Group #45699419 — subgroups

Subgroups — C₁₃₁×#13535 #45699419

Order 33536.

354 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₂×#365×C₁₃₁	C₂	1	2	yes	maximal, complemented
2	C₂×#361×C₁₃₁	C₂	1	2	yes	maximal
3	#1972×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1849×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1377×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
8	C₂×#108×C₁₃₁	—	2	1	no	—
9	C₂×#108×C₁₃₁	C₂²	1	2	yes	—
10	C₂×#106×C₁₃₁	C₂²	1	1	yes	characteristic
11	#423×C₁₃₁	—	2	1	no	—
12	C₂×#102×C₁₃₁	—	2	1	no	—
13	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
14	#390×C₁₃₁	—	2	1	no	—
15	#365×C₁₃₁	—	2	4	no	—
16	#361×C₁₃₁	—	2	4	no	—
17	C₂²×D₄×C₁₃₁	—	2	1	no	—
18	C₂×D₈×C₁₃₁	—	2	2	no	—
19	C₂×D₈×C₁₃₁	—	2	2	no	—
20	C₂×D₈×C₁₃₁	—	2	1	no	—
21	C₂×D₈×C₁₃₁	Q₈	1	2	yes	—
22	C₂×M₁₆×C₁₃₁	—	2	2	no	—
23	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
26	C₂×C₂²⋊C₄×C₁₃₁	—	4	2	no	—
27	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
28	M₃₂×C₁₃₁	—	2	2	no	—
29	M₃₂×C₁₃₁	D₄	1	4	yes	—
30	C₂×C₁₆×C₁₃₁	—	2	2	no	—
31	#108×C₁₃₁	—	2	2	no	—
32	#108×C₁₃₁	—	2	2	no	—
33	#108×C₁₃₁	—	4	2	no	—
34	#106×C₁₃₁	—	2	2	no	—
35	C₄⋊C₈×C₁₃₁	—	4	2	no	—
36	#102×C₁₃₁	—	4	2	no	—
37	#97×C₁₃₁	—	2	2	no	—
38	C₂⁴×C₁₃₁	—	4	1	no	—
39	C₂×D₄×C₁₃₁	—	2	2	no	—
40	C₂×D₄×C₁₃₁	—	4	1	no	—
41	C₂×D₄×C₁₃₁	—	4	1	no	—
42	C₂×D₄×C₁₃₁	—	4	1	no	—
43	C₂×D₄×C₁₃₁	—	4	1	no	—
44	C₂×D₄×C₁₃₁	—	4	1	no	—
45	C₂²×C₄×C₁₃₁	—	4	1	no	—
46	C₂²×C₄×C₁₃₁	SD₁₆	1	2	yes	—
47	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
48	D₈×C₁₃₁	—	2	4	no	—
49	D₈×C₁₃₁	—	4	2	no	—
50	M₁₆×C₁₃₁	—	4	2	no	—
51	M₁₆×C₁₃₁	—	4	2	no	—
52	C₂×C₈×C₁₃₁	—	2	1	no	—
53	C₂×C₈×C₁₃₁	—	2	1	no	—
54	C₂×C₈×C₁₃₁	—	4	2	no	—
55	C₂×C₈×C₁₃₁	SD₁₆	1	2	yes	—
56	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₁₃₁	#41	1	2	yes	—
59	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₁₃₁	—	4	1	no	—
61	C₄⋊C₄×C₁₃₁	—	4	1	no	—
62	C₂²⋊C₄×C₁₃₁	—	8	4	no	—
63	C₄²×C₁₃₁	—	2	1	no	—
64	C₄²×C₁₃₁	—	2	1	no	—
65	C₁₆×C₁₃₁	—	2	4	no	—
66	C₂³×C₁₃₁	—	4	2	no	—
67	C₂³×C₁₃₁	—	8	2	no	—
68	C₂³×C₁₃₁	—	8	2	no	—
69	C₂³×C₁₃₁	—	8	1	no	—
70	C₂³×C₁₃₁	—	8	1	no	—
71	C₂³×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
72	D₄×C₁₃₁	—	4	2	no	—
73	D₄×C₁₃₁	—	4	2	no	—
74	D₄×C₁₃₁	—	8	1	no	—
75	D₄×C₁₃₁	—	8	1	no	—
76	C₂×C₄×C₁₃₁	—	2	2	no	—
77	C₂×C₄×C₁₃₁	—	4	2	no	—
78	C₂×C₄×C₁₃₁	—	8	2	no	—
79	C₂×C₄×C₁₃₁	—	8	1	no	—
80	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃₁	#123	1	1	yes	characteristic
82	C₂×C₄×C₁₃₁	#127	1	1	yes	characteristic
83	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₈×C₁₃₁	—	2	2	no	—
87	C₈×C₁₃₁	—	8	2	no	—
88	C₈×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
89	C₈×C₁₃₁	#123	1	2	yes	—
90	C₂²×C₁₃₁	—	2	2	no	—
91	C₂²×C₁₃₁	—	8	4	no	—
92	C₂²×C₁₃₁	—	8	2	no	—
93	C₂²×C₁₃₁	—	16	1	no	—
94	C₂²×C₁₃₁	—	16	1	no	—
95	C₂²×C₁₃₁	—	16	2	no	—
96	C₂²×C₁₃₁	#476	1	1	yes	characteristic
97	C₂²×C₁₃₁	#481	1	1	yes	characteristic
98	C₂²×C₁₃₁	#492	1	1	yes	characteristic
99	C₄×C₁₃₁	—	4	2	no	—
100	C₄×C₁₃₁	—	16	1	no	—
101	C₄×C₁₃₁	#399	1	1	yes	characteristic
102	C₄×C₁₃₁	#476	1	1	yes	characteristic
103	C₄×C₁₃₁	#481	1	1	yes	characteristic
104	C₄×C₁₃₁	#496	1	1	yes	characteristic
105	C₂×C₁₃₁	—	2	1	no	—
106	C₂×C₁₃₁	—	2	1	no	—
107	C₂×C₁₃₁	—	16	2	no	—
108	C₂×C₁₃₁	#2094	1	1	yes	characteristic
