Group #41041163 — subgroups

Subgroups — C₁₀₇×#15199 #41041163

Order 27392.

1606 conjugacy classes of subgroups; only the 558 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 558 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×#391×C₁₀₇	C₂	1	3	yes	maximal
2	C₂×#384×C₁₀₇	C₂	1	3	yes	maximal
3	C₁₀₇×#1444	C₂	1	8	yes	maximal
4	C₄×#95×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₁₀₇	C₂²	1	3	yes	—
7	C₂²×C₄⋊C₄×C₁₀₇	C₂²	1	12	yes	—
8	C₂²×C₄²×C₁₀₇	C₂²	1	3	yes	—
9	#391×C₁₀₇	—	2	12	no	—
10	#384×C₁₀₇	—	2	12	no	—
11	C₂×#95×C₁₀₇	C₂²	1	12	yes	—
12	C₂×#95×C₁₀₇	C₂²	1	4	yes	—
13	C₂²×Q₈×C₁₀₇	—	2	6	no	—
14	C₂²×Q₈×C₁₀₇	—	2	6	no	—
15	C₂²×Q₈×C₁₀₇	Q₈	1	4	yes	—
16	C₂³×C₄×C₁₀₇	C₂³	1	6	yes	—
17	C₂³×C₄×C₁₀₇	C₂³	1	2	yes	—
18	C₂³×C₄×C₁₀₇	C₂³	1	6	yes	—
19	C₂³×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₀₇	—	2	24	no	—
21	C₂×C₄⋊C₄×C₁₀₇	—	2	12	no	—
22	C₂×C₄⋊C₄×C₁₀₇	—	2	6	no	—
23	C₂×C₄⋊C₄×C₁₀₇	—	4	24	no	—
24	C₂×C₄²×C₁₀₇	—	2	12	no	—
25	C₂×C₄²×C₁₀₇	D₄	1	12	yes	—
26	#95×C₁₀₇	—	2	24	no	—
27	#95×C₁₀₇	Q₈	1	16	yes	complemented
28	C₂⁴×C₁₀₇	C₂⁴	1	1	yes	characteristic
29	C₂×Q₈×C₁₀₇	—	2	24	no	—
30	C₂×Q₈×C₁₀₇	—	4	12	no	—
31	C₂×Q₈×C₁₀₇	—	4	4	no	—
32	C₂²×C₄×C₁₀₇	—	2	12	no	—
33	C₂²×C₄×C₁₀₇	—	2	12	no	—
34	C₂²×C₄×C₁₀₇	—	2	6	no	—
35	C₂²×C₄×C₁₀₇	—	2	6	no	—
36	C₂²×C₄×C₁₀₇	—	2	6	no	—
37	C₂²×C₄×C₁₀₇	—	2	12	no	—
38	C₂²×C₄×C₁₀₇	—	2	24	no	—
39	C₂²×C₄×C₁₀₇	C₂×D₄	1	12	yes	—
40	C₂²×C₄×C₁₀₇	C₂×D₄	1	6	yes	—
41	C₂²×C₄×C₁₀₇	C₂×Q₈	1	4	yes	—
42	C₂²×C₄×C₁₀₇	C₂×Q₈	1	24	yes	—
43	C₂²×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
44	C₂²×C₄×C₁₀₇	#41	1	6	yes	—
45	C₄⋊C₄×C₁₀₇	—	4	48	no	—
46	C₄⋊C₄×C₁₀₇	—	4	12	no	—
47	C₄²×C₁₀₇	—	2	24	no	—
48	C₂³×C₁₀₇	C₂²×D₄	1	3	yes	—
49	C₂³×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
50	C₂³×C₁₀₇	C₂²×Q₈	1	4	yes	—
51	C₂³×C₁₀₇	C₂×#41	1	3	yes	—
52	C₂³×C₁₀₇	H(5,𝔽₂)	1	4	yes	—
53	Q₈×C₁₀₇	—	4	16	no	—
54	C₂×C₄×C₁₀₇	—	2	12	no	—
55	C₂×C₄×C₁₀₇	—	2	12	no	—
56	C₂×C₄×C₁₀₇	—	2	48	no	—
57	C₂×C₄×C₁₀₇	—	2	12	no	—
58	C₂×C₄×C₁₀₇	—	4	6	no	—
59	C₂×C₄×C₁₀₇	—	4	2	no	—
60	C₂×C₄×C₁₀₇	—	4	6	no	—
61	C₂×C₄×C₁₀₇	—	4	24	no	—
62	C₂×C₄×C₁₀₇	—	4	12	no	—
63	C₂×C₄×C₁₀₇	C₂²⋊Q₈	1	12	yes	—
64	C₂×C₄×C₁₀₇	#123	1	12	yes	—
65	C₂×C₄×C₁₀₇	#127	1	4	yes	—
66	C₂×C₄×C₁₀₇	C₄⋊Q₈	1	24	yes	—
67	C₂²×C₁₀₇	C₂×C₂²⋊Q₈	1	3	yes	—
68	C₂²×C₁₀₇	C₂×#123	1	3	yes	—
69	C₂²×C₁₀₇	C₂×#127	1	1	yes	characteristic
70	C₂²×C₁₀₇	C₂×C₄⋊Q₈	1	6	yes	—
71	C₂²×C₁₀₇	#535	1	6	yes	—
72	C₂²×C₁₀₇	#554	1	12	yes	—
73	C₂²×C₁₀₇	Q₈²	1	4	yes	—
74	C₄×C₁₀₇	—	2	24	no	—
75	C₄×C₁₀₇	—	4	4	no	—
76	C₄×C₁₀₇	—	8	6	no	—
77	C₄×C₁₀₇	#399	1	8	yes	—
78	#15199	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×C₁₀₇	C₂×#399	1	1	yes	characteristic
