Finite Group Databasebeta

Subgroups — #34102033

Order 20096.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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	#23757711	C₂	1	1	yes	maximal, characteristic
2	#23757394	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757384	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757085	C₂	1	1	yes	maximal, characteristic, complemented
5	#23757082	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757050	C₂	1	1	yes	maximal, characteristic, complemented
7	#365×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#19097793	C₂²	1	1	yes	characteristic
9	C₂×D₁₂₅₆	C₂²	1	1	yes	characteristic
10	#19097779	—	2	1	no	—
11	#19097733	—	2	1	no	—
12	#19097716	C₄	1	2	yes	complemented
13	D₂₅₁₂	C₄	1	2	yes	complemented
14	#19097711	C₂²	1	1	yes	characteristic
15	#19097697	—	2	1	no	—
16	#19097691	—	2	1	no	—
17	#19097681	C₂²	1	1	yes	characteristic
18	M₃₂×C₁₅₇	C₂²	1	1	yes	characteristic
19	#108×C₁₅₇	C₂²	1	1	yes	characteristic
20	#106×C₁₅₇	C₂²	1	1	yes	characteristic
21	#14578954	—	2	1	no	—
22	C₂×D₆₂₈	—	2	1	no	—
23	C₂×C₄×D₁₅₇	—	4	1	no	—
24	Dic₆₂₈	C₂×C₄	1	1	yes	characteristic, complemented
25	#14578943	—	2	1	no	—
26	D₁₂₅₆	—	2	1	no	—
27	D₁₂₅₆	C₂×C₄	1	2	yes	—
28	D₁₂₅₆	C₂×C₄	1	1	yes	characteristic, complemented
29	#14578937	—	4	1	no	—
30	C₂×#11929434	D₄	1	1	yes	characteristic
31	C₈×D₁₅₇	—	4	1	no	—
32	#14578931	—	4	1	no	—
33	C₄×Dic₁₅₇	D₄	1	1	yes	characteristic
34	#14578929	—	2	1	no	—
35	M₁₆×C₁₅₇	—	2	1	no	—
36	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₅₇	—	2	1	no	—
38	C₁₆×C₁₅₇	C₂×C₄	1	2	yes	complemented
39	C₂²×D₁₅₇	—	4	1	no	—
40	Dic₃₁₄	—	2	1	no	—
41	D₆₂₈	—	2	2	no	—
42	D₆₂₈	—	2	1	no	—
43	D₆₂₈	—	4	1	no	—
44	C₂²⋊D₁₅₇	—	4	1	no	—
45	C₂×Dic₁₅₇	—	2	1	no	—
46	C₄×D₁₅₇	—	4	1	no	—
47	C₄×D₁₅₇	—	8	2	no	—
48	#11929434	—	2	1	no	—
49	C₂×C₄×C₁₅₇	—	4	1	no	—
50	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₅₇	—	4	1	no	—
52	C₈×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₁₅₇	#41	1	1	yes	characteristic
54	C₂²×C₁₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×D₁₅₇	—	4	1	no	—
56	C₂×D₁₅₇	—	4	2	no	—
57	C₂×D₁₅₇	—	8	1	no	—
58	C₂×D₁₅₇	—	8	1	no	—
59	C₄×C₁₅₇	—	8	1	no	—
60	C₄×C₁₅₇	C₄×D₄	1	1	yes	characteristic
61	C₄×C₁₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₅₇	—	4	1	no	—
63	Dic₁₅₇	—	4	1	no	—
64	C₂×C₁₅₇	—	2	1	no	—
65	C₂×C₁₅₇	#442	1	1	yes	characteristic
66	D₁₅₇	—	8	1	no	—
67	D₁₅₇	—	8	2	no	—
68	C₁₅₇	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
69	AGL(1,ℤ/16)	—	157	1	no	maximal, Sylow
70	#509	—	157	1	no	—
71	#444	—	157	1	no	—
72	#442	—	157	1	no	—
73	#365	D₁₅₇	1	1	yes	characteristic, complemented
74	#361	—	157	1	no	—
75	#360	—	157	1	no	—
76	#347	—	157	1	no	—
77	#135	—	157	1	no	—
78	C₂×D₈	—	157	1	no	—
79	#131	—	314	1	no	—
80	C₄×D₄	—	314	1	no	—
81	SD₃₂	—	157	2	no	—
82	D₁₆	—	157	2	no	—
83	M₃₂	—	157	1	no	—
84	M₃₂	C₂×D₁₅₇	1	1	yes	characteristic
85	#108	C₂×D₁₅₇	1	1	yes	characteristic
86	#106	C₂×D₁₅₇	1	1	yes	characteristic
87	#104	—	314	1	no	—
88	#102	—	314	1	no	—
89	#97	—	157	1	no	—
90	#41	—	314	1	no	—
91	C₂×D₄	—	314	1	no	—
92	C₂²×C₄	—	628	1	no	—
93	Q₁₆	—	157	1	no	—
94	SD₁₆	—	314	1	no	—
95	D₈	—	314	1	no	—
96	D₈	—	157	1	no	—
97	D₈	—	157	2	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	628	1	no	—
100	C₂×C₈	—	628	1	no	—
101	C₂×C₈	—	157	1	no	—
102	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	—	628	1	no	—
105	C₄²	—	157	1	no	—
106	C₁₆	—	314	1	no	—
107	C₁₆	C₄×D₁₅₇	1	2	yes	complemented
108	C₂³	—	628	1	no	—
109	Q₈	—	314	1	no	—
110	D₄	—	314	2	no	—
111	D₄	—	314	1	no	—
112	D₄	—	628	1	no	—
113	D₄	—	628	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	314	1	no	—
116	C₂×C₄	—	628	1	no	—
117	C₂×C₄	—	1256	2	no	—
118	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	314	1	no	—
121	C₈	C₂×C₄×D₁₅₇	1	1	yes	characteristic
122	C₈	#14578955	1	1	yes	characteristic
123	C₂²	—	628	1	no	—
124	C₂²	—	628	2	no	—
125	C₂²	—	1256	1	no	—
126	C₂²	—	1256	1	no	—
127	C₂²	#19097798	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	628	1	no	—
130	C₄	—	628	1	no	—
131	C₄	#19097733	1	1	yes	characteristic
132	C₄	#19097798	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	1256	2	no	—
135	C₂	—	1256	1	no	—
136	C₂	#23757384	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23757384
commutator	C₈×C₁₅₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅₇
Fitting	#365×C₁₅₇	C₂
socle	C₂×C₁₅₇	#442
prime radical	#23757711	C₂
cosocle	C₂×C₈×C₁₅₇	C₂³
hypercenter	#365	D₁₅₇
hypocenter	C₁₅₇	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

Complements

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

N	H ≅ G/N	type
2. #23757394	C₂	semidirect
3. #23757384	C₂	semidirect
4. #23757085	C₂	semidirect
5. #23757082	C₂	semidirect
6. #23757050	C₂	semidirect
7. #365×C₁₅₇	C₂	semidirect
12. #19097716	C₄	semidirect
13. D₂₅₁₂	C₄	semidirect
24. Dic₆₂₈	C₂×C₄	semidirect
28. D₁₂₅₆	C₂×C₄	semidirect
38. C₁₆×C₁₅₇	C₂×C₄	semidirect
68. C₁₅₇	AGL(1,ℤ/16)	semidirect
73. #365	D₁₅₇	semidirect
107. C₁₆	C₄×D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757711	2
#23757394	2
#23757384	2
#23757085	2
#23757082	2
#23757050	2
#365×C₁₅₇	2
AGL(1,ℤ/16)	157