Finite Group Databasebeta

Subgroups — #31918371

Order 18176.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590621	C₂	1	1	yes	maximal, characteristic, complemented
2	#22590221	C₂	1	1	yes	maximal, characteristic
3	#22586546	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586545	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586537	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586527	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1418	C₂	1	1	yes	maximal, characteristic
8	#17986403	—	2	1	no	—
9	#17986116	—	2	1	no	—
10	#17985833	C₂²	1	1	yes	characteristic
11	#17985829	C₂²	1	1	yes	characteristic
12	#17985827	C₂²	1	1	yes	characteristic
13	#17985826	—	2	1	no	—
14	#17985806	—	2	1	no	—
15	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
16	#508×C₇₁	C₂²	1	1	yes	characteristic
17	M₆₄×C₇₁	C₂²	1	1	yes	characteristic
18	#367×C₇₁	C₂²	1	1	yes	characteristic
19	#14556135	—	4	1	no	—
20	#14556127	—	4	1	no	—
21	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
22	#14556064	—	2	1	no	—
23	#14556062	—	2	1	no	—
24	#14556061	—	4	1	no	—
25	#14556055	—	4	1	no	—
26	C₈×Dic₇₁	D₄	1	1	yes	characteristic
27	C₇₁⋊C₃₂	—	2	1	no	—
28	#135×C₇₁	—	2	1	no	—
29	Q₃₂×C₇₁	Q₈	1	1	yes	characteristic, complemented
30	SD₃₂×C₇₁	—	2	1	no	—
31	D₁₆×C₇₁	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
33	#107×C₇₁	—	2	1	no	—
34	C₃₂×C₇₁	—	2	1	no	—
35	C₂×Dic₁₄₂	—	8	1	no	—
36	#11765849	—	4	1	no	—
37	#11765849	—	8	1	no	—
38	C₂×C₇₁⋊C₈	—	8	1	no	—
39	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
40	C₄⋊Dic₇₁	—	4	1	no	—
41	Dic₇₁⋊C₄	—	8	1	no	—
42	C₄×Dic₇₁	D₈	1	1	yes	characteristic
43	C₇₁⋊C₁₆	—	2	1	no	—
44	#41×C₇₁	—	4	1	no	—
45	Q₁₆×C₇₁	—	2	1	no	—
46	SD₁₆×C₇₁	—	4	1	no	—
47	D₈×C₇₁	—	2	1	no	—
48	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₄⋊C₄×C₇₁	—	4	1	no	—
50	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
51	C₁₆×C₇₁	#41	1	1	yes	characteristic
52	Q₈×C₇₁	—	4	1	no	—
53	D₄×C₇₁	—	4	1	no	—
54	D₄×C₇₁	—	8	1	no	—
55	C₂×C₄×C₇₁	—	8	1	no	—
56	C₂×C₄×C₇₁	—	8	1	no	—
57	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₇₁	—	2	1	no	—
59	C₂×Dic₇₁	—	8	1	no	—
60	Dic₁₄₂	—	16	2	no	—
61	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₇₁	#137	1	1	yes	characteristic
63	C₇₁⋊C₈	—	8	1	no	—
64	C₇₁⋊C₈	SD₃₂	1	1	yes	characteristic
65	C₇₁⋊C₈	SD₃₂	1	1	yes	characteristic
66	C₂²×C₇₁	—	8	1	no	—
67	C₂²×C₇₁	#509	1	1	yes	characteristic
68	C₄×C₇₁	—	8	1	no	—
69	C₄×C₇₁	—	16	1	no	—
70	C₄×C₇₁	#474	1	1	yes	characteristic
71	C₄×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
72	Dic₇₁	—	2	2	no	—
73	Dic₇₁	—	16	1	no	—
74	#13710	—	71	1	no	maximal, Sylow
75	C₂×C₇₁	—	2	1	no	—
76	C₂×C₇₁	—	16	1	no	—
77	C₂×C₇₁	#2226	1	1	yes	characteristic
