Finite Group Databasebeta

← #28338129

Subgroups — #28338129

Order 13696.

286 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 55 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
1#20293195C211yesmaximal, characteristic
2#99×D107C212yesmaximal, complemented
3#20292422C211yesmaximal, characteristic, complemented
4#20292419C212yesmaximal, complemented
5#354×C107C211yesmaximal, characteristic, complemented
6C2×#13133819C411yescharacteristic, complemented
7C2×D4×D107C2211yescharacteristic
8#16812106C411yescharacteristic, complemented
9#16812105C2211yescharacteristic
10#16812098C411yescharacteristic, complemented
11#16812082—22no—
12#16812079—22no—
13#16812066C411yescharacteristic, complemented
14C22⋊C4×D107—22no—
15#16812024C2212yes—
16#16812023—22no—
17#124×C107C2211yescharacteristic
18#99×C107C2212yes—
19C23×D107D412yes—
20#13133819—42no—
21C2×D428—21no—
22C2×D428—22no—
23C2×D428C2×C411yescharacteristic
24C2×D428C2×C411yescharacteristic, complemented
25C2×C22⋊D107—22no—
26C2×C22⋊D107D412yes—
27D4×D107—42no—
28C2×C4×D107—21no—
29C2×C4×D107—42no—
30C2×C4×D107C2×C411yescharacteristic
31Dic107⋊C4—21no—
32C22⋊Dic107—22no—
33(C2×D107)⋊C4—21no—
34(C2×D107)⋊C4—22no—
35(C2×D107)⋊C4—42no—
36C4×Dic107C2×C411yescharacteristic, complemented
37C2×Q8×C107C2×C411yescharacteristic, complemented
38C2×D4×C107C2311yescharacteristic
39C22⋊C4×C107—22no—
40C22⋊C4×C107—22no—
41C42×C107C2×C411yescharacteristic, complemented
42C23×C107C2×D412yes—
43C22×D107—22no—
44C22×D107—22no—
45C22×D107—21no—
46C22×D107—82no—
47C22×D107C22⋊C412yes—
48C22×D107C22⋊C411yescharacteristic
49Q8×C107—41no—
50D4×C107—41no—
51C2×C4×C107—21no—
52C2×C4×C107—21no—
53C2×C4×C107—42no—
54C2×C4×C107C22×C411yescharacteristic
55C22⋊D107—42no—
56C22⋊D107—82no—
57C2×Dic107—21no—
58C2×Dic107—42no—
59C2×Dic107C22⋊C411yescharacteristic
60D428—41no—
61D428—41no—
62D428—42no—
63D428—82no—
64C4×D107—22no—
65C4×D107—42no—
66C4×D107—84no—
67C22×C107—22no—
68C22×C107—42no—
69C22×C107C2×C22⋊C411yescharacteristic
70C2×D107—22no—
71C2×D107—41no—
72C2×D107—44no—
73C2×D107—42no—
74C2×D107—42no—
75C2×D107—41no—
76C2×D107—82no—
77C2×D107#9912yes—
78C4×C107—21no—
79C4×C107—41no—
80C4×C107—41no—
81C4×C107—82no—
82Dic107—21no—
83Dic107—41no—
84Dic107—82no—
85C2×C107—21no—
86C2×C107—42no—
87C2×C107C2×#9911yescharacteristic
88D107—22no—
89D107—42no—
90D107—81no—
91#2127—1071nomaximal, Sylow
92C107#212711yescharacteristic, complemented, Sylow
93#535—1071no—
94C2×#99—1072no—
95#354—1071no—
96#354D10711yescharacteristic, complemented
97#353—1072no—
98C2×#41—1071no—
99C22×D4—1071no—
100#127—1071no—
101#127—1071no—
102#124—1071no—
103#124C2×D10711yescharacteristic
104#121—2142no—
105#120—2142no—
106#117—1071no—
107C2×C22⋊C4—2142no—
108#99—1072no—
109#99—2142no—
110#99C2×D10712yes—
111C24—1072no—
112#41—4282no—
113C2×Q8C4×D10711yescharacteristic, complemented
114C2×D4—1072no—
115C2×D4—1071no—
116C2×D4—1071no—
117C2×D4—4282no—
118C2×D4—2142no—
119C2×D4—2141no—
120C2×D4—2142no—
121C2×D4C22×D10711yescharacteristic
122C22×C4—1071no—
123C22×C4—4282no—
124C22×C4—2141no—
125C4⋊C4—2141no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—4282no—
129C22⋊C4—2141no—
130C22⋊C4—2142no—
131C22⋊C4—2142no—
132C42—1071no—
133C42C4×D10711yescharacteristic, complemented
134C23—8562no—
135C23—1071no—
136C23—1072no—
137C23—2142no—
138C23—2142no—
139C23—2141no—
140C23D4×D10712yes—
141Q8—41no—
142D4—41no—
143D4—8562no—
144D4—8562no—
145D4—4281no—
146D4—4281no—
147D4—4282no—
148D4—4282no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—8564no—
153C2×C4—1071no—
154C2×C4—4282no—
155C2×C4—4282no—
156C2×C4—2142no—
157C2×C4—2141no—
158C2×C4C2×C4×D10711yescharacteristic
159C22—22no—
160C22—42no—
161C22—8562no—
162C22—1072no—
163C22—4282no—
164C22—4282no—
165C22—4281no—
166C22—4284no—
167C22—4281no—
168C22—2142no—
169C22C22⋊C4×D10711yescharacteristic
170C4—21no—
171C4—41no—
172C4—41no—
173C4—82no—
174C4—8562no—
175C4—4281no—
176C4—2141no—
177C2—21no—
178C2—42no—
179C2—8561no—
180C2—4282no—
181C2—2142no—
182C2#99×D10711yescharacteristic
subgroupG/H
centerC2#99×D107
commutatorC2×C4×C107C22×C4
FrattiniC2×D4C22×D107
Fitting#354×C107C2
socleC2×C107C2×#99
prime radical#20293195C2
cosocleC2×D4×C107C23
hypercenter#354D107
hypocenterC107#2127
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #99×D107C2semidirect
3. #20292422C2semidirect
4. #20292419C2semidirect
5. #354×C107C2semidirect
6. C2×#13133819C4semidirect
8. #16812106C4semidirect
10. #16812098C4semidirect
13. #16812066C4semidirect
24. C2×D428C2×C4semidirect
36. C4×Dic107C2×C4semidirect
37. C2×Q8×C107C2×C4semidirect
41. C42×C107C2×C4semidirect
92. C107#2127semidirect
96. #354D107semidirect
113. C2×Q8C4×D107semidirect
133. C42C4×D107semidirect

Maximal subgroups

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

subgroupindex
#202931952
#99×D1072
#202924222
#202924192
#354×C1072
#2127107