Subgroups — C₂×#23757998 #34113705

Order 20096.

1048 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757998	C₂	1	16	yes	maximal, direct product factor
2	C₂×#19097769	C₂	1	2	yes	maximal
3	C₂×#19097762	C₂	1	2	yes	maximal
4	C₂×#19097761	C₂	1	4	yes	maximal
5	C₂×Q₈×Dic₁₅₇	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₃₁₄	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₃₁₄	C₂	1	1	yes	maximal, characteristic
8	C₂×#19097736	C₂	1	2	yes	maximal
9	C₂×C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₃₁₄	C₂²	1	1	yes	characteristic
11	C₂²×Dic₃₁₄	C₂²	1	1	yes	characteristic
12	#19097769	C₂²	1	16	yes	—
13	#19097762	C₂²	1	16	yes	—
14	#19097761	C₂²	1	32	yes	—
15	Q₈×Dic₁₅₇	C₂²	1	16	yes	—
16	C₄×Dic₃₁₄	C₂²	1	8	yes	—
17	C₄×Dic₃₁₄	C₂²	1	8	yes	—
18	#19097736	C₂²	1	16	yes	—
19	C₂×C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
22	C₂×#14578933	C₂²	1	4	yes	—
23	C₂×#14578933	C₂²	1	2	yes	—
24	C₂×#14578933	C₂²	1	2	yes	—
25	C₂×#14578933	C₂²	1	4	yes	—
26	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₅₇	C₂²	1	4	yes	—
28	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₅₇	C₂²	1	2	yes	—
30	C₄⋊Q₈×C₁₅₇	C₂²	1	8	yes	—
31	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
34	C₂×Dic₃₁₄	—	2	4	no	—
35	C₂×Dic₃₁₄	Q₈	1	8	yes	complemented
36	C₂×Dic₃₁₄	C₂³	1	4	yes	—
37	C₂×Dic₃₁₄	C₂³	1	4	yes	—
38	C₂²×Dic₁₅₇	C₂³	1	2	yes	—
39	C₂²×Dic₁₅₇	C₂³	1	2	yes	—
40	C₂²×Dic₁₅₇	C₂³	1	2	yes	—
41	C₂²×Dic₁₅₇	C₂³	1	2	yes	—
42	C₄⋊Dic₁₅₇	C₂³	1	4	yes	—
43	C₄⋊Dic₁₅₇	C₂³	1	4	yes	—
44	C₄⋊Dic₁₅₇	C₂³	1	16	yes	—
45	#14578933	C₂³	1	16	yes	—
46	#14578933	C₂³	1	8	yes	—
47	#14578933	C₂³	1	8	yes	—
48	#14578933	C₂³	1	16	yes	—
49	C₄×Dic₁₅₇	C₂³	1	8	yes	—
50	C₄×Dic₁₅₇	C₂³	1	16	yes	—
51	C₄×Dic₁₅₇	C₂³	1	8	yes	—
52	C₂×Q₈×C₁₅₇	—	2	8	no	—
53	C₂×Q₈×C₁₅₇	C₂³	1	8	yes	—
54	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₅₇	C₂³	1	4	yes	—
56	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₅₇	C₂³	1	8	yes	—
59	C₄⋊C₄×C₁₅₇	C₂³	1	8	yes	—
60	C₄²×C₁₅₇	C₂³	1	4	yes	—
61	Dic₃₁₄	—	2	8	no	—
62	Dic₃₁₄	C₂×Q₈	1	16	yes	complemented
63	C₂×Dic₁₅₇	—	2	4	no	—
64	C₂×Dic₁₅₇	—	2	4	no	—
65	C₂×Dic₁₅₇	—	2	4	no	—
66	C₂×Dic₁₅₇	C₂×Q₈	1	8	yes	—
