Finite Group Databasebeta

← #28346279

Subgroups — #28346279

Order 13696.

384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C4×#13133818C211yesmaximal, characteristic
2#20292931C214yesmaximal, complemented
3#20292924C214yesmaximal, complemented
4#20292667C212yesmaximal, complemented
5#20292651C211yesmaximal, characteristic, complemented
6#20292626C212yesmaximal
7C2×#125×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C2×#13133802C2212yes—
10C4×Dic214C2212yes—
11C4×D428C2212yes—
12C4×C22⋊D107—22no—
13#16812068C2211yescharacteristic
14#16812066—21no—
15C42×D107—21no—
16#16812042C2214yes—
17#16812040C2214yes—
18C4⋊(C107⋊C8)C2214yes—
19#16812032C2214yes—
20#16812029C2214yes—
21#125×C107—21no—
22#125×C107—21no—
23#125×C107C2214yes—
24C2×C4⋊C4×C107—22no—
25C2×C4⋊C4×C107C2212yes—
26C2×C42×C107C2211yescharacteristic
27#13133818—22no—
28#13133818Q814yes—
29C2×Dic214C2311yescharacteristic
30C2×D428C2311yescharacteristic
31C2×C22⋊D107—21no—
32C2×C4×D107—21no—
33C2×C4×D107—22no—
34#13133802—24no—
35C2×C107⋊C8C2314yes—
36C4⋊Dic107C2312yes—
37Dic107⋊C4—22no—
38C22⋊Dic107—21no—
39(C2×D107)⋊C4—22no—
40C4×Dic107—21no—
41C4×Dic107—22no—
42C22×C4×C107—22no—
43C22×C4×C107D412yes—
44C22×C4×C107C2311yescharacteristic
45C4⋊C4×C107—24no—
46C4⋊C4×C107—24no—
47C4⋊C4×C107—22no—
48C4⋊C4×C107C2314yes—
49C42×C107D411yescharacteristic
50C42×C107D411yescharacteristic
51C42×C107C2312yes—
52C23×C107C2×D411yescharacteristic
53C22×D107—21no—
54C2×C4×C107—24no—
55C2×C4×C107—42no—
56C2×C4×C107—44no—
57C2×C4×C107C2×D412yes—
58C2×C4×C107C2×D412yes—
59C2×C4×C107C2×D411yescharacteristic
60C2×C4×C107C2×Q812yes—
61C2×C4×C107#4112yes—
62C2×C4×C107C2411yescharacteristic
63C22⋊D107—24no—
64C2×Dic107—21no—
65C2×Dic107—22no—
66D428—21no—
67D428C2×Q812yes—
68C4×D107—24no—
69C4×D107—44no—
70Dic214—21no—
71Dic214C2×Q812yes—
72C107⋊C8—24no—
73C22×C107—22no—
74C22×C107C22⋊Q812yes—
75C22×C107C22×D411yescharacteristic
76C2×D107—22no—
77C2×D107—42no—
78C4×C107—22no—
79C4×C107—22no—
80C4×C107—44no—
81C4×C107C22⋊Q812yes—
82C4×C107C22×Q811yescharacteristic
83C4×C107C2×#4111yescharacteristic
84Dic107—22no—
85Dic107—42no—
86C2×C107—22no—
87C2×C107C2×C22⋊Q811yescharacteristic
88C2×C107#57512yes—
89D107—42no—
90#3080—1071nomaximal, Sylow
91C107#308011yescharacteristic, complemented, Sylow
92C2×#125D10711yescharacteristic, complemented
93C4×#41—1071no—
94#479—1074no—
95#478—1074no—
96#428—1072no—
97#423—1071no—
98#417—1072no—
99C2×#41—1071no—
100C2×M16—1072no—
101#125—21no—
102#125—21no—
103#125C2×D10714yes—
104C4×Q8—1072no—
105C4×D4—1072no—
106C4×D4—2142no—
107#117—1071no—
108#117—2141no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4C2×D10712yes—
111C2×C42—2141no—
112C2×C42C2×D10711yescharacteristic
113#107—1074no—
114#106—1074no—
115C4⋊C8—1074no—
116#103—1074no—
117#102—1074no—
118#41—1074no—
119#41—2142no—
120C2×Q8—1071no—
121C2×D4—1071no—
122C2×D4—2141no—
123C22×C4—22no—
124C22×C4—2142no—
125C22×C4—2141no—
126C22×C4C22⋊D10712yes—
127C22×C4C22×D10711yescharacteristic
128M16—2144no—
129C2×C8—1074no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—1072no—
134C4⋊C4—2142no—
135C4⋊C4C22×D10714yes—
136C22⋊C4—2141no—
137C22⋊C4—2142no—
138C42—2141no—
139C42—2142no—
140C42C22⋊D10711yescharacteristic
141C42C22⋊D10711yescharacteristic
142C42C22×D10712yes—
143C23—2141no—
144C23C2×C22⋊D10711yescharacteristic
145Q8—1072no—
146Q8—2141no—
147D4—1072no—
148D4—2144no—
149D4—2141no—
150C2×C4—24no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—4284no—
154C2×C4—2142no—
155C2×C4—2141no—
156C2×C4—2144no—
157C2×C4C2×C22⋊D10712yes—
158C2×C4C2×C22⋊D10712yes—
159C2×C4C2×C22⋊D10711yescharacteristic
160C2×C4Q8×D10712yes—
161C2×C4#1313381912yes—
162C2×C4C23×D10711yescharacteristic
163C8—2144no—
164C22—22no—
165C22—4282no—
166C22—2142no—
167C22#1681208512yes—
168C22C22×C22⋊D10711yescharacteristic
169C4—22no—
170C4—22no—
171C4—44no—
172C4—4282no—
173C4—2142no—
174C4#1681208512yes—
175C4C2×Q8×D10711yescharacteristic
176C4C2×#1313381911yescharacteristic
177C2—22no—
178C2—4282no—
179C2C2×#1681208511yescharacteristic
180C2#2029342512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#125×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenterC2×#125D107
hypocenterC107#3080
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #20292931C2semidirect
3. #20292924C2semidirect
4. #20292667C2semidirect
5. #20292651C2semidirect
7. C2×#125×C107C2semidirect
91. C107#3080semidirect
92. C2×#125D107semidirect

Maximal subgroups

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

subgroupindex
C4×#131338182
#202929312
#202929242
#202926672
#202926512
#202926262
C2×#125×C1072
#3080107