Finite Group Databasebeta

← #28347003

Subgroups — #28347003

Order 13696.

380 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293184C211yesmaximal, characteristic
2#20293033C214yesmaximal, complemented
3#20293030C212yesmaximal, complemented
4#20293023C212yesmaximal, complemented
5#20292667C214yesmaximal
6C4×#13133802C211yesmaximal, characteristic
7C2×C4⋊Q8×C107C211yesmaximal, characteristic
8C2×#13133802C2212yes—
9#16812109C2211yescharacteristic
10#16812109C2211yescharacteristic
11#16812101C2212yes—
12#16812088—21no—
13#16812088—21no—
14C4×Dic214—21no—
15C4×Dic214—21no—
16#16812068C2212yes—
17#16812042C2218yes—
18#16812040C2218yes—
19#16812020C2212yes—
20C4×C107⋊C8C2212yes—
21C22×Q8×C107—21no—
22C4⋊Q8×C107—21no—
23C4⋊Q8×C107—21no—
24C4⋊Q8×C107C2212yes—
25C4⋊Q8×C107C2212yes—
26C2×C4⋊C4×C107—21no—
27C2×C4⋊C4×C107C2212yes—
28C2×C42×C107C2211yescharacteristic
29C2×Dic214—21no—
30C2×Dic214—21no—
31#13133802Q818yescomplemented
32C2×C107⋊C8C2314yes—
33C4⋊Dic107—21no—
34C4⋊Dic107—21no—
35C4⋊Dic107C2312yes—
36C4⋊Dic107C2312yes—
37Dic107⋊C4—22no—
38Dic107⋊C4—22no—
39C22⋊Dic107—22no—
40C4×Dic107—22no—
41C2×Q8×C107—22no—
42C2×Q8×C107—22no—
43C2×Q8×C107—44no—
44C22×C4×C107—22no—
45C22×C4×C107D411yescharacteristic
46C22×C4×C107D411yescharacteristic
47C22×C4×C107C2311yescharacteristic
48C4⋊C4×C107—22no—
49C4⋊C4×C107—22no—
50C4⋊C4×C107—22no—
51C4⋊C4×C107C2314yes—
52C42×C107D411yescharacteristic
53C42×C107D411yescharacteristic
54C42×C107C2311yescharacteristic
55C42×C107C2311yescharacteristic
56C23×C107C2×D411yescharacteristic
57Q8×C107—44no—
58Q8×C107—44no—
59C2×C4×C107—22no—
60C2×C4×C107—22no—
61C2×C4×C107—24no—
62C2×C4×C107—44no—
63C2×C4×C107C2×D411yescharacteristic
64C2×C4×C107C2×D411yescharacteristic
65C2×C4×C107C2×D411yescharacteristic
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107C2×Q814yes—
69C2×C4×C107C2411yescharacteristic
70C2×Dic107—22no—
71C2×Dic107—22no—
72Dic214—42no—
73Dic214—42no—
74C107⋊C8C2×Q818yescomplemented
75C22×C107—21no—
76C22×C107—21no—
77C22×C107C4⋊Q812yes—
78C22×C107C22×D411yescharacteristic
79C4×C107—21no—
80C4×C107—21no—
81C4×C107—44no—
82C4×C107C4⋊Q812yes—
83C4×C107AGL(1,ℤ/8)12yes—
84C4×C107#13712yes—
85C4×C107C22×Q812yes—
86Dic107—42no—
87Dic107—42no—
88C2×C107—22no—
89C2×C107C2×C4⋊Q811yescharacteristic
90C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
91C2×C107C2×#13711yescharacteristic
92#3165—1071nomaximal, Sylow
93C107#316511yescharacteristic, complemented, Sylow
94#533—1071no—
95C2×C4⋊Q8D10711yescharacteristic
96#501—1074no—
97#500—1072no—
98#498—1072no—
99#428—1074no—
100C4×M16—1071no—
101C22×Q8—21no—
102C2×M16—1072no—
103C4⋊Q8—21no—
104C4⋊Q8—21no—
105C4⋊Q8—1071no—
106C4⋊Q8—1071no—
107C4⋊Q8C2×D10712yes—
108C4⋊Q8C2×D10712yes—
109#125—1072no—
110C22⋊Q8—2141no—
111C22⋊Q8—2141no—
112C4×Q8—2141no—
113C4×Q8—2141no—
114#117—1072no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2×D10712yes—
117C2×C42C2×D10711yescharacteristic
118#107—1078no—
119#106—1078no—
120#97—1072no—
121C4×C8—1072no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—44no—
125C2×Q8—2141no—
126C2×Q8—2141no—
127C22×C4—22no—
128C22×C4C22⋊D10711yescharacteristic
129C22×C4C22⋊D10711yescharacteristic
130C22×C4C22×D10711yescharacteristic
131M16—1078no—
132C2×C8—1074no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—1072no—
137C4⋊C4—1072no—
138C4⋊C4—2142no—
139C4⋊C4—2142no—
140C4⋊C4—2141no—
141C4⋊C4—2141no—
142C4⋊C4C22×D10714yes—
143C22⋊C4—2142no—
144C42—2142no—
145C42C22⋊D10711yescharacteristic
146C42C22⋊D10711yescharacteristic
147C42C22×D10711yescharacteristic
148C42C22×D10711yescharacteristic
149C23C2×C22⋊D10711yescharacteristic
150Q8—44no—
151Q8—44no—
152Q8—4282no—
153Q8—4282no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—44no—
158C2×C4—2142no—
159C2×C4—2142no—
160C2×C4C2×C22⋊D10711yescharacteristic
161C2×C4C2×C22⋊D10711yescharacteristic
162C2×C4C2×C22⋊D10711yescharacteristic
163C2×C4C2×C22⋊D10711yescharacteristic
164C2×C4C2×C22⋊D10711yescharacteristic
165C2×C4Q8×D10714yes—
166C2×C4C23×D10711yescharacteristic
167C8—1078no—
168C22—21no—
169C22—21no—
170C22#1681210712yes—
171C22C22×C22⋊D10711yescharacteristic
172C4—21no—
173C4—21no—
174C4—44no—
175C4—4282no—
176C4—4282no—
177C4#1681210712yes—
178C4#1681213512yes—
179C4#1681214012yes—
180C4C2×Q8×D10712yes—
181C2—22no—
182C2C2×#1681210711yescharacteristic
183C2C2×#1681213511yescharacteristic
184C2C2×#1681214011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C4⋊Q8×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenterC2×C4⋊Q8D107
hypocenterC107#3165
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20293033C2semidirect
3. #20293030C2semidirect
4. #20293023C2semidirect
31. #13133802Q8semidirect
74. C107⋊C8C2×Q8semidirect
93. C107#3165semidirect

Maximal subgroups

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

subgroupindex
#202931842
#202930332
#202930302
#202930232
#202926672
C4×#131338022
C2×C4⋊Q8×C1072
#3165107