Group #26115317 — subgroups

Subgroups — #26115317

Order 12032.

478 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 163 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	#20251766	C₂	1	1	yes	maximal, characteristic
2	#20243505	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688871	C₂	1	2	yes	maximal
4	C₂×#15688868	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	C₂×#13094024	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	#15689029	—	2	2	no	—
10	#15689029	C₄	1	4	yes	—
11	#15688989	—	2	1	no	—
12	#15688989	—	2	1	no	—
13	#15688878	—	2	2	no	—
14	#15688871	—	2	4	no	—
15	#15688868	—	2	4	no	—
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	—	2	1	no	—
18	C₄³×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#336	—	2	1	no	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	#13094024	—	2	2	no	—
24	#13094024	—	2	2	no	—
25	#13094024	—	2	2	no	—
26	#13094024	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	no	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
31	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
33	C₄₇⋊#95	—	2	4	no	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₂²×C₈×C₄₇	—	2	2	no	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₄₇	D₄	1	2	yes	—
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₇	—	2	2	no	—
42	C₄⋊C₈×C₄₇	D₄	1	4	yes	—
43	C₄×C₈×C₄₇	—	4	2	no	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₄₇	—	2	2	no	—
50	C₂²×Dic₄₇	—	2	2	no	—
51	C₂×Dic₉₄	—	2	4	no	—
52	C₂×Dic₉₄	—	4	1	no	—
53	C₂×Dic₉₄	—	4	1	no	—
54	C₂×Dic₉₄	—	4	4	no	—
55	C₂×Dic₉₄	—	4	4	no	—
56	C₂×C₈×C₄₇	—	2	4	no	—
57	C₂×C₈×C₄₇	—	4	2	no	—
58	C₂×C₈×C₄₇	—	4	2	no	—
59	C₄²×C₄₇	—	2	4	no	—
60	C₄²×C₄₇	—	2	2	no	—
61	C₄²×C₄₇	—	4	2	no	—
62	C₄²×C₄₇	SD₁₆	1	2	yes	—
63	C₄²×C₄₇	Q₁₆	1	2	yes	—
64	C₄²×C₄₇	C₂×D₄	1	2	yes	—
65	C₄²×C₄₇	#41	1	2	yes	—
66	C₄⋊Dic₄₇	—	2	4	no	—
67	C₄⋊Dic₄₇	—	4	2	no	—
68	C₄⋊Dic₄₇	—	4	2	no	—
69	C₄⋊Dic₄₇	—	4	1	no	—
70	C₄⋊Dic₄₇	—	4	1	no	—
71	C₂×C₄₇⋊C₈	—	4	2	no	—
72	C₂×C₄₇⋊C₈	—	4	4	no	—
73	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	—	2	2	no	—
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	#103	1	4	yes	—
81	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₄₇	#121	1	2	yes	—
83	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
84	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₄₇	#135	1	4	yes	—
88	C₂×C₄×C₄₇	#137	1	2	yes	—
89	C₂×Dic₄₇	—	4	2	no	—
90	C₂×Dic₄₇	—	4	2	no	—
91	C₂×Dic₄₇	—	4	4	no	—
92	C₂×Dic₄₇	—	4	4	no	—
93	Dic₉₄	—	8	2	no	—
94	Dic₉₄	—	8	2	no	—
95	Dic₉₄	—	8	4	no	—
96	C₈×C₄₇	—	4	4	no	—
97	C₄₇⋊C₈	—	8	2	no	—
98	#9693	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	#390	1	1	yes	characteristic
100	C₂²×C₄₇	#413	1	2	yes	—
101	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
102	C₂²×C₄₇	#416	1	2	yes	—
103	C₂²×C₄₇	#419	1	1	yes	characteristic
104	C₄×C₄₇	—	2	4	no	—
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	—	4	2	no	—
107	C₄×C₄₇	#461	1	2	yes	—
108	C₄×C₄₇	#462	1	2	yes	—
109	C₄×C₄₇	#488	1	2	yes	—
110	C₄×C₄₇	#495	1	2	yes	—
111	Dic₄₇	—	8	2	no	—
112	Dic₄₇	—	8	2	no	—
113	#2862	—	47	1	no	—
114	#1831	—	47	1	no	—
