Finite Group Databasebeta

Subgroups — #31918337

Order 18176.

164 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 59 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	#22590614	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586542	C₂	1	1	yes	maximal, characteristic
4	#22586541	C₂	1	1	yes	maximal, characteristic
5	Dic₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	#22586521	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1416	C₂	1	1	yes	maximal, characteristic
8	#17986403	—	2	1	no	—
9	Q₁₆×Dic₇₁	—	2	1	no	—
10	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
11	#17985825	C₂²	1	1	yes	characteristic
12	#17985825	C₂²	1	1	yes	characteristic
13	#17985824	—	2	1	no	—
14	#17985803	—	2	1	no	—
15	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
18	#366×C₇₁	C₂²	1	1	yes	characteristic
19	#14556127	—	4	1	no	—
20	Q₈×Dic₇₁	—	4	1	no	—
21	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
22	#14556062	—	2	1	no	—
23	#14556062	—	2	1	no	—
24	#14556061	—	4	1	no	—
25	#14556052	—	4	1	no	—
26	C₈×Dic₇₁	D₄	1	1	yes	characteristic
27	C₇₁⋊C₃₂	—	2	1	no	—
28	C₂×Q₁₆×C₇₁	—	2	1	no	—
29	Q₃₂×C₇₁	—	2	1	no	—
30	Q₃₂×C₇₁	Q₈	1	2	yes	complemented
31	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
32	#107×C₇₁	—	2	1	no	—
33	C₃₂×C₇₁	—	2	1	no	—
34	C₂×Dic₁₄₂	—	8	1	no	—
35	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
36	C₄⋊Dic₇₁	—	4	1	no	—
37	C₄⋊Dic₇₁	—	4	1	no	—
38	C₄⋊Dic₇₁	—	8	1	no	—
39	Dic₇₁⋊C₄	—	8	1	no	—
40	C₄×Dic₇₁	—	8	1	no	—
41	C₄×Dic₇₁	D₈	1	1	yes	characteristic
42	C₇₁⋊C₁₆	—	2	1	no	—
43	C₂×Q₈×C₇₁	—	4	1	no	—
44	Q₁₆×C₇₁	—	2	2	no	—
45	Q₁₆×C₇₁	—	4	1	no	—
46	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₇₁	—	4	1	no	—
48	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
49	C₁₆×C₇₁	#41	1	1	yes	characteristic
50	Q₈×C₇₁	—	4	2	no	—
51	Q₈×C₇₁	—	8	1	no	—
52	C₂×C₄×C₇₁	—	8	1	no	—
53	C₂×C₄×C₇₁	—	8	1	no	—
54	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₇₁	—	8	1	no	—
56	C₂×Dic₇₁	—	8	1	no	—
57	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
58	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
59	Dic₁₄₂	—	16	2	no	—
60	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
61	C₈×C₇₁	#137	1	1	yes	characteristic
62	C₇₁⋊C₈	—	2	1	no	—
63	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₇₁	—	8	2	no	—
65	C₄×C₇₁	—	16	1	no	—
66	C₄×C₇₁	#474	1	1	yes	characteristic
67	C₄×C₇₁	#510	1	1	yes	characteristic
68	Dic₇₁	—	2	1	no	—
69	Dic₇₁	—	16	1	no	—
70	Dic₇₁	—	16	1	no	—
71	Dic₇₁	SD₆₄	1	2	yes	—
72	#13704	—	71	1	no	maximal, Sylow
73	C₂×C₇₁	#2224	1	1	yes	characteristic
74	C₂×C₇₁	C₂×SD₆₄	1	1	yes	characteristic
75	C₂×C₇₁	#2264	1	1	yes	characteristic
76	#2252	—	71	1	no	—
77	C₄×Q₃₂	—	71	1	no	—
78	#1424	—	71	1	no	—
79	#1424	—	71	1	no	—
80	C₄⋊C₃₂	—	71	1	no	—
81	#1416	—	71	1	no	—
82	#1416	D₇₁	1	1	yes	characteristic
83	C₇₁	#13704	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₃₂	C₂×D₇₁	1	1	yes	characteristic
85	#500	—	142	1	no	—
86	C₄×Q₁₆	—	142	1	no	—
87	C₂×C₃₂	—	71	1	no	—
88	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
89	#366	—	71	1	no	—
90	#366	—	71	1	no	—
91	#366	—	142	1	no	—
92	#366	C₂×D₇₁	1	1	yes	characteristic
93	#358	—	142	1	no	—
94	C₄×C₁₆	—	71	1	no	—
95	C₂×Q₁₆	—	2	1	no	—
96	C₄⋊Q₈	—	284	1	no	—
97	C₄×Q₈	—	284	1	no	—
98	Q₃₂	—	2	1	no	—
99	Q₃₂	Dic₁₄₂	1	2	yes	complemented
100	C₂×C₁₆	—	71	1	no	—
101	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
102	#107	—	2	1	no	—
103	#107	—	284	1	no	—
104	#107	—	142	1	no	—
105	#107	—	142	1	no	—
106	#103	—	284	1	no	—
107	C₄×C₈	—	71	1	no	—
108	C₃₂	—	2	1	no	—
109	C₃₂	—	142	1	no	—
110	C₂×Q₈	—	4	1	no	—
111	C₂×Q₈	—	568	1	no	—
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	4	1	no	—
114	C₂×C₈	—	71	1	no	—
115	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	284	1	no	—
118	C₄⋊C₄	—	284	1	no	—
119	C₄⋊C₄	—	568	1	no	—
120	C₄⋊C₄	—	568	1	no	—
121	C₄²	—	568	1	no	—
122	C₄²	—	71	1	no	—
123	C₁₆	—	142	1	no	—
124	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
125	C₁₆	#11765864	1	1	yes	characteristic
126	Q₈	—	4	2	no	—
127	Q₈	—	8	1	no	—
128	Q₈	—	1136	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	568	1	no	—
132	C₂×C₄	—	568	1	no	—
133	C₂×C₄	—	71	1	no	—
134	C₂×C₄	—	71	1	no	—
135	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
136	C₈	—	142	1	no	—
137	C₈	#14556105	1	1	yes	characteristic
138	C₈	#14556157	1	1	yes	characteristic
139	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	—	71	2	no	—
143	C₄	—	1136	1	no	—
144	C₄	—	1136	1	no	—
145	C₄	—	142	1	no	—
146	C₄	#17986277	1	1	yes	characteristic
147	C₄	#17986438	1	1	yes	characteristic
148	C₂	#22590475	1	1	yes	characteristic
149	C₂	SD₆₄×D₇₁	1	1	yes	characteristic
150	C₂	#22590659	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₁
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1416	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
hypercenter	#1416	D₇₁
hypocenter	C₇₁	#13704
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

Complements

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

N	H ≅ G/N	type
30. Q₃₂×C₇₁	Q₈	semidirect
83. C₇₁	#13704	semidirect
99. Q₃₂	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22590614	2
Q₃₂×Dic₇₁	2
#22586542	2
#22586541	2
Dic₇₁⋊C₃₂	2
#22586521	2
C₇₁×#1416	2
#13704	71