Group #45693521 — subgroups

Subgroups — C₁₃₁×#7637 #45693521

Order 33536.

636 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#127×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1745×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#1270×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	C₂×#127×C₁₃₁	C₄	1	1	yes	characteristic
5	C₂×C₄×D₄×C₁₃₁	—	2	2	no	—
6	C₂×C₄×D₄×C₁₃₁	C₄	1	2	yes	—
7	C₂×#97×C₁₃₁	—	2	2	no	—
8	#390×C₁₃₁	—	2	2	no	—
9	#390×C₁₃₁	C₂²	1	2	yes	—
10	#384×C₁₃₁	C₄	1	1	yes	characteristic
11	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
12	#336×C₁₃₁	—	2	2	no	—
13	#336×C₁₃₁	C₂²	1	4	yes	—
14	C₂²×D₄×C₁₃₁	—	2	2	no	—
15	C₂²×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
16	C₂³×C₄×C₁₃₁	—	2	2	no	—
17	C₂²×C₈×C₁₃₁	—	2	4	no	—
18	#127×C₁₃₁	—	2	4	no	—
19	C₄×D₄×C₁₃₁	—	4	8	no	—
20	C₄×D₄×C₁₃₁	—	4	4	no	—
21	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
24	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
25	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
28	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
29	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
30	#97×C₁₃₁	—	2	8	no	—
31	C₂⁴×C₁₃₁	—	2	2	no	—
32	C₂×D₄×C₁₃₁	—	2	8	no	—
33	C₂×D₄×C₁₃₁	—	2	2	no	—
34	C₂×D₄×C₁₃₁	—	4	2	no	—
35	C₂×D₄×C₁₃₁	—	4	2	no	—
36	C₂×D₄×C₁₃₁	—	4	4	no	—
37	C₂×D₄×C₁₃₁	—	4	4	no	—
38	C₂×D₄×C₁₃₁	M₁₆	1	4	yes	—
39	C₂²×C₄×C₁₃₁	—	2	2	no	—
40	C₂²×C₄×C₁₃₁	—	4	8	no	—
41	C₂²×C₄×C₁₃₁	—	4	4	no	—
42	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₁₃₁	—	2	8	no	—
48	C₂×C₈×C₁₃₁	—	4	4	no	—
49	C₂×C₈×C₁₃₁	—	4	4	no	—
50	C₄⋊C₄×C₁₃₁	—	4	2	no	—
51	C₄⋊C₄×C₁₃₁	—	4	4	no	—
52	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
53	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
54	C₄²×C₁₃₁	—	2	2	no	—
55	C₄²×C₁₃₁	—	2	4	no	—
56	C₄²×C₁₃₁	—	2	2	no	—
57	C₄²×C₁₃₁	—	2	4	no	—
58	C₄²×C₁₃₁	—	4	1	no	—
59	C₂³×C₁₃₁	—	2	4	no	—
60	C₂³×C₁₃₁	—	4	2	no	—
61	C₂³×C₁₃₁	—	4	2	no	—
62	C₂³×C₁₃₁	—	4	4	no	—
63	C₂³×C₁₃₁	—	4	4	no	—
64	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	D₄×C₁₃₁	—	4	4	no	—
66	D₄×C₁₃₁	—	4	4	no	—
67	D₄×C₁₃₁	—	4	16	no	—
68	C₂×C₄×C₁₃₁	—	2	2	no	—
69	C₂×C₄×C₁₃₁	—	2	2	no	—
70	C₂×C₄×C₁₃₁	—	2	1	no	—
71	C₂×C₄×C₁₃₁	—	2	1	no	—
72	C₂×C₄×C₁₃₁	—	2	2	no	—
73	C₂×C₄×C₁₃₁	—	2	2	no	—
74	C₂×C₄×C₁₃₁	—	2	2	no	—
75	C₂×C₄×C₁₃₁	—	2	1	no	—
76	C₂×C₄×C₁₃₁	—	2	2	no	—
77	C₂×C₄×C₁₃₁	—	4	2	no	—
78	C₂×C₄×C₁₃₁	—	4	4	no	—
79	C₂×C₄×C₁₃₁	—	8	8	no	—
80	C₂×C₄×C₁₃₁	#100	1	2	yes	—
81	C₂×C₄×C₁₃₁	#104	1	4	yes	—
82	C₂×C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
84	C₈×C₁₃₁	—	4	8	no	—
85	C₂²×C₁₃₁	—	4	4	no	—
86	C₂²×C₁₃₁	—	4	8	no	—
87	C₂²×C₁₃₁	—	8	2	no	—
88	C₂²×C₁₃₁	—	8	4	no	—
89	C₂²×C₁₃₁	—	8	2	no	—
90	C₂²×C₁₃₁	#407	1	1	yes	characteristic
91	C₂²×C₁₃₁	#410	1	1	yes	characteristic
92	C₂²×C₁₃₁	C₂×#100	1	1	yes	characteristic
93	C₂²×C₁₃₁	#418	1	2	yes	—
94	C₂²×C₁₃₁	C₂×#104	1	2	yes	—
95	C₄×C₁₃₁	—	2	2	no	—
96	C₄×C₁₃₁	—	2	2	no	—
97	C₄×C₁₃₁	—	2	4	no	—
98	C₄×C₁₃₁	—	2	4	no	—
99	C₄×C₁₃₁	—	4	1	no	—
100	C₄×C₁₃₁	—	8	2	no	—
101	C₂×C₁₃₁	—	8	4	no	—
102	C₂×C₁₃₁	#1466	1	1	yes	characteristic
103	C₂×C₁₃₁	#1490	1	2	yes	—
104	C₂×C₁₃₁	#1505	1	2	yes	—
