Group #26051080 — subgroups

Subgroups — C₄₇×#11382 #26051080

Order 12032.

1142 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 286 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	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#343	C₂	1	4	yes	maximal
3	C₄₇×#1728	C₂	1	2	yes	maximal
4	C₂³×Q₈×C₄₇	C₄	1	2	yes	—
5	C₂²×M₁₆×C₄₇	C₂²	1	4	yes	—
6	C₂×C₄⋊Q₈×C₄₇	—	2	8	no	—
7	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
8	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
9	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#343	—	2	8	no	—
12	C₄₇×#336	—	2	4	no	—
13	C₂²×Q₈×C₄₇	—	2	8	no	—
14	C₂²×Q₈×C₄₇	—	2	4	no	—
15	C₂²×Q₈×C₄₇	—	4	8	no	—
16	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₄₇	—	2	4	no	—
21	C₂×M₁₆×C₄₇	—	2	8	no	—
22	C₂×M₁₆×C₄₇	—	2	8	no	—
23	C₂×M₁₆×C₄₇	Q₈	1	8	yes	complemented
24	C₂²×C₈×C₄₇	—	2	4	no	—
25	C₄⋊Q₈×C₄₇	—	2	32	no	—
26	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
27	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
30	C₂×C₄⋊C₄×C₄₇	D₄	1	8	yes	—
31	C₂×C₄²×C₄₇	—	2	4	no	—
32	C₂×C₄²×C₄₇	D₄	1	4	yes	—
33	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×Q₈×C₄₇	—	2	16	no	—
35	C₂×Q₈×C₄₇	—	4	32	no	—
36	C₂×Q₈×C₄₇	—	4	8	no	—
37	C₂×Q₈×C₄₇	—	4	8	no	—
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	2	4	no	—
41	C₂²×C₄×C₄₇	—	2	4	no	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	4	8	no	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₄₇	#41	1	4	yes	—
52	M₁₆×C₄₇	—	2	16	no	—
53	M₁₆×C₄₇	—	4	8	no	—
54	C₂×C₈×C₄₇	—	2	8	no	—
55	C₂×C₈×C₄₇	—	4	4	no	—
56	C₂×C₈×C₄₇	—	4	4	no	—
57	C₄⋊C₄×C₄₇	—	2	16	no	—
58	C₄⋊C₄×C₄₇	—	2	16	no	—
59	C₄²×C₄₇	—	2	8	no	—
60	C₂³×C₄₇	—	2	2	no	—
61	C₂³×C₄₇	—	2	2	no	—
62	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
63	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
65	Q₈×C₄₇	—	4	32	no	—
66	C₂×C₄×C₄₇	—	2	8	no	—
67	C₂×C₄×C₄₇	—	2	8	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	8	no	—
73	C₂×C₄×C₄₇	—	4	8	no	—
74	C₂×C₄×C₄₇	—	4	8	no	—
75	C₂×C₄×C₄₇	—	4	8	no	—
76	C₂×C₄×C₄₇	#100	1	4	yes	—
77	C₂×C₄×C₄₇	#101	1	4	yes	—
78	C₂×C₄×C₄₇	#120	1	4	yes	—
79	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
80	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
81	C₂×C₄×C₄₇	#124	1	2	yes	—
82	C₂×C₄×C₄₇	#124	1	4	yes	—
83	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
84	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
85	C₈×C₄₇	—	4	8	no	—
86	#11382	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₄₇	—	2	8	no	—
88	C₂²×C₄₇	—	2	2	no	—
89	C₂²×C₄₇	—	2	2	no	—
90	C₂²×C₄₇	#379	1	1	yes	characteristic
91	C₂²×C₄₇	#391	1	4	yes	—
92	C₂²×C₄₇	#391	1	2	yes	—
93	C₂²×C₄₇	C₂×#100	1	2	yes	—
94	C₂²×C₄₇	C₂×#101	1	2	yes	—
95	C₄×C₄₇	—	2	4	no	—
96	C₄×C₄₇	—	2	4	no	—
97	C₄×C₄₇	—	4	8	no	—
98	C₄×C₄₇	#393	1	4	yes	—
99	C₄×C₄₇	#395	1	4	yes	—
100	C₂²×C₄⋊Q₈	C₂×C₄₇	1	1	yes	characteristic
101	C₂×#343	C₂×C₄₇	1	4	yes	—
102	#1728	C₂×C₄₇	1	2	yes	—
