Finite Group Databasebeta

← #28337510

Subgroups — #28337510

Order 13696.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#16812142C212yesmaximal, complemented
2#20293208C211yesmaximal, characteristic
3C2×#16812035C212yesmaximal, complemented
4#20292532C211yesmaximal, characteristic, complemented
5#343×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C2212yes—
7C22×Dic214C2211yescharacteristic
8#16812142—24no—
9#16812142—24no—
10C2×Dic428—22no—
11C2×#13133807—22no—
12#16812089—22no—
13#16812084—22no—
14#16812083—22no—
15C2×C4×Dic107C2211yescharacteristic
16#16812035—24no—
17C107⋊#95—22no—
18C2×M16×C107C2212yes—
19C2×C4⋊C4×C107C2211yescharacteristic
20#13133818—22no—
21#13133818—22no—
22#13133818—22no—
23#13133818—22no—
24C2×Dic214—22no—
25C2×Dic214—22no—
26C2×Dic214—21no—
27C2×Dic214D412yes—
28C2×Dic214D412yescomplemented
29C2×D428—22no—
30C2×D428D412yes—
31C2×C22⋊D107—22no—
32C22×Dic107—21no—
33C22×Dic107—21no—
34C2×C4×D107—22no—
35Dic428—28no—
36#13133807—28no—
37C4⋊Dic107—22no—
38(C2×D107)⋊C4—24no—
39C4×Dic107—22no—
40C22×C4×C107D412yes—
41C22×C4×C107C2311yescharacteristic
42M16×C107—22no—
43M16×C107D414yescomplemented
44C2×C8×C107—22no—
45C4⋊C4×C107—22no—
46C23×C107C2×D411yescharacteristic
47C22×D107—22no—
48C2×C4×C107—22no—
49C2×C4×C107—42no—
50C2×C4×C107C2×D411yescharacteristic
51C2×C4×C107C2×D412yes—
52C2×C4×C107C2×D412yes—
53C2×C4×C107#4111yescharacteristic
54C22⋊D107—42no—
55C22⋊D107—42no—
56C2×Dic107—22no—
57C2×Dic107—22no—
58C2×Dic107—22no—
59C2×Dic107—41no—
60C2×Dic107—41no—
61C2×Dic107—41no—
62C2×Dic107—41no—
63D428—22no—
64D428—22no—
65D428—82no—
66C4×D107—42no—
67C4×D107—42no—
68Dic214—24no—
69Dic214—22no—
70Dic214—22no—
71Dic214—42no—
72C8×C107—24no—
73C22×C107—21no—
74C22×C107—21no—
75C22×C107#12011yescharacteristic
76C22×C107#12111yescharacteristic
77C22×C107#12711yescharacteristic
78C2×D107—42no—
79C2×D107—42no—
80C2×D107—42no—
81C4×C107—42no—
82C4×C107#12012yes—
83C4×C107#12112yes—
84Dic107—42no—
85Dic107—42no—
86Dic107—42no—
87C2×C107—21no—
88C2×C107—21no—
89C2×C107#39211yescharacteristic
90C2×C107#45511yescharacteristic
91C2×C107#45611yescharacteristic
92D107—82no—
93#2007—1071nomaximal, Sylow
94C107#200711yescharacteristic, complemented, Sylow
95C2×#137—1072no—
96#537—1071no—
97C2×#104—1072no—
98#391—1071no—
99#343D10711yescharacteristic, complemented
100C2×#41—1072no—
101C22×Q8—1071no—
102#137—2144no—
103#137—2144no—
104C2×Q16—2142no—
105C2×SD16—2142no—
106C2×M16C2×D10712yes—
107C22⋊Q8—2142no—
108#121—2142no—
109#121—2142no—
110C2×C4⋊C4C2×D10711yescharacteristic
111C2×C42—1071no—
112#104—2144no—
113#95—2142no—
114#41—2142no—
115#41—2142no—
116#41—2142no—
117#41—2142no—
118C2×Q8—1072no—
119C2×Q8—1072no—
120C2×Q8—2142no—
121C2×Q8—2142no—
122C2×Q8—2141no—
123C2×D4—1072no—
124C2×D4—2142no—
125C2×D4—2142no—
126C22×C4—2141no—
127C22×C4—2141no—
128C22×C4—2142no—
129C22×C4D42812yes—
130C22×C4C22×D10711yescharacteristic
131Q16—2148no—
132SD16—2148no—
133M16—22no—
134M16D42814yescomplemented
135C2×C8—22no—
136C4⋊C4—22no—
137C4⋊C4—2142no—
138C22⋊C4—2144no—
139C42—2142no—
140C23—2142no—
141C23C2×D42811yescharacteristic
142Q8—4282no—
143Q8—2144no—
144Q8—2142no—
145Q8—2142no—
146D4—8562no—
147D4—4282no—
148D4—4282no—
149D4—2142no—
150D4—2142no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—4281no—
154C2×C4—4281no—
155C2×C4—4281no—
156C2×C4—4281no—
157C2×C4—4282no—
158C2×C4—4282no—
159C2×C4—2142no—
160C2×C4—2142no—
161C2×C4—2142no—
162C2×C4D4×D10711yescharacteristic
163C2×C4D4×D10712yes—
164C2×C4C2×D42812yes—
165C2×C4#1313381911yescharacteristic
166C8—24no—
167C22—21no—
168C22—21no—
169C22—4282no—
170C22—4282no—
171C22—4282no—
172C22#1681207911yescharacteristic
173C22#1681208411yescharacteristic
174C22#1681210611yescharacteristic
175C4—42no—
176C4—4282no—
177C4—4282no—
178C4—4282no—
179C4#1681207912yes—
180C4#1681208412yes—
181C2—21no—
182C2—21no—
183C2—8562no—
184C2#2029253711yescharacteristic
185C2#2029280711yescharacteristic
186C2#2029281211yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#343×C107C2
socleC22×C107#120
prime radical#20293208C2
hypercenter#343D107
hypocenterC107#2007
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
3. C2×#16812035C2semidirect
4. #20292532C2semidirect
5. #343×C107C2semidirect
28. C2×Dic214D4semidirect
43. M16×C107D4semidirect
94. C107#2007semidirect
99. #343D107semidirect
134. M16D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
#202932082
C2×#168120352
#202925322
#343×C1072
#2007107