80	C₂×C₁₀₇	#2502	1	6	yes	—
81	C₂×C₁₀₇	#2567	1	6	yes	—
82	C₂×C₁₀₇	#2657	1	1	yes	characteristic
83	C₂×C₁₀₇	#2868	1	1	yes	characteristic
84	C₂×#391	C₂×C₁₀₇	1	3	yes	—
85	C₂×#384	C₂×C₁₀₇	1	3	yes	—
86	#1444	C₂×C₁₀₇	1	8	yes	—
87	C₄×#95	C₂×C₁₀₇	1	1	yes	characteristic
88	C₁₀₇	#15199	1	1	yes	characteristic, direct product factor, Sylow
89	C₂³×Q₈	C₂²×C₁₀₇	1	1	yes	characteristic
90	C₂²×C₄⋊C₄	C₂²×C₁₀₇	1	3	yes	—
91	C₂²×C₄⋊C₄	C₂²×C₁₀₇	1	12	yes	—
92	C₂²×C₄²	C₂²×C₁₀₇	1	3	yes	—
93	#391	—	2	12	no	—
94	#384	—	2	12	no	—
95	C₂×#95	C₂²×C₁₀₇	1	12	yes	—
96	C₂×#95	C₂²×C₁₀₇	1	4	yes	—
97	C₂²×Q₈	—	2	6	no	—
98	C₂²×Q₈	—	2	6	no	—
99	C₂²×Q₈	Q₈×C₁₀₇	1	4	yes	—
100	C₂³×C₄	C₂³×C₁₀₇	1	6	yes	—
101	C₂³×C₄	C₂³×C₁₀₇	1	2	yes	—
102	C₂³×C₄	C₂³×C₁₀₇	1	6	yes	—
103	C₂³×C₄	C₂³×C₁₀₇	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	2	24	no	—
105	C₂×C₄⋊C₄	—	2	12	no	—
106	C₂×C₄⋊C₄	—	2	6	no	—
107	C₂×C₄⋊C₄	—	4	24	no	—
108	C₂×C₄²	—	2	12	no	—
109	C₂×C₄²	D₄×C₁₀₇	1	12	yes	—
110	#95	—	2	24	no	—
111	#95	Q₈×C₁₀₇	1	16	yes	complemented
112	C₂⁴	C₂⁴×C₁₀₇	1	1	yes	characteristic
113	C₂×Q₈	—	2	24	no	—
114	C₂×Q₈	—	4	12	no	—
115	C₂×Q₈	—	4	4	no	—
116	C₂²×C₄	—	2	12	no	—
117	C₂²×C₄	—	2	12	no	—
118	C₂²×C₄	—	2	6	no	—
119	C₂²×C₄	—	2	6	no	—
120	C₂²×C₄	—	2	6	no	—
121	C₂²×C₄	—	2	12	no	—
122	C₂²×C₄	—	2	24	no	—
123	C₂²×C₄	C₂×D₄×C₁₀₇	1	12	yes	—
124	C₂²×C₄	C₂×D₄×C₁₀₇	1	6	yes	—
125	C₂²×C₄	C₂×Q₈×C₁₀₇	1	4	yes	—
126	C₂²×C₄	C₂×Q₈×C₁₀₇	1	24	yes	—
127	C₂²×C₄	C₂×Q₈×C₁₀₇	1	2	yes	—
128	C₂²×C₄	#41×C₁₀₇	1	6	yes	—
129	C₄⋊C₄	—	4	48	no	—
130	C₄⋊C₄	—	4	12	no	—
131	C₄²	—	2	24	no	—
132	C₂³	C₂²×D₄×C₁₀₇	1	3	yes	—
133	C₂³	C₂²×Q₈×C₁₀₇	1	1	yes	characteristic
134	C₂³	C₂²×Q₈×C₁₀₇	1	4	yes	—
135	C₂³	C₂×#41×C₁₀₇	1	3	yes	—
136	C₂³	H(5,𝔽₂)×C₁₀₇	1	4	yes	—
137	Q₈	—	4	16	no	—
138	C₂×C₄	—	2	12	no	—
139	C₂×C₄	—	2	12	no	—
140	C₂×C₄	—	2	48	no	—
141	C₂×C₄	—	2	12	no	—
142	C₂×C₄	—	4	6	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	6	no	—
145	C₂×C₄	—	4	24	no	—
146	C₂×C₄	—	4	12	no	—
147	C₂×C₄	C₂²⋊Q₈×C₁₀₇	1	12	yes	—
148	C₂×C₄	#123×C₁₀₇	1	12	yes	—
149	C₂×C₄	#127×C₁₀₇	1	4	yes	—
150	C₂×C₄	C₄⋊Q₈×C₁₀₇	1	24	yes	—
151	C₂²	C₂×C₂²⋊Q₈×C₁₀₇	1	3	yes	—
152	C₂²	C₂×#123×C₁₀₇	1	3	yes	—
153	C₂²	C₂×#127×C₁₀₇	1	1	yes	characteristic
154	C₂²	C₂×C₄⋊Q₈×C₁₀₇	1	6	yes	—
155	C₂²	#535×C₁₀₇	1	6	yes	—
156	C₂²	#554×C₁₀₇	1	12	yes	—
157	C₂²	Q₈²×C₁₀₇	1	4	yes	—
158	C₄	—	2	24	no	—
159	C₄	—	4	4	no	—
160	C₄	—	8	6	no	—
161	C₄	#399×C₁₀₇	1	8	yes	—
162	C₂	C₂×#399×C₁₀₇	1	1	yes	characteristic
163	C₂	C₁₀₇×#2502	1	6	yes	—
164	C₂	C₁₀₇×#2567	1	6	yes	—
165	C₂	C₁₀₇×#2657	1	1	yes	characteristic
166	C₂	C₁₀₇×#2868	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#391×C₁₀₇