103	C₂×C₄₇	—	2	4	no	—
104	C₂×C₄₇	#1446	1	1	yes	characteristic
105	C₂×C₄₇	#1783	1	1	yes	characteristic
106	C₂×C₄₇	#1784	1	1	yes	characteristic
107	C₂×C₄₇	#1997	1	4	yes	—
108	C₂³×Q₈	C₄×C₄₇	1	2	yes	—
109	C₂²×M₁₆	C₂²×C₄₇	1	4	yes	—
110	C₂×C₄⋊Q₈	—	2	8	no	—
111	C₂×C₄⋊Q₈	—	2	4	no	—
112	C₂²×C₄⋊C₄	C₄×C₄₇	1	2	yes	—
113	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
114	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
115	#343	—	2	8	no	—
116	#336	—	2	4	no	—
117	C₄₇	#11382	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×Q₈	—	2	8	no	—
119	C₂²×Q₈	—	2	4	no	—
120	C₂²×Q₈	—	4	8	no	—
121	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
122	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
123	C₂³×C₄	C₂×C₄×C₄₇	1	4	yes	—
124	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
125	C₂×M₁₆	—	2	4	no	—
126	C₂×M₁₆	—	2	8	no	—
127	C₂×M₁₆	—	2	8	no	—
128	C₂×M₁₆	Q₈×C₄₇	1	8	yes	complemented
129	C₂²×C₈	—	2	4	no	—
130	C₄⋊Q₈	—	2	32	no	—
131	C₂×C₄⋊C₄	—	2	8	no	—
132	C₂×C₄⋊C₄	—	2	4	no	—
133	C₂×C₄⋊C₄	—	2	4	no	—
134	C₂×C₄⋊C₄	—	2	4	no	—
135	C₂×C₄⋊C₄	D₄×C₄₇	1	8	yes	—
136	C₂×C₄²	—	2	4	no	—
137	C₂×C₄²	D₄×C₄₇	1	4	yes	—
138	C₂⁴	C₂²×C₄×C₄₇	1	1	yes	characteristic
139	C₂×Q₈	—	2	16	no	—
140	C₂×Q₈	—	4	32	no	—
141	C₂×Q₈	—	4	8	no	—
142	C₂×Q₈	—	4	8	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	4	8	no	—
150	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
151	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
152	C₂²×C₄	C₂²⋊C₄×C₄₇	1	8	yes	—
153	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
154	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
155	C₂²×C₄	C₂×Q₈×C₄₇	1	4	yes	—
156	C₂²×C₄	#41×C₄₇	1	4	yes	—
157	M₁₆	—	2	16	no	—
158	M₁₆	—	4	8	no	—
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	—	4	4	no	—
161	C₂×C₈	—	4	4	no	—
162	C₄⋊C₄	—	2	16	no	—
163	C₄⋊C₄	—	2	16	no	—
164	C₄²	—	2	8	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
168	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
169	C₂³	C₄×Q₈×C₄₇	1	4	yes	—
170	Q₈	—	4	32	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	4	8	no	—
181	C₂×C₄	#100×C₄₇	1	4	yes	—
182	C₂×C₄	#101×C₄₇	1	4	yes	—
183	C₂×C₄	#120×C₄₇	1	4	yes	—
184	C₂×C₄	C₂²⋊Q₈×C₄₇	1	4	yes	—
185	C₂×C₄	C₂²⋊Q₈×C₄₇	1	8	yes	—
186	C₂×C₄	#124×C₄₇	1	2	yes	—
187	C₂×C₄	#124×C₄₇	1	4	yes	—
188	C₂×C₄	C₄⋊Q₈×C₄₇	1	4	yes	—
189	C₂×C₄	C₄⋊Q₈×C₄₇	1	2	yes	—
190	C₈	—	4	8	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	8	no	—
194	C₂²	C₄₇×#379	1	1	yes	characteristic
195	C₂²	C₄₇×#391	1	4	yes	—
196	C₂²	C₄₇×#391	1	2	yes	—
197	C₂²	C₂×#100×C₄₇	1	2	yes	—
198	C₂²	C₂×#101×C₄₇	1	2	yes	—
199	C₄	—	2	4	no	—
200	C₄	—	2	4	no	—
201	C₄	—	4	8	no	—
202	C₄	C₄₇×#393	1	4	yes	—
203	C₄	C₄₇×#395	1	4	yes	—
204	C₂	—	2	4	no	—
205	C₂	C₄₇×#1446	1	1	yes	characteristic
206	C₂	C₄₇×#1783	1	1	yes	characteristic
207	C₂	C₄₇×#1784	1	1	yes	characteristic
208	C₂	C₄₇×#1997	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄₇×#343

