Finite Group Databasebeta

← #28344511

Subgroups — #28344511

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×#16812097C211yesmaximal, characteristic
2#20292562C212yesmaximal, complemented
3#20292553C212yesmaximal, complemented
4#20292549C214yesmaximal, complemented
5#20292547C212yesmaximal, complemented
6#20292544C212yesmaximal
7#20292493C211yesmaximal, characteristic, complemented
8C2×#125×C107C211yesmaximal, characteristic, complemented
9C22×Dic214C2211yescharacteristic
10C22×D428C2211yescharacteristic
11#16812097—24no—
12C2×C4⋊Dic107C2212yes—
13C2×Dic107⋊C4C2214yes—
14C2×(C2×D107)⋊C4C2212yes—
15C2×(C2×D107)⋊C4C2212yes—
16C2×(C2×D107)⋊C4C2214yes—
17C2×(C2×D107)⋊C4C2214yes—
18C107⋊#95C2212yes—
19C107⋊#95C2212yes—
20C107⋊#95C2214yes—
21#125×C107—24no—
22C2×C4⋊C4×C107C2212yes—
23C2×C4⋊C4×C107C2212yes—
24C2×C4⋊C4×C107C2211yescharacteristic
25C2×C4⋊C4×C107C2211yescharacteristic
26C2×C42×C107C2211yescharacteristic
27C23×D107C2312yes—
28C2×Dic214—22no—
29C2×Dic214—42no—
30C2×D428—22no—
31C2×D428—42no—
32C22×Dic107C2312yes—
33C22×Dic107C2314yes—
34C4⋊Dic107—42no—
35Dic107⋊C4—44no—
36(C2×D107)⋊C4—24no—
37(C2×D107)⋊C4—24no—
38(C2×D107)⋊C4—44no—
39(C2×D107)⋊C4—44no—
40C22×C4×C107C2312yes—
41C22×C4×C107C2312yes—
42C22×C4×C107C2311yescharacteristic
43C22×C4×C107C2311yescharacteristic
44C22×C4×C107C2311yescharacteristic
45C4⋊C4×C107—24no—
46C4⋊C4×C107—22no—
47C4⋊C4×C107—22no—
48C4⋊C4×C107—24no—
49C42×C107D414yes—
50C23×C107C2411yescharacteristic
51C22×D107—22no—
52C22×D107—22no—
53C22×D107—24no—
54C22×D107—22no—
55C22×D107—24no—
56C2×C4×C107—21no—
57C2×C4×C107—22no—
58C2×C4×C107—21no—
59C2×C4×C107—22no—
60C2×C4×C107—22no—
61C2×C4×C107—22no—
62C2×C4×C107—22no—
63C2×C4×C107—24no—
64C2×C4×C107—22no—
65C2×C4×C107C2×D412yes—
66C2×C4×C107C2×D412yes—
67C2×C4×C107C2×D412yes—
68C2×Dic107—22no—
69C2×Dic107—24no—
70C2×Dic107—24no—
71C2×Dic107—24no—
72C2×Dic107—24no—
73D428—44no—
74Dic214—44no—
75C22×C107C22×D411yescharacteristic
76C22×C107H(5,𝔽2)12yes—
77C22×C107H(5,𝔽2)11yescharacteristic
78C22×C1072−1+411yescharacteristic
79C22×C1072−1+412yes—
80C2×D107—42no—
81C2×D107—42no—
82C2×D107—42no—
83C2×D107—44no—
84C2×D107—44no—
85C4×C107—22no—
86C4×C107—22no—
87C4×C107—22no—
88C4×C107—42no—
89C4×C107—42no—
90Dic107—42no—
91Dic107—44no—
92C2×C107#53511yescharacteristic
93C2×C107#53611yescharacteristic
94C2×C107#53711yescharacteristic
95C2×C107#56212yes—
96C2×C107#56312yes—
97D107—82no—
98#2820—1071nomaximal, Sylow
99C107#282011yescharacteristic, complemented, Sylow
100C2×#125D10711yescharacteristic, complemented
101C2×#124—1071no—
102#399—1072no—
103#397—1072no—
104#396—1074no—
105#395—1072no—
106#394—1072no—
107#382—1071no—
108C22×Q8—1071no—
109C22×D4—1071no—
110#125—24no—
111#124—2144no—
112C2×C4⋊C4—1074no—
113C2×C4⋊C4—1072no—
114C2×C4⋊C4C2×D10712yes—
115C2×C4⋊C4C2×D10712yes—
116C2×C4⋊C4C2×D10711yescharacteristic
117C2×C4⋊C4C2×D10711yescharacteristic
118C2×C22⋊C4—1074no—
119C2×C22⋊C4—1074no—
120C2×C22⋊C4—1072no—
121C2×C22⋊C4—1072no—
122C2×C42C2×D10711yescharacteristic
123#95—1072no—
124#95—1072no—
125#95—1074no—
126C24—1072no—
127C2×Q8—4282no—
128C2×Q8—2142no—
129C2×D4—4282no—
130C2×D4—2142no—
131C22×C4—1074no—
132C22×C4—1072no—
133C22×C4C22×D10712yes—
134C22×C4C22×D10712yes—
135C22×C4C22×D10711yescharacteristic
136C22×C4C22×D10711yescharacteristic
137C22×C4C22×D10711yescharacteristic
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—4284no—
143C4⋊C4—4282no—
144C22⋊C4—4284no—
145C22⋊C4—4284no—
146C22⋊C4—2144no—
147C22⋊C4—2144no—
148C42C22⋊D10714yes—
149C23—2142no—
150C23—2142no—
151C23—2144no—
152C23—2142no—
153C23—2144no—
154C23C23×D10711yescharacteristic
155Q8—4284no—
156D4—4284no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—2142no—
167C2×C4—2144no—
168C2×C4—2144no—
169C2×C4—2144no—
170C2×C4—2144no—
171C2×C4C2×C22⋊D10712yes—
172C2×C4C2×C22⋊D10712yes—
173C2×C4C2×C22⋊D10712yes—
174C22—4284no—
175C22—4282no—
176C22—4282no—
177C22—4282no—
178C22—4284no—
179C22C22×C22⋊D10711yescharacteristic
180C22#1681215512yes—
181C22#1681215511yescharacteristic
182C22#1681215611yescharacteristic
183C22#1681215712yes—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—42no—
188C4—42no—
189C4—4282no—
190C4—4284no—
191C2—8562no—
192C2#2029319111yescharacteristic
193C2#2029320111yescharacteristic
194C2#2029320511yescharacteristic
195C2#2029337512yes—
196C2#2029338112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#125×C107C2
prime radicalC22×D428C22
hypercenterC2×#125D107
hypocenterC107#2820
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #20292562C2semidirect
3. #20292553C2semidirect
4. #20292549C2semidirect
5. #20292547C2semidirect
7. #20292493C2semidirect
8. C2×#125×C107C2semidirect
99. C107#2820semidirect
100. C2×#125D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120972
#202925622
#202925532
#202925492
#202925472
#202925442
#202924932
C2×#125×C1072
#2820107