Group #26116088 — subgroups

Subgroups — #26116088

Order 12032.

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

normal subgroups only

This group has 694 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	C₂×#15689500	C₂	1	1	yes	maximal, characteristic
2	#20244193	C₂	1	2	yes	maximal, complemented
3	#20244193	C₂	1	2	yes	maximal, complemented
4	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₇₆	—	2	2	no	—
7	C₂×#13094021	C₂²	1	2	yes	—
8	#15689500	—	2	2	no	—
9	#15689500	C₄	1	4	yes	complemented
10	C₂×#13093944	—	2	4	no	—
11	C₂×#13093944	—	2	4	no	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	#15688845	—	2	2	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#384	C₂²	1	2	yes	—
17	C₄₇×#340	—	2	2	no	—
18	C₂²×D₁₈₈	—	2	2	no	—
19	C₂²×D₁₈₈	—	2	2	no	—
20	C₂²×D₁₈₈	—	2	2	no	—
21	C₂×D₃₇₆	—	2	8	no	—
22	C₂×D₃₇₆	—	4	8	no	—
23	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
24	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
25	#13094021	—	2	2	no	—
26	#13094021	—	2	2	no	—
27	#13094021	—	2	2	no	—
28	#13094021	C₂×C₄	1	4	yes	—
29	#13093944	—	4	8	no	—
30	#13093944	—	4	8	no	—
31	C₄×C₄₇⋊C₈	—	2	1	no	—
32	C₄×C₄₇⋊C₈	—	2	1	no	—
33	C₄×C₄₇⋊C₈	—	2	1	no	—
34	C₄×C₄₇⋊C₈	—	2	1	no	—
35	C₂²×C₈×C₄₇	D₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₄₇	D₄	1	2	yes	—
42	C₂³×D₄₇	—	4	2	no	—
43	C₂²×C₄×C₄₇	—	4	2	no	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×D₁₈₈	—	2	4	no	—
48	C₂×D₁₈₈	—	2	4	no	—
49	C₂×D₁₈₈	—	2	4	no	—
50	C₂×D₁₈₈	—	4	4	no	—
51	C₂×D₁₈₈	—	4	4	no	—
52	C₂×D₁₈₈	—	4	4	no	—
53	C₂×D₁₈₈	—	4	4	no	—
54	C₂×D₁₈₈	—	4	4	no	—
55	C₂×D₁₈₈	—	4	4	no	—
56	C₂×D₁₈₈	—	4	2	no	—
57	C₂×D₁₈₈	—	4	2	no	—
58	C₂×D₁₈₈	—	4	2	no	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
61	C₄⋊C₄×C₄₇	—	4	4	no	—
62	C₄⋊C₄×C₄₇	—	4	4	no	—
63	C₄⋊C₄×C₄₇	—	4	4	no	—
64	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₄₇	#41	1	1	yes	characteristic
67	C₄²×C₄₇	#41	1	1	yes	characteristic
68	C₂×C₄₇⋊C₈	—	2	2	no	—
69	C₂×C₄₇⋊C₈	—	2	2	no	—
70	C₂×C₄₇⋊C₈	—	2	2	no	—
71	C₂×C₄₇⋊C₈	—	2	4	no	—
72	C₂×C₄₇⋊C₈	D₈	1	4	yes	complemented
73	D₃₇₆	—	4	16	no	—
74	C₂³×C₄₇	#127	1	1	yes	characteristic
75	C₂²×D₄₇	—	4	4	no	—
76	C₂²×D₄₇	—	8	2	no	—
77	C₂²×D₄₇	—	8	2	no	—
78	C₂²×D₄₇	—	8	2	no	—
79	C₂²×D₄₇	—	8	2	no	—
80	C₂²×D₄₇	—	8	2	no	—
81	C₂²×D₄₇	—	8	2	no	—
