Subgroups — C₂×#428×C₁₅₇ #34097268

Order 20096.

536 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#428×C₁₅₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#107×C₁₅₇	C₂	1	2	yes	maximal, complemented
6	C₂×#106×C₁₅₇	C₂	1	2	yes	maximal, complemented
7	C₂³×C₄×C₁₅₇	—	2	1	no	—
8	C₂³×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₁₅₇	C₄	1	8	yes	complemented
10	C₂×M₁₆×C₁₅₇	C₂²	1	4	yes	—
11	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
12	#117×C₁₅₇	—	2	2	no	—
13	#117×C₁₅₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₁₅₇	—	2	2	no	—
15	C₂×C₄⋊C₄×C₁₅₇	—	2	1	no	—
16	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₁₅₇	—	2	1	no	—
21	C₂×C₄²×C₁₅₇	—	2	1	no	—
22	#107×C₁₅₇	C₂²	1	8	yes	complemented
23	#106×C₁₅₇	C₂²	1	8	yes	complemented
24	C₂⁴×C₁₅₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₅₇	—	2	4	no	—
26	C₂²×C₄×C₁₅₇	—	2	1	no	—
27	C₂²×C₄×C₁₅₇	—	2	1	no	—
28	C₂²×C₄×C₁₅₇	—	2	2	no	—
29	C₂²×C₄×C₁₅₇	—	4	4	no	—
30	C₂²×C₄×C₁₅₇	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
32	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₅₇	Q₈	1	4	yes	—
34	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₅₇	C₂×C₄	1	16	yes	complemented
37	C₂×C₈×C₁₅₇	C₂×C₄	1	8	yes	complemented
38	C₂×C₈×C₁₅₇	C₂³	1	4	yes	—
39	C₄⋊C₄×C₁₅₇	—	2	4	no	—
40	C₄⋊C₄×C₁₅₇	—	2	2	no	—
41	C₄⋊C₄×C₁₅₇	—	2	1	no	—
42	C₄⋊C₄×C₁₅₇	—	2	1	no	—
43	C₄⋊C₄×C₁₅₇	C₂³	1	2	yes	—
44	C₄⋊C₄×C₁₅₇	C₂³	1	4	yes	—
45	C₄⋊C₄×C₁₅₇	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₁₅₇	—	2	4	no	—
47	C₄²×C₁₅₇	—	2	4	no	—
48	C₂³×C₁₅₇	—	2	2	no	—
49	C₂³×C₁₅₇	—	2	2	no	—
50	C₂³×C₁₅₇	C₄⋊C₄	1	4	yes	—
51	C₂³×C₁₅₇	C₂×D₄	1	2	yes	—
52	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₅₇	—	2	2	no	—
54	C₂×C₄×C₁₅₇	—	2	2	no	—
55	C₂×C₄×C₁₅₇	—	2	4	no	—
56	C₂×C₄×C₁₅₇	—	4	8	no	—
57	C₂×C₄×C₁₅₇	—	4	2	no	—
58	C₂×C₄×C₁₅₇	—	4	4	no	—
59	C₂×C₄×C₁₅₇	—	4	2	no	—
60	C₂×C₄×C₁₅₇	C₄⋊C₄	1	8	yes	—
61	C₂×C₄×C₁₅₇	C₄⋊C₄	1	4	yes	—
62	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₅₇	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₅₇	C₂×Q₈	1	4	yes	—
67	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
69	C₈×C₁₅₇	C₂²×C₄	1	8	yes	complemented
70	C₂²×C₁₅₇	—	2	2	no	—
71	C₂²×C₁₅₇	—	2	2	no	—
72	C₂²×C₁₅₇	—	2	8	no	—
73	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₁₅₇	AGL(1,ℤ/8)	1	2	yes	—
76	C₂²×C₁₅₇	#137	1	2	yes	—
77	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₁₅₇	—	4	2	no	—
79	C₄×C₁₅₇	—	4	4	no	—
80	C₄×C₁₅₇	—	4	2	no	—
81	C₄×C₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₄×C₁₅₇	C₂×C₄⋊C₄	1	4	yes	—
83	C₄×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₁₅₇	—	2	4	no	—
86	C₂×C₁₅₇	#428	1	4	yes	direct product factor
87	C₂×C₁₅₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₁₅₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₁₅₇	C₂×#137	1	1	yes	characteristic
90	C₁₅₇	C₂×#428	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×#428	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×M₁₆	C₂×C₁₅₇	1	1	yes	characteristic
93	C₂×#117	C₂×C₁₅₇	1	1	yes	characteristic
94	C₂²×C₄⋊C₄	C₂×C₁₅₇	1	1	yes	characteristic
95	#428	C₂×C₁₅₇	1	8	yes	direct product factor
96	C₂×#107	C₂×C₁₅₇	1	2	yes	complemented
97	C₂×#106	C₂×C₁₅₇	1	2	yes	complemented
98	C₂³×C₄	—	2	1	no	—
99	C₂³×C₄	C₂²×C₁₅₇	1	1	yes	characteristic
100	C₂×M₁₆	C₄×C₁₅₇	1	8	yes	complemented
101	C₂×M₁₆	C₂²×C₁₅₇	1	4	yes	—
102	C₂²×C₈	C₂²×C₁₅₇	1	2	yes	—
