Finite Group Databasebeta

Subgroups — #45665109

Order 32896.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#30656786	C₂	1	1	yes	maximal, characteristic
2	#30656785	C₂	1	1	yes	maximal, characteristic
3	C₂×#22551527	C₂	1	1	yes	maximal, characteristic
4	C₂×#22551527	C₂	1	1	yes	maximal, characteristic
5	#30656693	C₂	1	1	yes	maximal, characteristic
6	#30656507	C₂	1	1	yes	maximal, characteristic
7	#344×C₂₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#16861602	C₂²	1	1	yes	characteristic
9	C₂×#16861602	C₂²	1	1	yes	characteristic
10	C₂×#16861602	C₂²	1	1	yes	characteristic
11	C₂²×#14523339	C₂²	1	1	yes	characteristic
12	#22551528	—	2	2	no	—
13	#22551527	—	2	2	no	—
14	#22551527	—	2	1	no	—
15	#22551527	—	2	1	no	—
16	#22551527	C₄	1	4	yes	complemented
17	#22551503	—	2	1	no	—
18	C₄×#14523339	—	2	1	no	—
19	C₂×M₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
21	#117×C₂₅₇	C₂²	1	1	yes	characteristic
22	#16861602	—	2	1	no	—
23	#16861602	—	2	1	no	—
24	#16861602	—	2	1	no	—
25	#16861602	—	2	1	no	—
26	#16861602	C₂×C₄	1	2	yes	—
27	#16861602	D₄	1	2	yes	—
28	C₂×#14523339	—	2	1	no	—
29	C₂×#14523339	—	2	1	no	—
30	C₂×#14523339	—	2	1	no	—
31	C₂×#14523339	C₂×C₄	1	2	yes	—
32	C₂×#14523339	D₄	1	1	yes	characteristic
33	C₂×#14523339	D₄	1	1	yes	characteristic
34	C₂×#14523339	Q₈	1	2	yes	—
35	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₂₅₇	—	2	1	no	—
37	M₁₆×C₂₅₇	—	2	1	no	—
38	M₁₆×C₂₅₇	—	2	1	no	—
39	M₁₆×C₂₅₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₂₅₇	—	2	1	no	—
41	C₂×C₈×C₂₅₇	—	2	1	no	—
42	C₄⋊C₄×C₂₅₇	—	2	1	no	—
43	C₂²⋊C₄×C₂₅₇	Q₈	1	2	yes	—
44	C₄²×C₂₅₇	—	2	1	no	—
45	#14523339	—	2	2	no	—
46	#14523339	—	2	2	no	—
47	#14523339	—	4	1	no	—
48	#14523339	C₄⋊C₄	1	2	yes	—
49	#14523339	C₄⋊C₄	1	2	yes	—
50	C₂³×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₂₅₇	—	2	2	no	—
52	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
57	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
58	C₈×C₂₅₇	—	2	2	no	—
59	C₈×C₂₅₇	—	4	1	no	—
60	C₂²×C₂₅₇	—	4	1	no	—
61	C₂²×C₂₅₇	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₂₅₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₂₅₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₄×C₂₅₇	—	4	2	no	—
65	C₄×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₂₅₇	C₄×D₄	1	1	yes	characteristic
67	C₄×C₂₅₇	#121	1	1	yes	characteristic
68	C₄×C₂₅₇	#125	1	1	yes	characteristic
69	C₂×C₂₅₇	—	2	1	no	—
70	C₂×C₂₅₇	—	2	1	no	—
71	C₂×C₂₅₇	—	2	1	no	—
72	C₂×C₂₅₇	#389	1	1	yes	characteristic
73	C₂₅₇	#1953	1	1	yes	characteristic, complemented, Sylow
74	#1953	—	257	1	no	maximal, Sylow
75	#430	—	257	1	no	—
76	#430	—	257	1	no	—
77	C₂×#108	—	257	1	no	—
78	C₂×#108	—	257	1	no	—
79	#405	—	257	1	no	—
80	#344	—	257	1	no	—
81	#344	D₂₅₇	1	1	yes	characteristic
82	C₂×M₁₆	—	257	1	no	—
83	C₂×M₁₆	—	257	1	no	—
84	C₂×M₁₆	—	257	1	no	—
85	C₂×M₁₆	C₂×D₂₅₇	1	1	yes	characteristic
86	C₂×M₁₆	C₂×D₂₅₇	1	1	yes	characteristic
87	C₂²×C₈	—	257	1	no	—
88	#117	C₂×D₂₅₇	1	1	yes	characteristic
89	#108	—	257	4	no	—
90	#108	—	514	1	no	—
91	#108	—	514	1	no	—
92	#108	—	514	2	no	—
93	#108	—	514	2	no	—
94	#97	—	514	1	no	—
95	C₄×C₈	—	514	1	no	—
96	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
97	M₁₆	—	257	2	no	—
98	M₁₆	—	257	2	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	2	1	no	—
102	M₁₆	—	514	1	no	—
103	M₁₆	—	514	1	no	—
104	M₁₆	—	514	1	no	—
105	M₁₆	—	514	1	no	—
106	M₁₆	C₄×D₂₅₇	1	2	yes	—
107	C₂×C₈	—	257	2	no	—
108	C₂×C₈	—	257	1	no	—
109	C₂×C₈	—	257	1	no	—
110	C₂×C₈	—	257	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	514	1	no	—
114	C₂×C₈	—	514	1	no	—
115	C₂×C₈	—	514	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Dic₅₁₄	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₂³	C₂×Dic₅₁₄	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
122	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
123	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₂₅₇	1	1	yes	characteristic
125	C₂×C₄	#16861617	1	1	yes	characteristic
126	C₂×C₄	#16861618	1	1	yes	characteristic
127	C₈	—	257	2	no	—
128	C₈	—	257	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	514	2	no	—
131	C₈	—	514	2	no	—
132	C₈	—	4	1	no	—
133	C₈	—	1028	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₄×Dic₅₁₄	1	1	yes	characteristic
136	C₂²	#22551569	1	1	yes	characteristic
137	C₂²	#22551591	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	C₄⋊C₄×D₂₅₇	1	1	yes	characteristic
140	C₄	#22551552	1	1	yes	characteristic
141	C₄	#22551565	1	1	yes	characteristic
142	C₄	#22551582	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#30656634	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#22551565
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#344×C₂₅₇	C₂
socle	C₂×C₂₅₇	#389
prime radical	C₂³×C₂₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#344	D₂₅₇
hypocenter	C₂₅₇	#1953
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
16. #22551527	C₄	semidirect
73. C₂₅₇	#1953	semidirect

Maximal subgroups

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

subgroup	index
#30656786	2
#30656785	2
C₂×#22551527	2
#30656693	2
#30656507	2
#344×C₂₅₇	2
#1953	257