Finite Group Databasebeta

← #28335760

Subgroups — #28335760

Order 13696.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#20293377C211yesmaximal, characteristic, complemented
2#20293017C211yesmaximal, characteristic
3#20292992C212yesmaximal, complemented
4#20292368C212yesmaximal, complemented
5#329×C107C211yesmaximal, characteristic, complemented
6C2×#13133805—21no—
7C2×#13133803—21no—
8#16812107—21no—
9#16812106C2211yescharacteristic
10#16812102—22no—
11#16812101C2212yes—
12#16812090—21no—
13#16812086—22no—
14#16812031—22no—
15#16812031—22no—
16#16812020C2211yescharacteristic
17#124×C107C2211yescharacteristic
18#97×C107C2212yes—
19C2×Dic214—21no—
20C2×D428—21no—
21C2×D428—22no—
22C22×Dic107—21no—
23#13133805—42no—
24#13133803—42no—
25C2×C107⋊C8—21no—
26C4⋊Dic107D412yes—
27C4⋊Dic107D412yes—
28Dic107⋊C4—22no—
29Dic107⋊C4—22no—
30Dic107⋊C4—22no—
31C22⋊Dic107—22no—
32C22⋊Dic107—21no—
33C4×Dic107—21no—
34C2×Q8×C107D411yescharacteristic
35C2×D4×C107D411yescharacteristic
36C2×C8×C107—22no—
37C22⋊C4×C107—22no—
38C42×C107C2311yescharacteristic
39C23×C107—21no—
40C22×D107—41no—
41Q8×C107—41no—
42D4×C107—41no—
43C2×C4×C107—21no—
44C2×C4×C107C2×D412yes—
45C2×C4×C107C2×D411yescharacteristic
46C2×Dic107—22no—
47C2×Dic107—22no—
48C2×Dic107—81no—
49D428—41no—
50D428—41no—
51D428—44no—
52Dic214—81no—
53C8×C107—42no—
54C107⋊C8—22no—
55C22×C107—41no—
56C22×C107—42no—
57C22×C107#12011yescharacteristic
58C2×D107—81no—
59C2×D107—82no—
60C4×C107—21no—
61C4×C107—22no—
62C4×C107—41no—
63Dic107—42no—
64Dic107—42no—
65C2×C107—81no—
66C2×C107#45311yescharacteristic
67C2×C107#45412yes—
68D107—161no—
69#1657—1071nomaximal, Sylow
70C107#165711yescharacteristic, complemented, Sylow
71#563—1071no—
72#496—1071no—
73#490—1072no—
74#330—1072no—
75#329D10711yescharacteristic, complemented
76C2×SD16—2141no—
77C2×D8—2141no—
78C4⋊Q8—2141no—
79#127—1071no—
80#126—2142no—
81#125—1072no—
82#124C2×D10711yescharacteristic
83#123—2141no—
84C22⋊Q8—2142no—
85#102—2142no—
86#102—2142no—
87#97—1071no—
88#97C2×D10712yes—
89C2×Q8—2141no—
90C2×Q8D42811yescharacteristic
91C2×D4—2142no—
92C2×D4—2141no—
93C2×D4D42811yescharacteristic
94C22×C4—2141no—
95SD16—4282no—
96D8—4282no—
97C2×C8—22no—
98C2×C8—2141no—
99C4⋊C4—1072no—
100C4⋊C4—1072no—
101C4⋊C4—2142no—
102C4⋊C4—2142no—
103C4⋊C4—2142no—
104C22⋊C4—22no—
105C22⋊C4—2142no—
106C22⋊C4—2141no—
107C42—2141no—
108C42C22×D10711yescharacteristic
109C23—21no—
110C23—4281no—
111Q8—41no—
112Q8—8561no—
113D4—41no—
114D4—4281no—
115D4—4281no—
116D4—4284no—
117C2×C4—21no—
118C2×C4—8561no—
119C2×C4—2142no—
120C2×C4—2142no—
121C2×C4D4×D10712yes—
122C2×C4C2×D42811yescharacteristic
123C8—42no—
124C8—2142no—
125C22—41no—
126C22—42no—
127C22—8561no—
128C22—8562no—
129C22#1681207911yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—4282no—
134C4—4282no—
135C2—81no—
136C2—17121no—
137C2#2029279711yescharacteristic
138C2#2029280112yes—
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#329×C107C2
socleC22×C107#120
prime radical#20293017C2
hypercenter#329D107
hypocenterC107#1657
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #20293377C2semidirect
3. #20292992C2semidirect
4. #20292368C2semidirect
5. #329×C107C2semidirect
70. C107#1657semidirect
75. #329D107semidirect

Maximal subgroups

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

subgroupindex
#202933772
#202930172
#202929922
#202923682
#329×C1072
#1657107