Finite Group Databasebeta

← #28344512

Subgroups — #28344512

Order 13696.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812101C211yesmaximal, characteristic, complemented
2#20292563C212yesmaximal, complemented
3#20292552C212yesmaximal, complemented
4#20292550C214yesmaximal, complemented
5#20292546C212yesmaximal, complemented
6#20292541C212yesmaximal
7#20292493C211yesmaximal, characteristic, complemented
8C2×#124×C107C211yesmaximal, characteristic
9#16812101—24no—
10C2×C4⋊Dic107C2211yescharacteristic
11C2×C4⋊Dic107C2211yescharacteristic
12C2×C4⋊Dic107C2212yes—
13C2×Dic107⋊C4C2212yes—
14C2×Dic107⋊C4C2212yes—
15C2×Dic107⋊C4C2214yes—
16C2×C22⋊Dic107C2214yes—
17C2×C22⋊Dic107C2214yes—
18C107⋊#95C2212yes—
19C107⋊#95C2212yes—
20C107⋊#95C2214yes—
21C22×Q8×C107C2211yescharacteristic
22C22×D4×C107C2211yescharacteristic
23#124×C107—24no—
24C2×C22⋊C4×C107C2212yes—
25C2×C22⋊C4×C107C2212yes—
26C2×C42×C107C2211yescharacteristic
27C22×Dic107C2312yes—
28C22×Dic107C2312yes—
29C22×Dic107C2314yes—
30C4⋊Dic107—22no—
31C4⋊Dic107—22no—
32C4⋊Dic107—42no—
33Dic107⋊C4—24no—
34Dic107⋊C4—24no—
35Dic107⋊C4—44no—
36C22⋊Dic107—44no—
37C22⋊Dic107—44no—
38C24×C107C2312yes—
39C2×Q8×C107—22no—
40C2×Q8×C107—42no—
41C2×D4×C107—22no—
42C2×D4×C107—42no—
43C22×C4×C107C2312yes—
44C22×C4×C107C2311yescharacteristic
45C22×C4×C107C2311yescharacteristic
46C22×C4×C107C2311yescharacteristic
47C22⋊C4×C107—24no—
48C22⋊C4×C107—24no—
49C42×C107D414yes—
50C23×C107—22no—
51C23×C107—24no—
52C23×C107—22no—
53C23×C107—24no—
54C23×C107—22no—
55C23×C107C2411yescharacteristic
56Q8×C107—44no—
57D4×C107—44no—
58C2×C4×C107—22no—
59C2×C4×C107—21no—
60C2×C4×C107—22no—
61C2×C4×C107—21no—
62C2×C4×C107—22no—
63C2×C4×C107—24no—
64C2×C4×C107C2×D412yes—
65C2×C4×C107C2×D412yes—
66C2×C4×C107C2×D412yes—
67C2×Dic107—22no—
68C2×Dic107—22no—
69C2×Dic107—22no—
70C2×Dic107—24no—
71C2×Dic107—24no—
72C2×Dic107—24no—
73C2×Dic107—22no—
74C2×Dic107—24no—
75C22×C107—42no—
76C22×C107—42no—
77C22×C107—42no—
78C22×C107—44no—
79C22×C107—44no—
80C22×C107C22×D411yescharacteristic
81C22×C107H(5,𝔽2)12yes—
82C22×C107H(5,𝔽2)11yescharacteristic
83C22×C1072−1+412yes—
84C22×C1072−1+411yescharacteristic
85C4×C107—22no—
86C4×C107—22no—
87C4×C107—22no—
88C4×C107—42no—
89Dic107—42no—
90Dic107—42no—
91Dic107—44no—
92C2×C107—82no—
93C2×C107#53511yescharacteristic
94C2×C107#53611yescharacteristic
95C2×C107#53711yescharacteristic
96C2×C107#56212yes—
97C2×C107#56312yes—
98#2820—1071nomaximal, Sylow
99C107#282011yescharacteristic, complemented, Sylow
100C2×#125—1071no—
101C2×#124D10711yescharacteristic
102#399—1072no—
103#397—1072no—
104#396—1074no—
105#395—1072no—
106#394—1072no—
107#382—1071no—
108C22×Q8C2×D10711yescharacteristic
109C22×D4C2×D10711yescharacteristic
110#125—2144no—
111#124—24no—
112C2×C4⋊C4—1074no—
113C2×C4⋊C4—1072no—
114C2×C4⋊C4—1072no—
115C2×C4⋊C4—1072no—
116C2×C4⋊C4—1071no—
117C2×C4⋊C4—1071no—
118C2×C22⋊C4—1074no—
119C2×C22⋊C4—1074no—
120C2×C22⋊C4C2×D10712yes—
121C2×C22⋊C4C2×D10712yes—
122C2×C42C2×D10711yescharacteristic
123#95—1072no—
124#95—1072no—
125#95—1074no—
126C24C22×D10712yes—
127C2×Q8—22no—
128C2×Q8—42no—
129C2×D4—22no—
130C2×D4—42no—
131C22×C4—1074no—
132C22×C4—1072no—
133C22×C4—1072no—
134C22×C4C22×D10712yes—
135C22×C4C22×D10711yescharacteristic
136C22×C4C22×D10711yescharacteristic
137C22×C4C22×D10711yescharacteristic
138C4⋊C4—4284no—
139C4⋊C4—4282no—
140C4⋊C4—2144no—
141C4⋊C4—2144no—
142C4⋊C4—2142no—
143C4⋊C4—2142no—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—4284no—
147C22⋊C4—4284no—
148C42C22⋊D10714yes—
149C23—22no—
150C23—24no—
151C23—22no—
152C23—22no—
153C23—24no—
154C23C23×D10711yescharacteristic
155Q8—44no—
156D4—44no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—2142no—
164C2×C4—2142no—
165C2×C4—2144no—
166C2×C4—2144no—
167C2×C4—2142no—
168C2×C4—2144no—
169C2×C4—2142no—
170C2×C4—2144no—
171C2×C4C2×C22⋊D10712yes—
172C2×C4C2×C22⋊D10712yes—
173C2×C4C2×C22⋊D10712yes—
174C22—44no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22—44no—
179C22C22×C22⋊D10711yescharacteristic
180C22#1681215412yes—
181C22#1681215511yescharacteristic
182C22#1681215611yescharacteristic
183C22#1681215612yes—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—42no—
188C4—4282no—
189C4—4282no—
190C4—4284no—
191C2—82no—
192C2#2029319411yescharacteristic
193C2#2029319811yescharacteristic
194C2#2029320511yescharacteristic
195C2#2029337112yes—
196C2#2029337712yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#124×C107C2
prime radicalC22×D4×C107C22
hypercenterC2×#124D107
hypocenterC107#2820
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16812101C2semidirect
2. #20292563C2semidirect
3. #20292552C2semidirect
4. #20292550C2semidirect
5. #20292546C2semidirect
7. #20292493C2semidirect
99. C107#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#168121012
#202925632
#202925522
#202925502
#202925462
#202925412
#202924932
C2×#124×C1072
#2820107