Finite Group Databasebeta

Subgroups — #45666203

Order 32896.

144 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 53 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	#30657167	C₂	1	1	yes	maximal, characteristic
2	#30656854	C₂	1	1	yes	maximal, characteristic, complemented
3	#30656853	C₂	1	1	yes	maximal, characteristic, complemented
4	#30656566	C₂	1	1	yes	maximal, characteristic, complemented
5	#30656545	C₂	1	1	yes	maximal, characteristic, complemented
6	#30656512	C₂	1	1	yes	maximal, characteristic, complemented
7	#359×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#22551614	C₂²	1	1	yes	characteristic
9	#22551600	—	2	1	no	—
10	#22551599	—	2	1	no	—
11	#22551538	C₄	1	1	yes	characteristic, complemented
12	#22551536	C₄	1	1	yes	characteristic, complemented
13	#22551535	C₄	1	1	yes	characteristic, complemented
14	#22551533	C₄	1	1	yes	characteristic, complemented
15	C₂×#16861593	C₂²	1	1	yes	characteristic
16	#22551528	C₂²	1	1	yes	characteristic
17	#22551519	—	2	1	no	—
18	#22551518	—	2	1	no	—
19	C₈×Dic₂₅₇	C₂²	1	1	yes	characteristic
20	#135×C₂₅₇	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
22	#108×C₂₅₇	C₂²	1	1	yes	characteristic
23	#16861618	—	2	1	no	—
24	Dic₁₀₂₈	C₂×C₄	1	1	yes	characteristic
25	#16861607	—	2	1	no	—
26	D₂₀₅₆	C₂×C₄	1	1	yes	characteristic
27	#16861602	—	2	1	no	—
28	#16861602	—	4	1	no	—
29	#16861601	—	4	1	no	—
30	C₂×#14523339	—	4	1	no	—
31	C₂×#14523339	D₄	1	1	yes	characteristic
32	C₈×D₂₅₇	—	4	1	no	—
33	C₄×Dic₂₅₇	D₄	1	1	yes	characteristic
34	#16861593	C₂×C₄	1	1	yes	characteristic
35	#16861593	C₂×C₄	1	1	yes	characteristic
36	#41×C₂₅₇	—	2	1	no	—
37	Q₁₆×C₂₅₇	C₂×C₄	1	1	yes	characteristic
38	SD₁₆×C₂₅₇	—	2	1	no	—
39	D₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₂₅₇	—	2	1	no	—
41	C₂×C₈×C₂₅₇	C₂³	1	1	yes	characteristic
42	C₁₆×C₂₅₇	—	2	1	no	—
43	Dic₅₁₄	—	2	1	no	—
44	D₁₀₂₈	—	2	1	no	—
45	C₂²⋊D₂₅₇	—	4	1	no	—
46	C₂×Dic₂₅₇	—	2	1	no	—
47	C₄×D₂₅₇	—	4	1	no	—
48	#14523339	—	4	1	no	—
49	#14523339	D₈	1	1	yes	characteristic
50	#14523339	D₈	1	1	yes	characteristic
51	Q₈×C₂₅₇	—	2	1	no	—
52	D₄×C₂₅₇	—	2	1	no	—
53	D₄×C₂₅₇	—	4	1	no	—
54	C₂×C₄×C₂₅₇	—	4	1	no	—
55	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
56	C₈×C₂₅₇	—	4	1	no	—
57	C₈×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
58	C₈×C₂₅₇	#41	1	1	yes	characteristic
59	C₂²×C₂₅₇	—	4	1	no	—
60	C₂²×C₂₅₇	#135	1	1	yes	characteristic
61	C₂×D₂₅₇	—	4	1	no	—
62	C₄×C₂₅₇	—	4	1	no	—
63	C₄×C₂₅₇	C₄×D₄	1	1	yes	characteristic
64	C₄×C₂₅₇	C₂×D₈	1	1	yes	characteristic
65	Dic₂₅₇	—	4	1	no	—
66	Dic₂₅₇	—	4	1	no	—
67	C₂×C₂₅₇	—	2	1	no	—
68	C₂×C₂₅₇	—	8	1	no	—
69	C₂×C₂₅₇	C₄×D₈	1	1	yes	characteristic
70	D₂₅₇	—	8	1	no	—
71	C₂₅₇	#2179	1	1	yes	characteristic, complemented, Sylow