67	C₂×Dic₁₅₇	C₂⁴	1	4	yes	—
68	C₂×Dic₁₅₇	C₂⁴	1	4	yes	—
69	C₂×Dic₁₅₇	C₂⁴	1	4	yes	—
70	C₂×Dic₁₅₇	C₂⁴	1	4	yes	—
71	C₂³×C₁₅₇	C₂⁴	1	1	yes	characteristic
72	Q₈×C₁₅₇	—	2	16	no	—
73	C₂×C₄×C₁₅₇	—	2	8	no	—
74	C₂×C₄×C₁₅₇	—	2	2	no	—
75	C₂×C₄×C₁₅₇	C₂×Q₈	1	4	yes	—
76	C₂×C₄×C₁₅₇	#41	1	4	yes	—
77	C₂×C₄×C₁₅₇	C₂⁴	1	2	yes	—
78	C₂×C₄×C₁₅₇	C₂⁴	1	8	yes	—
79	C₂×C₄×C₁₅₇	C₂⁴	1	2	yes	—
80	C₂×C₄×C₁₅₇	C₂⁴	1	2	yes	—
81	C₂²×C₁₅₇	C₂²×Q₈	1	2	yes	—
82	C₂²×C₁₅₇	C₂×#41	1	2	yes	—
83	C₂²×C₁₅₇	2₋¹⁺⁴	1	2	yes	—
84	C₂²×C₁₅₇	C₂⁵	1	1	yes	characteristic
85	C₄×C₁₅₇	—	2	8	no	—
86	C₄×C₁₅₇	—	2	2	no	—
87	C₄×C₁₅₇	C₂²×Q₈	1	4	yes	—
88	C₄×C₁₅₇	C₂×#41	1	4	yes	—
89	Dic₁₅₇	—	2	4	no	—
90	Dic₁₅₇	—	2	4	no	—
91	Dic₁₅₇	—	2	4	no	—
92	Dic₁₅₇	C₂²×Q₈	1	8	yes	—
93	C₂×C₁₅₇	#557	1	4	yes	complemented
94	C₂×C₁₅₇	C₂³×Q₈	1	1	yes	characteristic
95	C₂×C₁₅₇	C₂²×#41	1	1	yes	characteristic
96	C₂×C₁₅₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
97	C₁₅₇	C₂×#557	1	1	yes	characteristic, complemented, Sylow
98	C₂×#557	—	157	1	no	maximal, Sylow
99	#557	—	157	16	no	—
100	C₂×C₄⋊Q₈	—	157	2	no	—
101	C₂×C₄⋊Q₈	D₁₅₇	1	1	yes	characteristic
102	C₂×#125	—	157	4	no	—
103	C₂×#125	—	157	2	no	—
104	C₂×C₄×Q₈	—	157	2	no	—
105	C₂×C₄×Q₈	—	157	2	no	—
106	C₂×C₄×Q₈	—	157	1	no	—
107	C₂×C₄×Q₈	—	157	1	no	—
108	C₂²×Q₈	—	157	1	no	—
109	C₂²×Q₈	—	157	1	no	—
110	C₂²×Q₈	C₂×D₁₅₇	1	2	yes	—
111	C₄⋊Q₈	—	157	16	no	—
112	C₄⋊Q₈	C₂×D₁₅₇	1	8	yes	—
113	#125	—	157	32	no	—
114	#125	—	157	16	no	—
115	C₄×Q₈	—	157	16	no	—
116	C₄×Q₈	—	157	16	no	—
117	C₄×Q₈	—	157	8	no	—
118	C₄×Q₈	—	157	8	no	—
119	C₂×C₄⋊C₄	—	157	1	no	—
120	C₂×C₄⋊C₄	—	157	2	no	—
121	C₂×C₄⋊C₄	—	157	2	no	—
122	C₂×C₄⋊C₄	—	157	4	no	—
123	C₂×C₄⋊C₄	—	157	1	no	—
124	C₂×C₄⋊C₄	—	157	4	no	—
125	C₂×C₄⋊C₄	—	157	4	no	—
126	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
127	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
128	C₂×C₄²	—	157	2	no	—
129	C₂×C₄²	—	157	4	no	—
130	C₂×C₄²	—	157	2	no	—
131	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	314	4	no	—
134	C₂×Q₈	—	157	8	no	—
135	C₂×Q₈	—	157	4	no	—
136	C₂×Q₈	—	157	4	no	—
137	C₂×Q₈	C₂²×D₁₅₇	1	8	yes	—
138	C₂²×C₄	—	157	2	no	—
139	C₂²×C₄	—	157	2	no	—
140	C₂²×C₄	—	157	2	no	—
141	C₂²×C₄	—	157	2	no	—