C₂

yes

maximal

C₄₇×#1728

C₂

yes

maximal

C₂³×Q₈×C₄₇

C₄

yes

—

C₂²×M₁₆×C₄₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₄₇

—

C₂×C₄⋊Q₈×C₄₇

—

C₂²×C₄⋊C₄×C₄₇

C₄

yes

—

C₂²×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₄₇×#343

—

C₄₇×#336

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

Q₈

yes

complemented

C₂²×C₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂⁴×C₄₇

C₂²×C₄

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂²×C₄×C₄₇

#41

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₄×Q₈

yes

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#100

yes

—

C₂×C₄×C₄₇

#101

yes

—

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₈×C₄₇

—

#11382

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#379

yes

characteristic

C₂²×C₄₇

#391

yes

—

C₂²×C₄₇

#391

yes

—

C₂²×C₄₇

C₂×#100

yes

—

C₂²×C₄₇

C₂×#101

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#393

yes

—

C₄×C₄₇

#395

yes

—

100

C₂²×C₄⋊Q₈

C₂×C₄₇

yes

characteristic

101

C₂×#343

C₂×C₄₇

yes

—

102

#1728

C₂×C₄₇

yes

—

103

C₂×C₄₇

—

104

C₂×C₄₇

#1446

yes

characteristic

105

C₂×C₄₇

#1783

yes

characteristic

106

C₂×C₄₇

#1784

yes

characteristic

107

C₂×C₄₇

#1997

yes

—

108

C₂³×Q₈

C₄×C₄₇

yes

—

109

C₂²×M₁₆

C₂²×C₄₇

yes

—

110

C₂×C₄⋊Q₈

—

111

C₂×C₄⋊Q₈

—

112

C₂²×C₄⋊C₄

C₄×C₄₇

yes

—

113

C₂²×C₄⋊C₄

C₂²×C₄₇

yes

—

114

C₂²×C₄²

C₂²×C₄₇

yes

characteristic

115

#343

—

116

#336

—

117

C₄₇

#11382

yes

characteristic, direct product factor, Sylow

118

C₂²×Q₈

—

119

C₂²×Q₈

—

120

C₂²×Q₈

—

121

C₂³×C₄

C₂×C₄×C₄₇

yes

characteristic

122

C₂³×C₄

C₂×C₄×C₄₇

yes

characteristic

123

C₂³×C₄

C₂×C₄×C₄₇

yes

—

124

C₂³×C₄

C₂³×C₄₇

yes

characteristic

125

C₂×M₁₆

—

126

C₂×M₁₆

—

127

C₂×M₁₆

—

128

C₂×M₁₆

Q₈×C₄₇

yes

complemented

129

C₂²×C₈

—

130

C₄⋊Q₈

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

D₄×C₄₇

yes

—

136

C₂×C₄²

—

137

C₂×C₄²

D₄×C₄₇

yes

—

138

C₂⁴

C₂²×C₄×C₄₇

yes

characteristic

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂²⋊C₄×C₄₇

yes

—

151

C₂²×C₄

C₂²⋊C₄×C₄₇

yes

—

152

C₂²×C₄

C₂²⋊C₄×C₄₇

