Group #38789719 — subgroups

Subgroups — C₁₀₁×#7568 #38789719

Order 25856.

444 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 170 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×#125×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#1744	C₂	1	2	yes	maximal
3	C₁₀₁×#1271	C₂	1	4	yes	maximal
4	C₂×#125×C₁₀₁	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₁₀₁	—	2	2	no	—
6	#389×C₁₀₁	C₂²	1	2	yes	—
7	#387×C₁₀₁	—	2	2	no	—
8	#382×C₁₀₁	C₄	1	1	yes	characteristic
9	C₄×C₄⋊C₄×C₁₀₁	—	2	2	no	—
10	C₄×C₄⋊C₄×C₁₀₁	C₄	1	2	yes	—
11	C₄³×C₁₀₁	C₂²	1	1	yes	characteristic
12	#336×C₁₀₁	—	2	2	no	—
13	#336×C₁₀₁	C₂²	1	4	yes	—
14	C₂²×C₈×C₁₀₁	—	2	4	no	—
15	#125×C₁₀₁	—	2	2	no	—
16	#125×C₁₀₁	C₈	1	4	yes	complemented
17	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₀₁	—	2	2	no	—
22	C₂×C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₀₁	D₄	1	2	yes	—
25	C₂×C₄²×C₁₀₁	D₄	1	2	yes	—
26	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
27	C₄×C₈×C₁₀₁	—	2	8	no	—
28	#95×C₁₀₁	—	2	2	no	—
29	#95×C₁₀₁	—	2	2	no	—
30	#95×C₁₀₁	—	2	2	no	—
31	C₂²×C₄×C₁₀₁	—	2	2	no	—
32	C₂²×C₄×C₁₀₁	—	2	2	no	—
33	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₁₀₁	—	2	8	no	—
39	C₂×C₈×C₁₀₁	—	4	4	no	—
40	C₂×C₈×C₁₀₁	—	4	4	no	—
41	C₄⋊C₄×C₁₀₁	—	2	8	no	—
42	C₄⋊C₄×C₁₀₁	—	2	2	no	—
43	C₄⋊C₄×C₁₀₁	—	4	2	no	—
44	C₄⋊C₄×C₁₀₁	—	4	4	no	—
45	C₄⋊C₄×C₁₀₁	C₂×C₈	1	4	yes	—
46	C₄²×C₁₀₁	—	2	2	no	—
47	C₄²×C₁₀₁	—	2	4	no	—
48	C₄²×C₁₀₁	—	2	4	no	—
49	C₄²×C₁₀₁	—	4	1	no	—
50	C₄²×C₁₀₁	—	4	4	no	—
51	C₄²×C₁₀₁	C₂×C₈	1	2	yes	—
52	C₄²×C₁₀₁	M₁₆	1	2	yes	—
53	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₁	—	2	1	no	—
55	C₂×C₄×C₁₀₁	—	2	2	no	—
56	C₂×C₄×C₁₀₁	—	2	1	no	—
57	C₂×C₄×C₁₀₁	—	2	2	no	—
58	C₂×C₄×C₁₀₁	—	2	2	no	—
59	C₂×C₄×C₁₀₁	—	2	2	no	—
60	C₂×C₄×C₁₀₁	—	2	4	no	—
61	C₂×C₄×C₁₀₁	—	2	1	no	—
62	C₂×C₄×C₁₀₁	—	2	2	no	—
63	C₂×C₄×C₁₀₁	—	4	2	no	—
64	C₂×C₄×C₁₀₁	—	4	2	no	—
65	C₂×C₄×C₁₀₁	—	4	2	no	—
66	C₂×C₄×C₁₀₁	—	4	2	no	—
67	C₂×C₄×C₁₀₁	—	4	2	no	—
68	C₂×C₄×C₁₀₁	C₂²⋊C₈	1	4	yes	—
69	C₂×C₄×C₁₀₁	#101	1	2	yes	—
70	C₂×C₄×C₁₀₁	#104	1	4	yes	—
71	C₂×C₄×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₀₁	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₀₁	#135	1	4	yes	—
74	C₈×C₁₀₁	—	4	8	no	—
75	C₂²×C₁₀₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
76	C₂²×C₁₀₁	#410	1	1	yes	characteristic
77	C₂²×C₁₀₁	C₂×#101	1	1	yes	characteristic
78	C₂²×C₁₀₁	#416	1	2	yes	—
79	C₂²×C₁₀₁	C₂×#104	1	2	yes	—
80	C₄×C₁₀₁	—	2	1	no	—
81	C₄×C₁₀₁	—	2	1	no	—
82	C₄×C₁₀₁	—	2	4	no	—
83	C₄×C₁₀₁	—	2	4	no	—
84	C₄×C₁₀₁	—	4	1	no	—
85	C₄×C₁₀₁	—	4	4	no	—
86	C₄×C₁₀₁	—	8	2	no	—
87	C₄×C₁₀₁	#330	1	4	yes	—
88	#7568	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₁₀₁	#1461	1	1	yes	characteristic
90	C₂×C₁₀₁	#1476	1	2	yes	—
91	C₂×C₁₀₁	#1500	1	2	yes	—
92	C₂×C₁₀₁	C₂×#330	1	1	yes	characteristic
93	C₂×C₁₀₁	#1544	1	1	yes	characteristic
94	C₄×#125	C₂×C₁₀₁	1	1	yes	characteristic
95	#1744	C₂×C₁₀₁	1	2	yes	—
96	#1271	C₂×C₁₀₁	1	4	yes	—
97	C₁₀₁	#7568	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×#125	C₄×C₁₀₁	1	1	yes	characteristic
99	C₂×C₄×C₈	—	2	2	no	—
100	#389	C₂²×C₁₀₁	1	2	yes	—
101	#387	—	2	2	no	—
102	#382	C₄×C₁₀₁	1	1	yes	characteristic
103	C₄×C₄⋊C₄	—	2	2	no	—
104	C₄×C₄⋊C₄	C₄×C₁₀₁	1	2	yes	—
105	C₄³	C₂²×C₁₀₁	1	1	yes	characteristic
106	#336	—	2	2	no	—
107	#336	C₂²×C₁₀₁	1	4	yes	—
108	C₂²×C₈	—	2	4	no	—
109	#125	—	2	2	no	—
110	#125	C₈×C₁₀₁	1	4	yes	complemented
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
115	C₂×C₄²	—	2	2	no	—
116	C₂×C₄²	C₂×C₄×C₁₀₁	1	1	yes	characteristic
117	C₂×C₄²	C₂×C₄×C₁₀₁	1	1	yes	characteristic
118	C₂×C₄²	D₄×C₁₀₁	1	2	yes	—
119	C₂×C₄²	D₄×C₁₀₁	1	2	yes	—
120	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
121	C₄×C₈	—	2	8	no	—
122	#95	—	2	2	no	—
123	#95	—	2	2	no	—
124	#95	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
129	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
130	C₂²×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
131	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
132	C₂×C₈	—	2	8	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	—	4	4	no	—
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	C₂×C₈×C₁₀₁	1	4	yes	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	4	no	—
142	C₄²	—	2	4	no	—
143	C₄²	—	4	1	no	—
144	C₄²	—	4	4	no	—
145	C₄²	C₂×C₈×C₁₀₁	1	2	yes	—
146	C₄²	M₁₆×C₁₀₁	1	2	yes	—
147	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	C₂²⋊C₈×C₁₀₁	1	4	yes	—
163	C₂×C₄	#101×C₁₀₁	1	2	yes	—
164	C₂×C₄	#104×C₁₀₁	1	4	yes	—
165	C₂×C₄	C₂²×C₈×C₁₀₁	1	1	yes	characteristic
166	C₂×C₄	C₂×M₁₆×C₁₀₁	1	1	yes	characteristic
167	C₂×C₄	#135×C₁₀₁	1	4	yes	—
168	C₈	—	4	8	no	—
169	C₂²	C₂×C₂²⋊C₈×C₁₀₁	1	1	yes	characteristic
170	C₂²	#410×C₁₀₁	1	1	yes	characteristic
171	C₂²	C₂×#101×C₁₀₁	1	1	yes	characteristic
172	C₂²	#416×C₁₀₁	1	2	yes	—
173	C₂²	C₂×#104×C₁₀₁	1	2	yes	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	4	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	4	no	—
179	C₄	—	4	1	no	—
180	C₄	—	8	2	no	—
181	C₄	#330×C₁₀₁	1	4	yes	—
182	C₂	C₁₀₁×#1461	1	1	yes	characteristic
183	C₂	C₁₀₁×#1476	1	2	yes	—
184	C₂	C₁₀₁×#1500	1	2	yes	—
185	C₂	C₂×#330×C₁₀₁	1	1	yes	characteristic
186	C₂	C₁₀₁×#1544	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#125×C₁₀₁

