Finite Group Databasebeta

← #28344519

Subgroups — #28344519

Order 13696.

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

normal subgroups only

This group has 430 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×#16812109C211yesmaximal, characteristic, complemented
2#20292566C212yesmaximal, complemented
3#20292563C212yesmaximal
4#20292557C212yesmaximal, complemented
5#20292552C212yesmaximal
6#20292547C212yesmaximal, complemented
7#20292539C212yesmaximal
8#20292495C211yesmaximal, characteristic, complemented
9C2×#126×C107C211yesmaximal, characteristic
10C22×Dic214C2212yes—
11#16812109—24no—
12C2×C4⋊Dic107C2212yes—
13C2×C4⋊Dic107C2212yes—
14C2×Dic107⋊C4C2214yes—
15C2×C22⋊Dic107C2214yes—
16C107⋊#95C2212yes—
17C107⋊#95C2212yes—
18C107⋊#95C2212yes—
19C107⋊#95C2214yes—
20C107⋊#95C2214yes—
21#126×C107—24no—
22C2×C4⋊C4×C107C2211yescharacteristic
23C2×C4⋊C4×C107C2211yescharacteristic
24C2×C4⋊C4×C107C2211yescharacteristic
25C2×C22⋊C4×C107C2211yescharacteristic
26C2×C22⋊C4×C107C2211yescharacteristic
27C2×C22⋊C4×C107C2211yescharacteristic
28C2×C42×C107C2211yescharacteristic
29C2×Dic214—24no—
30C2×Dic214—44no—
31C22×Dic107C2314yes—
32C22×Dic107C2314yes—
33C4⋊Dic107—24no—
34C4⋊Dic107—24no—
35Dic107⋊C4—44no—
36C22⋊Dic107—44no—
37C24×C107C2311yescharacteristic
38C22×C4×C107C2311yescharacteristic
39C22×C4×C107C2311yescharacteristic
40C22×C4×C107C2311yescharacteristic
41C22×C4×C107C2311yescharacteristic
42C22×C4×C107C2311yescharacteristic
43C22×C4×C107C2311yescharacteristic
44C4⋊C4×C107—22no—
45C4⋊C4×C107—22no—
46C4⋊C4×C107—22no—
47C22⋊C4×C107—22no—
48C22⋊C4×C107—22no—
49C22⋊C4×C107—22no—
50C42×C107D414yes—
51C23×C107—21no—
52C23×C107—22no—
53C23×C107—22no—
54C23×C107—22no—
55C23×C107C2411yescharacteristic
56C2×C4×C107—22no—
57C2×C4×C107—22no—
58C2×C4×C107—22no—
59C2×C4×C107—21no—
60C2×C4×C107—21no—
61C2×C4×C107—21no—
62C2×C4×C107—22no—
63C2×C4×C107—22no—
64C2×C4×C107—22no—
65C2×C4×C107C2×D412yes—
66C2×C4×C107C2×D412yes—
67C2×C4×C107C2×D412yes—
68C2×Dic107—24no—
69C2×Dic107—24no—
70C2×Dic107—24no—
71C2×Dic107—24no—
72C2×Dic107—24no—
73C2×Dic107—24no—
74Dic214—48no—
75C22×C107—41no—
76C22×C107—41no—
77C22×C107—41no—
78C22×C107—44no—
79C22×C107C22×D411yescharacteristic
80C22×C107H(5,𝔽2)12yes—
81C22×C1072−1+412yes—
82C22×C1072−1+412yes—
83C4×C107—22no—
84C4×C107—22no—
85C4×C107—22no—
86C4×C107—41no—
87C4×C107—41no—
88C4×C107—41no—
89Dic107—44no—
90Dic107—44no—
91C2×C107—81no—
92C2×C107#53511yescharacteristic
93C2×C107#53611yescharacteristic
94C2×C107#53611yescharacteristic
95C2×C107#56314yes—
96#2821—1071nomaximal, Sylow
97C107#282111yescharacteristic, complemented, Sylow
98C2×C4⋊Q8—1071no—
99C2×#126D10711yescharacteristic
100#400—1072no—
101#399—1072no—
102#398—1072no—
103#397—1072no—
104#395—1072no—
105#393—1072no—
106#383—1071no—
107C22×Q8—1072no—
108C4⋊Q8—2144no—
109#126—24no—
110C2×C4⋊C4—1074no—
111C2×C4⋊C4—1072no—
112C2×C4⋊C4—1072no—
113C2×C4⋊C4C2×D10711yescharacteristic
114C2×C4⋊C4C2×D10711yescharacteristic
115C2×C4⋊C4C2×D10711yescharacteristic
116C2×C22⋊C4—1074no—
117C2×C22⋊C4C2×D10711yescharacteristic
118C2×C22⋊C4C2×D10711yescharacteristic
119C2×C22⋊C4C2×D10711yescharacteristic
120C2×C42C2×D10711yescharacteristic
121#95—1072no—
122#95—1072no—
123#95—1072no—
124#95—1074no—
125#95—1074no—
126C24C22×D10711yescharacteristic
127C2×Q8—4284no—
128C2×Q8—2144no—
129C22×C4—1074no—
130C22×C4—1074no—
131C22×C4C22×D10711yescharacteristic
132C22×C4C22×D10711yescharacteristic
133C22×C4C22×D10711yescharacteristic
134C22×C4C22×D10711yescharacteristic
135C22×C4C22×D10711yescharacteristic
136C22×C4C22×D10711yescharacteristic
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—4284no—
141C4⋊C4—2144no—
142C4⋊C4—2144no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—4284no—
147C42C22⋊D10714yes—
148C23—21no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23C23×D10711yescharacteristic
153Q8—4288no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—2144no—
164C2×C4—2144no—
165C2×C4—2144no—
166C2×C4—2144no—
167C2×C4—2144no—
168C2×C4—2144no—
169C2×C4C2×C22⋊D10712yes—
170C2×C4C2×C22⋊D10712yes—
171C2×C4C2×C22⋊D10712yes—
172C22—44no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C22×C22⋊D10711yescharacteristic
177C22#1681215512yes—
178C22#1681215612yes—
179C22#1681215612yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—41no—
184C4—41no—
185C4—41no—
186C4—4284no—
187C4—4284no—
188C2—81no—
189C2#2029319111yescharacteristic
190C2#2029319811yescharacteristic
191C2#2029319811yescharacteristic
192C2#2029337914yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#126×C107C2
prime radicalC24×C107C23
hypercenterC2×#126D107
hypocenterC107#2821
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#16812109C2semidirect
2. #20292566C2semidirect
4. #20292557C2semidirect
6. #20292547C2semidirect
8. #20292495C2semidirect
97. C107#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#168121092
#202925662
#202925632
#202925572
#202925522
#202925472
#202925392
#202924952
C2×#126×C1072
#2821107