Finite Group Databasebeta

Subgroups — #45666499

Order 32896.

150 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 47 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	#30656970	C₂	1	1	yes	maximal, characteristic
2	#107×D₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#30656563	C₂	1	1	yes	maximal, characteristic, complemented
4	#30656562	C₂	1	1	yes	maximal, characteristic, complemented
5	#30656538	C₂	1	1	yes	maximal, characteristic, complemented
6	#30656516	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₂₅₇	C₂²	1	1	yes	characteristic
9	#22551563	—	2	1	no	—
10	C₄⋊C₄×D₂₅₇	—	2	1	no	—
11	C₂×#16861593	C₂²	1	1	yes	characteristic
12	#22551526	C₂²	1	1	yes	characteristic
13	#22551526	C₂²	1	1	yes	characteristic
14	#22551525	—	2	1	no	—
15	#22551513	—	2	1	no	—
16	C₂×D₈×C₂₅₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
18	#107×C₂₅₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊D₂₅₇	—	4	1	no	—
20	C₂×C₄×D₂₅₇	—	4	1	no	—
21	C₂×C₄×D₂₅₇	D₄	1	1	yes	characteristic
22	C₂×#14523339	D₄	1	1	yes	characteristic
23	C₈×D₂₅₇	—	2	1	no	—
24	C₈×D₂₅₇	—	2	1	no	—
25	C₄⋊Dic₂₅₇	—	2	1	no	—
26	C₄⋊Dic₂₅₇	—	2	1	no	—
27	#16861597	—	4	1	no	—
28	C₂²⋊Dic₂₅₇	—	4	1	no	—
29	#16861593	—	2	1	no	—
30	C₂×D₄×C₂₅₇	—	2	1	no	—
31	D₈×C₂₅₇	—	2	1	no	—
32	D₈×C₂₅₇	—	2	1	no	—
33	C₂×C₈×C₂₅₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₂₅₇	—	2	1	no	—
35	C₁₆×C₂₅₇	—	2	1	no	—
36	C₂²×D₂₅₇	D₈	1	1	yes	characteristic
37	C₂²⋊D₂₅₇	—	8	2	no	—
38	C₂×Dic₂₅₇	—	4	1	no	—
39	C₂×Dic₂₅₇	—	4	1	no	—
40	C₂×Dic₂₅₇	D₈	1	1	yes	characteristic
41	C₄×D₂₅₇	—	2	1	no	—
42	C₄×D₂₅₇	—	2	1	no	—
43	C₄×D₂₅₇	—	8	2	no	—
44	#14523339	—	2	1	no	—
45	C₂³×C₂₅₇	—	4	1	no	—
46	D₄×C₂₅₇	—	4	1	no	—
47	D₄×C₂₅₇	—	4	1	no	—
48	C₂×C₄×C₂₅₇	—	4	1	no	—
49	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
50	C₈×C₂₅₇	#41	1	1	yes	characteristic
51	C₈×C₂₅₇	#41	1	1	yes	characteristic
52	C₂²×C₂₅₇	—	8	1	no	—
53	C₂²×C₂₅₇	—	8	1	no	—
54	C₂²×C₂₅₇	—	8	1	no	—
55	C₂²×C₂₅₇	C₂×D₈	1	1	yes	characteristic
56	C₂×D₂₅₇	—	2	1	no	—
57	C₂×D₂₅₇	—	2	1	no	—
58	C₂×D₂₅₇	D₁₆	1	2	yes	—
59	C₄×C₂₅₇	—	8	1	no	—
60	C₄×C₂₅₇	#123	1	1	yes	characteristic
61	C₄×C₂₅₇	#137	1	1	yes	characteristic
62	Dic₂₅₇	—	2	1	no	—
63	Dic₂₅₇	—	8	1	no	—
64	Dic₂₅₇	—	8	1	no	—
65	C₂×C₂₅₇	—	16	1	no	—
66	C₂×C₂₅₇	#480	1	1	yes	characteristic
67	C₂×C₂₅₇	C₂×D₁₆	1	1	yes	characteristic
68	C₂×C₂₅₇	#510	1	1	yes	characteristic
69	D₂₅₇	—	2	2	no	—
70	C₂₅₇	#2232	1	1	yes	characteristic, complemented, Sylow
71	#2232	—	257	1	no	maximal, Sylow
72	#466	—	257	1	no	—
73	C₂×#107	—	257	1	no	—
74	#366	—	257	1	no	—
