Finite Group Databasebeta

← #28335403

Subgroups — #28335403

Order 13696.

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

normal subgroups only

This group has 266 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#20293258C211yesmaximal, characteristic, complemented
2#20292780C212yesmaximal
3#20292544C211yesmaximal, characteristic
4#20292365C212yesmaximal, complemented
5#324×C107C211yesmaximal, characteristic, complemented
6C22×D428C2211yescharacteristic
7C2×#13133807—22no—
8#16812108—22no—
9#16812100—22no—
10#16812088—21no—
11#16812088C2212yes—
12#16812068—21no—
13C2×(C2×D107)⋊C4—21no—
14C2×(C2×D107)⋊C4—21no—
15#16812031—22no—
16C107⋊#95C2211yescharacteristic
17C2×C4⋊C4×C107C2211yescharacteristic
18C22⋊C8×C107C2212yes—
19C23×D107—21no—
20C2×Dic214—21no—
21C2×Dic214D412yes—
22C2×D428—22no—
23C2×D428—42no—
24C22×Dic107—21no—
25#13133807—44no—
26C4⋊Dic107—21no—
27C4⋊Dic107—21no—
28C4⋊Dic107—21no—
29C4⋊Dic107D412yes—
30Dic107⋊C4—22no—
31Dic107⋊C4—22no—
32C22⋊Dic107—22no—
33(C2×D107)⋊C4—42no—
34(C2×D107)⋊C4—42no—
35C4×Dic107—22no—
36C22×C4×C107D411yescharacteristic
37C22×C4×C107D411yescharacteristic
38C22×C4×C107C2311yescharacteristic
39C2×C8×C107—22no—
40C4⋊C4×C107—22no—
41C23×C107C2×D411yescharacteristic
42C22×D107—22no—
43C22×D107—41no—
44C22×D107—41no—
45C22×D107—41no—
46C22×D107—41no—
47C22×D107—41no—
48C22×D107—41no—
49C2×C4×C107—21no—
50C2×C4×C107—21no—
51C2×C4×C107—21no—
52C2×C4×C107—41no—
53C2×C4×C107—41no—
54C2×C4×C107SD1612yescomplemented
55C2×C4×C107C2×D412yes—
56C2×Dic107—22no—
57C2×Dic107—22no—
58C2×Dic107—41no—
59C2×Dic107—41no—
60C2×Dic107—41no—
61D428—24no—
62D428—42no—
63Dic214—42no—
64Dic214—81no—
65C8×C107—42no—
66C22×C107—21no—
67C22×C107—21no—
68C22×C107#12011yescharacteristic
69C22×C107C2×SD1611yescharacteristic
70C22×C107#13711yescharacteristic
71C2×D107—42no—
72C2×D107—42no—
73C2×D107—42no—
74C2×D107—81no—
75C2×D107—81no—
76C2×D107—81no—
77C2×D107—81no—
78C4×C107—22no—
79C4×C107—22no—
80C4×C107—41no—
81Dic107—42no—
82Dic107—42no—
83Dic107—81no—
84C2×C107—21no—
85C2×C107—21no—
86C2×C107#44811yescharacteristic
87C2×C107#45411yescharacteristic
88C2×C107B(4,𝔽2)11yescharacteristic
89D107—82no—
90#1599—1071nomaximal, Sylow
91C107#159911yescharacteristic, complemented, Sylow
92#544—1071no—
93#450—1072no—
94#394—1071no—
95#328—1072no—
96#324D10711yescharacteristic, complemented
97C22×D4—1071no—
98C2×SD16—2142no—
99C4⋊Q8—2142no—
100#125—2142no—
101C22⋊Q8—1072no—
102C22⋊Q8—2141no—
103#117—2141no—
104C2×C4⋊C4C2×D10711yescharacteristic
105C2×C22⋊C4—2141no—
106C2×C22⋊C4—2141no—
107#102—2142no—
108C22⋊C8C2×D10712yes—
109#95—1071no—
110C24—2141no—
111C2×Q8—1072no—
112C2×Q8—2141no—
113C2×D4—4282no—
114C2×D4—2142no—
115C22×C4—2141no—
116C22×C4D42811yescharacteristic
117C22×C4D42811yescharacteristic
118C22×C4C22×D10711yescharacteristic
119SD16—4284no—
120C2×C8—22no—
121C4⋊C4—22no—
122C4⋊C4—1072no—
123C4⋊C4—2141no—
124C4⋊C4—2141no—
125C4⋊C4—2141no—
126C4⋊C4—2142no—
127C4⋊C4—2142no—
128C22⋊C4—4282no—
129C22⋊C4—4282no—
130C22⋊C4—2142no—
131C42—2142no—
132C23—4281no—
133C23—4281no—
134C23—4281no—
135C23—4281no—
136C23—4281no—
137C23—4281no—
138C23—2142no—
139C23C2×D42811yescharacteristic
140Q8—8561no—
141Q8—4282no—
142D4—4282no—
143D4—2144no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—4281no—
150C2×C4—4281no—
151C2×C4—4281no—
152C2×C4—2142no—
153C2×C4—2142no—
154C2×C4#1313380712yescomplemented
155C2×C4D4×D10712yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—8561no—
160C22—8561no—
161C22—8561no—
162C22—8561no—
163C22—4282no—
164C22—4282no—
165C22—4282no—
166C22#1681207911yescharacteristic
167C22C2×#1313380711yescharacteristic
168C22#1681214211yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—8561no—
173C4—4282no—
174C4—4282no—
175C2—21no—
176C2—21no—
177C2—8562no—
178C2#2029276811yescharacteristic
179C2#2029280211yescharacteristic
180C2#2029281511yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#324×C107C2
socleC22×C107#120
prime radicalC22×D428C22
hypercenter#324D107
hypocenterC107#1599
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #20293258C2semidirect
4. #20292365C2semidirect
5. #324×C107C2semidirect
54. C2×C4×C107SD16semidirect
91. C107#1599semidirect
96. #324D107semidirect
154. C2×C4#13133807semidirect

Maximal subgroups

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

subgroupindex
#202932582
#202927802
#202925442
#202923652
#324×C1072
#1599107