Finite Group Databasebeta

Subgroups — #45663136

Order 32896.

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

normal subgroups only

This group has 160 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	Q₈×#14523339	C₂	1	1	yes	maximal, characteristic
2	#30656861	C₂	1	1	yes	maximal, characteristic
3	C₄×#16861608	C₂	1	1	yes	maximal, characteristic
4	#30656487	C₂	1	1	yes	maximal, characteristic
5	#30656469	C₂	1	1	yes	maximal, characteristic
6	#30656459	C₂	1	1	yes	maximal, characteristic
7	Q₈⋊C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#16861608	C₄	1	1	yes	characteristic
9	C₄×Dic₅₁₄	C₂²	1	1	yes	characteristic
10	#22551525	C₄	1	1	yes	characteristic
11	C₄⋊#14523339	—	2	1	no	—
12	C₄⋊#14523339	C₂²	1	1	yes	characteristic
13	#22551521	—	2	1	no	—
14	#22551516	C₄	1	1	yes	characteristic
15	#22551515	C₄	1	1	yes	characteristic
16	C₄×#14523339	—	2	1	no	—
17	C₄×#14523339	C₂²	1	1	yes	characteristic
18	C₄×#14523339	C₂²	1	1	yes	characteristic
19	C₈×Dic₂₅₇	—	2	1	no	—
20	C₄×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₂₅₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₅₁₄	C₂×C₄	1	1	yes	characteristic
24	#16861608	C₈	1	4	yes	complemented
25	C₂×#14523339	—	2	1	no	—
26	C₂×#14523339	C₂×C₄	1	1	yes	characteristic
27	C₂×#14523339	C₂×C₄	1	1	yes	characteristic
28	C₂×#14523339	D₄	1	1	yes	characteristic
29	C₂×#14523339	D₄	1	1	yes	characteristic
30	C₄⋊Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
31	#16861597	—	2	1	no	—
32	C₄×Dic₂₅₇	—	2	1	no	—
33	C₂×Q₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₂₅₇	—	2	1	no	—
35	C₂×C₈×C₂₅₇	—	2	1	no	—
36	C₄⋊C₄×C₂₅₇	—	2	1	no	—
37	C₄⋊C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₂₅₇	—	2	1	no	—
39	C₄²×C₂₅₇	C₂³	1	1	yes	characteristic
40	Dic₅₁₄	—	2	1	no	—
41	Dic₅₁₄	C₂×C₈	1	2	yes	—
42	C₂×Dic₂₅₇	—	2	1	no	—
43	C₂×Dic₂₅₇	—	2	1	no	—
44	#14523339	—	2	1	no	—
45	#14523339	—	2	1	no	—
46	#14523339	—	2	1	no	—
47	#14523339	—	4	1	no	—
48	#14523339	C₂×C₈	1	2	yes	—
49	Q₈×C₂₅₇	—	2	1	no	—
50	Q₈×C₂₅₇	C₂×C₈	1	2	yes	—
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	—	4	1	no	—
57	C₈×C₂₅₇	—	4	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	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₂₅₇	#131	1	1	yes	characteristic
64	C₄×C₂₅₇	#137	1	1	yes	characteristic
65	C₄×C₂₅₇	#137	1	1	yes	characteristic
66	Dic₂₅₇	—	2	2	no	—
67	Dic₂₅₇	—	4	1	no	—
68	C₂×C₂₅₇	C₈×D₄	1	1	yes	characteristic
69	C₂×C₂₅₇	#441	1	1	yes	characteristic
70	C₂×C₂₅₇	#444	1	1	yes	characteristic
71	C₂₅₇	#1579	1	1	yes	characteristic, complemented, Sylow
72	#1579	—	257	1	no	maximal, Sylow
73	C₈×Q₈	—	257	1	no	—
74	C₈×Q₈	—	257	1	no	—
75	C₄×Q₁₆	—	257	1	no	—
76	#334	—	257	1	no	—
77	Q₈⋊C₈	—	257	1	no	—
78	Q₈⋊C₈	D₂₅₇	1	1	yes	characteristic
79	#322	—	257	1	no	—
80	C₂×Q₁₆	—	257	1	no	—
81	C₄×Q₈	—	257	1	no	—
82	C₄×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
83	#107	—	257	1	no	—
84	C₄⋊C₈	—	257	1	no	—
85	C₄⋊C₈	—	514	1	no	—
86	C₄⋊C₈	—	514	1	no	—
87	C₄⋊C₈	C₂×D₂₅₇	1	1	yes	characteristic
88	#103	—	257	1	no	—
89	#103	—	257	1	no	—
90	C₄×C₈	—	257	1	no	—
91	C₄×C₈	—	257	1	no	—
92	C₄×C₈	—	514	1	no	—
93	C₄×C₈	—	514	1	no	—
94	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
95	C₂×Q₈	—	257	1	no	—
96	C₂×Q₈	C₄×D₂₅₇	1	1	yes	characteristic
97	Q₁₆	—	257	4	no	—
98	C₂×C₈	—	257	1	no	—
99	C₂×C₈	—	257	1	no	—
100	C₂×C₈	—	257	1	no	—
101	C₂×C₈	—	257	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	514	1	no	—
105	C₄⋊C₄	—	257	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	514	1	no	—
108	C₄⋊C₄	C₄×D₂₅₇	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	—	514	1	no	—
111	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
112	Q₈	—	257	2	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	514	1	no	—
115	Q₈	C₈×D₂₅₇	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	514	1	no	—
119	C₂×C₄	—	514	1	no	—
120	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
121	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
122	C₂×C₄	#16861617	1	1	yes	characteristic
123	C₈	—	257	2	no	—
124	C₈	—	514	1	no	—
125	C₈	—	514	1	no	—
126	C₈	—	514	1	no	—
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	1028	1	no	—
130	C₂²	#22551552	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	514	2	no	—
134	C₄	—	4	1	no	—
135	C₄	—	1028	1	no	—
136	C₄	C₂×C₈×D₂₅₇	1	1	yes	characteristic
137	C₄	#22551600	1	1	yes	characteristic
138	C₄	#22551621	1	1	yes	characteristic
139	C₄	#22551624	1	1	yes	characteristic
140	C₂	#30656800	1	1	yes	characteristic
141	C₂	#30656841	1	1	yes	characteristic
142	C₂	#30656856	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₅₇
commutator	C₄×C₂₅₇	C₂²×C₈
Frattini	C₄²	C₂²×D₂₅₇
Fitting	Q₈⋊C₈×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	C₄×D₄
cosocle	C₄²×C₂₅₇	C₂³
hypercenter	Q₈⋊C₈	D₂₅₇
hypocenter	C₂₅₇	#1579
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
24. #16861608	C₈	semidirect
71. C₂₅₇	#1579	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#14523339	2
#30656861	2
C₄×#16861608	2
#30656487	2
#30656469	2
#30656459	2
Q₈⋊C₈×C₂₅₇	2
#1579	257