	4	no	—
51	C₂×Q₈×C₇₁	—	2	4	no	—
52	C₂×Q₈×C₇₁	—	4	2	no	—
53	C₂×Q₈×C₇₁	—	4	4	no	—
54	C₂×Q₈×C₇₁	—	4	4	no	—
55	C₂²×C₄×C₇₁	—	4	2	no	—
56	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
60	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
61	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
62	C₄⋊C₄×C₇₁	—	2	4	no	—
63	C₄⋊C₄×C₇₁	—	2	4	no	—
64	C₄⋊C₄×C₇₁	—	4	2	no	—
65	C₄⋊C₄×C₇₁	—	4	2	no	—
66	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₇₁	#41	1	1	yes	characteristic
69	C₄²×C₇₁	#41	1	1	yes	characteristic
70	C₂³×C₇₁	#127	1	1	yes	characteristic
71	Q₈×C₇₁	—	4	8	no	—
72	Q₈×C₇₁	—	8	4	no	—
73	C₂×C₄×C₇₁	—	4	4	no	—
74	C₂×C₄×C₇₁	—	8	2	no	—
75	C₂×C₄×C₇₁	—	8	2	no	—
76	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
84	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
85	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₇₁	#135	1	2	yes	—
87	C₂×C₄×C₇₁	#135	1	2	yes	—
88	C₂×Dic₇₁	—	8	2	no	—
89	C₂×Dic₇₁	—	8	2	no	—
90	C₂×Dic₇₁	—	8	2	no	—
91	C₈×C₇₁	#103	1	8	yes	complemented
92	C₇₁⋊C₈	—	2	4	no	—
93	C₇₁⋊C₈	—	2	4	no	—
94	C₂²×C₇₁	#390	1	1	yes	characteristic
95	C₂²×C₇₁	#492	1	1	yes	characteristic
96	C₂²×C₇₁	#493	1	1	yes	characteristic
97	C₂²×C₇₁	#494	1	1	yes	characteristic
98	C₂²×C₇₁	#494	1	1	yes	characteristic
99	C₂²×C₇₁	#495	1	1	yes	characteristic
100	C₂²×C₇₁	#495	1	1	yes	characteristic
101	C₄×C₇₁	—	8	4	no	—
102	C₄×C₇₁	C₂×#103	1	2	yes	—
103	C₄×C₇₁	C₄×D₈	1	2	yes	—
104	C₄×C₇₁	C₄×Q₁₆	1	2	yes	—
105	C₄×C₇₁	#466	1	2	yes	—
106	C₄×C₇₁	#467	1	2	yes	—
107	C₄×C₇₁	#487	1	2	yes	—
108	Dic₇₁	—	16	2	no	—
109	#9823	—	71	1	no	maximal, Sylow
110	C₂×C₇₁	#1706	1	1	yes	characteristic
111	C₂×C₇₁	#1707	1	1	yes	characteristic
112	C₂×C₇₁	#1710	1	1	yes	characteristic
113	C₂×C₇₁	#1713	1	1	yes	characteristic
114	C₂×C₇₁	#1956	1	1	yes	characteristic
115	C₂×C₇₁	#1969	1	1	yes	characteristic
116	C₂×C₇₁	#1970	1	1	yes	characteristic
117	C₂×#500	D₇₁	1	1	yes	characteristic
118	#1990	—	71	2	no	—
119	#1956	—	71	2	no	—
120	#1845	—	71	1	no	—
121	C₂×C₈²	—	71	1	no	—
122	C₇₁	#9823	1	1	yes	characteristic, complemented, Sylow
123	C₂×C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
124	#500	—	2	2	no	—
125	#500	Dic₇₁	1	4	yes	complemented
126	C₂×#107	—	2	2	no	—
127	C₂×#107	—	142	2	no	—
128	C₂×#103	—	142	4	no	—
129	C₂×C₄×C₈	—	71	1	no	—
130	C₂×C₄×C₈	—	71	1	no	—
131	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
132	#384	—	71	2	no	—
133	#340	—	142	4	no	—
134	C₈²	—	142	2	no	—
135	C₂²×Q₈	—	2	2	no	—
136	C₂²×C₈	—	71	2	no	—
137	C₂²×C₈	—	71	2	no	—
138	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	C₂×Dic₇₁	1	4	yes	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	142	2	no	—
146	C₂×C₄⋊C₄	—	142	2	no	—
147	C₂×C₄⋊C₄	—	142	2	no	—
148	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
149	#107	—	2	8	no	—
150	#107	—	284	4	no	—
151	#103	—	284	8	no	—
152	C₄×C₈	—	142	1	no	—
153	C₄×C₈	—	142	1	no	—
154	C₄×C₈	—	142	1	no	—
