Subgroups — #31916034

Order 18176.

366 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556062	C₂	1	1	yes	maximal, characteristic
3	#22586431	C₂	1	2	yes	maximal
4	#22586431	C₂	1	2	yes	maximal
5	C₂×#499×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14556062	C₂²	1	2	yes	—
8	C₄×C₄⋊Dic₇₁	—	2	1	no	—
9	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
10	#17985762	—	2	1	no	—
11	C₂×#125×C₇₁	—	2	1	no	—
12	#499×C₇₁	—	2	2	no	—
13	#499×C₇₁	C₄	1	4	yes	complemented
14	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#106×C₇₁	—	2	1	no	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₇₁	—	2	2	no	—
19	C₂×C₄×Dic₇₁	—	4	1	no	—
20	C₂×C₇₁⋊C₁₆	—	2	2	no	—
21	C₂×C₇₁⋊C₁₆	—	2	4	no	—
22	#14556062	—	2	2	no	—
23	#14556062	Q₈	1	4	yes	—
24	C₇₁⋊#95	—	4	1	no	—
25	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
27	#125×C₇₁	—	2	4	no	—
28	#125×C₇₁	—	4	2	no	—
29	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
32	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
33	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
34	#107×C₇₁	—	2	1	no	—
35	#107×C₇₁	—	2	1	no	—
36	#107×C₇₁	C₂×C₄	1	2	yes	—
37	#107×C₇₁	C₂×C₄	1	2	yes	—
38	#106×C₇₁	—	2	4	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	C₄⋊Dic₇₁	—	2	4	no	—
43	C₄⋊Dic₇₁	—	4	2	no	—
44	C₄×Dic₇₁	—	8	2	no	—
45	C₇₁⋊C₁₆	—	4	2	no	—
46	C₂²×C₄×C₇₁	—	4	1	no	—
47	C₂²×C₄×C₇₁	—	4	1	no	—
48	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	—	2	2	no	—
52	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₇₁	#41	1	2	yes	—
57	C₄⋊C₄×C₇₁	—	2	2	no	—
58	C₄⋊C₄×C₇₁	—	2	2	no	—
59	C₄⋊C₄×C₇₁	—	4	4	no	—
60	C₄⋊C₄×C₇₁	—	4	4	no	—
61	C₄⋊C₄×C₇₁	—	4	1	no	—
62	C₄⋊C₄×C₇₁	—	4	1	no	—
63	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₇₁	SD₁₆	1	2	yes	—
65	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	—	2	1	no	—
68	C₂×C₄×C₇₁	—	2	1	no	—
69	C₂×C₄×C₇₁	—	4	2	no	—
70	C₂×C₄×C₇₁	—	4	2	no	—
71	C₂×C₄×C₇₁	—	8	2	no	—
72	C₂×C₄×C₇₁	—	8	1	no	—
73	C₂×C₄×C₇₁	—	8	1	no	—
74	C₂×C₄×C₇₁	#102	1	2	yes	—
75	C₂×C₄×C₇₁	#102	1	2	yes	—
76	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#137	1	2	yes	—
80	C₂×Dic₇₁	—	4	4	no	—
81	C₂×Dic₇₁	—	8	2	no	—
82	C₂×Dic₇₁	—	8	2	no	—
83	C₈×C₇₁	—	2	2	no	—
84	C₈×C₇₁	C₄×Q₈	1	2	yes	—
85	C₈×C₇₁	#124	1	1	yes	characteristic
86	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
87	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
88	C₂²×C₇₁	#474	1	2	yes	—
89	C₂²×C₇₁	#508	1	2	yes	—
90	C₂²×C₇₁	#509	1	1	yes	characteristic
91	C₂²×C₇₁	#510	1	1	yes	characteristic
92	C₄×C₇₁	—	2	2	no	—
93	C₄×C₇₁	—	2	2	no	—
94	C₄×C₇₁	—	8	2	no	—
95	C₄×C₇₁	—	8	2	no	—
96	C₄×C₇₁	#391	1	1	yes	characteristic
97	C₄×C₇₁	#419	1	1	yes	characteristic
98	C₄×C₇₁	#475	1	2	yes	—
99	Dic₇₁	—	8	4	no	—
100	#13277	—	71	1	no	maximal, Sylow
101	C₂×C₇₁	#1988	1	1	yes	characteristic
102	C₂×C₇₁	#2138	1	1	yes	characteristic
103	C₂×C₇₁	#2139	1	1	yes	characteristic
104	C₂×C₇₁	#2228	1	2	yes	—
105	C₂×C₇₁	#2229	1	2	yes	—
106	C₂×#499	D₇₁	1	1	yes	characteristic
107	C₂×C₄⋊C₁₆	—	71	1	no	—