115	C₄×C₄⋊C₈	D₄₇	1	1	yes	characteristic
116	C₂×#333	—	47	2	no	—
117	C₂×#332	—	47	2	no	—
118	C₂×C₄₇	#1542	1	2	yes	—
119	C₂×C₄₇	#1554	1	2	yes	—
120	C₂×C₄₇	#1958	1	1	yes	characteristic
121	C₂×C₄₇	#1967	1	1	yes	characteristic
122	C₂×C₄₇	#1981	1	1	yes	characteristic
123	C₂×C₄⋊Q₈	—	47	1	no	—
124	C₂×C₄⋊Q₈	—	47	1	no	—
125	C₂×C₄⋊C₈	—	47	2	no	—
126	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
127	C₂×C₄×C₈	—	2	1	no	—
128	#391	—	47	4	no	—
129	#391	—	94	2	no	—
130	#382	—	94	1	no	—
131	#382	—	94	1	no	—
132	C₄³	C₂×D₄₇	1	1	yes	characteristic
133	#336	—	2	1	no	—
134	#336	—	94	2	no	—
135	#333	—	94	4	no	—
136	#332	—	94	4	no	—
137	C₄₇	#9693	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	47	2	no	—
139	C₂²×Q₈	—	94	1	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	94	2	no	—
142	C₄⋊Q₈	—	94	2	no	—
143	C₄⋊Q₈	—	94	2	no	—
144	C₄⋊Q₈	—	94	2	no	—
145	C₄⋊Q₈	—	94	2	no	—
146	C₂×C₄⋊C₄	—	47	2	no	—
147	C₂×C₄⋊C₄	—	94	1	no	—
148	C₂×C₄⋊C₄	—	94	1	no	—
149	C₂×C₄⋊C₄	—	94	1	no	—
150	C₂×C₄²	C₄×D₄₇	1	2	yes	—
151	C₂×C₄²	D₁₈₈	1	2	yes	—
152	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	94	2	no	—
157	C₄⋊C₈	—	94	2	no	—
158	C₄⋊C₈	D₁₈₈	1	4	yes	—
159	C₄×C₈	—	4	2	no	—
160	#95	—	94	4	no	—
161	#95	—	94	4	no	—
162	C₂×Q₈	—	94	4	no	—
163	C₂×Q₈	—	188	4	no	—
164	C₂×Q₈	—	188	1	no	—
165	C₂×Q₈	—	188	1	no	—
166	C₂×Q₈	—	188	4	no	—
167	C₂²×C₄	—	94	2	no	—
168	C₂²×C₄	—	94	2	no	—
169	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
170	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
171	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	188	2	no	—
178	C₂×C₈	—	188	4	no	—
179	C₄⋊C₄	—	94	4	no	—
180	C₄⋊C₄	—	188	2	no	—
181	C₄⋊C₄	—	188	2	no	—
182	C₄⋊C₄	—	188	1	no	—
183	C₄⋊C₄	—	188	1	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	#10625335	1	2	yes	—
188	C₄²	#10625337	1	2	yes	—
189	C₄²	C₂×D₁₈₈	1	2	yes	—
190	C₄²	#10625350	1	2	yes	—
191	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
192	Q₈	—	376	4	no	—
193	Q₈	—	376	2	no	—
194	Q₈	—	376	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	188	2	no	—
202	C₂×C₄	—	188	2	no	—
203	C₂×C₄	—	188	4	no	—
204	C₂×C₄	—	188	4	no	—
205	C₂×C₄	#13093947	1	4	yes	—
206	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
207	C₂×C₄	#13093998	1	2	yes	—
208	C₂×C₄	#13094012	1	1	yes	characteristic
209	C₂×C₄	#13094021	1	1	yes	characteristic
210	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
211	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
212	C₂×C₄	#13094046	1	4	yes	—
213	C₂×C₄	#13094054	1	2	yes	—
214	C₈	—	4	4	no	—
215	C₈	—	376	2	no	—
216	C₂²	#15689025	1	1	yes	characteristic
217	C₂²	#15689105	1	2	yes	—
218	C₂²	C₂×#13093947	1	1	yes	characteristic
219	C₂²	#15689120	1	2	yes	—
220	C₂²	#15689131	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	—	376	2	no	—
225	C₄	—	376	2	no	—
226	C₄	#15689328	1	2	yes	—
227	C₄	#15689333	1	2	yes	—
228	C₄	#15689478	1	2	yes	—
229	C₄	#15689507	1	2	yes	—
230	C₂	#20242046	1	2	yes	—
231	C₂	#20242096	1	2	yes	—
232	C₂	#20244199	1	1	yes	characteristic
233	C₂	#20244237	1	1	yes	characteristic
234	C₂	#20244311	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20251766