109	C₂×C₁₃₁	#2238	1	2	yes	—
110	#13535	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₁₃₁	#13535	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×#365	C₂×C₁₃₁	1	2	yes	complemented
113	C₂×#361	C₂×C₁₃₁	1	2	yes	—
114	#1972	C₂×C₁₃₁	1	1	yes	characteristic
115	#1849	C₂×C₁₃₁	1	1	yes	characteristic, complemented
116	#1377	C₂×C₁₃₁	1	1	yes	characteristic, complemented
117	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
118	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
119	C₂×#108	—	2	1	no	—
120	C₂×#108	C₂²×C₁₃₁	1	2	yes	—
121	C₂×#106	C₂²×C₁₃₁	1	1	yes	characteristic
122	#423	—	2	1	no	—
123	C₂×#102	—	2	1	no	—
124	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
125	#390	—	2	1	no	—
126	#365	—	2	4	no	—
127	#361	—	2	4	no	—
128	C₂²×D₄	—	2	1	no	—
129	C₂×D₈	—	2	2	no	—
130	C₂×D₈	—	2	2	no	—
131	C₂×D₈	—	2	1	no	—
132	C₂×D₈	Q₈×C₁₃₁	1	2	yes	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₂²⋊C₄	—	4	2	no	—
138	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
139	M₃₂	—	2	2	no	—
140	M₃₂	D₄×C₁₃₁	1	4	yes	—
141	C₂×C₁₆	—	2	2	no	—
142	#108	—	2	2	no	—
143	#108	—	2	2	no	—
144	#108	—	4	2	no	—
145	#106	—	2	2	no	—
146	C₄⋊C₈	—	4	2	no	—
147	#102	—	4	2	no	—
148	#97	—	2	2	no	—
149	C₂⁴	—	4	1	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	4	1	no	—
152	C₂×D₄	—	4	1	no	—
153	C₂×D₄	—	4	1	no	—
154	C₂×D₄	—	4	1	no	—
155	C₂×D₄	—	4	1	no	—
156	C₂²×C₄	—	4	1	no	—
157	C₂²×C₄	SD₁₆×C₁₃₁	1	2	yes	—
158	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
159	D₈	—	2	4	no	—
160	D₈	—	4	2	no	—
161	M₁₆	—	4	2	no	—
162	M₁₆	—	4	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	SD₁₆×C₁₃₁	1	2	yes	—
167	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
168	C₂×C₈	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
169	C₂×C₈	#41×C₁₃₁	1	2	yes	—
170	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
171	C₄⋊C₄	—	4	1	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₂²⋊C₄	—	8	4	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₁₆	—	2	4	no	—
177	C₂³	—	4	2	no	—
178	C₂³	—	8	2	no	—
179	C₂³	—	8	2	no	—
180	C₂³	—	8	1	no	—
181	C₂³	—	8	1	no	—
182	C₂³	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
183	D₄	—	4	2	no	—
184	D₄	—	4	2	no	—
185	D₄	—	8	1	no	—
186	D₄	—	8	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
192	C₂×C₄	#123×C₁₃₁	1	1	yes	characteristic
193	C₂×C₄	#127×C₁₃₁	1	1	yes	characteristic
194	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
195	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
196	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	—	8	2	no	—
199	C₈	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
200	C₈	#123×C₁₃₁	1	2	yes	—
201	C₂²	—	2	2	no	—
202	C₂²	—	8	4	no	—
203	C₂²	—	8	2	no	—
204	C₂²	—	16	1	no	—
205	C₂²	—	16	1	no	—
206	C₂²	—	16	2	no	—
207	C₂²	#476×C₁₃₁	1	1	yes	characteristic
208	C₂²	#481×C₁₃₁	1	1	yes	characteristic
209	C₂²	#492×C₁₃₁	1	1	yes	characteristic
210	C₄	—	4	2	no	—
211	C₄	—	16	1	no	—
212	C₄	#399×C₁₃₁	1	1	yes	characteristic
213	C₄	#476×C₁₃₁	1	1	yes	characteristic
214	C₄	#481×C₁₃₁	1	1	yes	characteristic
215	C₄	#496×C₁₃₁	1	1	yes	characteristic
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	—	16	2	no	—
219	C₂	#2094×C₁₃₁	1	1	yes	characteristic
220	C₂	#2238×C₁₃₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#365×C₁₃₁

