Subgroups — #2792×C₁₅₇ #34097150

Order 20096.

384 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#398×C₁₅₇	C₂	1	1	yes	maximal, characteristic
2	#398×C₁₅₇	C₂	1	2	yes	maximal
3	#395×C₁₅₇	C₂	1	2	yes	maximal
4	#395×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	#389×C₁₅₇	C₂	1	2	yes	maximal
6	#389×C₁₅₇	C₂	1	2	yes	maximal
7	#387×C₁₅₇	C₂	1	2	yes	maximal
8	#387×C₁₅₇	C₂	1	2	yes	maximal
9	#384×C₁₅₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₅₇	C₂²	1	2	yes	—
23	#95×C₁₅₇	C₂²	1	2	yes	—
24	#95×C₁₅₇	C₂²	1	2	yes	—
25	#95×C₁₅₇	C₂²	1	1	yes	characteristic
26	#95×C₁₅₇	C₂²	1	2	yes	—
27	#95×C₁₅₇	C₂²	1	2	yes	—
28	#95×C₁₅₇	C₂²	1	2	yes	—
29	#95×C₁₅₇	C₂²	1	2	yes	—
30	#95×C₁₅₇	C₂²	1	1	yes	characteristic
31	C₂×Q₈×C₁₅₇	—	4	1	no	—
32	C₂×Q₈×C₁₅₇	—	4	2	no	—
33	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₁₅₇	—	2	4	no	—
43	C₄⋊C₄×C₁₅₇	—	4	2	no	—
44	C₄⋊C₄×C₁₅₇	—	4	2	no	—
45	C₄⋊C₄×C₁₅₇	—	4	2	no	—
46	C₄⋊C₄×C₁₅₇	—	4	2	no	—
47	C₄⋊C₄×C₁₅₇	—	4	1	no	—
48	C₄⋊C₄×C₁₅₇	—	4	1	no	—
49	C₄⋊C₄×C₁₅₇	—	4	1	no	—
50	C₄⋊C₄×C₁₅₇	—	4	2	no	—
51	C₄⋊C₄×C₁₅₇	Q₈	1	8	yes	complemented
52	C₄²×C₁₅₇	—	2	2	no	—
53	C₄²×C₁₅₇	—	4	2	no	—
54	C₂³×C₁₅₇	C₂⁴	1	1	yes	characteristic
55	Q₈×C₁₅₇	—	4	4	no	—
56	C₂×C₄×C₁₅₇	—	2	2	no	—
57	C₂×C₄×C₁₅₇	—	2	2	no	—
58	C₂×C₄×C₁₅₇	—	2	2	no	—
59	C₂×C₄×C₁₅₇	—	2	2	no	—
60	C₂×C₄×C₁₅₇	—	2	1	no	—
61	C₂×C₄×C₁₅₇	—	2	2	no	—
62	C₂×C₄×C₁₅₇	—	2	1	no	—
63	C₂×C₄×C₁₅₇	—	2	2	no	—
64	C₂×C₄×C₁₅₇	—	2	2	no	—
65	C₂×C₄×C₁₅₇	—	2	2	no	—
66	C₂×C₄×C₁₅₇	—	2	2	no	—
67	C₂×C₄×C₁₅₇	—	2	2	no	—
68	C₂×C₄×C₁₅₇	—	2	1	no	—
69	C₂×C₄×C₁₅₇	—	2	2	no	—
70	C₂×C₄×C₁₅₇	—	2	2	no	—
71	C₂×C₄×C₁₅₇	—	2	1	no	—
72	C₂×C₄×C₁₅₇	—	2	2	no	—
73	C₂×C₄×C₁₅₇	—	2	2	no	—
74	C₂×C₄×C₁₅₇	—	2	2	no	—
75	C₂×C₄×C₁₅₇	—	2	2	no	—
76	C₂×C₄×C₁₅₇	—	2	2	no	—
77	C₂×C₄×C₁₅₇	C₂×Q₈	1	4	yes	—
78	C₂×C₄×C₁₅₇	C₂×Q₈	1	4	yes	—
79	C₂×C₄×C₁₅₇	C₂×Q₈	1	4	yes	—
80	C₂×C₄×C₁₅₇	#41	1	2	yes	—
81	C₂²×C₁₅₇	C₂²×Q₈	1	2	yes	—
82	C₂²×C₁₅₇	C₂×#41	1	1	yes	characteristic
83	C₂²×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₁₅₇	H(5,𝔽₂)	1	2	yes	—
85	C₂²×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
86	C₄×C₁₅₇	—	2	2	no	—
87	C₄×C₁₅₇	—	2	4	no	—
88	C₄×C₁₅₇	—	2	4	no	—
89	C₄×C₁₅₇	—	2	4	no	—
90	C₄×C₁₅₇	—	4	2	no	—
91	C₄×C₁₅₇	—	4	2	no	—
92	C₄×C₁₅₇	—	4	1	no	—
93	C₄×C₁₅₇	—	4	1	no	—
94	C₄×C₁₅₇	—	4	2	no	—
95	C₂×C₁₅₇	#539	1	1	yes	characteristic
96	C₂×C₁₅₇	#544	1	2	yes	—
97	C₂×C₁₅₇	#554	1	2	yes	—
98	C₂×C₁₅₇	Q₈²	1	1	yes	characteristic
99	C₂×C₁₅₇	#560	1	1	yes	characteristic
100	C₁₅₇	#2792	1	1	yes	characteristic, direct product factor, Sylow