C₂

yes

maximal, characteristic

C₁₀₁×#1744

C₂

yes

maximal

C₁₀₁×#1271

C₂

yes

maximal

C₂×#125×C₁₀₁

C₄

yes

characteristic

C₂×C₄×C₈×C₁₀₁

—

#389×C₁₀₁

C₂²

yes

—

#387×C₁₀₁

—

#382×C₁₀₁

C₄

yes

characteristic

C₄×C₄⋊C₄×C₁₀₁

—

C₄×C₄⋊C₄×C₁₀₁

C₄

yes

—

C₄³×C₁₀₁

C₂²

yes

characteristic

#336×C₁₀₁

—

#336×C₁₀₁

C₂²

yes

—

C₂²×C₈×C₁₀₁

—

#125×C₁₀₁

—

#125×C₁₀₁

C₈

yes

complemented

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₂×C₄²×C₁₀₁

—

C₂×C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₀₁

D₄

yes

—

C₂×C₄²×C₁₀₁

D₄

yes

—

C₂×C₄²×C₁₀₁

C₂³

yes

characteristic

C₄×C₈×C₁₀₁

—

#95×C₁₀₁

—

#95×C₁₀₁

—

#95×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

C₂×C₈

yes

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₈

yes

—

C₄²×C₁₀₁

M₁₆

yes

—

C₂³×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₈

yes

—

C₂×C₄×C₁₀₁

#101

yes

—

C₂×C₄×C₁₀₁

#104

yes

—

C₂×C₄×C₁₀₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₀₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₀₁