yes

—

153

C₂²×C₄

C₂×D₄×C₄₇

yes

—

154

C₂²×C₄

C₂×D₄×C₄₇

yes

—

155

C₂²×C₄

C₂×Q₈×C₄₇

yes

—

156

C₂²×C₄

#41×C₄₇

yes

—

157

M₁₆

—

158

M₁₆

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄²

—

165

C₂³

—

166

C₂³

—

167

C₂³

C₂×C₂²⋊C₄×C₄₇

yes

—

168

C₂³

C₂×C₂²⋊C₄×C₄₇

yes

characteristic

169

C₂³

C₄×Q₈×C₄₇

yes

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

#100×C₄₇

yes

—

182

C₂×C₄

#101×C₄₇

yes

—

183

C₂×C₄

#120×C₄₇

yes

—

184

C₂×C₄

C₂²⋊Q₈×C₄₇

yes

—

185

C₂×C₄

C₂²⋊Q₈×C₄₇

yes

—

186

C₂×C₄

#124×C₄₇

yes

—

187

C₂×C₄

#124×C₄₇

yes

—

188

C₂×C₄

C₄⋊Q₈×C₄₇

yes

—

189

C₂×C₄

C₄⋊Q₈×C₄₇

yes

—

190

C₈

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

C₄₇×#379

yes

characteristic

195

C₂²

C₄₇×#391

yes

—

196

C₂²

C₄₇×#391

yes

—

197

C₂²

C₂×#100×C₄₇

yes

—

198

C₂²

C₂×#101×C₄₇

yes

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

C₄₇×#393

yes

—

203

C₄

C₄₇×#395

yes

—

204

C₂

—

205

C₂

C₄₇×#1446

yes

characteristic

206

C₂

C₄₇×#1783

yes

characteristic

207

C₂

C₄₇×#1784

yes

characteristic

208

C₂

C₄₇×#1997

yes

—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
prime radical	C₂⁴×C₄₇	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂²×C₄²×C₄₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₄₇

C₂×C₂²⋊C₄

commutator

C₂⁴

C₂²×C₄×C₄₇

Frattini, cosocle

C₂³×C₄

C₂³×C₄₇

prime radical

C₂⁴×C₄₇

C₂²×C₄

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₄₇ ◁ C₂²×C₄²×C₄₇ ◁ G

N	H ≅ G/N	type
23. C₂×M₁₆×C₄₇	Q₈	semidirect
86. #11382	C₄₇	direct (N × H)
117. C₄₇	#11382	direct (N × H)
128. C₂×M₁₆	Q₈×C₄₇	semidirect

H ≅ G/N

type

23. C₂×M₁₆×C₄₇

Q₈

semidirect

86. #11382

C₄₇

direct (N × H)

117. C₄₇

#11382

direct (N × H)

128. C₂×M₁₆

Q₈×C₄₇

semidirect

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

subgroup	index
C₂²×C₄⋊Q₈×C₄₇	2
C₂×C₄₇×#343	2
C₄₇×#1728	2
#11382	47

subgroup

index

C₂²×C₄⋊Q₈×C₄₇

C₂×C₄₇×#343

C₄₇×#1728

#11382

Subgroups — C₄₇×#11382 #26051080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47×#11382 #26051080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#11382 #26051080