Finite Group Databasebeta

Subgroups — #34102210

Order 20096.

170 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 45 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	#23757708	C₂	1	1	yes	maximal, characteristic, complemented
2	#23757648	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757390	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757093	C₂	1	1	yes	maximal, characteristic, complemented
5	#23757087	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757086	C₂	1	1	yes	maximal, characteristic, complemented
7	#509×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₂₅₆	C₂²	1	1	yes	characteristic
9	C₂×#14578939	—	2	1	no	—
10	#19097778	—	2	1	no	—
11	#19097767	—	2	1	no	—
12	#19097714	—	2	1	no	—
13	#19097712	—	2	1	no	—
14	#19097711	C₂²	1	1	yes	characteristic
15	#19097707	C₂²	1	1	yes	characteristic
16	#19097698	—	2	1	no	—
17	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
18	#135×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₁₅₇	C₂²	1	1	yes	characteristic
20	SD₃₂×C₁₅₇	—	2	1	no	—
21	D₁₆×C₁₅₇	—	2	1	no	—
22	M₃₂×C₁₅₇	C₂²	1	1	yes	characteristic
23	C₂×D₆₂₈	—	2	1	no	—
24	C₂×C₂²⋊D₁₅₇	—	4	2	no	—
25	D₁₂₅₆	—	2	1	no	—
26	D₁₂₅₆	—	2	1	no	—
27	#14578939	—	4	1	no	—
28	#14578939	—	4	1	no	—
29	#14578938	—	2	1	no	—
30	#14578938	—	4	1	no	—
31	C₂×#11929434	—	4	1	no	—
32	C₂×#11929434	D₄	1	1	yes	characteristic
33	C₄×Dic₁₅₇	D₄	1	1	yes	characteristic
34	#14578929	—	2	1	no	—
35	#41×C₁₅₇	—	2	1	no	—
36	C₂×D₄×C₁₅₇	—	2	1	no	—
37	Q₁₆×C₁₅₇	D₄	1	1	yes	characteristic, complemented
38	SD₁₆×C₁₅₇	—	2	1	no	—
39	D₈×C₁₅₇	—	2	1	no	—
40	D₈×C₁₅₇	—	2	1	no	—
41	D₈×C₁₅₇	D₄	1	1	yes	characteristic, complemented
42	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
43	C₁₆×C₁₅₇	—	2	1	no	—
44	C₂²×D₁₅₇	—	4	1	no	—
45	D₆₂₈	—	4	1	no	—
46	D₆₂₈	—	4	1	no	—
47	C₂²⋊D₁₅₇	—	8	4	no	—
48	C₂×Dic₁₅₇	—	2	1	no	—
49	#11929434	—	4	1	no	—
50	#11929434	D₈	1	1	yes	characteristic
51	#11929434	D₈	1	1	yes	characteristic
52	C₂³×C₁₅₇	—	4	1	no	—
53	Q₈×C₁₅₇	—	2	1	no	—
54	D₄×C₁₅₇	—	2	1	no	—
55	D₄×C₁₅₇	—	4	1	no	—
56	D₄×C₁₅₇	—	4	1	no	—
57	D₄×C₁₅₇	—	4	1	no	—
58	C₂×C₄×C₁₅₇	—	4	1	no	—
59	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
60	C₈×C₁₅₇	C₂×D₄	1	1	yes	characteristic
61	C₈×C₁₅₇	#41	1	1	yes	characteristic
62	C₂²×C₁₅₇	—	4	1	no	—
63	C₂²×C₁₅₇	—	8	2	no	—
64	C₂²×C₁₅₇	—	8	1	no	—
65	C₂²×C₁₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂×D₁₅₇	—	8	2	no	—
67	C₂×D₁₅₇	—	8	1	no	—
68	C₄×C₁₅₇	—	4	1	no	—
69	C₄×C₁₅₇	#121	1	1	yes	characteristic
70	C₄×C₁₅₇	C₂×D₈	1	1	yes	characteristic
71	Dic₁₅₇	—	2	2	no	—
72	C₂×C₁₅₇	—	2	1	no	—
73	C₂×C₁₅₇	—	8	1	no	—
74	C₂×C₁₅₇	—	16	1	no	—
75	C₂×C₁₅₇	#459	1	1	yes	characteristic
76	D₁₅₇	—	16	1	no	—
77	C₁₅₇	#2213	1	1	yes	characteristic, complemented, Sylow
78	#2213	—	157	1	no	maximal, Sylow
79	#509	—	157	1	no	—