105	C₂×C₁₃₁	#1525	1	1	yes	characteristic
106	C₂×C₁₃₁	#1555	1	1	yes	characteristic
107	#7637	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₁₃₁	#7637	1	1	yes	characteristic, direct product factor, Sylow
109	C₄×#127	C₂×C₁₃₁	1	1	yes	characteristic
110	#1745	C₂×C₁₃₁	1	2	yes	complemented
111	#1270	C₂×C₁₃₁	1	4	yes	complemented
112	C₂×#127	C₄×C₁₃₁	1	1	yes	characteristic
113	C₂×C₄×D₄	—	2	2	no	—
114	C₂×C₄×D₄	C₄×C₁₃₁	1	2	yes	—
115	C₂×#97	—	2	2	no	—
116	#390	—	2	2	no	—
117	#390	C₂²×C₁₃₁	1	2	yes	—
118	#384	C₄×C₁₃₁	1	1	yes	characteristic
119	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
120	#336	—	2	2	no	—
121	#336	C₂²×C₁₃₁	1	4	yes	—
122	C₂²×D₄	—	2	2	no	—
123	C₂²×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
124	C₂³×C₄	—	2	2	no	—
125	C₂²×C₈	—	2	4	no	—
126	#127	—	2	4	no	—
127	C₄×D₄	—	4	8	no	—
128	C₄×D₄	—	4	4	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
131	C₂×C₂²⋊C₄	—	2	2	no	—
132	C₂×C₂²⋊C₄	—	2	4	no	—
133	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
134	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
135	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
136	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
137	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
138	#97	—	2	8	no	—
139	C₂⁴	—	2	2	no	—
140	C₂×D₄	—	2	8	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	—	4	2	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂×D₄	—	4	4	no	—
145	C₂×D₄	—	4	4	no	—
146	C₂×D₄	M₁₆×C₁₃₁	1	4	yes	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	4	8	no	—
149	C₂²×C₄	—	4	4	no	—
150	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
151	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
152	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
153	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	4	4	no	—
157	C₂×C₈	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₂²⋊C₄	—	4	8	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	4	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	4	1	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	4	2	no	—
169	C₂³	—	4	2	no	—
170	C₂³	—	4	4	no	—
171	C₂³	—	4	4	no	—
172	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
173	D₄	—	4	4	no	—
174	D₄	—	4	4	no	—
175	D₄	—	4	16	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	8	8	no	—
188	C₂×C₄	#100×C₁₃₁	1	2	yes	—
189	C₂×C₄	#104×C₁₃₁	1	4	yes	—
190	C₂×C₄	C₂×M₁₆×C₁₃₁	1	2	yes	—
191	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
192	C₈	—	4	8	no	—
193	C₂²	—	4	4	no	—
194	C₂²	—	4	8	no	—
195	C₂²	—	8	4	no	—
196	C₂²	—	8	2	no	—
197	C₂²	—	8	2	no	—
198	C₂²	#407×C₁₃₁	1	1	yes	characteristic
199	C₂²	#410×C₁₃₁	1	1	yes	characteristic
200	C₂²	C₂×#100×C₁₃₁	1	1	yes	characteristic
201	C₂²	#418×C₁₃₁	1	2	yes	—
202	C₂²	C₂×#104×C₁₃₁	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	2	4	no	—
206	C₄	—	2	2	no	—
207	C₄	—	4	1	no	—
208	C₄	—	8	2	no	—
209	C₂	—	8	4	no	—
210	C₂	#1466×C₁₃₁	1	1	yes	characteristic
211	C₂	#1490×C₁₃₁	1	2	yes	—
212	C₂	#1505×C₁₃₁	1	2	yes	—
213	C₂	#1525×C₁₃₁	1	1	yes	characteristic
214	C₂	#1555×C₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#127×C₁₃₁