101	#2792	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	#398	C₂×C₁₅₇	1	2	yes	—
103	#398	C₂×C₁₅₇	1	1	yes	characteristic
104	#395	C₂×C₁₅₇	1	1	yes	characteristic
105	#395	C₂×C₁₅₇	1	2	yes	—
106	#389	C₂×C₁₅₇	1	2	yes	—
107	#389	C₂×C₁₅₇	1	2	yes	—
108	#387	C₂×C₁₅₇	1	2	yes	—
109	#387	C₂×C₁₅₇	1	2	yes	—
110	#384	C₂×C₁₅₇	1	1	yes	characteristic
111	C₂²×Q₈	C₂²×C₁₅₇	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	2	yes	—
114	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	2	yes	—
122	C₂×C₄²	C₂²×C₁₅₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²×C₁₅₇	1	2	yes	—
124	#95	C₂²×C₁₅₇	1	1	yes	characteristic
125	#95	C₂²×C₁₅₇	1	2	yes	—
126	#95	C₂²×C₁₅₇	1	2	yes	—
127	#95	C₂²×C₁₅₇	1	2	yes	—
128	#95	C₂²×C₁₅₇	1	2	yes	—
129	#95	C₂²×C₁₅₇	1	2	yes	—
130	#95	C₂²×C₁₅₇	1	2	yes	—
131	#95	C₂²×C₁₅₇	1	1	yes	characteristic
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂²×C₄	C₂³×C₁₅₇	1	2	yes	—
135	C₂²×C₄	C₂³×C₁₅₇	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₁₅₇	1	2	yes	—
137	C₂²×C₄	C₂³×C₁₅₇	1	2	yes	—
138	C₂²×C₄	C₂³×C₁₅₇	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₁₅₇	1	2	yes	—
140	C₂²×C₄	C₂³×C₁₅₇	1	2	yes	—
141	C₂²×C₄	C₂³×C₁₅₇	1	2	yes	—
142	C₂²×C₄	C₂³×C₁₅₇	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	Q₈×C₁₅₇	1	8	yes	complemented
153	C₄²	—	2	2	no	—
154	C₄²	—	4	2	no	—
155	C₂³	C₂⁴×C₁₅₇	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₂×Q₈×C₁₅₇	1	4	yes	—
179	C₂×C₄	C₂×Q₈×C₁₅₇	1	4	yes	—
180	C₂×C₄	C₂×Q₈×C₁₅₇	1	4	yes	—
181	C₂×C₄	#41×C₁₅₇	1	2	yes	—
182	C₂²	C₂²×Q₈×C₁₅₇	1	2	yes	—
183	C₂²	C₂×#41×C₁₅₇	1	1	yes	characteristic
184	C₂²	H(5,𝔽₂)×C₁₅₇	1	1	yes	characteristic
185	C₂²	H(5,𝔽₂)×C₁₅₇	1	2	yes	—
186	C₂²	2₋¹⁺⁴×C₁₅₇	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	4	2	no	—
192	C₄	—	4	1	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	1	no	—
196	C₂	#539×C₁₅₇	1	1	yes	characteristic
197	C₂	#544×C₁₅₇	1	2	yes	—
198	C₂	#554×C₁₅₇	1	2	yes	—
199	C₂	Q₈²×C₁₅₇	1	1	yes	characteristic
200	C₂	#560×C₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₅₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₅₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
51. C₄⋊C₄×C₁₅₇	Q₈	semidirect
100. C₁₅₇	#2792	direct (N × H)
101. #2792	C₁₅₇	direct (N × H)
152. C₄⋊C₄	Q₈×C₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#398×C₁₅₇	2
#395×C₁₅₇	2
#389×C₁₅₇	2
#387×C₁₅₇	2
#384×C₁₅₇	2
#2792	157

Subgroups — #2792×C157 #34097150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2792×C₁₅₇ #34097150