75	#366	—	257	1	no	—
76	#357	—	257	1	no	—
77	#357	D₂₅₇	1	1	yes	characteristic, complemented
78	C₂²⋊C₁₆	—	257	1	no	—
79	C₂×D₈	C₂×D₂₅₇	1	1	yes	characteristic
80	C₂²×C₈	—	257	1	no	—
81	#121	—	514	1	no	—
82	C₂×C₄⋊C₄	—	514	1	no	—
83	C₂×C₁₆	—	257	1	no	—
84	C₂×C₁₆	C₂×D₂₅₇	1	1	yes	characteristic
85	#107	—	257	1	no	—
86	#107	—	257	1	no	—
87	#107	—	514	1	no	—
88	#107	C₂×D₂₅₇	1	1	yes	characteristic
89	#102	—	514	1	no	—
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	—	1028	1	no	—
92	C₂²×C₄	—	257	1	no	—
93	C₂²×C₄	—	1028	1	no	—
94	D₈	—	2	1	no	—
95	D₈	—	2	1	no	—
96	C₂×C₈	—	257	1	no	—
97	C₂×C₈	—	514	1	no	—
98	C₂×C₈	—	514	1	no	—
99	C₂×C₈	C₂²×D₂₅₇	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	514	1	no	—
102	C₄⋊C₄	—	514	1	no	—
103	C₄⋊C₄	—	1028	1	no	—
104	C₂²⋊C₄	—	1028	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	514	1	no	—
107	C₂³	—	257	1	no	—
108	C₂³	—	4	1	no	—
109	D₄	—	4	1	no	—
110	D₄	—	4	1	no	—
111	D₄	—	2056	2	no	—
112	C₂×C₄	—	257	1	no	—
113	C₂×C₄	—	514	1	no	—
114	C₂×C₄	—	514	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	1028	1	no	—
117	C₂×C₄	—	1028	1	no	—
118	C₂×C₄	—	2056	2	no	—
119	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
120	C₈	—	514	1	no	—
121	C₈	#16861617	1	1	yes	characteristic
122	C₈	#16861618	1	1	yes	characteristic
123	C₂²	—	257	2	no	—
124	C₂²	—	514	1	no	—
125	C₂²	—	514	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	—	8	1	no	—
129	C₂²	D₈×D₂₅₇	1	1	yes	characteristic
130	C₄	—	514	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	2056	1	no	—
133	C₄	—	2056	1	no	—
134	C₄	#22551574	1	1	yes	characteristic
135	C₄	#22551621	1	1	yes	characteristic
136	C₂	—	514	2	no	—
137	C₂	—	16	1	no	—
138	C₂	#30657049	1	1	yes	characteristic
139	C₂	D₁₆×D₂₅₇	1	1	yes	characteristic
140	C₂	#30657176	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₅₇
commutator, cosocle	C₂×C₈×C₂₅₇	C₂³
Frattini	C₂×C₈	C₂²×D₂₅₇
Fitting	#357×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×D₈
prime radical	#30656970	C₂
hypercenter	#357	D₂₅₇
hypocenter	C₂₅₇	#2232
LCS	G ▷ C₂×C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

Complements

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

N	H ≅ G/N	type
2. #107×D₂₅₇	C₂	semidirect
3. #30656563	C₂	semidirect
4. #30656562	C₂	semidirect
5. #30656538	C₂	semidirect
6. #30656516	C₂	semidirect
7. #357×C₂₅₇	C₂	semidirect
70. C₂₅₇	#2232	semidirect
77. #357	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#30656970	2
#107×D₂₅₇	2
#30656563	2
#30656562	2
#30656538	2
#30656516	2
#357×C₂₅₇	2
#2232	257