Finite Group Databasebeta

Subgroups — #45663778

Order 32896.

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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	#30657088	C₂	1	1	yes	maximal, characteristic
2	#30657032	C₂	1	1	yes	maximal, characteristic
3	#30657022	C₂	1	1	yes	maximal, characteristic
4	#30656490	C₂	1	1	yes	maximal, characteristic
5	#30656485	C₂	1	1	yes	maximal, characteristic
6	#30656484	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
8	#22551592	C₂²	1	1	yes	characteristic
9	#22551592	C₂²	1	1	yes	characteristic
10	#22551525	—	2	1	no	—
11	#22551523	—	2	1	no	—
12	C₄⋊#14523339	C₂²	1	1	yes	characteristic
13	C₄⋊#14523339	C₂²	1	1	yes	characteristic
14	#22551517	—	2	1	no	—
15	#22551517	—	2	1	no	—
16	#22551516	—	2	1	no	—
17	#22551516	—	2	1	no	—
18	C₄×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₂₅₇	—	2	1	no	—
20	C₄⋊C₈×C₂₅₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₂₅₇	—	2	1	no	—
22	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₅₁₄	—	2	1	no	—
24	C₂×Dic₅₁₄	—	2	1	no	—
25	C₂×#14523339	—	2	1	no	—
26	C₂×#14523339	—	2	1	no	—
27	C₄⋊Dic₂₅₇	—	2	1	no	—
28	C₄⋊Dic₂₅₇	—	2	1	no	—
29	C₄⋊Dic₂₅₇	—	2	1	no	—
30	C₄⋊Dic₂₅₇	—	2	1	no	—
31	C₂×Q₈×C₂₅₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₅₇	—	2	1	no	—
33	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₂₅₇	—	2	1	no	—
36	C₄⋊C₄×C₂₅₇	D₄	1	1	yes	characteristic
37	C₄²×C₂₅₇	—	2	1	no	—
38	C₄²×C₂₅₇	C₂³	1	1	yes	characteristic
39	Dic₅₁₄	—	4	1	no	—
40	Dic₅₁₄	—	4	1	no	—
41	Dic₅₁₄	—	8	1	no	—
42	C₂×Dic₂₅₇	—	4	1	no	—
43	C₂×Dic₂₅₇	—	4	1	no	—
44	#14523339	—	4	1	no	—
45	#14523339	—	4	1	no	—
46	Q₈×C₂₅₇	—	2	1	no	—
47	Q₈×C₂₅₇	Q₁₆	1	2	yes	—
48	C₂×C₄×C₂₅₇	—	2	1	no	—
49	C₂×C₄×C₂₅₇	—	2	1	no	—
50	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
53	C₈×C₂₅₇	—	2	1	no	—
54	C₈×C₂₅₇	—	2	1	no	—
55	C₈×C₂₅₇	—	4	1	no	—
56	C₂²×C₂₅₇	#121	1	1	yes	characteristic
57	C₄×C₂₅₇	—	2	2	no	—
58	C₄×C₂₅₇	—	2	1	no	—
59	C₄×C₂₅₇	—	4	1	no	—
60	C₄×C₂₅₇	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₂₅₇	#135	1	1	yes	characteristic
62	C₄×C₂₅₇	#135	1	1	yes	characteristic
63	C₄×C₂₅₇	#137	1	1	yes	characteristic
64	Dic₂₅₇	—	8	1	no	—
65	Dic₂₅₇	—	8	1	no	—
66	C₂×C₂₅₇	#464	1	1	yes	characteristic
67	C₂×C₂₅₇	#467	1	1	yes	characteristic
68	C₂×C₂₅₇	#471	1	1	yes	characteristic
69	C₂₅₇	#1684	1	1	yes	characteristic, complemented, Sylow
70	#1684	—	257	1	no	maximal, Sylow
71	#487	—	257	1	no	—
72	#477	—	257	1	no	—
73	#475	—	257	1	no	—
74	C₈×Q₈	D₂₅₇	1	1	yes	characteristic
75	#335	—	257	1	no	—
76	#333	—	257	1	no	—
77	#332	—	257	1	no	—
78	C₄⋊Q₈	—	257	1	no	—
79	C₄⋊Q₈	—	257	1	no	—
80	C₄×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
81	#107	—	514	1	no	—
82	#106	—	514	1	no	—
83	C₄⋊C₈	—	257	1	no	—
84	C₄⋊C₈	—	257	1	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	C₂×D₂₅₇	1	1	yes	characteristic
87	#103	—	514	1	no	—
88	#103	—	514	1	no	—
89	#103	—	514	1	no	—
90	#103	—	514	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
93	C₂×Q₈	—	514	1	no	—
94	C₂×Q₈	—	514	1	no	—
95	C₂×Q₈	D₁₀₂₈	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	514	1	no	—
98	C₂×C₈	—	514	1	no	—
99	C₂×C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
100	C₂×C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	514	1	no	—
103	C₄⋊C₄	—	514	1	no	—
104	C₄⋊C₄	—	514	1	no	—
105	C₄⋊C₄	—	514	1	no	—
106	C₄⋊C₄	D₁₀₂₈	1	1	yes	characteristic
107	C₄²	—	2	1	no	—
108	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
109	Q₈	—	2	1	no	—
110	Q₈	—	1028	1	no	—
111	Q₈	—	1028	1	no	—
112	Q₈	—	2056	1	no	—
113	Q₈	Dic₁₀₂₈	1	2	yes	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	1028	1	no	—
117	C₂×C₄	—	1028	1	no	—
118	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
119	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
120	C₂×C₄	#16861618	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	1028	1	no	—
125	C₈	—	1028	1	no	—
126	C₂²	#22551566	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	2056	1	no	—
131	C₄	—	2056	1	no	—
132	C₄	C₂×Dic₁₀₂₈	1	1	yes	characteristic
133	C₄	#22551612	1	1	yes	characteristic
134	C₄	#22551614	1	1	yes	characteristic
135	C₄	#22551623	1	1	yes	characteristic
136	C₂	#30656959	1	1	yes	characteristic
137	C₂	#30656978	1	1	yes	characteristic
138	C₂	#30657000	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22551566
commutator, cosocle	C₄²×C₂₅₇	C₂³
Frattini	C₄²	C₂²×D₂₅₇
Fitting	C₈×Q₈×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	#121
hypercenter	C₈×Q₈	D₂₅₇
hypocenter	C₂₅₇	#1684
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
69. C₂₅₇	#1684	semidirect

Maximal subgroups

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

subgroup	index
#30657088	2
#30657032	2
#30657022	2
#30656490	2
#30656485	2
#30656484	2
C₈×Q₈×C₂₅₇	2
#1684	257