Finite Group Databasebeta

← #28344496

Subgroups — #28344496

Order 13696.

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

normal subgroups only

This group has 444 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×#16812088C211yesmaximal, characteristic
2#20292566C212yesmaximal, complemented
3#20292565C212yesmaximal, complemented
4#20292555C214yesmaximal
5#20292547C212yesmaximal, complemented
6#20292546C212yesmaximal, complemented
7#20292490C211yesmaximal, characteristic
8C2×C22⋊Q8×C107C211yesmaximal, characteristic
9C22×Dic214C2211yescharacteristic
10#16812088—24no—
11C2×C4⋊Dic107C2211yescharacteristic
12C2×Dic107⋊C4C2212yes—
13C2×Dic107⋊C4C2214yes—
14C2×C22⋊Dic107C2212yes—
15C2×C22⋊Dic107C2212yes—
16C107⋊#95C2212yes—
17C107⋊#95C2212yes—
18C107⋊#95C2214yes—
19C107⋊#95C2214yes—
20C107⋊#95C2214yes—
21C22×Q8×C107C2211yescharacteristic
22C23×C4×C107C2211yescharacteristic
23C22⋊Q8×C107—24no—
24C2×C4⋊C4×C107C2212yes—
25C2×C4⋊C4×C107C2211yescharacteristic
26C2×C22⋊C4×C107C2212yes—
27C2×Dic214—22no—
28C2×Dic214—42no—
29C22×Dic107C2312yes—
30C22×Dic107C2312yes—
31C22×Dic107C2314yes—
32C4⋊Dic107—22no—
33Dic107⋊C4—24no—
34Dic107⋊C4—44no—
35C22⋊Dic107—24no—
36C22⋊Dic107—42no—
37C24×C107C2311yescharacteristic
38C2×Q8×C107—22no—
39C2×Q8×C107—42no—
40C22×C4×C107—42no—
41C22×C4×C107D414yes—
42C22×C4×C107C2312yes—
43C22×C4×C107C2312yes—
44C22×C4×C107C2311yescharacteristic
45C22×C4×C107C2311yescharacteristic
46C4⋊C4×C107—22no—
47C4⋊C4×C107—24no—
48C22⋊C4×C107—24no—
49C23×C107—22no—
50C23×C107—22no—
51C23×C107—22no—
52C23×C107C2×D412yes—
53C23×C107C2411yescharacteristic
54Q8×C107—44no—
55C2×C4×C107—22no—
56C2×C4×C107—22no—
57C2×C4×C107—22no—
58C2×C4×C107—22no—
59C2×C4×C107—24no—
60C2×C4×C107—24no—
61C2×C4×C107—44no—
62C2×C4×C107C2×D412yes—
63C2×C4×C107C2×D412yes—
64C2×Dic107—22no—
65C2×Dic107—22no—
66C2×Dic107—24no—
67C2×Dic107—24no—
68C2×Dic107—24no—
69C2×Dic107—24no—
70C2×Dic107—24no—
71Dic214—44no—
72C22×C107—22no—
73C22×C107—22no—
74C22×C107—22no—
75C22×C107—44no—
76C22×C107C22×D411yescharacteristic
77C22×C107H(5,𝔽2)12yes—
78C22×C1072−1+412yes—
79C22×C1072−1+412yes—
80C4×C107—22no—
81C4×C107—22no—
82C4×C107—42no—
83C4×C107—42no—
84Dic107—42no—
85Dic107—42no—
86Dic107—44no—
87C2×C107—42no—
88C2×C107#53411yescharacteristic
89C2×C107#53611yescharacteristic
90C2×C107#53611yescharacteristic
91C2×C107#56314yes—
92#2817—1071nomaximal, Sylow
93C107#281711yescharacteristic, complemented, Sylow
94C2×C22⋊Q8—1071no—
95C2×C22⋊Q8D10711yescharacteristic
96#400—1072no—
97#400—1072no—
98#397—1074no—
99#395—1072no—
100#395—1072no—
101#381—1071no—
102C22×Q8—1071no—
103C22×Q8C2×D10711yescharacteristic
104C23×C4C2×D10711yescharacteristic
105C22⋊Q8—24no—
106C22⋊Q8—2144no—
107C2×C4⋊C4—1074no—
108C2×C4⋊C4—1072no—
109C2×C4⋊C4—1071no—
110C2×C4⋊C4C2×D10712yes—
111C2×C4⋊C4C2×D10711yescharacteristic
112C2×C22⋊C4—1072no—
113C2×C22⋊C4—1072no—
114C2×C22⋊C4C2×D10712yes—
115#95—1072no—
116#95—1072no—
117#95—1074no—
118#95—1074no—
119#95—1074no—
120C24C22×D10711yescharacteristic
121C2×Q8—22no—
122C2×Q8—42no—
123C2×Q8—4282no—
124C2×Q8—2142no—
125C22×C4—42no—
126C22×C4—1074no—
127C22×C4—1072no—
128C22×C4—1072no—
129C22×C4C22⋊D10714yes—
130C22×C4C22×D10712yes—
131C22×C4C22×D10712yes—
132C22×C4C22×D10711yescharacteristic
133C22×C4C22×D10711yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—24no—
136C4⋊C4—4284no—
137C4⋊C4—2144no—
138C4⋊C4—2142no—
139C22⋊C4—24no—
140C22⋊C4—4282no—
141C22⋊C4—2144no—
142C23—22no—
143C23—22no—
144C23—22no—
145C23C2×C22⋊D10712yes—
146C23C23×D10711yescharacteristic
147Q8—44no—
148Q8—4284no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—24no—
155C2×C4—44no—
156C2×C4—2142no—
157C2×C4—2142no—
158C2×C4—2144no—
159C2×C4—2144no—
160C2×C4—2144no—
161C2×C4—2144no—
162C2×C4—2144no—
163C2×C4C2×C22⋊D10712yes—
164C2×C4C2×C22⋊D10712yes—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—44no—
169C22C22×C22⋊D10711yescharacteristic
170C22#1681215412yes—
171C22#1681215612yes—
172C22#1681215712yes—
173C4—22no—
174C4—22no—
175C4—42no—
176C4—42no—
177C4—4284no—
178C4—4282no—
179C4—4282no—
180C2—42no—
181C2#2029318611yescharacteristic
182C2#2029319811yescharacteristic
183C2#2029320111yescharacteristic
184C2#2029337814yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C22⋊Q8×C107C2
prime radicalC24×C107C23
hypercenterC2×C22⋊Q8D107
hypocenterC107#2817
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20292566C2semidirect
3. #20292565C2semidirect
5. #20292547C2semidirect
6. #20292546C2semidirect
93. C107#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#168120882
#202925662
#202925652
#202925552
#202925472
#202925462
#202924902
C2×C22⋊Q8×C1072
#2817107