Finite Group Databasebeta

← #28334774

Subgroups — #28334774

Order 13696.

264 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16812067C211yesmaximal, characteristic
2#20292677C212yesmaximal, complemented
3#20292353C211yesmaximal, characteristic, complemented
4#20292350C212yesmaximal
5#324×C107C211yesmaximal, characteristic, complemented
6C22×C4×D107C411yescharacteristic
7#16812067—22no—
8#16812067—22no—
9C2×C4⋊Dic107C411yescharacteristic
10C2×(C2×D107)⋊C4C411yescharacteristic
11C2×(C2×D107)⋊C4C2211yescharacteristic
12C2×C4×Dic107C411yescharacteristic
13C2×C4×Dic107C2211yescharacteristic
14Dic107⋊C8—22no—
15C22⋊(C107⋊C8)C2212yes—
16C8×Dic107—22no—
17C2×C4⋊C4×C107C2211yescharacteristic
18C22⋊C8×C107C2212yes—
19C23×D107C2×C411yescharacteristic
20C22×Dic107C2×C411yescharacteristic
21C22×Dic107C2×C411yescharacteristic
22C22×Dic107C2×C411yescharacteristic
23C2×C4×D107—21no—
24C2×C4×D107—21no—
25C2×C4×D107—22no—
26C2×C4×D107C814yescomplemented
27C2×C107⋊C8—22no—
28C4⋊Dic107—22no—
29(C2×D107)⋊C4—22no—
30(C2×D107)⋊C4—22no—
31C4×Dic107—21no—
32C4×Dic107—21no—
33C4×Dic107D412yes—
34C4×Dic107D412yes—
35C22×C4×C107C2×C411yescharacteristic
36C22×C4×C107C2×C411yescharacteristic
37C22×C4×C107C2311yescharacteristic
38C2×C8×C107—22no—
39C4⋊C4×C107—22no—
40C23×C107C22×C411yescharacteristic
41C22×D107—21no—
42C22×D107—21no—
43C22×D107—41no—
44C22×D107—41no—
45C22×D107C2×C812yes—
46C2×C4×C107—21no—
47C2×C4×C107—21no—
48C2×C4×C107—21no—
49C2×C4×C107—41no—
50C2×C4×C107—41no—
51C2×C4×C107C2×C812yes—
52C2×C4×C107C2×D412yes—
53C2×Dic107—21no—
54C2×Dic107—41no—
55C2×Dic107—41no—
56C2×Dic107C22⋊C412yes—
57C2×Dic107C22⋊C411yescharacteristic
58C2×Dic107C22⋊C411yescharacteristic
59C2×Dic107C2×C812yes—
60C2×Dic107M1612yes—
61C4×D107—24no—
62C4×D107—24no—
63C8×C107—42no—
64C107⋊C8—42no—
65C22×C107—21no—
66C22×C107—21no—
67C22×C107C2×C22⋊C411yescharacteristic
68C22×C107C22×C811yescharacteristic
69C22×C107C2×M1611yescharacteristic
70C2×D107—22no—
71C2×D107—42no—
72C2×D107—41no—
73C2×D107—41no—
74C2×D107—42no—
75C4×C107—22no—
76C4×C107—22no—
77C4×C107—41no—
78Dic107—22no—
79Dic107—41no—
80Dic107C22⋊C812yes—
81Dic107C22⋊C812yes—
82C2×C107—21no—
83C2×C107—21no—
84C2×C107C2×C22⋊C811yescharacteristic
85C2×C107#41011yescharacteristic
86C2×C107#41311yescharacteristic
87D107—42no—
88#1458—1071nomaximal, Sylow
89C107#145811yescharacteristic, complemented, Sylow
90C2×#117—1071no—
91#431—1072no—
92#324—1071no—
93#324D10711yescharacteristic, complemented
94#323—1072no—
95C23×C4—1071no—
96#117—2142no—
97#117—2142no—
98C2×C4⋊C4—1071no—
99C2×C4⋊C4C2×D10711yescharacteristic
100C2×C22⋊C4—1071no—
101C2×C22⋊C4—1071no—
102C2×C42—1071no—
103C2×C42—1071no—
104C4⋊C8—2142no—
105C22⋊C8—1072no—
106C22⋊C8C2×D10712yes—
107C4×C8—2142no—
108C24—1071no—
109C22×C4—1071no—
110C22×C4—1071no—
111C22×C4—1074no—
112C22×C4—1071no—
113C22×C4—2141no—
114C22×C4—2141no—
115C22×C4—2142no—
116C22×C4C4×D10711yescharacteristic
117C22×C4C4×D10711yescharacteristic
118C22×C4C22×D10711yescharacteristic
119C2×C8—22no—
120C2×C8—2142no—
121C4⋊C4—22no—
122C4⋊C4—2142no—
123C22⋊C4—2142no—
124C22⋊C4—2142no—
125C42—1072no—
126C42—1072no—
127C42—2141no—
128C42—2141no—
129C23—1072no—
130C23—4281no—
131C23—4281no—
132C23—2141no—
133C23—2141no—
134C23C2×C4×D10711yescharacteristic
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—1071no—
141C2×C4—1071no—
142C2×C4—1072no—
143C2×C4—1072no—
144C2×C4—1072no—
145C2×C4—4281no—
146C2×C4—4281no—
147C2×C4—2144no—
148C2×C4—2144no—
149C2×C4—2141no—
150C2×C4C8×D10712yes—
151C2×C4D4×D10712yes—
152C8—42no—
153C8—4282no—
154C22—21no—
155C22—21no—
156C22—4282no—
157C22—4282no—
158C22—4281no—
159C22—4281no—
160C22—2142no—
161C22C22⋊C4×D10711yescharacteristic
162C22C2×C8×D10711yescharacteristic
163C22M16×D10711yescharacteristic
164C4—22no—
165C4—22no—
166C4—41no—
167C4—1072no—
168C4—1072no—
169C4—4281no—
170C4—2142no—
171C2—21no—
172C2—21no—
173C2—4282no—
174C2C22⋊C8×D10711yescharacteristic
175C2#2029259911yescharacteristic
176C2#2029261011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D107
commutatorC22×C107C22×C8
FrattiniC22×C4C22×D107
Fitting#324×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC23×D107C2×C4
cosocleC22×C4×C107C23
hypercenter#324D107
hypocenterC107#1458
LCSG ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #20292677C2semidirect
3. #20292353C2semidirect
5. #324×C107C2semidirect
26. C2×C4×D107C8semidirect
89. C107#1458semidirect
93. #324D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120672
#202926772
#202923532
#202923502
#324×C1072
#1458107