78	#2254	—	71	1	no	—
79	#2178	—	71	1	no	—
80	#1426	—	71	1	no	—
81	#1426	—	71	1	no	—
82	#1422	—	71	1	no	—
83	#1418	—	71	1	no	—
84	#1418	D₇₁	1	1	yes	characteristic
85	C₇₁	#13710	1	1	yes	characteristic, complemented, Sylow
86	#508	C₂×D₇₁	1	1	yes	characteristic
87	#500	—	142	1	no	—
88	#443	—	142	1	no	—
89	M₆₄	—	71	1	no	—
90	M₆₄	C₂×D₇₁	1	1	yes	characteristic
91	#368	—	71	1	no	—
92	#367	—	71	1	no	—
93	#367	—	142	1	no	—
94	#367	C₂×D₇₁	1	1	yes	characteristic
95	#359	—	142	1	no	—
96	C₄×C₁₆	—	71	1	no	—
97	#135	—	2	1	no	—
98	#131	—	284	1	no	—
99	C₄⋊Q₈	—	284	1	no	—
100	Q₃₂	Dic₁₄₂	1	1	yes	characteristic, complemented
101	SD₃₂	—	2	1	no	—
102	D₁₆	Dic₁₄₂	1	1	yes	characteristic, complemented
103	C₂×C₁₆	—	71	1	no	—
104	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
105	#108	—	142	1	no	—
106	#107	—	2	1	no	—
107	#107	—	284	1	no	—
108	#107	—	142	1	no	—
109	#104	—	284	1	no	—
110	C₄×C₈	—	71	1	no	—
111	C₃₂	—	2	1	no	—
112	C₃₂	—	142	1	no	—
113	#41	—	4	1	no	—
114	C₂×Q₈	—	568	1	no	—
115	Q₁₆	—	2	1	no	—
116	SD₁₆	—	4	1	no	—
117	D₈	—	2	1	no	—
118	M₁₆	—	284	1	no	—
119	M₁₆	—	568	1	no	—
120	C₂×C₈	—	568	1	no	—
121	C₂×C₈	—	71	1	no	—
122	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
123	C₄⋊C₄	—	4	1	no	—
124	C₄⋊C₄	—	284	1	no	—
125	C₄⋊C₄	—	568	1	no	—
126	C₄²	—	71	1	no	—
127	C₁₆	—	142	1	no	—
128	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
129	C₁₆	#11765864	1	1	yes	characteristic
130	Q₈	—	4	1	no	—
131	Q₈	—	1136	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	568	1	no	—
137	C₂×C₄	—	142	1	no	—
138	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
139	C₈	—	568	1	no	—
140	C₈	—	71	1	no	—
141	C₈	—	71	1	no	—
142	C₈	#14556105	1	1	yes	characteristic
143	C₈	#14556157	1	1	yes	characteristic
144	C₂²	—	8	1	no	—
145	C₂²	#17986433	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	16	1	no	—
148	C₄	—	1136	1	no	—
149	C₄	—	142	2	no	—
150	C₄	#17986277	1	1	yes	characteristic
151	C₄	SD₃₂×D₇₁	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	16	1	no	—
154	C₂	#22590485	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22590485
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1418	C₂
socle	C₂×C₇₁	#2226
prime radical	#508×C₇₁	C₂²
hypercenter	#1418	D₇₁
hypocenter	C₇₁	#13710
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
1. #22590621	C₂	semidirect
3. #22586546	C₂	semidirect
4. #22586545	C₂	semidirect
5. #22586537	C₂	semidirect
29. Q₃₂×C₇₁	Q₈	semidirect
31. D₁₆×C₇₁	Q₈	semidirect
85. C₇₁	#13710	semidirect
100. Q₃₂	Dic₁₄₂	semidirect
102. D₁₆	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22590621	2
#22590221	2
#22586546	2
#22586545	2
#22586537	2
#22586527	2
C₇₁×#1418	2
#13710	71