82	C₂×C₄×C₄₇	—	8	2	no	—
83	C₂×C₄×C₄₇	—	8	2	no	—
84	C₂×C₄×C₄₇	—	8	2	no	—
85	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₄₇	#135	1	2	yes	—
93	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
94	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
95	D₁₈₈	—	4	8	no	—
96	D₁₈₈	—	4	8	no	—
97	D₁₈₈	—	4	8	no	—
98	D₁₈₈	—	8	4	no	—
99	D₁₈₈	—	8	4	no	—
100	D₁₈₈	—	8	4	no	—
101	C₈×C₄₇	—	2	4	no	—
102	C₄₇⋊C₈	—	2	4	no	—
103	C₄₇⋊C₈	—	4	2	no	—
104	#9832	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	#390	1	1	yes	characteristic
106	C₂²×C₄₇	#493	1	1	yes	characteristic
107	C₂²×C₄₇	#495	1	1	yes	characteristic
108	C₂²×C₄₇	#496	1	2	yes	—
109	C₂²×C₄₇	#496	1	2	yes	—
110	C₂×D₄₇	—	8	4	no	—
111	C₂×D₄₇	—	8	4	no	—
112	C₂×D₄₇	—	8	4	no	—
113	C₂×D₄₇	—	16	2	no	—
114	C₂×D₄₇	—	16	2	no	—
115	C₂×D₄₇	—	16	2	no	—
116	C₂×D₄₇	—	16	2	no	—
117	C₄×C₄₇	—	16	2	no	—
118	C₄×C₄₇	#418	1	2	yes	—
119	C₄×C₄₇	C₄×D₈	1	2	yes	—
120	C₄×C₄₇	#442	1	2	yes	—
121	C₄×C₄₇	#466	1	2	yes	—
122	C₄×C₄₇	#469	1	2	yes	—
123	C₄×C₄₇	#490	1	2	yes	—
124	C₂×#493	—	47	1	no	—
125	#1957	—	47	2	no	—
126	#1957	—	47	2	no	—
127	#1834	D₄₇	1	1	yes	characteristic, complemented
128	C₂×#322	—	47	1	no	—
129	C₂×C₄₇	#1717	1	2	yes	—
130	C₂×C₄₇	#1721	1	2	yes	—
131	C₂×C₄₇	#1961	1	1	yes	characteristic
132	C₂×C₄₇	#1970	1	1	yes	characteristic
133	C₂×C₄₇	#1972	1	1	yes	characteristic
134	D₄₇	—	16	4	no	—
135	C₂²×D₈	—	94	2	no	—
136	C₂×#127	—	47	2	no	—
137	#493	—	47	4	no	—
138	#493	—	94	2	no	—
139	C₂×#102	—	94	4	no	—
140	C₂×#102	—	94	4	no	—
141	C₂×C₄×C₈	—	47	1	no	—
142	C₂×C₄×C₈	—	47	1	no	—
143	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
144	#384	C₂×D₄₇	1	2	yes	—
145	#340	—	2	2	no	—
146	#322	—	94	2	no	—
147	C₄₇	#9832	1	1	yes	characteristic, complemented, Sylow
148	C₂²×D₄	—	94	2	no	—
149	C₂²×D₄	—	94	2	no	—
150	C₂²×D₄	—	94	2	no	—
151	C₂×D₈	—	94	8	no	—
152	C₂×D₈	—	188	8	no	—
153	C₂²×C₈	—	47	2	no	—
154	C₂²×C₈	—	47	2	no	—
155	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
156	#127	—	47	4	no	—
157	#127	—	94	2	no	—
158	#127	—	94	2	no	—
159	#127	—	94	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
164	#102	—	188	8	no	—
165	#102	—	188	8	no	—
166	C₄×C₈	—	94	1	no	—
167	C₄×C₈	—	94	1	no	—
168	C₄×C₈	—	94	1	no	—
169	C₄×C₈	—	94	1	no	—
170	C₄×C₈	C₄×D₄₇	1	2	yes	—
171	C₄×C₈	D₁₈₈	1	2	yes	—
