Finite Group Databasebeta

Subgroups — #45663748

Order 32896.

150 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 55 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	#30657087	C₂	1	1	yes	maximal, characteristic
2	#30656947	C₂	1	1	yes	maximal, characteristic
3	Q₈×#14523339	C₂	1	1	yes	maximal, characteristic
4	#30656487	C₂	1	1	yes	maximal, characteristic
5	#30656484	C₂	1	1	yes	maximal, characteristic
6	#30656483	C₂	1	1	yes	maximal, characteristic
7	#475×C₂₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#16861608	—	2	1	no	—
9	#22551592	C₂²	1	1	yes	characteristic
10	C₄⋊#14523339	—	2	1	no	—
11	C₄⋊#14523339	C₂²	1	1	yes	characteristic
12	C₄⋊#14523339	C₂²	1	1	yes	characteristic
13	#22551516	—	2	1	no	—
14	#22551516	—	2	1	no	—
15	#22551515	—	2	1	no	—
16	C₄×#14523339	—	2	1	no	—
17	C₄×#14523339	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
20	#106×C₂₅₇	—	2	1	no	—
21	C₄⋊C₈×C₂₅₇	C₂²	1	1	yes	characteristic
22	#103×C₂₅₇	—	2	1	no	—
23	C₂×Dic₅₁₄	—	2	1	no	—
24	#16861608	—	4	2	no	—
25	C₂×#14523339	—	2	1	no	—
26	C₂×#14523339	—	2	1	no	—
27	C₂×#14523339	D₄	1	1	yes	characteristic
28	C₂×#14523339	D₄	1	1	yes	characteristic
29	C₄⋊Dic₂₅₇	—	2	1	no	—
30	C₄⋊Dic₂₅₇	—	2	1	no	—
31	C₂×Q₈×C₂₅₇	—	2	1	no	—
32	C₂×Q₈×C₂₅₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₂₅₇	—	2	1	no	—
34	C₄⋊C₄×C₂₅₇	—	2	1	no	—
35	C₄⋊C₄×C₂₅₇	—	2	1	no	—
36	C₄⋊C₄×C₂₅₇	—	2	1	no	—
37	C₄⋊C₄×C₂₅₇	D₄	1	1	yes	characteristic
38	C₄²×C₂₅₇	—	2	1	no	—
39	C₄²×C₂₅₇	C₂³	1	1	yes	characteristic
40	Dic₅₁₄	—	4	1	no	—
41	Dic₅₁₄	—	4	1	no	—
42	C₂×Dic₂₅₇	—	4	1	no	—
43	#14523339	—	2	1	no	—
44	#14523339	—	2	1	no	—
45	#14523339	—	4	1	no	—
46	#14523339	—	4	1	no	—
47	Q₈×C₂₅₇	—	2	1	no	—
48	Q₈×C₂₅₇	—	4	1	no	—
49	Q₈×C₂₅₇	—	4	1	no	—
50	Q₈×C₂₅₇	SD₁₆	1	2	yes	—
51	C₂×C₄×C₂₅₇	—	2	1	no	—
52	C₂×C₄×C₂₅₇	—	2	1	no	—
53	C₂×C₄×C₂₅₇	—	4	1	no	—
54	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
57	C₈×C₂₅₇	—	4	1	no	—
58	C₂²×C₂₅₇	#121	1	1	yes	characteristic
59	C₄×C₂₅₇	—	2	2	no	—
60	C₄×C₂₅₇	—	2	1	no	—
61	C₄×C₂₅₇	—	4	1	no	—
62	C₄×C₂₅₇	—	8	1	no	—
63	C₄×C₂₅₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₄×C₂₅₇	#135	1	1	yes	characteristic
65	C₄×C₂₅₇	#135	1	1	yes	characteristic
66	C₄×C₂₅₇	#137	1	1	yes	characteristic
67	Dic₂₅₇	—	8	1	no	—
68	C₂×C₂₅₇	#464	1	1	yes	characteristic
69	C₂×C₂₅₇	#465	1	1	yes	characteristic
70	C₂×C₂₅₇	#473	1	1	yes	characteristic
71	C₂₅₇	#1680	1	1	yes	characteristic, complemented, Sylow
72	#1680	—	257	1	no	maximal, Sylow
73	#487	—	257	1	no	—
74	#475	D₂₅₇	1	1	yes	characteristic
75	#462	—	257	1	no	—
76	C₈×Q₈	—	257	1	no	—
77	#334	—	257	1	no	—
78	#332	—	257	1	no	—
79	#332	—	257	1	no	—
80	C₂×Q₁₆	—	514	1	no	—
81	C₄⋊Q₈	—	257	1	no	—
82	C₄⋊Q₈	C₂×D₂₅₇	1	1	yes	characteristic
83	C₄×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
84	#106	—	2	1	no	—
85	C₄⋊C₈	—	257	1	no	—
86	C₄⋊C₈	—	257	1	no	—
87	C₄⋊C₈	—	514	1	no	—
88	C₄⋊C₈	C₂×D₂₅₇	1	1	yes	characteristic
89	#103	—	2	1	no	—
90	#103	—	514	1	no	—
91	#103	—	514	1	no	—
92	#103	—	514	1	no	—
93	C₄×C₈	—	257	1	no	—
94	C₄×C₈	—	514	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	514	1	no	—
97	C₂×Q₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
98	Q₁₆	—	1028	2	no	—
99	C₂×C₈	—	257	1	no	—
100	C₂×C₈	—	257	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	514	1	no	—
103	C₂×C₈	—	514	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	514	1	no	—
108	C₄⋊C₄	—	514	1	no	—
109	C₄⋊C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
110	C₄²	—	2	1	no	—
111	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	4	1	no	—
115	Q₈	—	1028	1	no	—
116	Q₈	—	1028	1	no	—
117	Q₈	#16861605	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	1028	1	no	—
122	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
124	C₂×C₄	#16861617	1	1	yes	characteristic
125	C₈	—	514	1	no	—
126	C₈	—	514	1	no	—
127	C₈	—	4	1	no	—
128	C₈	—	1028	1	no	—
129	C₈	—	1028	1	no	—
130	C₂²	#22551564	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	2056	1	no	—
136	C₄	C₂×#16861605	1	1	yes	characteristic
137	C₄	#22551612	1	1	yes	characteristic
138	C₄	#22551613	1	1	yes	characteristic
139	C₄	#22551621	1	1	yes	characteristic
140	C₂	#30656957	1	1	yes	characteristic
141	C₂	#30656968	1	1	yes	characteristic
142	C₂	#30657010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22551564
commutator, cosocle	C₄²×C₂₅₇	C₂³
Frattini	C₄²	C₂²×D₂₅₇
Fitting	#475×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	#121
hypercenter	#475	D₂₅₇
hypocenter	C₂₅₇	#1680
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

Complements

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

N	H ≅ G/N	type
71. C₂₅₇	#1680	semidirect

Maximal subgroups

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

subgroup	index
#30657087	2
#30656947	2
Q₈×#14523339	2
#30656487	2
#30656484	2
#30656483	2
#475×C₂₅₇	2
#1680	257