Subgroups — #34102543

Order 20096.

656 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097730	C₂	1	2	yes	maximal
2	#23757143	C₂	1	2	yes	maximal, complemented
3	#23757139	C₂	1	2	yes	maximal, complemented
4	#23757133	C₂	1	4	yes	maximal
5	C₄×C₄⋊Dic₁₅₇	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄²×C₁₅₇	C₂	1	1	yes	maximal, characteristic
7	#19097730	—	2	8	no	—
8	C₂×C₄⋊Dic₁₅₇	C₂²	1	8	yes	—
9	C₂×C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₁₅₇	C₂²	1	4	yes	—
11	C₂×C₄×Dic₁₅₇	C₂²	1	4	yes	—
12	#19097678	C₂²	1	8	yes	—
13	C₂³×C₄×C₁₅₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₅₇	C₄	1	8	yes	complemented
16	C₂×C₄²×C₁₅₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂²×Dic₁₅₇	C₂³	1	8	yes	—
20	C₄⋊Dic₁₅₇	—	2	8	no	—
21	C₄⋊Dic₁₅₇	—	2	16	no	—
22	C₂²⋊Dic₁₅₇	—	2	8	no	—
23	C₄×Dic₁₅₇	—	2	8	no	—
24	C₂⁴×C₁₅₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₅₇	—	2	8	no	—
26	C₂²×C₄×C₁₅₇	C₂×C₄	1	8	yes	—
27	C₂²×C₄×C₁₅₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₁₅₇	D₄	1	4	yes	—
29	C₂²×C₄×C₁₅₇	Q₈	1	4	yes	—
30	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
34	C₄²×C₁₅₇	C₂×C₄	1	4	yes	—
35	C₄²×C₁₅₇	C₂×C₄	1	8	yes	—
36	C₄²×C₁₅₇	C₂×C₄	1	4	yes	—
37	C₂×Dic₁₅₇	—	2	8	no	—
38	C₂×Dic₁₅₇	—	2	16	no	—
39	C₂³×C₁₅₇	—	2	4	no	—
40	C₂³×C₁₅₇	C₂²×C₄	1	2	yes	—
41	C₂³×C₁₅₇	C₂×D₄	1	2	yes	—
42	C₂³×C₁₅₇	C₂×Q₈	1	2	yes	—
43	C₂³×C₁₅₇	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	—	2	4	no	—
45	C₂×C₄×C₁₅₇	—	2	16	no	—
46	C₂×C₄×C₁₅₇	C₄⋊C₄	1	16	yes	complemented
47	C₂×C₄×C₁₅₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₅₇	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₁₅₇	#41	1	8	yes	—
56	C₂²×C₁₅₇	—	2	4	no	—
57	C₂²×C₁₅₇	—	2	4	no	—
58	C₂²×C₁₅₇	—	2	4	no	—
59	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	4	yes	—
60	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
61	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
62	C₂²×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
63	C₂²×C₁₅₇	C₂×#41	1	4	yes	—
64	C₄×C₁₅₇	—	2	4	no	—
65	C₄×C₁₅₇	C₂×C₄⋊C₄	1	4	yes	—
66	C₄×C₁₅₇	C₂×C₄⋊C₄	1	4	yes	—
67	C₄×C₁₅₇	#117	1	8	yes	complemented
68	Dic₁₅₇	—	4	8	no	—
69	C₂×C₁₅₇	—	2	4	no	—
70	C₂×C₁₅₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₁₅₇	C₂×#117	1	2	yes	—
72	C₂×C₁₅₇	#532	1	2	yes	—
73	C₂×C₁₅₇	#533	1	2	yes	—
74	C₁₅₇	#2285	1	1	yes	characteristic, complemented, Sylow
75	#2285	—	157	1	no	maximal, Sylow
76	C₂×#117	—	157	2	no	—
77	C₂²×C₄²	D₁₅₇	1	1	yes	characteristic
78	#384	—	157	2	no	—
79	#382	—	157	2	no	—
80	#380	—	157	4	no	—
81	C₄×C₄⋊C₄	—	157	4	no	—
82	C₂³×C₄	C₂×D₁₅₇	1	2	yes	—
83	C₂³×C₄	C₂×D₁₅₇	1	1	yes	characteristic
84	#117	—	314	8	no	—
85	C₂×C₄⋊C₄	—	157	4	no	—
86	C₂×C₄⋊C₄	—	157	8	no	—
87	C₂×C₂²⋊C₄	—	157	4	no	—