C₂

yes

maximal, characteristic

#20243505

C₂

yes

maximal, characteristic

C₂×#15688871

C₂

yes

maximal

C₂×#15688868

C₂

yes

maximal

C₄×C₄⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂×#13094024

C₂²

yes

characteristic

C₂×#13094024

C₂²

yes

characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#15689029

—

#15689029

C₄

yes

—

—

—

—

—

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×Dic₉₄

—

C₂²×Dic₉₄

C₂×C₄

yes

—

C₂²×C₄₇⋊C₈

—

—

—

—

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

100

C₂²×C₄₇

#413

yes

—

101

C₂²×C₄₇

C₂×#103

yes

characteristic

102

C₂²×C₄₇

#416

yes

—

103

C₂²×C₄₇

#419

yes

characteristic

104

C₄×C₄₇

—

105

C₄×C₄₇

—

106

C₄×C₄₇

—

107

C₄×C₄₇

#461

yes

—

108

C₄×C₄₇

#462

yes

—

109

C₄×C₄₇

#488

yes

—

110

C₄×C₄₇

#495

yes

—

111

Dic₄₇

—

112

Dic₄₇

—

113

#2862

—

114

#1831

—

115

C₄×C₄⋊C₈

D₄₇

yes

characteristic

116

C₂×#333

—

117

C₂×#332

—

118

C₂×C₄₇

#1542

yes

—

119

C₂×C₄₇

#1554

yes

—

120

C₂×C₄₇

#1958

yes

characteristic

121

C₂×C₄₇

#1967

yes

characteristic

122

C₂×C₄₇

#1981

yes

characteristic

123

C₂×C₄⋊Q₈

—

124

C₂×C₄⋊Q₈

—

125

C₂×C₄⋊C₈

—

126

C₂×C₄⋊C₈

C₂×D₄₇

yes

—

127

C₂×C₄×C₈

—

128

#391

—

129

#391

—

130

#382

—

131

#382

—

132

C₄³

C₂×D₄₇

yes

characteristic

133

#336

—

134

#336

—

135

#333

—

136

#332

—

137

C₄₇

#9693

yes

characteristic, complemented, Sylow

138

C₂²×Q₈

—

139

C₂²×Q₈

—

140

C₂²×C₈

—

141

C₂²×C₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄²

C₄×D₄₇

yes

—

151

C₂×C₄²

D₁₈₈

yes

—

152

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

153

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

154

C₂×C₄²

C₂²×D₄₇

yes

characteristic

155

C₄⋊C₈

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

158

C₄⋊C₈

D₁₈₈

yes

—

159

C₄×C₈

—

160

#95

—

161

#95

—

162

C₂×Q₈

—

163

C₂×Q₈

—

188

—

164

C₂×Q₈

—

188

—

165

C₂×Q₈

—

188

—

166

C₂×Q₈

—

188

—

167

C₂²×C₄

—

168

C₂²×C₄

—

169

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

170

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

171

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

172

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

173

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

188

—

178

C₂×C₈

—

188

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

188

—

181

C₄⋊C₄

—

188

—

182

C₄⋊C₄

—

188

—

183

C₄⋊C₄

—

188

—

184

C₄²

—

185

C₄²

—

186

C₄²

—

187

C₄²

#10625335

yes

—

188

C₄²

#10625337

yes

—

189

C₄²

C₂×D₁₈₈

yes

—

190

C₄²

#10625350

yes

—

191

C₂³

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

192

Q₈

—

376

—

193

Q₈

—

376

—

194

Q₈

—

376

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

188

—

202

C₂×C₄

—

188

—

203

C₂×C₄

—

188

—

204

C₂×C₄

—

188

—

205

C₂×C₄

#13093947

yes

—

206

C₂×C₄

C₄×D₁₈₈

yes

—

207

C₂×C₄

#13093998

yes

—

208

C₂×C₄

#13094012

yes

characteristic

209

C₂×C₄

#13094021

yes

characteristic

210

C₂×C₄

C₂×#10625335

yes

characteristic

211

C₂×C₄

C₂×#10625337

yes

characteristic

212

C₂×C₄

#13094046

yes

—

213

C₂×C₄

#13094054

yes

—

214

C₈

—

215

C₈

—

376

—

216

C₂²

#15689025

yes

characteristic

217

C₂²

#15689105

yes

—

218

C₂²

C₂×#13093947

yes

characteristic

219

C₂²

#15689120

yes

—

220

C₂²

#15689131

yes

characteristic

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

—

376

—

225

C₄

—

376

—

226

C₄

#15689328

yes

—

227

C₄

#15689333

yes

—

228

C₄

#15689478

yes

—

229

C₄

#15689507

yes

—

230

C₂

#20242046

yes

—

231

C₂

#20242096

yes

—

232

C₂

#20244199

yes

characteristic

233

C₂

#20244237

yes

characteristic

234

C₂

#20244311

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#9693
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄×C₄⋊C₈×C₄₇

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

hypercenter

C₄×C₄⋊C₈

D₄₇

hypocenter

C₄₇

#9693

LCS

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

N	H ≅ G/N	type
137. C₄₇	#9693	semidirect

H ≅ G/N

type

137. C₄₇

#9693

semidirect

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

subgroup	index
#20251766	2
#20243505	2
C₂×#15688871	2
C₂×#15688868	2
C₄×C₄⋊C₈×C₄₇	2
#9693	47

subgroup

index

C₄×C₄⋊C₈×C₄₇

Subgroups — #26115317

Conjugacy classes of subgroups

Complements

Maximal subgroups