Finite Group Databasebeta

Subgroups — #45663192

Order 32896.

152 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 71 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	#30656868	C₂	1	1	yes	maximal, characteristic
2	Q₈×#14523339	C₂	1	1	yes	maximal, characteristic
3	#30656488	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#14523339	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#14523339	C₂	1	1	yes	maximal, characteristic
6	#30656459	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#14523339	—	2	1	no	—
9	C₄⋊#14523339	—	2	1	no	—
10	C₄⋊#14523339	C₂²	1	1	yes	characteristic
11	C₄⋊#14523339	C₂²	1	1	yes	characteristic
12	#22551503	—	2	1	no	—
13	C₄×#14523339	—	2	1	no	—
14	C₄×#14523339	C₂²	1	1	yes	characteristic
15	C₄×#14523339	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₂₅₇	C₄	1	1	yes	characteristic
17	C₄×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
19	#107×C₂₅₇	C₄	1	1	yes	characteristic
20	#103×C₂₅₇	C₄	1	1	yes	characteristic
21	#103×C₂₅₇	C₄	1	1	yes	characteristic
22	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
23	C₂×#14523339	—	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	C₂×Q₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
29	C₂×Q₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
30	Q₁₆×C₂₅₇	—	2	2	no	—
31	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₂₅₇	—	2	1	no	—
34	C₄⋊C₄×C₂₅₇	—	2	1	no	—
35	C₄⋊C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₂₅₇	—	2	1	no	—
38	C₄²×C₂₅₇	—	2	1	no	—
39	C₄²×C₂₅₇	C₂³	1	1	yes	characteristic
40	#14523339	—	2	2	no	—
41	#14523339	—	2	1	no	—
42	#14523339	—	2	1	no	—
43	#14523339	—	4	1	no	—
44	#14523339	—	4	1	no	—
45	Q₈×C₂₅₇	—	2	1	no	—
46	Q₈×C₂₅₇	—	2	1	no	—
47	Q₈×C₂₅₇	—	2	1	no	—
48	Q₈×C₂₅₇	M₁₆	1	2	yes	—
49	C₂×C₄×C₂₅₇	—	2	1	no	—
50	C₂×C₄×C₂₅₇	—	2	1	no	—
51	C₂×C₄×C₂₅₇	—	2	1	no	—
52	C₂×C₄×C₂₅₇	—	2	1	no	—
53	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
56	C₈×C₂₅₇	—	2	1	no	—
57	C₈×C₂₅₇	—	2	1	no	—
58	C₂²×C₂₅₇	C₄×D₄	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	—	4	1	no	—
63	C₄×C₂₅₇	—	4	1	no	—
64	C₄×C₂₅₇	C₂×M₁₆	1	1	yes	characteristic
65	C₄×C₂₅₇	#131	1	1	yes	characteristic
66	C₄×C₂₅₇	#137	1	1	yes	characteristic
67	C₄×C₂₅₇	#137	1	1	yes	characteristic
68	C₂×C₂₅₇	#435	1	1	yes	characteristic
69	C₂×C₂₅₇	#441	1	1	yes	characteristic
70	C₂×C₂₅₇	#443	1	1	yes	characteristic
71	C₂₅₇	#1587	1	1	yes	characteristic, complemented, Sylow
72	#1587	—	257	1	no	maximal, Sylow
73	#446	—	257	1	no	—
74	C₈×Q₈	—	257	1	no	—
75	C₄×Q₁₆	D₂₅₇	1	1	yes	characteristic
76	#334	—	257	1	no	—
77	Q₈⋊C₈	—	257	1	no	—
78	Q₈⋊C₈	—	257	1	no	—
79	#322	—	257	1	no	—
80	C₂×Q₁₆	Dic₂₅₇	1	1	yes	characteristic
81	C₄×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
82	C₄×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
83	#107	Dic₂₅₇	1	1	yes	characteristic
84	C₄⋊C₈	—	257	1	no	—
85	C₄⋊C₈	—	257	1	no	—
86	C₄⋊C₈	—	514	1	no	—
87	C₄⋊C₈	—	514	1	no	—
88	#103	Dic₂₅₇	1	1	yes	characteristic
89	#103	Dic₂₅₇	1	1	yes	characteristic
90	#97	—	514	1	no	—
91	C₄×C₈	—	257	1	no	—
92	C₄×C₈	—	257	1	no	—
93	C₄×C₈	—	514	1	no	—
94	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
95	C₂×Q₈	C₂×Dic₂₅₇	1	1	yes	characteristic
96	C₂×Q₈	C₂×Dic₂₅₇	1	1	yes	characteristic
97	Q₁₆	—	2	2	no	—
98	C₂×C₈	—	257	1	no	—
99	C₂×C₈	—	257	1	no	—
100	C₂×C₈	—	514	1	no	—
101	C₂×C₈	—	514	1	no	—
102	C₂×C₈	—	514	1	no	—
103	C₂×C₈	C₂×Dic₂₅₇	1	1	yes	characteristic
104	C₂×C₈	C₂×Dic₂₅₇	1	1	yes	characteristic
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	C₂×Dic₂₅₇	1	1	yes	characteristic
108	C₄⋊C₄	C₂×Dic₂₅₇	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	—	2	1	no	—
111	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	2	1	no	—
115	Q₈	#16861602	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	C₂²×Dic₂₅₇	1	1	yes	characteristic
121	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
122	C₂×C₄	#16861617	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	514	2	no	—
126	C₈	—	514	1	no	—
127	C₈	—	514	1	no	—
128	C₈	—	1028	1	no	—
129	C₈	—	1028	1	no	—
130	C₂²	D₄×Dic₂₅₇	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	4	1	no	—
136	C₄	C₂×#16861602	1	1	yes	characteristic
137	C₄	#22551599	1	1	yes	characteristic
138	C₄	#22551624	1	1	yes	characteristic
139	C₄	#22551624	1	1	yes	characteristic
140	C₂	#30656809	1	1	yes	characteristic
141	C₂	#30656844	1	1	yes	characteristic
142	C₂	#30656854	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₅₇
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₄²	C₂²×D₂₅₇
Fitting	C₄×Q₁₆×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	C₄×D₄
cosocle	C₄²×C₂₅₇	C₂³
hypercenter	C₄×Q₁₆	D₂₅₇
hypocenter	C₂₅₇	#1587
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
71. C₂₅₇	#1587	semidirect

Maximal subgroups

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

subgroup	index
#30656868	2
Q₈×#14523339	2
#30656488	2
Q₈⋊#14523339	2
#30656459	2
C₄×Q₁₆×C₂₅₇	2
#1587	257