Finite Group Databasebeta

Subgroups — #45663734

Order 32896.

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

normal subgroups only

This group has 160 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	#30657083	C₂	1	1	yes	maximal, characteristic
2	#30657032	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×#14523339	C₂	1	1	yes	maximal, characteristic, complemented
4	#30656487	C₂	1	1	yes	maximal, characteristic, complemented
5	#30656485	C₂	1	1	yes	maximal, characteristic, complemented
6	#30656481	C₂	1	1	yes	maximal, characteristic
7	#460×C₂₅₇	C₂	1	1	yes	maximal, characteristic
8	#22551592	C₂²	1	1	yes	characteristic
9	#22551525	—	2	1	no	—
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	#22551513	—	2	1	no	—
15	#22551513	—	2	1	no	—
16	C₄×#14523339	—	2	1	no	—
17	C₄×#14523339	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₂₅₇	—	2	1	no	—
19	#127×C₂₅₇	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₂₅₇	C₂²	1	1	yes	characteristic
22	#102×C₂₅₇	—	2	1	no	—
23	C₂×Dic₅₁₄	—	2	1	no	—
24	C₂×#14523339	—	2	1	no	—
25	C₂×#14523339	—	2	1	no	—
26	C₂×#14523339	D₄	1	1	yes	characteristic
27	C₂×#14523339	D₄	1	1	yes	characteristic
28	C₄⋊Dic₂₅₇	—	2	1	no	—
29	C₄⋊Dic₂₅₇	—	2	1	no	—
30	C₂×Q₈×C₂₅₇	D₄	1	1	yes	characteristic
31	C₂×D₄×C₂₅₇	—	2	1	no	—
32	C₂×D₄×C₂₅₇	—	2	1	no	—
33	C₂×D₄×C₂₅₇	—	2	1	no	—
34	SD₁₆×C₂₅₇	—	4	2	no	—
35	C₂×C₈×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	—	8	1	no	—
41	C₂×Dic₂₅₇	—	4	1	no	—
42	#14523339	—	2	1	no	—
43	#14523339	—	2	1	no	—
44	#14523339	—	4	1	no	—
45	#14523339	—	4	1	no	—
46	C₂³×C₂₅₇	—	4	1	no	—
47	Q₈×C₂₅₇	—	2	1	no	—
48	Q₈×C₂₅₇	SD₁₆	1	2	yes	—
49	D₄×C₂₅₇	—	4	1	no	—
50	D₄×C₂₅₇	—	4	1	no	—
51	D₄×C₂₅₇	—	4	1	no	—
52	D₄×C₂₅₇	—	4	1	no	—
53	D₄×C₂₅₇	—	8	1	no	—
54	C₂×C₄×C₂₅₇	—	2	1	no	—
55	C₂×C₄×C₂₅₇	—	2	1	no	—
56	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
59	C₈×C₂₅₇	—	4	1	no	—
60	C₂²×C₂₅₇	—	8	1	no	—
61	C₂²×C₂₅₇	—	8	1	no	—
62	C₂²×C₂₅₇	—	8	1	no	—
63	C₂²×C₂₅₇	#121	1	1	yes	characteristic
64	C₄×C₂₅₇	—	2	2	no	—
65	C₄×C₂₅₇	—	2	1	no	—
66	C₄×C₂₅₇	—	4	1	no	—
67	C₄×C₂₅₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₂₅₇	#135	1	1	yes	characteristic
69	C₄×C₂₅₇	#135	1	1	yes	characteristic
70	C₄×C₂₅₇	#137	1	1	yes	characteristic
71	Dic₂₅₇	—	8	1	no	—
72	C₂×C₂₅₇	—	16	1	no	—
73	C₂×C₂₅₇	#464	1	1	yes	characteristic
74	C₂×C₂₅₇	#465	1	1	yes	characteristic
75	C₂×C₂₅₇	#472	1	1	yes	characteristic
76	C₂₅₇	#1678	1	1	yes	characteristic, complemented, Sylow
77	#1678	—	257	1	no	maximal, Sylow
78	#486	—	257	1	no	—
79	#477	—	257	1	no	—
80	#460	D₂₅₇	1	1	yes	characteristic
81	C₈×Q₈	—	257	1	no	—
82	#334	—	257	1	no	—
83	#333	—	257	1	no	—
84	#331	—	257	1	no	—
85	C₂×SD₁₆	—	2	1	no	—
86	C₄⋊Q₈	—	257	1	no	—
87	#127	C₂×D₂₅₇	1	1	yes	characteristic
88	C₄×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
89	#107	—	514	1	no	—
90	C₄⋊C₈	—	257	1	no	—
91	C₄⋊C₈	—	257	1	no	—
92	C₄⋊C₈	—	514	1	no	—
93	C₄⋊C₈	C₂×D₂₅₇	1	1	yes	characteristic
94	#103	—	514	1	no	—
95	#102	—	2	1	no	—
96	#102	—	514	1	no	—
97	#102	—	514	1	no	—
98	C₄×C₈	—	257	1	no	—
99	C₄×C₈	—	514	1	no	—
100	C₂×Q₈	—	514	1	no	—
101	C₂×Q₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	2	1	no	—
105	SD₁₆	—	4	2	no	—
106	C₂×C₈	—	257	1	no	—
107	C₂×C₈	—	257	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	514	1	no	—
110	C₂×C₈	—	514	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	514	1	no	—
113	C₄⋊C₄	—	514	1	no	—
114	C₄⋊C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
115	C₄²	—	2	1	no	—
116	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
117	C₂³	—	4	1	no	—
118	Q₈	—	2	1	no	—
119	Q₈	—	2056	1	no	—
120	Q₈	#16861606	1	2	yes	—
121	D₄	—	4	1	no	—
122	D₄	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	8	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	1028	1	no	—
129	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
131	C₂×C₄	#16861617	1	1	yes	characteristic
132	C₈	—	514	1	no	—
133	C₈	—	514	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	1028	1	no	—
136	C₈	—	1028	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	#22551564	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	1	no	—
143	C₄	—	4	1	no	—
144	C₄	—	2056	1	no	—
145	C₄	C₂×#16861606	1	1	yes	characteristic
146	C₄	#22551612	1	1	yes	characteristic
147	C₄	#22551613	1	1	yes	characteristic
148	C₄	#22551621	1	1	yes	characteristic
149	C₂	—	16	1	no	—
150	C₂	#30656957	1	1	yes	characteristic
151	C₂	#30656964	1	1	yes	characteristic
152	C₂	#30657005	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22551564
commutator, cosocle	C₄²×C₂₅₇	C₂³
Frattini	C₄²	C₂²×D₂₅₇
Fitting	#460×C₂₅₇	C₂
socle	C₂²×C₂₅₇	#121
prime radical	#127×C₂₅₇	C₂²
hypercenter	#460	D₂₅₇
hypocenter	C₂₅₇	#1678
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

Complements

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

N	H ≅ G/N	type
2. #30657032	C₂	semidirect
3. Q₈×#14523339	C₂	semidirect
4. #30656487	C₂	semidirect
5. #30656485	C₂	semidirect
76. C₂₅₇	#1678	semidirect

Maximal subgroups

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

subgroup	index
#30657083	2
#30657032	2
Q₈×#14523339	2
#30656487	2
#30656485	2
#30656481	2
#460×C₂₅₇	2
#1678	257