72	#2179	—	257	1	no	maximal, Sylow
73	#508	—	257	1	no	—
74	#443	—	257	1	no	—
75	#443	—	257	1	no	—
76	#368	—	257	1	no	—
77	#359	—	257	1	no	—
78	#359	D₂₅₇	1	1	yes	characteristic, complemented
79	#346	—	257	1	no	—
80	#135	—	257	1	no	—
81	#135	C₂×D₂₅₇	1	1	yes	characteristic
82	#131	—	514	1	no	—
83	#131	—	514	1	no	—
84	Q₃₂	—	257	1	no	—
85	SD₃₂	—	257	1	no	—
86	SD₃₂	—	257	1	no	—
87	D₁₆	—	257	1	no	—
88	C₂×C₁₆	—	257	1	no	—
89	C₂×C₁₆	C₂×D₂₅₇	1	1	yes	characteristic
90	#108	—	257	1	no	—
91	#108	C₂×D₂₅₇	1	1	yes	characteristic
92	#104	—	514	1	no	—
93	#104	—	514	1	no	—
94	C₄×C₈	—	257	1	no	—
95	#41	—	2	1	no	—
96	#41	—	514	1	no	—
97	Q₁₆	—	257	1	no	—
98	Q₁₆	C₄×D₂₅₇	1	1	yes	characteristic
99	SD₁₆	—	2	1	no	—
100	SD₁₆	—	514	1	no	—
101	D₈	—	257	1	no	—
102	D₈	C₄×D₂₅₇	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	514	1	no	—
105	M₁₆	—	1028	1	no	—
106	M₁₆	—	1028	1	no	—
107	C₂×C₈	—	257	1	no	—
108	C₂×C₈	—	1028	1	no	—
109	C₂×C₈	—	1028	1	no	—
110	C₂×C₈	C₂²×D₂₅₇	1	1	yes	characteristic
111	C₄²	—	257	1	no	—
112	C₁₆	—	257	1	no	—
113	C₁₆	—	257	1	no	—
114	C₁₆	—	2	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	514	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	514	1	no	—
119	D₄	—	4	1	no	—
120	D₄	—	1028	1	no	—
121	C₂×C₄	—	514	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	1028	1	no	—
124	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
125	C₈	—	257	1	no	—
126	C₈	—	257	1	no	—
127	C₈	—	4	1	no	—
128	C₈	—	1028	1	no	—
129	C₈	C₂×C₄×D₂₅₇	1	1	yes	characteristic
130	C₈	#16861617	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	1028	1	no	—
133	C₂²	#22551613	1	1	yes	characteristic
134	C₄	—	4	1	no	—
135	C₄	—	1028	1	no	—
136	C₄	—	1028	1	no	—
137	C₄	#22551552	1	1	yes	characteristic
138	C₄	D₈×D₂₅₇	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	2056	1	no	—
142	C₂	#30656817	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₂₅₇
commutator	C₈×C₂₅₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₅₇
Fitting	#359×C₂₅₇	C₂
socle	C₂×C₂₅₇	C₄×D₈
prime radical	#30657167	C₂
cosocle	C₂×C₈×C₂₅₇	C₂³
hypercenter	#359	D₂₅₇
hypocenter	C₂₅₇	#2179
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

Complements

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

N	H ≅ G/N	type
2. #30656854	C₂	semidirect
3. #30656853	C₂	semidirect
4. #30656566	C₂	semidirect
5. #30656545	C₂	semidirect
6. #30656512	C₂	semidirect
7. #359×C₂₅₇	C₂	semidirect
11. #22551538	C₄	semidirect
12. #22551536	C₄	semidirect
13. #22551535	C₄	semidirect
14. #22551533	C₄	semidirect
71. C₂₅₇	#2179	semidirect
78. #359	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#30657167	2
#30656854	2
#30656853	2
#30656566	2
#30656545	2
#30656512	2
#359×C₂₅₇	2
#2179	257