Subgroups — #34109074

Order 20096.

348 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757826	C₂	1	1	yes	maximal, characteristic
2	#23757403	C₂	1	4	yes	maximal, complemented
3	#23757401	C₂	1	4	yes	maximal, complemented
4	#23757301	C₂	1	1	yes	maximal, characteristic
5	#23757300	C₂	1	1	yes	maximal, characteristic
6	#23757296	C₂	1	1	yes	maximal, characteristic
7	#23757228	C₂	1	1	yes	maximal, characteristic
8	C₂×#19097681	C₂	1	1	yes	maximal, characteristic
9	#405×C₁₅₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#14578938	C₂²	1	1	yes	characteristic
11	C₂²×#11929434	C₂²	1	1	yes	characteristic
12	#19097770	C₄	1	2	yes	—
13	#19097762	C₄	1	2	yes	—
14	#19097748	C₄	1	4	yes	—
15	C₄×Dic₃₁₄	C₂²	1	4	yes	—
16	#19097730	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
18	C₄⋊#11929434	C₂²	1	4	yes	—
19	#19097700	C₂²	1	4	yes	—
20	#19097700	C₂²	1	4	yes	—
21	#19097681	C₂²	1	2	yes	—
22	#19097681	C₂²	1	2	yes	—
23	#19097681	C₂²	1	2	yes	—
24	C₈×Dic₁₅₇	C₂²	1	2	yes	—
25	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
27	#117×C₁₅₇	C₂²	1	1	yes	characteristic
28	#97×C₁₅₇	C₂²	1	2	yes	—
29	C₄×C₈×C₁₅₇	C₂²	1	2	yes	—
30	C₂×Dic₃₁₄	C₂×C₄	1	4	yes	—
31	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
32	#14578938	—	2	2	no	—
33	C₂×#11929434	—	2	2	no	—
34	C₂×#11929434	C₂³	1	2	yes	—
35	C₂×#11929434	C₂³	1	2	yes	—
36	C₄⋊Dic₁₅₇	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₁₅₇	C₂×C₄	1	4	yes	—
38	#14578933	C₂×C₄	1	4	yes	—
39	#14578933	C₂×C₄	1	4	yes	—
40	C₂²⋊Dic₁₅₇	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₁₅₇	C₂³	1	2	yes	—
44	C₄×Dic₁₅₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
46	M₁₆×C₁₅₇	—	2	2	no	—
47	C₂×C₈×C₁₅₇	Q₈	1	4	yes	complemented
48	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
49	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
53	C₄²×C₁₅₇	C₂³	1	2	yes	—
54	Dic₃₁₄	—	4	4	no	—
55	C₂×Dic₁₅₇	—	4	2	no	—
56	C₂×Dic₁₅₇	C₂²×C₄	1	2	yes	—
57	C₂×Dic₁₅₇	C₂²×C₄	1	2	yes	—
58	C₂×Dic₁₅₇	C₂²×C₄	1	4	yes	—
59	#11929434	—	2	2	no	—
60	#11929434	—	2	2	no	—
61	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₅₇	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₁₅₇	#41	1	2	yes	—
66	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
67	C₈×C₁₅₇	—	2	2	no	—
68	C₈×C₁₅₇	C₂×Q₈	1	2	yes	—
69	C₈×C₁₅₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₁₅₇	—	2	2	no	—
71	C₂²×C₁₅₇	C₄×Q₈	1	2	yes	—
72	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₁₅₇	—	2	4	no	—
74	C₄×C₁₅₇	C₄×Q₈	1	2	yes	—
75	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
76	C₄×C₁₅₇	C₂×#41	1	1	yes	characteristic
77	Dic₁₅₇	—	2	2	no	—
78	Dic₁₅₇	—	2	2	no	—
79	Dic₁₅₇	—	2	4	no	—
80	C₂×C₁₅₇	—	2	2	no	—
81	C₂×C₁₅₇	C₂×C₄×Q₈	1	1	yes	characteristic
82	C₂×C₁₅₇	#568	1	2	yes	—
83	C₁₅₇	#2961	1	1	yes	characteristic, complemented, Sylow
84	#2961	—	157	1	no	maximal, Sylow
85	#533	—	157	1	no	—