C₂

yes

maximal, characteristic

#1745×C₁₃₁

C₂

yes

maximal, complemented

#1270×C₁₃₁

C₂

yes

maximal, complemented

C₂×#127×C₁₃₁

C₄

yes

characteristic

C₂×C₄×D₄×C₁₃₁

—

C₂×C₄×D₄×C₁₃₁

C₄

yes

—

C₂×#97×C₁₃₁

—

#390×C₁₃₁

—

#390×C₁₃₁

C₂²

yes

—

#384×C₁₃₁

C₄

yes

characteristic

C₄³×C₁₃₁

C₂²

yes

characteristic

#336×C₁₃₁

—

#336×C₁₃₁

C₂²

yes

—

C₂²×D₄×C₁₃₁

—

C₂²×D₄×C₁₃₁

C₂×C₄

yes

—

C₂³×C₄×C₁₃₁

—

C₂²×C₈×C₁₃₁

—

#127×C₁₃₁

—

C₄×D₄×C₁₃₁

—

C₄×D₄×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₁₃₁

—

C₂×C₂²⋊C₄×C₁₃₁

—

C₂×C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃₁

D₄

yes

—

C₂×C₄²×C₁₃₁

D₄

yes

—

C₂×C₄²×C₁₃₁

C₂³

yes

characteristic

#97×C₁₃₁

—

C₂⁴×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

M₁₆

yes

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₂²⋊C₄×C₁₃₁

—

C₂²⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

C₂×C₂²⋊C₄

yes

characteristic

D₄×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

#100

yes

—

C₂×C₄×C₁₃₁

#104

yes

—

C₂×C₄×C₁₃₁

C₂×M₁₆

yes

—

C₂×C₄×C₁₃₁

AGL(1,ℤ/8)

