Finite Group Databasebeta

← #28336983

Subgroups — #28336983

Order 13696.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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#20293198C211yesmaximal, characteristic
2#20292637C212yesmaximal, complemented
3#20292601C211yesmaximal, characteristic
4#20292384C212yesmaximal, complemented
5#568×C107C211yesmaximal, characteristic
6C22×Dic214C2211yescharacteristic
7#16812107C412yescomplemented
8#16812096C412yescomplemented
9#16812090—22no—
10#16812088—22no—
11#16812068C2212yes—
12#16812068C2211yescharacteristic
13#16812034—24no—
14#16812024—22no—
15C2×#41×C107C2211yescharacteristic
16#131×C107—24no—
17C2×M16×C107—22no—
18C2×M16×C107C2212yes—
19C2×Dic214—22no—
20C2×Dic214—22no—
21C2×Dic214—21no—
22C2×Dic214C2×C412yes—
23C22×Dic107—21no—
24C4⋊Dic107—21no—
25C4⋊Dic107—22no—
26Dic107⋊C4—24no—
27C22⋊Dic107—24no—
28C22⋊Dic107—21no—
29C22⋊Dic107—22no—
30C4×Dic107—22no—
31C4×Dic107C2×C412yes—
32#41×C107—22no—
33#41×C107—22no—
34C2×Q8×C107C2×C411yescharacteristic
35C2×D4×C107C2×C411yescharacteristic
36C2×D4×C107D412yes—
37C22×C4×C107D412yes—
38C22×C4×C107C2311yescharacteristic
39M16×C107—24no—
40M16×C107—22no—
41M16×C107—22no—
42C2×C8×C107—24no—
43C2×C8×C107D414yes—
44C23×C107C22⋊C412yes—
45C23×C107C2×D411yescharacteristic
46Q8×C107—22no—
47D4×C107—24no—
48D4×C107—22no—
49C2×C4×C107—22no—
50C2×C4×C107—22no—
51C2×C4×C107C22⋊C412yes—
52C2×C4×C107C22×C411yescharacteristic
53C2×C4×C107C2×D412yes—
54C2×C4×C107C2×D411yescharacteristic
55C2×C4×C107#4112yes—
56C2×Dic107—22no—
57C2×Dic107—22no—
58C2×Dic107—42no—
59C2×Dic107—42no—
60Dic214—24no—
61Dic214—41no—
62Dic214—41no—
63C8×C107—24no—
64C22×C107—22no—
65C22×C107—41no—
66C22×C107—42no—
67C22×C107C2×C22⋊C411yescharacteristic
68C22×C107#12112yes—
69C4×C107—22no—
70C4×C107C4×D412yes—
71C4×C107#12011yescharacteristic
72C4×C107#12311yescharacteristic
73Dic107—42no—
74Dic107—42no—
75Dic107—82no—
76C2×C107—21no—
77C2×C107—22no—
78C2×C107—42no—
79C2×C107#38611yescharacteristic
80#1909—1071nomaximal, Sylow
81C107#190911yescharacteristic, complemented, Sylow
82#568D10711yescharacteristic
83#536—1071no—
84#419—1072no—
85#410—1071no—
86H(3,ℤ/4)—1072no—
87C2×#41C2×D10711yescharacteristic
88C22×Q8—1071no—
89#131—24no—
90C2×M16—22no—
91C2×M16C2×D10712yes—
92C4⋊Q8—1072no—
93#124—1072no—
94#123—2142no—
95C22⋊Q8—2142no—
96#117—1071no—
97#117—1072no—
98#103—2144no—
99#99—2142no—
100#41—22no—
101#41—22no—
102C2×Q8—1072no—
103C2×Q8—2142no—
104C2×Q8—2142no—
105C2×Q8—2141no—
106C2×Q8C4×D10711yescharacteristic
107C2×D4C4×D10711yescharacteristic
108C2×D4D42812yes—
109C22×C4—2141no—
110C22×C4C22⋊D10712yes—
111C22×C4C22×D10711yescharacteristic
112M16—24no—
113M16—22no—
114M16—22no—
115C2×C8—24no—
116C2×C8C22⋊D10714yes—
117C4⋊C4—2142no—
118C4⋊C4—2144no—
119C4⋊C4—2141no—
120C22⋊C4—2142no—
121C22⋊C4—2144no—
122C22⋊C4—2141no—
123C42—1072no—
124C42—2142no—
125C23(C2×D107)⋊C412yes—
126C23C2×D42811yescharacteristic
127Q8—22no—
128Q8—4281no—
129Q8—4281no—
130Q8—2144no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—4282no—
136C2×C4—4282no—
137C2×C4—2142no—
138C2×C4—2142no—
139C2×C4(C2×D107)⋊C412yes—
140C2×C4C2×C4×D10711yescharacteristic
141C2×C4C2×C22⋊D10712yes—
142C2×C4C2×C22⋊D10711yescharacteristic
143C2×C4#1313381812yes—
144C8—24no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×(C2×D107)⋊C411yescharacteristic
149C22#1681208312yes—
150C4—22no—
151C4—8562no—
152C4—4282no—
153C4—4282no—
154C4C4×C22⋊D10712yes—
155C4#1681207811yescharacteristic
156C4#1681209211yescharacteristic
157C2—22no—
158C2—21no—
159C2—42no—
160C2#2029250611yescharacteristic
subgroupG/H
centerC2#20292506
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#568×C107C2
socleC2×C107#386
prime radicalC2×#41×C107C22
cosocleC22×C4×C107C23
hypercenter#568D107
hypocenterC107#1909
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20292637C2semidirect
4. #20292384C2semidirect
7. #16812107C4semidirect
8. #16812096C4semidirect
81. C107#1909semidirect

Maximal subgroups

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

subgroupindex
#202931982
#202926372
#202926012
#202923842
#568×C1072
#1909107