172	C₂⁴	—	188	2	no	—
173	C₂×D₄	—	94	4	no	—
174	C₂×D₄	—	94	4	no	—
175	C₂×D₄	—	94	4	no	—
176	C₂×D₄	—	188	2	no	—
177	C₂×D₄	—	188	2	no	—
178	C₂×D₄	—	188	2	no	—
179	C₂×D₄	—	188	4	no	—
180	C₂×D₄	—	188	4	no	—
181	C₂×D₄	—	188	4	no	—
182	C₂×D₄	—	188	4	no	—
183	C₂×D₄	—	188	4	no	—
184	C₂×D₄	—	188	4	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
187	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
189	D₈	—	188	16	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	47	4	no	—
192	C₂×C₈	—	94	2	no	—
193	C₂×C₈	—	94	2	no	—
194	C₂×C₈	—	94	2	no	—
195	C₂×C₈	—	94	4	no	—
196	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
201	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
202	C₄²	#10625354	1	1	yes	characteristic
203	C₄²	#10625354	1	1	yes	characteristic
204	C₂³	—	376	2	no	—
205	C₂³	—	376	2	no	—
206	C₂³	—	376	2	no	—
207	C₂³	—	376	2	no	—
208	C₂³	—	376	2	no	—
209	C₂³	—	376	2	no	—
210	C₂³	—	188	4	no	—
211	C₂³	#13094020	1	1	yes	characteristic
212	D₄	—	376	4	no	—
213	D₄	—	376	4	no	—
214	D₄	—	376	4	no	—
215	D₄	—	188	8	no	—
216	D₄	—	188	8	no	—
217	D₄	—	188	8	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
222	C₂×C₄	#13093986	1	1	yes	characteristic
223	C₂×C₄	#13093986	1	1	yes	characteristic
224	C₂×C₄	#13093999	1	1	yes	characteristic
225	C₂×C₄	#13093999	1	1	yes	characteristic
226	C₂×C₄	#13094013	1	1	yes	characteristic
227	C₂×C₄	D₈×D₄₇	1	2	yes	—
228	C₂×C₄	#13094047	1	2	yes	—
229	C₂×C₄	#13094049	1	4	yes	—
230	C₂×C₄	#13094051	1	4	yes	—
231	C₈	—	2	4	no	—
232	C₈	—	94	4	no	—
233	C₈	—	188	2	no	—
234	C₂²	—	376	4	no	—
235	C₂²	—	376	4	no	—
236	C₂²	—	376	4	no	—
237	C₂²	—	752	2	no	—
238	C₂²	—	752	2	no	—
239	C₂²	—	752	2	no	—
240	C₂²	—	752	2	no	—
241	C₂²	#15689024	1	1	yes	characteristic
242	C₂²	#15689498	1	1	yes	characteristic
243	C₂²	#15689508	1	1	yes	characteristic
244	C₂²	#15689509	1	2	yes	—
245	C₂²	#15689513	1	2	yes	—
246	C₄	—	16	2	no	—
247	C₄	#15689127	1	2	yes	—
248	C₄	#15689205	1	2	yes	—
249	C₄	#15689234	1	2	yes	—
250	C₄	#15689360	1	2	yes	—
251	C₄	#15689377	1	2	yes	—
252	C₄	#15689489	1	2	yes	—
253	C₂	—	752	4	no	—
254	C₂	#20243030	1	2	yes	—
255	C₂	#20243056	1	2	yes	—
256	C₂	#20244211	1	1	yes	characteristic
257	C₂	#20244252	1	1	yes	characteristic
258	C₂	#20244262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689500