#135

yes

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₈

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂²×C₁₀₁

C₂×#104

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#330

yes

—

#7568

C₁₀₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₁

#1461

yes

characteristic

C₂×C₁₀₁

#1476

yes

—

C₂×C₁₀₁

#1500

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

characteristic

#1744

C₂×C₁₀₁

yes

—

#1271

C₂×C₁₀₁

yes

—

C₁₀₁

#7568

yes

characteristic, direct product factor, Sylow

C₂×#125

C₄×C₁₀₁

yes

characteristic

C₂×C₄×C₈

—

100

#389

C₂²×C₁₀₁

yes

—

101

#387

—

102

#382

C₄×C₁₀₁

yes

characteristic

103

C₄×C₄⋊C₄

—

104

C₄×C₄⋊C₄

C₄×C₁₀₁

yes

—

105

C₄³

C₂²×C₁₀₁

yes

characteristic

106

#336

—

107

#336

C₂²×C₁₀₁

yes

—

108

C₂²×C₈

—

109

#125

—

110

#125

C₈×C₁₀₁

yes

complemented

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂×C₄×C₁₀₁

yes

—

114

C₂×C₄⋊C₄

C₂×C₄×C₁₀₁

yes

—

115

C₂×C₄²

—

116

C₂×C₄²

C₂×C₄×C₁₀₁

yes

characteristic

117

C₂×C₄²

C₂×C₄×C₁₀₁

yes

characteristic

118

C₂×C₄²

D₄×C₁₀₁

yes

—

119

C₂×C₄²

D₄×C₁₀₁

yes

—

120

C₂×C₄²

C₂³×C₁₀₁

yes

characteristic

121

C₄×C₈

—

122

#95

—

123

#95

—

124

#95

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂²⋊C₄×C₁₀₁

yes

characteristic

128

C₂²×C₄

C₂²⋊C₄×C₁₀₁

yes

—

129

C₂²×C₄

C₂²⋊C₄×C₁₀₁

yes

characteristic

130

C₂²×C₄

C₂²×C₄×C₁₀₁

yes

characteristic

131

C₂²×C₄

C₂×D₄×C₁₀₁

yes

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

C₂×C₈×C₁₀₁

yes

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

C₂×C₈×C₁₀₁

yes

—

146

C₄²

M₁₆×C₁₀₁

yes

—

147

C₂³

C₂×C₂²⋊C₄×C₁₀₁

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₂²⋊C₈×C₁₀₁

yes

—

163

C₂×C₄

#101×C₁₀₁

yes

—

164

C₂×C₄

#104×C₁₀₁

yes

—

165

C₂×C₄

C₂²×C₈×C₁₀₁

yes

characteristic

166

C₂×C₄

C₂×M₁₆×C₁₀₁

yes

characteristic

167

C₂×C₄

#135×C₁₀₁

yes

—

168

C₈

—

169

C₂²

C₂×C₂²⋊C₈×C₁₀₁

yes

characteristic

170

C₂²

#410×C₁₀₁

yes

characteristic

171

C₂²

C₂×#101×C₁₀₁

yes

characteristic

172

C₂²

#416×C₁₀₁

yes

—

173

C₂²

C₂×#104×C₁₀₁

yes

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

#330×C₁₀₁

yes

—

182

C₂

C₁₀₁×#1461

yes

characteristic

183

C₂

C₁₀₁×#1476

yes

—

184

C₂

C₁₀₁×#1500

yes

—

185

C₂

C₂×#330×C₁₀₁

yes

characteristic

186

C₂

C₁₀₁×#1544

yes

characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₁	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₀₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₁
socle, prime radical	C₂³×C₁₀₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₄³×C₁₀₁ ◁ G

subgroup

G/H

center

C₂²×C₄×C₁₀₁

C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₈×C₁₀₁

Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₁

socle, prime radical

C₂³×C₁₀₁

C₂×C₂²⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₄³×C₁₀₁ ◁ G

N	H ≅ G/N	type
16. #125×C₁₀₁	C₈	semidirect
88. #7568	C₁₀₁	direct (N × H)
97. C₁₀₁	#7568	direct (N × H)
110. #125	C₈×C₁₀₁	semidirect

H ≅ G/N

type

16. #125×C₁₀₁

C₈

semidirect

88. #7568

C₁₀₁

direct (N × H)

97. C₁₀₁

#7568

direct (N × H)

110. #125

C₈×C₁₀₁

semidirect

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

subgroup	index
C₄×#125×C₁₀₁	2
C₁₀₁×#1744	2
C₁₀₁×#1271	2
#7568	101

subgroup

index

C₄×#125×C₁₀₁

C₁₀₁×#1744

C₁₀₁×#1271

#7568

101

Subgroups — C₁₀₁×#7568 #38789719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C101×#7568 #38789719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#7568 #38789719