80	#509	D₁₅₇	1	1	yes	characteristic, complemented
81	#493	—	157	1	no	—
82	#443	—	157	1	no	—
83	#364	—	157	1	no	—
84	#361	—	157	1	no	—
85	#361	—	157	1	no	—
86	#135	C₂×D₁₅₇	1	1	yes	characteristic
87	C₂×D₈	—	314	1	no	—
88	C₂×D₈	—	157	1	no	—
89	C₂×D₈	C₂×D₁₅₇	1	1	yes	characteristic
90	#131	—	314	1	no	—
91	#127	—	314	1	no	—
92	SD₃₂	—	2	1	no	—
93	SD₃₂	—	314	1	no	—
94	D₁₆	—	2	1	no	—
95	D₁₆	—	314	1	no	—
96	M₃₂	—	157	1	no	—
97	M₃₂	C₂×D₁₅₇	1	1	yes	characteristic
98	#108	—	157	1	no	—
99	#104	—	314	1	no	—
100	C₄×C₈	—	157	1	no	—
101	#41	—	2	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	314	1	no	—
104	C₂×D₄	—	628	2	no	—
105	Q₁₆	C₂²⋊D₁₅₇	1	1	yes	characteristic, complemented
106	SD₁₆	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	2	1	no	—
109	D₈	—	314	1	no	—
110	D₈	—	314	1	no	—
111	D₈	—	628	1	no	—
112	D₈	—	628	1	no	—
113	D₈	C₂²⋊D₁₅₇	1	1	yes	characteristic, complemented
114	M₁₆	—	314	1	no	—
115	M₁₆	—	628	1	no	—
116	C₂×C₈	—	628	1	no	—
117	C₂×C₈	—	157	1	no	—
118	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
119	C₄²	—	157	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	314	1	no	—
122	C₂³	—	4	1	no	—
123	C₂³	—	628	1	no	—
124	Q₈	—	2	1	no	—
125	D₄	—	2	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	628	1	no	—
130	D₄	—	628	1	no	—
131	D₄	—	1256	4	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	314	1	no	—
134	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
135	C₈	—	628	1	no	—
136	C₈	—	157	1	no	—
137	C₈	—	157	1	no	—
138	C₈	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
139	C₈	#14578953	1	1	yes	characteristic
140	C₂²	—	4	1	no	—
141	C₂²	—	8	2	no	—
142	C₂²	—	8	1	no	—
143	C₂²	—	1256	2	no	—
144	C₂²	—	1256	1	no	—
145	C₂²	#19097795	1	1	yes	characteristic
146	C₄	—	4	1	no	—
147	C₄	—	314	2	no	—
148	C₄	#19097743	1	1	yes	characteristic
149	C₄	D₈×D₁₅₇	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	8	1	no	—
152	C₂	—	16	1	no	—
153	C₂	—	2512	1	no	—
154	C₂	#23757466	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23757466
commutator, cosocle	C₂×C₈×C₁₅₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₅₇
Fitting	#509×C₁₅₇	C₂
socle	C₂×C₁₅₇	#459
hypercenter	#509	D₁₅₇
hypocenter	C₁₅₇	#2213
LCS	G ▷ C₂×C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #509

Complements

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

N	H ≅ G/N	type
1. #23757708	C₂	semidirect
2. #23757648	C₂	semidirect
3. #23757390	C₂	semidirect
4. #23757093	C₂	semidirect
5. #23757087	C₂	semidirect
6. #23757086	C₂	semidirect
7. #509×C₁₅₇	C₂	semidirect
37. Q₁₆×C₁₅₇	D₄	semidirect
41. D₈×C₁₅₇	D₄	semidirect
77. C₁₅₇	#2213	semidirect
80. #509	D₁₅₇	semidirect
105. Q₁₆	C₂²⋊D₁₅₇	semidirect
113. D₈	C₂²⋊D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757708	2
#23757648	2
#23757390	2
#23757093	2
#23757087	2
#23757086	2
#509×C₁₅₇	2
#2213	157