86	#446	—	157	4	no	—
87	#446	—	157	4	no	—
88	#424	—	157	1	no	—
89	#424	—	157	1	no	—
90	#423	—	157	1	no	—
91	#405	—	157	1	no	—
92	#405	D₁₅₇	1	1	yes	characteristic
93	C₂×#97	—	157	1	no	—
94	C₂×M₁₆	—	157	1	no	—
95	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
96	C₂²×C₈	—	157	1	no	—
97	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
98	C₄⋊Q₈	—	157	2	no	—
99	#125	—	157	2	no	—
100	C₂²⋊Q₈	—	157	4	no	—
101	C₄×Q₈	—	157	4	no	—
102	#117	—	157	1	no	—
103	#117	C₂×D₁₅₇	1	1	yes	characteristic
104	C₂×C₄²	—	157	1	no	—
105	C₄⋊C₈	—	157	4	no	—
106	C₄⋊C₈	—	157	4	no	—
107	C₄⋊C₈	—	157	4	no	—
108	#97	—	157	2	no	—
109	#97	—	157	2	no	—
110	#97	—	157	2	no	—
111	#97	C₂×D₁₅₇	1	2	yes	—
112	C₄×C₈	—	157	2	no	—
113	C₄×C₈	C₂×D₁₅₇	1	2	yes	—
114	C₂×Q₈	—	157	4	no	—
115	C₂²×C₄	—	157	2	no	—
116	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	314	2	no	—
119	C₂×C₈	—	314	2	no	—
120	C₂×C₈	—	157	2	no	—
121	C₂×C₈	—	157	2	no	—
122	C₂×C₈	Dic₃₁₄	1	4	yes	complemented
123	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
124	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
126	C₄⋊C₄	—	157	2	no	—
127	C₄⋊C₄	—	157	4	no	—
128	C₄⋊C₄	—	157	4	no	—
129	C₄⋊C₄	—	157	4	no	—
130	C₄⋊C₄	C₄×D₁₅₇	1	2	yes	—
131	C₂²⋊C₄	—	157	2	no	—
132	C₂²⋊C₄	C₄×D₁₅₇	1	2	yes	—
133	C₄²	—	157	1	no	—
134	C₄²	—	157	1	no	—
135	C₄²	—	157	2	no	—
136	C₄²	—	157	2	no	—
137	C₄²	C₂²×D₁₅₇	1	2	yes	—
138	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
139	Q₈	—	628	4	no	—
140	C₂×C₄	—	628	2	no	—
141	C₂×C₄	—	157	2	no	—
142	C₂×C₄	—	157	2	no	—
143	C₂×C₄	—	157	4	no	—
144	C₂×C₄	C₂×C₄×D₁₅₇	1	4	yes	—
145	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
146	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
147	C₂×C₄	#14578954	1	2	yes	—
148	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	314	2	no	—
151	C₈	—	314	2	no	—
152	C₈	C₂×Dic₃₁₄	1	2	yes	—
153	C₈	C₂×Dic₃₁₄	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₃₁₄	1	2	yes	—
156	C₂²	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	314	2	no	—
159	C₄	—	314	2	no	—
160	C₄	—	314	4	no	—
161	C₄	C₄×Dic₃₁₄	1	2	yes	—
162	C₄	C₂²×Dic₃₁₄	1	1	yes	characteristic
163	C₄	C₂×#14578954	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₃₁₄	1	1	yes	characteristic
166	C₂	#23758038	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₇
commutator, socle	C₂²×C₁₅₇	C₂³×C₄
Fitting	#405×C₁₅₇	C₂
prime radical	C₂³×C₁₅₇	C₂²×C₄
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	#405	D₁₅₇
hypocenter	C₁₅₇	#2961
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #23757403	C₂	semidirect
3. #23757401	C₂	semidirect
47. C₂×C₈×C₁₅₇	Q₈	semidirect
83. C₁₅₇	#2961	semidirect
122. C₂×C₈	Dic₃₁₄	semidirect

Maximal subgroups

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

subgroup	index
#23757826	2
#23757403	2
#23757401	2
#23757301	2
#23757300	2
#23757296	2
#23757228	2
C₂×#19097681	2
#405×C₁₅₇	2
#2961	157