C₂

yes

maximal, complemented

C₂×#361×C₁₃₁

C₂

yes

maximal

#1972×C₁₃₁

C₂

yes

maximal, characteristic

#1849×C₁₃₁

C₂

yes

maximal, characteristic, complemented

#1377×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₁₃₁

C₂²

yes

characteristic

C₂×M₃₂×C₁₃₁

C₂²

yes

—

C₂×#108×C₁₃₁

—

C₂×#108×C₁₃₁

C₂²

yes

—

C₂×#106×C₁₃₁

C₂²

yes

characteristic

#423×C₁₃₁

—

C₂×#102×C₁₃₁

—

C₂×#97×C₁₃₁

C₂²

yes

characteristic

#390×C₁₃₁

—

#365×C₁₃₁

—

#361×C₁₃₁

—

C₂²×D₄×C₁₃₁

—

C₂×D₈×C₁₃₁

—

C₂×D₈×C₁₃₁

—

C₂×D₈×C₁₃₁

—

C₂×D₈×C₁₃₁

Q₈

yes

—

C₂×M₁₆×C₁₃₁

—

C₂²×C₈×C₁₃₁

D₄

yes

characteristic

C₂²×C₈×C₁₃₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₂²⋊C₄×C₁₃₁

—

C₂×C₄²×C₁₃₁

D₄

yes

characteristic

M₃₂×C₁₃₁

—

M₃₂×C₁₃₁

D₄

yes

—

C₂×C₁₆×C₁₃₁

—

#108×C₁₃₁

—

#108×C₁₃₁

—

#108×C₁₃₁

—

#106×C₁₃₁

—

C₄⋊C₈×C₁₃₁

—

#102×C₁₃₁

—

#97×C₁₃₁

—

C₂⁴×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

SD₁₆

yes

—

C₂²×C₄×C₁₃₁

C₂×D₄

yes

characteristic

D₈×C₁₃₁

—

D₈×C₁₃₁

—

M₁₆×C₁₃₁

—

M₁₆×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

SD₁₆

yes

—

C₂×C₈×C₁₃₁

C₂×D₄

yes

—

C₂×C₈×C₁₃₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃₁

#41

yes

—

C₂×C₈×C₁₃₁

#41

yes

characteristic

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₂²⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