88	C₂×C₄²	—	157	4	no	—
89	C₂×C₄²	Dic₁₅₇	1	8	yes	complemented
90	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
91	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₁₅₇	1	2	yes	—
93	#95	—	157	8	no	—
94	C₂⁴	C₂²×D₁₅₇	1	1	yes	characteristic
95	C₂²×C₄	—	2	8	no	—
96	C₂²×C₄	—	157	8	no	—
97	C₂²×C₄	C₂×Dic₁₅₇	1	8	yes	—
98	C₂²×C₄	C₂×Dic₁₅₇	1	4	yes	—
99	C₂²×C₄	D₆₂₈	1	4	yes	—
100	C₂²×C₄	Dic₃₁₄	1	4	yes	—
101	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
102	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
104	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
105	C₄⋊C₄	—	314	8	no	—
106	C₄⋊C₄	—	314	16	no	—
107	C₂²⋊C₄	—	314	8	no	—
108	C₄²	—	314	8	no	—
109	C₄²	C₂×Dic₁₅₇	1	4	yes	—
110	C₄²	C₂×Dic₁₅₇	1	8	yes	—
111	C₄²	C₂×Dic₁₅₇	1	4	yes	—
112	C₂³	—	2	4	no	—
113	C₂³	C₂²×Dic₁₅₇	1	2	yes	—
114	C₂³	C₂×D₆₂₈	1	2	yes	—
115	C₂³	C₂×Dic₃₁₄	1	2	yes	—
116	C₂³	C₂³×D₁₅₇	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	16	no	—
119	C₂×C₄	—	314	8	no	—
120	C₂×C₄	—	314	16	no	—
121	C₂×C₄	C₄⋊Dic₁₅₇	1	16	yes	complemented
122	C₂×C₄	C₂²×Dic₁₅₇	1	4	yes	—
123	C₂×C₄	C₂²×Dic₁₅₇	1	2	yes	—
124	C₂×C₄	C₂²×Dic₁₅₇	1	2	yes	—
125	C₂×C₄	C₂²×Dic₁₅₇	1	4	yes	—
126	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
127	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
128	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
129	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
130	C₂×C₄	#14578954	1	8	yes	—
131	C₂²	—	2	4	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	2	4	no	—
134	C₂²	C₂×C₄⋊Dic₁₅₇	1	4	yes	—
135	C₂²	C₂³×Dic₁₅₇	1	1	yes	characteristic
136	C₂²	C₂²×D₆₂₈	1	1	yes	characteristic
137	C₂²	C₂²×Dic₃₁₄	1	1	yes	characteristic
138	C₂²	C₂×#14578954	1	4	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	628	8	no	—
141	C₄	C₂×C₄⋊Dic₁₅₇	1	4	yes	—
142	C₄	C₂×C₄⋊Dic₁₅₇	1	4	yes	—
143	C₄	#19097730	1	8	yes	complemented
144	C₂	—	2	4	no	—
145	C₂	C₂²×C₄⋊Dic₁₅₇	1	1	yes	characteristic
146	C₂	C₂×#19097730	1	2	yes	—
147	C₂	#23757823	1	2	yes	—
148	C₂	#23757830	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₅₇
commutator	C₂²×C₁₅₇	C₂³×C₄
Frattini	C₂³	C₂³×D₁₅₇
Fitting	C₂²×C₄²×C₁₅₇	C₂
socle, cosocle	C₂³×C₁₅₇	C₂⁴
prime radical	C₂⁴×C₁₅₇	C₂³
hypercenter	C₂²×C₄²	D₁₅₇
hypocenter	C₁₅₇	#2285
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. #23757143	C₂	semidirect
3. #23757139	C₂	semidirect
5. C₄×C₄⋊Dic₁₅₇	C₂	semidirect
15. C₂×C₄²×C₁₅₇	C₄	semidirect
46. C₂×C₄×C₁₅₇	C₄⋊C₄	semidirect
67. C₄×C₁₅₇	#117	semidirect
74. C₁₅₇	#2285	semidirect
89. C₂×C₄²	Dic₁₅₇	semidirect
121. C₂×C₄	C₄⋊Dic₁₅₇	semidirect
143. C₄	#19097730	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19097730	2
#23757143	2
#23757139	2
#23757133	2
C₄×C₄⋊Dic₁₅₇	2
C₂²×C₄²×C₁₅₇	2
#2285	157