Finite Group Databasebeta

← #28338134

Subgroups — #28338134

Order 13696.

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

normal subgroups only

This group has 238 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#20293197C211yesmaximal, characteristic
2#20292599C212yesmaximal
3#20292422C211yesmaximal, characteristic, complemented
4#20292421C212yesmaximal, complemented
5#354×C107C211yesmaximal, characteristic, complemented
6C2×#13133817C2211yescharacteristic
7C2×Q8×D107C411yescharacteristic, complemented
8#16812109C411yescharacteristic, complemented
9#16812107C411yescharacteristic, complemented
10#16812096C2211yescharacteristic
11#16812091—22no—
12#16812086—22no—
13#16812066C411yescharacteristic, complemented
14#16812065—22no—
15#16812024C2212yes—
16#16812023—22no—
17#124×C107C2211yescharacteristic
18#99×C107C2212yes—
19#13133817—42no—
20C2×Dic214—21no—
21C2×Dic214C2×C411yescharacteristic
22C2×Dic214C2×C411yescharacteristic
23Q8×D107—42no—
24C2×C22⋊D107D412yes—
25C22×Dic107D412yes—
26C2×C4×D107—21no—
27C2×C4×D107C2×C411yescharacteristic
28C4⋊Dic107—22no—
29Dic107⋊C4—21no—
30Dic107⋊C4—22no—
31Dic107⋊C4—42no—
32C22⋊Dic107—22no—
33C22⋊Dic107—22no—
34(C2×D107)⋊C4—21no—
35C4×Dic107—42no—
36C4×Dic107C2×C411yescharacteristic
37C2×Q8×C107C2×C411yescharacteristic
38C2×D4×C107C2311yescharacteristic
39C22⋊C4×C107—22no—
40C22⋊C4×C107—22no—
41C42×C107C2×C411yescharacteristic
42C23×C107C2×D412yes—
43C22×D107C22⋊C411yescharacteristic
44Q8×C107—41no—
45D4×C107—41no—
46C2×C4×C107—21no—
47C2×C4×C107—21no—
48C2×C4×C107—42no—
49C2×C4×C107C22×C411yescharacteristic
50C22⋊D107—42no—
51C2×Dic107—22no—
52C2×Dic107—22no—
53C2×Dic107—21no—
54C2×Dic107—21no—
55C2×Dic107—42no—
56C2×Dic107C22⋊C412yes—
57C2×Dic107C22⋊C411yescharacteristic
58C4×D107—22no—
59C4×D107—42no—
60Dic214—41no—
61Dic214—42no—
62Dic214—41no—
63C22×C107—22no—
64C22×C107—42no—
65C22×C107C2×C22⋊C411yescharacteristic
66C2×D107—21no—
67C2×D107—41no—
68C4×C107—21no—
69C4×C107—41no—
70C4×C107—41no—
71C4×C107—82no—
72Dic107—22no—
73Dic107—41no—
74Dic107—41no—
75Dic107—82no—
76Dic107#9912yes—
77C2×C107—21no—
78C2×C107—42no—
79C2×C107C2×#9911yescharacteristic
80D107—41no—
81#2129—1071nomaximal, Sylow
82C107#212911yescharacteristic, complemented, Sylow
83#536—1071no—
84#410—1072no—
85#354—1071no—
86#354—1072no—
87#354D10711yescharacteristic, complemented
88C2×#41—1071no—
89C22×Q8—1071no—
90C4⋊Q8—1071no—
91C4⋊Q8—1071no—
92#124—1071no—
93#124C2×D10711yescharacteristic
94#123—2142no—
95C22⋊Q8—2142no—
96#117—1071no—
97#117—2142no—
98#99—1072no—
99#99—2142no—
100#99C2×D10712yes—
101#41—4282no—
102C2×Q8—1071no—
103C2×Q8—1071no—
104C2×Q8—4282no—
105C2×Q8—2141no—
106C2×Q8C4×D10711yescharacteristic
107C2×D4—1072no—
108C2×D4C22×D10711yescharacteristic
109C22×C4—1072no—
110C22×C4—1071no—
111C22×C4—2141no—
112C4⋊C4—4282no—
113C4⋊C4—2141no—
114C4⋊C4—2142no—
115C4⋊C4—2142no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—2141no—
119C22⋊C4—2142no—
120C22⋊C4—2142no—
121C42—1071no—
122C42—4282no—
123C42C4×D10711yescharacteristic
124C23—1071no—
125C23D4×D10712yes—
126Q8—41no—
127Q8—4281no—
128Q8—4281no—
129Q8—4282no—
130D4—41no—
131D4—4282no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—1071no—
136C2×C4—1072no—
137C2×C4—4282no—
138C2×C4—4282no—
139C2×C4—2142no—
140C2×C4—2142no—
141C2×C4—2142no—
142C2×C4—2141no—
143C2×C4—2141no—
144C2×C4C2×C4×D10711yescharacteristic
145C22—22no—
146C22—42no—
147C22—4281no—
148C22—2141no—
149C22C22⋊C4×D10711yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—8562no—
155C4—1072no—
156C4—4281no—
157C4—4281no—
158C4—2142no—
159C2—21no—
160C2—42no—
161C2—4281no—
162C2#99×D10711yescharacteristic
subgroupG/H
centerC2#99×D107
commutatorC2×C4×C107C22×C4
FrattiniC2×D4C22×D107
Fitting#354×C107C2
socleC2×C107C2×#99
prime radicalC2×#13133817C22
cosocleC2×D4×C107C23
hypercenter#354D107
hypocenterC107#2129
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
3. #20292422C2semidirect
4. #20292421C2semidirect
5. #354×C107C2semidirect
7. C2×Q8×D107C4semidirect
8. #16812109C4semidirect
9. #16812107C4semidirect
13. #16812066C4semidirect
82. C107#2129semidirect
87. #354D107semidirect

Maximal subgroups

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

subgroupindex
#202931972
#202925992
#202924222
#202924212
#354×C1072
#2129107