C₂

yes

maximal, characteristic

#20244193

C₂

yes

maximal, complemented

#20244193

C₂

yes

maximal, complemented

C₂×#15688845

C₂

yes

maximal, characteristic, complemented

C₄₇×#1834

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₇₆

—

C₂×#13094021

C₂²

yes

—

#15689500

—

#15689500

C₄

yes

complemented

C₂×#13093944

—

C₂×#13093944

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#15688845

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

—

C₄₇×#340

—

C₂²×D₁₈₈

—

C₂²×D₁₈₈

—

C₂²×D₁₈₈

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂²×C₄₇⋊C₈

D₄

yes

—

C₂²×C₄₇⋊C₈

D₄

yes

—

—

—

—

C₂×C₄

yes

—

#13093944

—

#13093944

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₈

yes

complemented

D₃₇₆

—

C₂³×C₄₇

#127

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

100

D₁₈₈

—

101

C₈×C₄₇

—

102

C₄₇⋊C₈

—

103

C₄₇⋊C₈

—

104

#9832

—

maximal, Sylow

105

C₂²×C₄₇

#390

yes

characteristic

106

C₂²×C₄₇

#493

yes

characteristic

107

C₂²×C₄₇

#495

yes

characteristic

108

C₂²×C₄₇

#496

yes

—

109

C₂²×C₄₇

#496

yes

—

110

C₂×D₄₇

—

111

C₂×D₄₇

—

112

C₂×D₄₇

—

113

C₂×D₄₇

—

114

C₂×D₄₇

—

115

C₂×D₄₇

—

116

C₂×D₄₇

—

117

C₄×C₄₇

—

118

C₄×C₄₇

#418

yes

—

119

C₄×C₄₇

C₄×D₈

yes

—

120

C₄×C₄₇

#442

yes

—

121

C₄×C₄₇

#466

yes

—

122

C₄×C₄₇

#469

yes

—

123

C₄×C₄₇

#490

yes

—

124

C₂×#493

—

125

#1957

—

126

#1957

—

127

#1834

D₄₇

yes

characteristic, complemented

128

C₂×#322

—

129

C₂×C₄₇

#1717

yes

—

130

C₂×C₄₇

#1721

yes

—

131

C₂×C₄₇

#1961

yes

characteristic

132

C₂×C₄₇

#1970

yes

characteristic

133

C₂×C₄₇

#1972

yes

characteristic

134

D₄₇

—

135

C₂²×D₈

—

136

C₂×#127

—

137

#493

—

138

#493

—

139

C₂×#102

—

140

C₂×#102

—

141

C₂×C₄×C₈

—

142

C₂×C₄×C₈

—

143

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

144

#384

C₂×D₄₇

yes

—

145

#340

—

146

#322

—

147

C₄₇

#9832

yes

characteristic, complemented, Sylow

148

C₂²×D₄

—

149

C₂²×D₄

—

150

C₂²×D₄

—

151

C₂×D₈

—

152

C₂×D₈

—

188

—

153

C₂²×C₈

—

154

C₂²×C₈

—

155

C₂²×C₈

C₂²⋊D₄₇

yes

—

156

#127

—

157

#127

—

158

#127

—

159

#127

—

160

C₂×C₄⋊C₄

—

161

C₂×C₄⋊C₄

—

162

C₂×C₄⋊C₄

—

163

C₂×C₄²

C₂²×D₄₇

yes

characteristic

164

#102

—

188

—

165

#102

—

188

—

166

C₄×C₈

—

167

C₄×C₈

—

168

C₄×C₈

—

169

C₄×C₈

—

170

C₄×C₈

C₄×D₄₇

yes

—

171

C₄×C₈

D₁₈₈

yes

—

172

C₂⁴

—

188

—

173

C₂×D₄

—

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂×D₄

—

188

—

177

C₂×D₄

—

188

—

178

C₂×D₄

—

188

—

179

C₂×D₄

—

188

—

180

C₂×D₄

—

188

—

181

C₂×D₄

—

188

—

182

C₂×D₄

—

188

—

183

C₂×D₄

—

188

—

184

C₂×D₄

—

188

—

185

C₂²×C₄

—

186

C₂²×C₄

D₄×D₄₇

yes

characteristic

187

C₂²×C₄

D₄×D₄₇

yes

characteristic

188

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

