Finite Group Databasebeta

← #28337626

Subgroups — #28337626

Order 13696.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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#20293253C211yesmaximal, characteristic, complemented
2#20292482C212yesmaximal
3#20292482C211yesmaximal, characteristic
4#20292393C212yesmaximal, complemented
5#342×C107C211yesmaximal, characteristic, complemented
6C24×D107C2211yescharacteristic
7#16812088—22no—
8#16812086—24no—
9C2×C22⋊Dic107C2211yescharacteristic
10C2×C22⋊Dic107C2212yes—
11C2×(C2×D107)⋊C4—24no—
12C2×(C2×D107)⋊C4—22no—
13C2×C22⋊C4×C107C2212yes—
14C2×C22⋊C4×C107C2211yescharacteristic
15C23×D107—22no—
16C23×D107—21no—
17C23×D107—21no—
18C23×D107—22no—
19C23×D107—44no—
20C23×D107Q812yescomplemented
21C2×Dic214—22no—
22C22×Dic107—21no—
23C22×Dic107D414yes—
24C4⋊Dic107—22no—
25Dic107⋊C4—24no—
26C22⋊Dic107—22no—
27C22⋊Dic107—22no—
28C22⋊Dic107—22no—
29(C2×D107)⋊C4—48no—
30(C2×D107)⋊C4—44no—
31C24×C107C2311yescharacteristic
32C22×C4×C107—22no—
33C22×C4×C107D412yes—
34C22⋊C4×C107—22no—
35C22⋊C4×C107—22no—
36C22⋊C4×C107—22no—
37C23×C107—42no—
38C23×C107C2×D412yes—
39C23×C107C2×D411yescharacteristic
40C23×C107C2×Q811yescharacteristic
41C23×C107#4111yescharacteristic
42C23×C107#4112yes—
43C22×D107—28no—
44C22×D107—24no—
45C22×D107—24no—
46C22×D107—22no—
47C22×D107—44no—
48C22×D107—44no—
49C22×D107—48no—
50C22×D107—44no—
51C22×D107—42no—
52C22×D107—41no—
53C22×D107—42no—
54C22×D107—41no—
55C2×C4×C107—22no—
56C2×C4×C107—42no—
57C2×C4×C107—44no—
58C2×C4×C107—42no—
59C2×Dic107—24no—
60C2×Dic107—44no—
61C2×Dic107—42no—
62C2×Dic107—41no—
63Dic214—82no—
64C22×C107—24no—
65C22×C107—22no—
66C22×C107—41no—
67C22×C107—41no—
68C22×C107—42no—
69C22×C107#12011yescharacteristic
70C22×C107C22⋊Q812yes—
71C22×C107C22⋊Q811yescharacteristic
72C22×C107#12412yes—
73C22×C107#12411yescharacteristic
74C2×D107—28no—
75C2×D107—24no—
76C2×D107—44no—
77C2×D107—48no—
78C2×D107—44no—
79C2×D107—42no—
80C2×D107—42no—
81C2×D107—44no—
82C4×C107—42no—
83C4×C107—82no—
84Dic107—44no—
85Dic107—81no—
86C2×C107—21no—
87C2×C107—22no—
88C2×C107—22no—
89C2×C107—21no—
90C2×C107#39311yescharacteristic
91C2×C107B(4,𝔽2)12yes—
92D107—44no—
93#2029—1071nomaximal, Sylow
94C107#202911yescharacteristic, complemented, Sylow
95#543—1071no—
96#379—1072no—
97#379—1071no—
98#342—1072no—
99#342D10711yescharacteristic, complemented
100C25—1071no—
101C22⋊Q8—2144no—
102C22⋊Q8—2142no—
103C2×C22⋊C4—1072no—
104C2×C22⋊C4—1071no—
105C2×C22⋊C4—2144no—
106C2×C22⋊C4—2142no—
107C2×C22⋊C4C2×D10712yes—
108C2×C22⋊C4C2×D10711yescharacteristic
109C24—1072no—
110C24—4284no—
111C24—2141no—
112C24—2142no—
113C24—2142no—
114C24—2141no—
115C24C22×D10711yescharacteristic
116C2×Q8—2142no—
117C22×C4—22no—
118C22×C4—1074no—
119C22×C4—2141no—
120C22×C4D42812yes—
121C4⋊C4—2144no—
122C4⋊C4—2142no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—4284no—
127C22⋊C4—4288no—
128C22⋊C4—2142no—
129C22⋊C4—2142no—
130C22⋊C4—2142no—
131C23—42no—
132C23—4284no—
133C23—4282no—
134C23—4284no—
135C23—4288no—
136C23—4284no—
137C23—4281no—
138C23—4282no—
139C23—4281no—
140C23—2148no—
141C23—2144no—
142C23—2144no—
143C23—2142no—
144C23D4×D10712yes—
145C23C2×D42811yescharacteristic
146C23Q8×D10711yescharacteristic
147C23#1313381711yescharacteristic
148C23#1313381812yes—
149Q8—8562no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—4282no—
155C2×C4—4281no—
156C2×C4—4284no—
157C2×C4—2144no—
158C22—24no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22—42no—
163C22—4288no—
164C22—4284no—
165C22—4284no—
166C22—4282no—
167C22—4282no—
168C22—4284no—
169C22—2148no—
170C22—2144no—
171C22#1681207911yescharacteristic
172C22#1681208712yes—
173C22#1681208911yescharacteristic
174C22#1681209512yes—
175C22#1681209711yescharacteristic
176C4—42no—
177C4—82no—
178C4—8561no—
179C4—4284no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—4284no—
185C2#2029254011yescharacteristic
186C2#2029281512yes—
subgroupG/H
centerC22#16812079
commutator, cosocleC24×C107C23
FrattiniC24C22×D107
Fitting#342×C107C2
socleC22×C107#120
prime radicalC24×D107C22
hypercenter#342D107
hypocenterC107#2029
LCSG ▷ C24×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #20293253C2semidirect
4. #20292393C2semidirect
5. #342×C107C2semidirect
20. C23×D107Q8semidirect
94. C107#2029semidirect
99. #342D107semidirect

Maximal subgroups

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

subgroupindex
#202932532
#202924822
#202923932
#342×C1072
#2029107