C₂

yes

maximal

C₂×#384×C₁₀₇

C₂

yes

maximal

C₁₀₇×#1444

C₂

yes

maximal

C₄×#95×C₁₀₇

C₂

yes

maximal, characteristic

C₂³×Q₈×C₁₀₇

C₂²

yes

characteristic

C₂²×C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂²×C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂²×C₄²×C₁₀₇

C₂²

yes

—

#391×C₁₀₇

—

#384×C₁₀₇

—

C₂×#95×C₁₀₇

C₂²

yes

—

C₂×#95×C₁₀₇

C₂²

yes

—

C₂²×Q₈×C₁₀₇

—

C₂²×Q₈×C₁₀₇

—

C₂²×Q₈×C₁₀₇

Q₈

yes

—

C₂³×C₄×C₁₀₇

C₂³

yes

—

C₂³×C₄×C₁₀₇

C₂³

yes

—

C₂³×C₄×C₁₀₇

C₂³

yes

—

C₂³×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄²×C₁₀₇

—

C₂×C₄²×C₁₀₇

D₄

yes

—

#95×C₁₀₇

—

#95×C₁₀₇

Q₈

yes

complemented

C₂⁴×C₁₀₇

C₂⁴

yes

characteristic

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂×D₄

yes

—

C₂²×C₄×C₁₀₇

C₂×D₄

yes

—

C₂²×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂²×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂²×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂²×C₄×C₁₀₇

#41

yes

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₂³×C₁₀₇

C₂²×D₄

yes

—

C₂³×C₁₀₇

C₂²×Q₈

yes

characteristic

C₂³×C₁₀₇

C₂²×Q₈

yes

—

C₂³×C₁₀₇

C₂×#41

yes

—

C₂³×C₁₀₇

H(5,𝔽₂)