—

C₁₆×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

C₂×SD₁₆

yes

characteristic

D₄×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂²⋊Q₈

yes

C₂×C₄×C₁₃₁

yes

C₂×C₄×C₁₃₁

yes

C₂×C₄×C₁₃₁

C₂×SD₁₆

yes

C₂×C₄×C₁₃₁

yes

C₂×C₄×C₁₃₁

yes

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

C₂²⋊Q₈

yes

—

C₈×C₁₃₁

#123

yes

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₄×C₁₃₁

—

100

C₄×C₁₃₁

—

101

C₄×C₁₃₁

#399

yes

characteristic

102

C₄×C₁₃₁

#476

yes

characteristic

103

C₄×C₁₃₁

#481

yes

characteristic

104

C₄×C₁₃₁

#496

yes

characteristic

105

C₂×C₁₃₁

—

106

C₂×C₁₃₁

—

107

C₂×C₁₃₁

—

108

C₂×C₁₃₁

#2094

yes

characteristic

109

C₂×C₁₃₁

#2238

yes

—

110

#13535

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

111

C₁₃₁

#13535

yes

characteristic, direct product factor, Sylow

112

C₂×#365

C₂×C₁₃₁

yes

complemented

113

C₂×#361

C₂×C₁₃₁

yes

—

114

#1972

C₂×C₁₃₁

yes

characteristic

115

#1849

C₂×C₁₃₁

yes

characteristic, complemented

116

#1377

C₂×C₁₃₁

yes

characteristic, complemented

117

C₂²×D₈

C₂²×C₁₃₁

yes

characteristic

118

C₂×M₃₂

C₂²×C₁₃₁

yes

—

119

C₂×#108

—

120

C₂×#108

C₂²×C₁₃₁

yes

—

121

C₂×#106

C₂²×C₁₃₁

yes

characteristic

122

#423

—

123

C₂×#102

—

124

C₂×#97

C₂²×C₁₃₁

yes

characteristic

125

#390

—

126

#365

—

127

#361

—

128

C₂²×D₄

—

129

C₂×D₈

—

130

C₂×D₈

—

131

C₂×D₈

—

132

C₂×D₈

Q₈×C₁₃₁

yes

—

133

C₂×M₁₆

—

134

C₂²×C₈

D₄×C₁₃₁

yes

characteristic

135

C₂²×C₈

C₂³×C₁₃₁

yes

characteristic

136

C₂×C₄⋊C₄

—

137

C₂×C₂²⋊C₄

—

138

C₂×C₄²

D₄×C₁₃₁

yes

characteristic

139

M₃₂

—

140

M₃₂

D₄×C₁₃₁

yes

—

141

C₂×C₁₆

—

142

#108

—

143

#108

—

144

#108

—

145

#106

—

146

C₄⋊C₈

—

147

#102

—

148

#97

—

149

C₂⁴

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂²×C₄

—

157

C₂²×C₄

SD₁₆×C₁₃₁

yes

—

158

C₂²×C₄

C₂×D₄×C₁₃₁

yes

characteristic

159

D₈

—

160

D₈

—

161

M₁₆

—

162

M₁₆

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

SD₁₆×C₁₃₁

yes

—

167

C₂×C₈

C₂×D₄×C₁₃₁

yes

—

168

C₂×C₈

C₂×Q₈×C₁₃₁

yes

characteristic

169

C₂×C₈

#41×C₁₃₁

yes

—

170

C₂×C₈

#41×C₁₃₁

yes

characteristic

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₂²⋊C₄

—

174

C₄²

—

175