155	C₄×C₈	—	142	1	no	—
156	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
157	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	284	2	no	—
164	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
165	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
167	C₂×C₈	—	71	4	no	—
168	C₂×C₈	—	71	4	no	—
169	C₂×C₈	—	142	2	no	—
170	C₂×C₈	—	142	2	no	—
171	C₂×C₈	—	142	2	no	—
172	C₂×C₈	—	142	2	no	—
173	C₂×C₈	C₂²⋊Dic₇₁	1	4	yes	—
174	C₂×C₈	#11765852	1	4	yes	—
175	C₂×C₈	#11765855	1	4	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	284	4	no	—
181	C₄⋊C₄	—	284	4	no	—
182	C₄⋊C₄	—	284	4	no	—
183	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
184	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
185	C₄²	#11765864	1	1	yes	characteristic
186	C₄²	#11765864	1	1	yes	characteristic
187	C₂³	#14556124	1	1	yes	characteristic
188	Q₈	—	4	8	no	—
189	Q₈	—	8	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	568	2	no	—
194	C₂×C₄	—	568	2	no	—
195	C₂×C₄	—	568	2	no	—
196	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
197	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
198	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
199	C₂×C₄	#14556099	1	1	yes	characteristic
200	C₂×C₄	#14556099	1	1	yes	characteristic
201	C₂×C₄	#14556115	1	1	yes	characteristic
202	C₂×C₄	D₈×D₇₁	1	2	yes	—
203	C₂×C₄	C₂×#11765852	1	2	yes	—
204	C₂×C₄	Q₁₆×D₇₁	1	2	yes	—
205	C₂×C₄	C₂×#11765855	1	2	yes	—
206	C₂×C₄	#14556147	1	2	yes	—
207	C₂×C₄	#14556151	1	2	yes	—
208	C₈	—	142	4	no	—
209	C₈	—	142	4	no	—
210	C₈	#14556052	1	8	yes	complemented
211	C₂²	#17985902	1	1	yes	characteristic
212	C₂²	#17986373	1	1	yes	characteristic
213	C₂²	#17986374	1	1	yes	characteristic
214	C₂²	#17986377	1	1	yes	characteristic
215	C₂²	#17986378	1	1	yes	characteristic
216	C₂²	#17986382	1	1	yes	characteristic
217	C₂²	#17986384	1	1	yes	characteristic
218	C₄	—	8	4	no	—
219	C₄	—	1136	2	no	—
220	C₄	C₂×#14556052	1	2	yes	—
221	C₄	D₈×Dic₇₁	1	2	yes	—
222	C₄	Q₁₆×Dic₇₁	1	2	yes	—
223	C₄	#17986232	1	2	yes	—
224	C₄	#17986237	1	2	yes	—
225	C₄	#17986349	1	2	yes	—
226	C₂	#22587933	1	1	yes	characteristic
227	C₂	#22587938	1	1	yes	characteristic
228	C₂	#22587949	1	1	yes	characteristic
229	C₂	#22587957	1	1	yes	characteristic
230	C₂	#22589143	1	1	yes	characteristic
231	C₂	#22589202	1	1	yes	characteristic
232	C₂	#22589207	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#500×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#127
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#500	D₇₁
hypocenter	C₇₁	#9823
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
15. #500×C₇₁	C₄	semidirect
91. C₈×C₇₁	#103	semidirect
122. C₇₁	#9823	semidirect
125. #500	Dic₇₁	semidirect
210. C₈	#14556052	semidirect

Maximal subgroups

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

subgroup	index
#22589302	2
#22589143	2
#22588500	2
C₂×C₈×C₇₁⋊C₈	2
C₂×#500×C₇₁	2
#9823	71