yes

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₀₇

#123

yes

—

C₂×C₄×C₁₀₇

#127

yes

—

C₂×C₄×C₁₀₇

C₄⋊Q₈

yes

—

C₂²×C₁₀₇

C₂×C₂²⋊Q₈

yes

—

C₂²×C₁₀₇

C₂×#123

yes

—

C₂²×C₁₀₇

C₂×#127

yes

characteristic

C₂²×C₁₀₇

C₂×C₄⋊Q₈

yes

—

C₂²×C₁₀₇

#535

yes

—

C₂²×C₁₀₇

#554

yes

—

C₂²×C₁₀₇

Q₈²

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#399

yes

—

#15199

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

C₂×#399

yes

characteristic

C₂×C₁₀₇

#2502

yes

—

C₂×C₁₀₇

#2567

yes

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

C₂×#384

C₂×C₁₀₇

yes

—

#1444

C₂×C₁₀₇

yes

—

C₄×#95

C₂×C₁₀₇

yes

characteristic

C₁₀₇

#15199

yes

characteristic, direct product factor, Sylow

C₂³×Q₈

C₂²×C₁₀₇

yes

characteristic

C₂²×C₄⋊C₄

C₂²×C₁₀₇

yes

—

C₂²×C₄⋊C₄

C₂²×C₁₀₇

yes

—

C₂²×C₄²

C₂²×C₁₀₇

yes

—

#391

—

#384

—

C₂×#95

C₂²×C₁₀₇

yes

—

C₂×#95

C₂²×C₁₀₇

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

Q₈×C₁₀₇

yes

—

100

C₂³×C₄

C₂³×C₁₀₇

yes

—

101

C₂³×C₄

C₂³×C₁₀₇

yes

—

102

C₂³×C₄

C₂³×C₁₀₇

yes

—

103

C₂³×C₄

C₂³×C₁₀₇

yes

characteristic

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

—

109

C₂×C₄²

D₄×C₁₀₇

yes

—

110

#95

—

111

#95

Q₈×C₁₀₇

yes

complemented

112

C₂⁴

C₂⁴×C₁₀₇

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×D₄×C₁₀₇

yes

—

124

C₂²×C₄

C₂×D₄×C₁₀₇

yes

—

125

C₂²×C₄

C₂×Q₈×C₁₀₇

yes

—

126

C₂²×C₄

C₂×Q₈×C₁₀₇

yes

—

127

C₂²×C₄

C₂×Q₈×C₁₀₇

yes

—

128

C₂²×C₄

#41×C₁₀₇

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄²

—

132

C₂³

C₂²×D₄×C₁₀₇

yes

—

133

C₂³

C₂²×Q₈×C₁₀₇

yes

characteristic

134

C₂³

C₂²×Q₈×C₁₀₇

yes

—

135

C₂³

C₂×#41×C₁₀₇

yes

—

136

C₂³

H(5,𝔽₂)×C₁₀₇

yes

—

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₂²⋊Q₈×C₁₀₇

yes

—

148

C₂×C₄

#123×C₁₀₇

yes

—

149

C₂×C₄

#127×C₁₀₇

yes

—

150

C₂×C₄

C₄⋊Q₈×C₁₀₇

yes

—

151

C₂²

C₂×C₂²⋊Q₈×C₁₀₇

yes

—

152

C₂²

C₂×#123×C₁₀₇

yes

—

153

C₂²

C₂×#127×C₁₀₇

yes

characteristic

154

C₂²

C₂×C₄⋊Q₈×C₁₀₇

yes

—

155

C₂²

#535×C₁₀₇

yes

—

156

C₂²

#554×C₁₀₇

yes

—

157

C₂²

Q₈²×C₁₀₇

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#399×C₁₀₇

yes

—

162

C₂

C₂×#399×C₁₀₇

yes

characteristic

163

C₂

C₁₀₇×#2502

yes

—

164

C₂

C₁₀₇×#2567

yes

—

165

C₂

C₁₀₇×#2657

yes

characteristic

166

C₂

C₁₀₇×#2868

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₀₇	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₀₇
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂⁴×C₁₀₇

C₂⁴

commutator, Frattini, cosocle

C₂⁴

C₂⁴×C₁₀₇

LCS

G ▷ C₂⁴ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₁₀₇ ◁ G

N	H ≅ G/N	type
27. #95×C₁₀₇	Q₈	semidirect
78. #15199	C₁₀₇	direct (N × H)
88. C₁₀₇	#15199	direct (N × H)
111. #95	Q₈×C₁₀₇	semidirect

H ≅ G/N

type

27. #95×C₁₀₇

Q₈

semidirect

78. #15199

C₁₀₇

direct (N × H)

88. C₁₀₇

#15199

direct (N × H)

111. #95

Q₈×C₁₀₇

semidirect

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

subgroup	index
C₂×#391×C₁₀₇	2
C₂×#384×C₁₀₇	2
C₁₀₇×#1444	2
C₄×#95×C₁₀₇	2
#15199	107

subgroup

index

C₂×#391×C₁₀₇

C₂×#384×C₁₀₇

C₁₀₇×#1444

C₄×#95×C₁₀₇

#15199

107

Subgroups — C₁₀₇×#15199 #41041163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#15199 #41041163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#15199 #41041163