C₄²

—

176

C₁₆

—

177

C₂³

—

178

C₂³

—

179

C₂³

—

180

C₂³

—

181

C₂³

—

182

C₂³

C₂×SD₁₆×C₁₃₁

yes

characteristic

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

C₂²⋊Q₈×C₁₃₁

yes

characteristic

192

C₂×C₄

#123×C₁₃₁

yes

characteristic

193

C₂×C₄

#127×C₁₃₁

yes

characteristic

194

C₂×C₄

C₂×SD₁₆×C₁₃₁

yes

characteristic

195

C₂×C₄

AGL(1,ℤ/8)×C₁₃₁

yes

characteristic

196

C₂×C₄

AGL(1,ℤ/8)×C₁₃₁

yes

characteristic

197

C₈

—

198

C₈

—

199

C₈

C₂²⋊Q₈×C₁₃₁

yes

—

200

C₈

#123×C₁₃₁

yes

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

#476×C₁₃₁

yes

characteristic

208

C₂²

#481×C₁₃₁

yes

characteristic

209

C₂²

#492×C₁₃₁

yes

characteristic

210

C₄

—

211

C₄

—

212

C₄

#399×C₁₃₁

yes

characteristic

213

C₄

#476×C₁₃₁

yes

characteristic

214

C₄

#481×C₁₃₁

yes

characteristic

215

C₄

#496×C₁₃₁

yes

characteristic

216

C₂

—

217

C₂

—

218

C₂

—

219

C₂

#2094×C₁₃₁

yes

characteristic

220

C₂

#2238×C₁₃₁

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂²×D₈×C₁₃₁	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×#97×C₁₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃₁

#481

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₃₁

prime radical

C₂²×D₈×C₁₃₁

C₂²

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×#97×C₁₃₁ ◁ G

N	H ≅ G/N	type
1. C₂×#365×C₁₃₁	C₂	semidirect
4. #1849×C₁₃₁	C₂	semidirect
5. #1377×C₁₃₁	C₂	semidirect
110. #13535	C₁₃₁	direct (N × H)
111. C₁₃₁	#13535	direct (N × H)
112. C₂×#365	C₂×C₁₃₁	semidirect
115. #1849	C₂×C₁₃₁	semidirect
116. #1377	C₂×C₁₃₁	semidirect

H ≅ G/N

type

1. C₂×#365×C₁₃₁

C₂

semidirect

4. #1849×C₁₃₁

C₂

semidirect

5. #1377×C₁₃₁

C₂

semidirect

110. #13535

C₁₃₁

direct (N × H)

111. C₁₃₁

#13535

direct (N × H)

112. C₂×#365

C₂×C₁₃₁

semidirect

115. #1849

C₂×C₁₃₁

semidirect

116. #1377

C₂×C₁₃₁

semidirect

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

subgroup	index
C₂×#365×C₁₃₁	2
C₂×#361×C₁₃₁	2
#1972×C₁₃₁	2
#1849×C₁₃₁	2
#1377×C₁₃₁	2
#13535	131

subgroup

index

C₂×#365×C₁₃₁

C₂×#361×C₁₃₁

#1972×C₁₃₁

#1849×C₁₃₁

#1377×C₁₃₁

#13535

131

Subgroups — C₁₃₁×#13535 #45699419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C131×#13535 #45699419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13535 #45699419