Finite Group Databasebeta

Subgroups — #30665544

Order 16768.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#22553663	C₂	1	1	yes	maximal, characteristic
2	#22553608	C₂	1	1	yes	maximal, characteristic
3	#22553598	C₂	1	1	yes	maximal, characteristic
4	#22553442	C₂	1	1	yes	maximal, characteristic
5	#22553060	C₂	1	1	yes	maximal, characteristic
6	#22553057	C₂	1	1	yes	maximal, characteristic
7	#334×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#16862246	C₂²	1	1	yes	characteristic
9	#16862246	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
11	#16862180	—	2	1	no	—
12	#16862178	—	2	1	no	—
13	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₁₃₁⋊C₈	—	2	1	no	—
15	#16862171	—	2	1	no	—
16	#16862171	—	2	1	no	—
17	#16862171	—	2	1	no	—
18	#16862171	—	2	1	no	—
19	#16862156	—	2	1	no	—
20	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₆₂	—	2	1	no	—
24	C₂×Dic₂₆₂	—	2	1	no	—
25	C₂×Dic₂₆₂	D₄	1	1	yes	characteristic
26	C₂×C₁₃₁⋊C₈	—	2	1	no	—
27	C₄⋊Dic₁₃₁	—	2	1	no	—
28	C₄⋊Dic₁₃₁	—	2	1	no	—
29	C₄⋊Dic₁₃₁	—	2	1	no	—
30	C₄⋊Dic₁₃₁	—	2	1	no	—
31	C₄⋊Dic₁₃₁	D₄	1	1	yes	characteristic
32	Dic₁₃₁⋊C₄	—	2	1	no	—
33	C₄×Dic₁₃₁	—	2	1	no	—
34	C₂×C₈×C₁₃₁	—	2	1	no	—
35	C₂×C₈×C₁₃₁	—	2	1	no	—
36	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
38	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
39	Dic₂₆₂	—	2	1	no	—
40	Dic₂₆₂	—	2	1	no	—
41	Dic₂₆₂	—	4	1	no	—
42	Dic₂₆₂	—	4	1	no	—
43	Dic₂₆₂	—	8	1	no	—
44	C₂×Dic₁₃₁	—	2	1	no	—
45	C₂×Dic₁₃₁	—	2	1	no	—
46	C₂×Dic₁₃₁	—	4	1	no	—
47	C₂×Dic₁₃₁	—	4	1	no	—
48	C₁₃₁⋊C₈	—	4	1	no	—
49	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₁	#41	1	1	yes	characteristic
52	C₈×C₁₃₁	—	2	1	no	—
53	C₈×C₁₃₁	—	2	1	no	—
54	C₈×C₁₃₁	—	4	1	no	—
55	C₈×C₁₃₁	—	4	1	no	—
56	C₂²×C₁₃₁	#121	1	1	yes	characteristic
57	C₄×C₁₃₁	—	2	1	no	—
58	C₄×C₁₃₁	#135	1	1	yes	characteristic
59	C₄×C₁₃₁	#137	1	1	yes	characteristic
60	C₄×C₁₃₁	#137	1	1	yes	characteristic
61	C₄×C₁₃₁	#137	1	1	yes	characteristic
62	Dic₁₃₁	—	4	1	no	—
63	Dic₁₃₁	—	4	1	no	—
64	Dic₁₃₁	—	8	1	no	—
65	Dic₁₃₁	—	8	1	no	—
66	C₂×C₁₃₁	#464	1	1	yes	characteristic
67	C₂×C₁₃₁	#470	1	1	yes	characteristic
68	C₂×C₁₃₁	#471	1	1	yes	characteristic
69	C₁₃₁	#1704	1	1	yes	characteristic, complemented, Sylow
70	#1704	—	131	1	no	maximal, Sylow
71	#487	—	131	1	no	—
72	#477	—	131	1	no	—
73	#475	—	131	1	no	—
74	#446	—	131	1	no	—
75	#334	D₁₃₁	1	1	yes	characteristic
76	#333	—	131	1	no	—
77	#332	—	131	1	no	—
78	C₄⋊Q₈	—	131	1	no	—
79	C₄⋊Q₈	—	131	1	no	—
80	C₄×Q₈	—	131	1	no	—
81	#107	—	262	1	no	—
82	#106	—	262	1	no	—
83	C₄⋊C₈	—	262	1	no	—
84	C₄⋊C₈	—	131	1	no	—
85	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
87	#103	—	262	1	no	—
88	#103	—	262	1	no	—
89	#103	—	262	1	no	—
90	#103	—	262	1	no	—
91	#97	—	262	1	no	—
92	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
93	C₂×Q₈	—	262	1	no	—
94	C₂×Q₈	—	262	1	no	—
95	C₂×Q₈	—	131	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	262	1	no	—
99	C₂×C₈	D₅₂₄	1	1	yes	characteristic
100	C₂×C₈	D₅₂₄	1	1	yes	characteristic
101	C₄⋊C₄	—	262	1	no	—
102	C₄⋊C₄	—	262	1	no	—
103	C₄⋊C₄	—	262	1	no	—
104	C₄⋊C₄	—	262	1	no	—
105	C₄⋊C₄	—	262	1	no	—
106	C₄⋊C₄	—	131	1	no	—
107	C₄²	—	262	1	no	—
108	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
109	Q₈	—	262	1	no	—
110	Q₈	—	262	1	no	—
111	Q₈	—	524	1	no	—
112	Q₈	—	524	1	no	—
113	Q₈	—	1048	1	no	—
114	C₂×C₄	—	262	1	no	—
115	C₂×C₄	—	262	1	no	—
116	C₂×C₄	—	524	1	no	—
117	C₂×C₄	—	524	1	no	—
118	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
119	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
120	C₂×C₄	#14523514	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	524	1	no	—
126	C₂²	#16862221	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	524	1	no	—
129	C₄	—	524	1	no	—
130	C₄	—	1048	1	no	—
131	C₄	—	1048	1	no	—
132	C₄	#16862268	1	1	yes	characteristic
133	C₄	#16862275	1	1	yes	characteristic
134	C₄	#16862275	1	1	yes	characteristic
135	C₄	#16862279	1	1	yes	characteristic
136	C₂	#22553535	1	1	yes	characteristic
137	C₂	#22553571	1	1	yes	characteristic
138	C₂	#22553576	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16862221
commutator, cosocle	C₄²×C₁₃₁	C₂³
Frattini	C₄²	C₂²×D₁₃₁
Fitting	#334×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#121
hypercenter	#334	D₁₃₁
hypocenter	C₁₃₁	#1704
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

Complements

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

N	H ≅ G/N	type
69. C₁₃₁	#1704	semidirect

Maximal subgroups

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

subgroup	index
#22553663	2
#22553608	2
#22553598	2
#22553442	2
#22553060	2
#22553057	2
#334×C₁₃₁	2
#1704	131