108	C₄×#107	—	71	1	no	—
109	#1380	—	71	2	no	—
110	#1380	—	71	2	no	—
111	C₇₁	#13277	1	1	yes	characteristic, complemented, Sylow
112	C₂×#125	—	2	1	no	—
113	C₂²×C₁₆	—	71	2	no	—
114	#499	—	2	2	no	—
115	#499	Dic₇₁	1	4	yes	complemented
116	C₂×#107	—	71	2	no	—
117	C₂×#107	C₂×D₇₁	1	1	yes	characteristic
118	C₂×#107	C₂×D₇₁	1	1	yes	characteristic
119	C₂×#106	—	2	1	no	—
120	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
121	C₄×C₄⋊C₄	—	142	1	no	—
122	C₄⋊C₁₆	—	142	2	no	—
123	#340	—	142	1	no	—
124	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
126	#125	—	2	4	no	—
127	#125	—	4	2	no	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄⋊C₄	—	142	2	no	—
133	C₂×C₄²	—	284	1	no	—
134	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
135	C₂×C₁₆	—	142	2	no	—
136	C₂×C₁₆	—	142	4	no	—
137	#107	—	2	1	no	—
138	#107	—	2	1	no	—
139	#107	—	71	4	no	—
140	#107	—	142	2	no	—
141	#107	C₂×Dic₇₁	1	2	yes	—
142	#107	C₂×Dic₇₁	1	2	yes	—
143	#106	—	2	4	no	—
144	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
145	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
146	#95	—	284	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	—	284	2	no	—
150	C₂²×C₄	#11765850	1	1	yes	characteristic
151	C₂²×C₄	#11765850	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
155	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
157	C₂×C₈	Q₈×D₇₁	1	2	yes	—
158	C₂×C₈	#11765866	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄⋊C₄	—	284	2	no	—
166	C₄⋊C₄	—	142	4	no	—
167	C₄²	—	568	2	no	—
168	C₄²	C₂²⋊Dic₇₁	1	2	yes	—
169	C₄²	#11765853	1	2	yes	—
170	C₁₆	—	284	2	no	—
171	C₂³	C₂×#11765850	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	—	284	4	no	—
181	C₂×C₄	—	568	2	no	—
182	C₂×C₄	—	568	2	no	—
183	C₂×C₄	#14556049	1	2	yes	—
184	C₂×C₄	#14556049	1	2	yes	—
185	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
186	C₂×C₄	#14556104	1	2	yes	—
187	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
188	C₂×C₄	#14556159	1	2	yes	—
189	C₈	—	2	2	no	—
190	C₈	Q₈×Dic₇₁	1	2	yes	—
191	C₈	#14556117	1	1	yes	characteristic
192	C₈	#14556126	1	1	yes	characteristic
193	C₂²	C₂×#14556049	1	1	yes	characteristic
194	C₂²	#17986276	1	2	yes	—
195	C₂²	#17986429	1	2	yes	—
196	C₂²	#17986434	1	1	yes	characteristic
197	C₂²	#17986439	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	8	2	no	—
201	C₄	—	8	2	no	—
202	C₄	—	568	4	no	—
203	C₄	#17985906	1	1	yes	characteristic
204	C₄	#17986008	1	1	yes	characteristic
205	C₄	#17986281	1	2	yes	—
206	C₂	#22589293	1	1	yes	characteristic
207	C₂	#22590047	1	1	yes	characteristic
208	C₂	#22590052	1	1	yes	characteristic
209	C₂	#22590494	1	2	yes	—
210	C₂	#22590501	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#499×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#499	D₇₁
hypocenter	C₇₁	#13277
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
13. #499×C₇₁	C₄	semidirect
111. C₇₁	#13277	semidirect
115. #499	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₄×#14556062	2
#22586431	2
C₂×#499×C₇₁	2
#13277	71