Finite Group Databasebeta

← #28335648

Subgroups — #28335648

Order 13696.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293363C211yesmaximal, characteristic
2#20292973C211yesmaximal, characteristic
3#20292913C211yesmaximal, characteristic, complemented
4#20292838C211yesmaximal, characteristic, complemented
5#20292357C211yesmaximal, characteristic, complemented
6#20292357C211yesmaximal, characteristic, complemented
7#331×C107C211yesmaximal, characteristic
8C2×Dic428—21no—
9#16812109C2211yescharacteristic
10#16812096—21no—
11#16812096—21no—
12#16812090—22no—
13C4×Dic214C2211yescharacteristic
14C4×Dic214C2211yescharacteristic
15D4×Dic107—21no—
16D4×Dic107—21no—
17#16812041—21no—
18#16812034—21no—
19#16812034—21no—
20#16812030—22no—
21C4×C107⋊C8C2211yescharacteristic
22#127×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4⋊C8×C107C2211yescharacteristic
25C2×Dic214—21no—
26C2×Dic214D411yescharacteristic
27C2×Dic214D411yescharacteristic
28C22×Dic107—22no—
29Dic428—42no—
30C2×C107⋊C8—21no—
31C4⋊Dic107—21no—
32C4⋊Dic107—21no—
33C4⋊Dic107D411yescharacteristic
34C4⋊Dic107D411yescharacteristic
35Dic107⋊C4—21no—
36Dic107⋊C4—21no—
37C22⋊Dic107—22no—
38C22⋊Dic107—22no—
39C22⋊Dic107—22no—
40C4×Dic107—21no—
41C4×Dic107—21no—
42C2×D4×C107—21no—
43C2×D4×C107—21no—
44C2×D4×C107D412yes—
45C2×C8×C107—21no—
46C2×C8×C107—21no—
47C42×C107C2311yescharacteristic
48C23×C107—22no—
49D4×C107—42no—
50D4×C107—41no—
51D4×C107—41no—
52D4×C107—41no—
53D4×C107—41no—
54C2×C4×C107C2×D411yescharacteristic
55C2×C4×C107C2×D411yescharacteristic
56C2×C4×C107C2×D411yescharacteristic
57C2×Dic107—21no—
58C2×Dic107—21no—
59C2×Dic107—21no—
60C2×Dic107—21no—
61C2×Dic107—41no—
62C2×Dic107—82no—
63Dic214—21no—
64Dic214—21no—
65Dic214—21no—
66Dic214—21no—
67Dic214—41no—
68Dic214—41no—
69C8×C107—41no—
70C8×C107—41no—
71C107⋊C8—41no—
72C22×C107—42no—
73C22×C107—42no—
74C22×C107—42no—
75C22×C107#12011yescharacteristic
76C4×C107—21no—
77C4×C107#13511yescharacteristic
78C4×C107#13511yescharacteristic
79C4×C107AGL(1,ℤ/8)11yescharacteristic
80C4×C107#13711yescharacteristic
81Dic107—41no—
82Dic107—41no—
83Dic107—41no—
84Dic107—41no—
85Dic107—81no—
86C2×C107—82no—
87C2×C107#44911yescharacteristic
88C2×C107#45211yescharacteristic
89C2×C107#45511yescharacteristic
90#1640—1071nomaximal, Sylow
91C107#164011yescharacteristic, complemented, Sylow
92#559—1071no—
93#486—1071no—
94#475—1071no—
95#462—1071no—
96#331D10711yescharacteristic
97Q8⋊C8—1071no—
98Q8⋊C8—1071no—
99C2×Q16—2141no—
100C4⋊Q8—1071no—
101#127C2×D10711yescharacteristic
102#124—2141no—
103#124—2141no—
104#123—2142no—
105C4×Q8—1071no—
106C4×Q8—1071no—
107C4×D4—2141no—
108C4×D4—2141no—
109#106—2141no—
110C4⋊C8C2×D10711yescharacteristic
111C4⋊C8C2×D10711yescharacteristic
112#103—2141no—
113#103—2141no—
114#102—2142no—
115C4×C8—1071no—
116C2×Q8—1071no—
117C2×Q8—1071no—
118C2×Q8—2141no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4D42812yes—
122C22×C4—2142no—
123Q16—4282no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—2141no—
127C4⋊C4—1071no—
128C4⋊C4—1071no—
129C4⋊C4—2141no—
130C4⋊C4—2141no—
131C4⋊C4—2141no—
132C4⋊C4—2141no—
133C22⋊C4—2142no—
134C22⋊C4—2142no—
135C22⋊C4—2142no—
136C42—2141no—
137C42—2141no—
138C42C22×D10711yescharacteristic
139C23—22no—
140Q8—4281no—
141Q8—4281no—
142Q8—2141no—
143Q8—2141no—
144Q8—2141no—
145Q8—2141no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—8562no—
152C2×C4—4281no—
153C2×C4—2141no—
154C2×C4—2141no—
155C2×C4—2141no—
156C2×C4—2141no—
157C2×C4D4×D10711yescharacteristic
158C2×C4D4×D10711yescharacteristic
159C2×C4C2×D42811yescharacteristic
160C8—41no—
161C8—41no—
162C8—4281no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1681207911yescharacteristic
167C4—21no—
168C4—8561no—
169C4—4281no—
170C4—4281no—
171C4—4281no—
172C4—4281no—
173C4#1681212811yescharacteristic
174C4#1681213011yescharacteristic
175C4#1681213311yescharacteristic
176C4#1681213811yescharacteristic
177C2—82no—
178C2#2029277311yescharacteristic
179C2#2029279011yescharacteristic
180C2#2029280711yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#331×C107C2
socleC22×C107#120
prime radical#127×C107C22
hypercenter#331D107
hypocenterC107#1640
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
3. #20292913C2semidirect
4. #20292838C2semidirect
5. #20292357C2semidirect
6. #20292357C2semidirect
91. C107#1640semidirect

Maximal subgroups

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

subgroupindex
#202933632
#202929732
#202929132
#202928382
#202923572
#331×C1072
#1640107