yes

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

—

C₂²×C₁₃₁

C₂×#104

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

100

C₄×C₁₃₁

—

101

C₂×C₁₃₁

—

102

C₂×C₁₃₁

#1466

yes

characteristic

103

C₂×C₁₃₁

#1490

yes

—

104

C₂×C₁₃₁

#1505

yes

—

105

C₂×C₁₃₁

#1525

yes

characteristic

106

C₂×C₁₃₁

#1555

yes

characteristic

107

#7637

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

108

C₁₃₁

#7637

yes

characteristic, direct product factor, Sylow

109

C₄×#127

C₂×C₁₃₁

yes

characteristic

110

#1745

C₂×C₁₃₁

yes

complemented

111

#1270

C₂×C₁₃₁

yes

complemented

112

C₂×#127

C₄×C₁₃₁

yes

characteristic

113

C₂×C₄×D₄

—

114

C₂×C₄×D₄

C₄×C₁₃₁

yes

—

115

C₂×#97

—

116

#390

—

117

#390

C₂²×C₁₃₁

yes

—

118

#384

C₄×C₁₃₁

yes

characteristic

119

C₄³

C₂²×C₁₃₁

yes

characteristic

120

#336

—

121

#336

C₂²×C₁₃₁

yes

—

122

C₂²×D₄

—

123

C₂²×D₄

C₂×C₄×C₁₃₁

yes

—

124

C₂³×C₄

—

125

C₂²×C₈

—

126

#127

—

127

C₄×D₄

—

128

C₄×D₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

C₂×C₄×C₁₃₁

yes

—

131

C₂×C₂²⋊C₄

—

132

C₂×C₂²⋊C₄

—

133

C₂×C₄²

C₂×C₄×C₁₃₁

yes

characteristic

134

C₂×C₄²

C₂×C₄×C₁₃₁

yes

characteristic

135

C₂×C₄²

D₄×C₁₃₁

yes

—

136

C₂×C₄²

D₄×C₁₃₁

yes

—

137

C₂×C₄²

C₂³×C₁₃₁

yes

characteristic

138

#97

—

139

C₂⁴

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

M₁₆×C₁₃₁

yes

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂²⋊C₄×C₁₃₁

yes

characteristic

151

C₂²×C₄

C₂²⋊C₄×C₁₃₁

yes

characteristic

152

C₂²×C₄

C₂²⋊C₄×C₁₃₁

yes

—

153

C₂²×C₄

C₂²×C₄×C₁₃₁

yes

characteristic

154

C₂²×C₄

C₂×D₄×C₁₃₁

yes

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

C₂×C₂²⋊C₄×C₁₃₁

yes

characteristic

173

D₄

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

#100×C₁₃₁

yes

—

189

C₂×C₄

#104×C₁₃₁

yes

—

190

C₂×C₄

C₂×M₁₆×C₁₃₁

yes

—

191

C₂×C₄

AGL(1,ℤ/8)×C₁₃₁

yes

—

192

C₈

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#407×C₁₃₁

yes

characteristic

199

C₂²

#410×C₁₃₁

yes

characteristic

200

C₂²

C₂×#100×C₁₃₁

yes

characteristic

201

C₂²

#418×C₁₃₁

yes

—

202

C₂²

C₂×#104×C₁₃₁

yes

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₂

—

210

C₂

#1466×C₁₃₁

yes

characteristic

211

C₂

#1490×C₁₃₁

yes

—

212

C₂

#1505×C₁₃₁

yes

—

213

C₂

#1525×C₁₃₁

yes

characteristic

214

C₂

#1555×C₁₃₁

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
prime radical	C₂×#127×C₁₃₁	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₃₁

C₂×C₂²⋊C₄

commutator

C₂²×C₄

C₂²×C₄×C₁₃₁

Frattini, cosocle

C₂×C₄²

C₂³×C₁₃₁

prime radical

C₂×#127×C₁₃₁

C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

N	H ≅ G/N	type
2. #1745×C₁₃₁	C₂	semidirect
3. #1270×C₁₃₁	C₂	semidirect
107. #7637	C₁₃₁	direct (N × H)
108. C₁₃₁	#7637	direct (N × H)
110. #1745	C₂×C₁₃₁	semidirect
111. #1270	C₂×C₁₃₁	semidirect

H ≅ G/N

type

2. #1745×C₁₃₁

C₂

semidirect

3. #1270×C₁₃₁

C₂

semidirect

107. #7637

C₁₃₁

direct (N × H)

108. C₁₃₁

#7637

direct (N × H)

110. #1745

C₂×C₁₃₁

semidirect

111. #1270

C₂×C₁₃₁

semidirect

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

subgroup	index
C₄×#127×C₁₃₁	2
#1745×C₁₃₁	2
#1270×C₁₃₁	2
#7637	131

subgroup

index

C₄×#127×C₁₃₁

#1745×C₁₃₁

#1270×C₁₃₁

#7637

131

Subgroups — C₁₃₁×#7637 #45693521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C131×#7637 #45693521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7637 #45693521