103	#117	—	2	2	no	—
104	#117	C₂²×C₁₅₇	1	4	yes	—
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	2	yes	—
108	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	4	yes	—
109	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₄²	—	2	1	no	—
113	#107	C₂²×C₁₅₇	1	8	yes	complemented
114	#106	C₂²×C₁₅₇	1	8	yes	complemented
115	C₂⁴	D₄×C₁₅₇	1	1	yes	characteristic
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	4	4	no	—
121	C₂²×C₄	C₂×C₄×C₁₅₇	1	4	yes	—
122	C₂²×C₄	D₄×C₁₅₇	1	1	yes	characteristic
123	C₂²×C₄	D₄×C₁₅₇	1	2	yes	—
124	C₂²×C₄	Q₈×C₁₅₇	1	4	yes	—
125	C₂²×C₄	C₂³×C₁₅₇	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₁₅₇	1	2	yes	—
127	M₁₆	C₂×C₄×C₁₅₇	1	16	yes	complemented
128	C₂×C₈	C₂×C₄×C₁₅₇	1	8	yes	complemented
129	C₂×C₈	C₂³×C₁₅₇	1	4	yes	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	C₂³×C₁₅₇	1	2	yes	—
135	C₄⋊C₄	C₂³×C₁₅₇	1	4	yes	—
136	C₄⋊C₄	C₂³×C₁₅₇	1	2	yes	—
137	C₂²⋊C₄	—	2	4	no	—
138	C₄²	—	2	4	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	C₄⋊C₄×C₁₅₇	1	4	yes	—
142	C₂³	C₂×D₄×C₁₅₇	1	2	yes	—
143	C₂³	C₂×D₄×C₁₅₇	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	4	8	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	C₄⋊C₄×C₁₅₇	1	8	yes	—
152	C₂×C₄	C₄⋊C₄×C₁₅₇	1	4	yes	—
153	C₂×C₄	C₂²×C₄×C₁₅₇	1	2	yes	—
154	C₂×C₄	C₂²×C₄×C₁₅₇	1	4	yes	—
155	C₂×C₄	C₂×D₄×C₁₅₇	1	2	yes	—
156	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
157	C₂×C₄	C₂×Q₈×C₁₅₇	1	4	yes	—
158	C₂×C₄	C₂×Q₈×C₁₅₇	1	2	yes	—
159	C₂×C₄	C₂⁴×C₁₅₇	1	1	yes	characteristic
160	C₈	C₂²×C₄×C₁₅₇	1	8	yes	complemented
161	C₂²	—	2	8	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	C₂×C₄⋊C₄×C₁₅₇	1	4	yes	—
165	C₂²	C₂×C₄⋊C₄×C₁₅₇	1	2	yes	—
166	C₂²	AGL(1,ℤ/8)×C₁₅₇	1	2	yes	—
167	C₂²	#137×C₁₅₇	1	2	yes	—
168	C₂²	C₂²×D₄×C₁₅₇	1	1	yes	characteristic
169	C₄	—	4	4	no	—
170	C₄	—	4	2	no	—
171	C₄	—	4	2	no	—
172	C₄	C₂×C₄⋊C₄×C₁₅₇	1	2	yes	—
173	C₄	C₂×C₄⋊C₄×C₁₅₇	1	4	yes	—
174	C₄	C₂³×C₄×C₁₅₇	1	1	yes	characteristic
175	C₄	C₂²×Q₈×C₁₅₇	1	1	yes	characteristic
176	C₂	—	2	4	no	—
177	C₂	#428×C₁₅₇	1	4	yes	direct product factor
178	C₂	C₂²×C₄⋊C₄×C₁₅₇	1	1	yes	characteristic
179	C₂	C₂×AGL(1,ℤ/8)×C₁₅₇	1	1	yes	characteristic
180	C₂	C₂×#137×C₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₅₇	C₂×D₄
commutator	C₄	C₂³×C₄×C₁₅₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₅₇
prime radical	C₂⁴×C₁₅₇	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₅₇ ◁ C₂³×C₄×C₁₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #428×C₁₅₇	C₂	direct (N × H)
5. C₂×#107×C₁₅₇	C₂	semidirect
6. C₂×#106×C₁₅₇	C₂	semidirect
9. C₂×M₁₆×C₁₅₇	C₄	semidirect
22. #107×C₁₅₇	C₂²	semidirect
23. #106×C₁₅₇	C₂²	semidirect
36. M₁₆×C₁₅₇	C₂×C₄	semidirect
37. C₂×C₈×C₁₅₇	C₂×C₄	semidirect
69. C₈×C₁₅₇	C₂²×C₄	semidirect
86. C₂×C₁₅₇	#428	direct (N × H)
90. C₁₅₇	C₂×#428	direct (N × H)
91. C₂×#428	C₁₅₇	direct (N × H)
95. #428	C₂×C₁₅₇	direct (N × H)
96. C₂×#107	C₂×C₁₅₇	semidirect
97. C₂×#106	C₂×C₁₅₇	semidirect
100. C₂×M₁₆	C₄×C₁₅₇	semidirect
113. #107	C₂²×C₁₅₇	semidirect
114. #106	C₂²×C₁₅₇	semidirect
127. M₁₆	C₂×C₄×C₁₅₇	semidirect
128. C₂×C₈	C₂×C₄×C₁₅₇	semidirect
160. C₈	C₂²×C₄×C₁₅₇	semidirect
177. C₂	#428×C₁₅₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₁₅₇	2
C₂×#117×C₁₅₇	2
C₂²×C₄⋊C₄×C₁₅₇	2
#428×C₁₅₇	2
C₂×#107×C₁₅₇	2
C₂×#106×C₁₅₇	2
C₂×#428	157

Subgroups — C2×#428×C157 #34097268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#428×C₁₅₇ #34097268