189

D₈

—

188

—

190

C₂×C₈

—

191

C₂×C₈

—

192

C₂×C₈

—

193

C₂×C₈

—

194

C₂×C₈

—

195

C₂×C₈

—

196

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

197

C₄⋊C₄

—

198

C₄⋊C₄

—

199

C₄⋊C₄

—

200

C₄²

C₂×C₄×D₄₇

yes

characteristic

201

C₄²

C₂×D₁₈₈

yes

characteristic

202

C₄²

#10625354

yes

characteristic

203

C₄²

#10625354

yes

characteristic

204

C₂³

—

376

—

205

C₂³

—

376

—

206

C₂³

—

376

—

207

C₂³

—

376

—

208

C₂³

—

376

—

209

C₂³

—

376

—

210

C₂³

—

188

—

211

C₂³

#13094020

yes

characteristic

212

D₄

—

376

—

213

D₄

—

376

—

214

D₄

—

376

—

215

D₄

—

188

—

216

D₄

—

188

—

217

D₄

—

188

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

222

C₂×C₄

#13093986

yes

characteristic

223

C₂×C₄

#13093986

yes

characteristic

224

C₂×C₄

#13093999

yes

characteristic

225

C₂×C₄

#13093999

yes

characteristic

226

C₂×C₄

#13094013

yes

characteristic

227

C₂×C₄

D₈×D₄₇

yes

—

228

C₂×C₄

#13094047

yes

—

229

C₂×C₄

#13094049

yes

—

230

C₂×C₄

#13094051

yes

—

231

C₈

—

232

C₈

—

233

C₈

—

188

—

234

C₂²

—

376

—

235

C₂²

—

376

—

236

C₂²

—

376

—

237

C₂²

—

752

—

238

C₂²

—

752

—

239

C₂²

—

752

—

240

C₂²

—

752

—

241

C₂²

#15689024

yes

characteristic

242

C₂²

#15689498

yes

characteristic

243

C₂²

#15689508

yes

characteristic

244

C₂²

#15689509

yes

—

245

C₂²

#15689513

yes

—

246

C₄

—

247

C₄

#15689127

yes

—

248

C₄

#15689205

yes

—

249

C₄

#15689234

yes

—

250

C₄

#15689360

yes

—

251

C₄

#15689377

yes

—

252

C₄

#15689489

yes

—

253

C₂

—

752

—

254

C₂

#20243030

yes

—

255

C₂

#20243056

yes

—

256

C₂

#20244211

yes

characteristic

257

C₂

#20244252

yes

characteristic

258

C₂

#20244262

yes

characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1834	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#15689500	C₂
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1834	D₄₇
hypocenter	C₄₇	#9832
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1834

C₂

C₂³×C₄₇

C₂

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
2. #20244193	C₂	semidirect
3. #20244193	C₂	semidirect
4. C₂×#15688845	C₂	semidirect
5. C₄₇×#1834	C₂	semidirect
9. #15689500	C₄	semidirect
72. C₂×C₄₇⋊C₈	D₈	semidirect
127. #1834	D₄₇	semidirect
147. C₄₇	#9832	semidirect

H ≅ G/N

type

2. #20244193

C₂

semidirect

3. #20244193

C₂

semidirect

4. C₂×#15688845

C₂

semidirect

5. C₄₇×#1834

C₂

semidirect

9. #15689500

C₄

semidirect

72. C₂×C₄₇⋊C₈

D₈

semidirect

127. #1834

D₄₇

semidirect

147. C₄₇

#9832

semidirect

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

subgroup	index
C₂×#15689500	2
#20244193	2
C₂×#15688845	2
C₄₇×#1834	2
#9832	47

subgroup

index

C₄₇×#1834

Subgroups — #26116088

Conjugacy classes of subgroups

Complements

Maximal subgroups