142	C₂²×C₄	C₂²×D₁₅₇	1	4	yes	—
143	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
146	C₄⋊C₄	—	157	4	no	—
147	C₄⋊C₄	—	157	8	no	—
148	C₄⋊C₄	—	157	8	no	—
149	C₄⋊C₄	—	157	16	no	—
150	C₄⋊C₄	—	157	4	no	—
151	C₄⋊C₄	—	157	16	no	—
152	C₄⋊C₄	—	157	16	no	—
153	C₄⋊C₄	C₂²×D₁₅₇	1	8	yes	—
154	C₄⋊C₄	C₂²×D₁₅₇	1	8	yes	—
155	C₄²	—	157	8	no	—
156	C₄²	—	157	16	no	—
157	C₄²	—	157	8	no	—
158	C₄²	C₂²×D₁₅₇	1	4	yes	—
159	C₂³	C₂³×D₁₅₇	1	1	yes	characteristic
160	Q₈	—	2	16	no	—
161	Q₈	—	314	8	no	—
162	Q₈	—	157	16	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	314	4	no	—
166	C₂×C₄	—	314	4	no	—
167	C₂×C₄	—	314	4	no	—
168	C₂×C₄	—	157	4	no	—
169	C₂×C₄	—	157	4	no	—
170	C₂×C₄	—	157	4	no	—
171	C₂×C₄	—	157	8	no	—
172	C₂×C₄	—	157	4	no	—
173	C₂×C₄	Q₈×D₁₅₇	1	4	yes	—
174	C₂×C₄	#14578953	1	4	yes	—
175	C₂×C₄	C₂³×D₁₅₇	1	8	yes	—
176	C₂×C₄	C₂³×D₁₅₇	1	2	yes	—
177	C₂×C₄	C₂³×D₁₅₇	1	2	yes	—
178	C₂×C₄	C₂³×D₁₅₇	1	2	yes	—
179	C₂²	C₂×Q₈×D₁₅₇	1	2	yes	—
180	C₂²	C₂×#14578953	1	2	yes	—
181	C₂²	#19097819	1	2	yes	—
182	C₂²	C₂⁴×D₁₅₇	1	1	yes	characteristic
183	C₄	—	2	8	no	—
184	C₄	—	2	2	no	—
185	C₄	—	314	4	no	—
186	C₄	—	314	4	no	—
187	C₄	—	314	4	no	—
188	C₄	—	157	8	no	—
189	C₄	C₂×Q₈×D₁₅₇	1	4	yes	—
190	C₄	C₂×#14578953	1	4	yes	—
191	C₂	#23757998	1	4	yes	direct product factor
192	C₂	C₂²×Q₈×D₁₅₇	1	1	yes	characteristic
193	C₂	C₂²×#14578953	1	1	yes	characteristic
194	C₂	C₂×#19097819	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₅₇
commutator, cosocle	C₂²×C₁₅₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅₇
Fitting	C₂×C₄⋊Q₈×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	C₂×#557
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #23757998	C₂	direct (N × H)
35. C₂×Dic₃₁₄	Q₈	semidirect
62. Dic₃₁₄	C₂×Q₈	semidirect
93. C₂×C₁₅₇	#557	semidirect
97. C₁₅₇	C₂×#557	semidirect
191. C₂	#23757998	direct (N × H)

Maximal subgroups

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

subgroup	index
#23757998	2
C₂×#19097769	2
C₂×#19097762	2
C₂×#19097761	2
C₂×Q₈×Dic₁₅₇	2
C₂×C₄×Dic₃₁₄	2
C₂×#19097736	2
C₂×C₄⋊Q₈×C₁₅₇	2
C₂×#557	157

Subgroups — C2×#23757998 #34113705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23757998 #34113705