Finite Group Databasebeta

← #28335757

Subgroups — #28335757

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#20292991C212yesmaximal, complemented
3#20292368C212yesmaximal, complemented
4#20292367C211yesmaximal, characteristic, complemented
5#496×C107C211yesmaximal, characteristic
6#16812107—21no—
7#16812102—22no—
8#16812101C2212yes—
9#16812090—21no—
10#16812086—22no—
11#16812030—22no—
12#16812030—22no—
13#16812020C2212yes—
14C2×SD16×C107—21no—
15C2×D8×C107—21no—
16#127×C107C2211yescharacteristic
17#124×C107C2211yescharacteristic
18#97×C107C2211yescharacteristic
19C2×Dic214—21no—
20C22×Dic107—21no—
21C2×C107⋊C8—22no—
22C4⋊Dic107D412yes—
23C4⋊Dic107D412yes—
24Dic107⋊C4—22no—
25Dic107⋊C4—22no—
26Dic107⋊C4—22no—
27C22⋊Dic107—22no—
28C22⋊Dic107—21no—
29C4×Dic107—21no—
30C2×Q8×C107D411yescharacteristic
31C2×D4×C107—21no—
32C2×D4×C107—22no—
33C2×D4×C107D411yescharacteristic
34SD16×C107—42no—
35D8×C107—42no—
36C2×C8×C107—21no—
37C22⋊C4×C107—22no—
38C42×C107C2311yescharacteristic
39C23×C107—21no—
40C23×C107—41no—
41Q8×C107—41no—
42D4×C107—41no—
43D4×C107—41no—
44D4×C107—41no—
45D4×C107—44no—
46C2×C4×C107—21no—
47C2×C4×C107C2×D412yes—
48C2×C4×C107C2×D411yescharacteristic
49C2×Dic107—22no—
50C2×Dic107—22no—
51C2×Dic107—81no—
52Dic214—81no—
53C8×C107—22no—
54C107⋊C8—42no—
55C22×C107—41no—
56C22×C107—42no—
57C22×C107—81no—
58C22×C107—82no—
59C22×C107#12011yescharacteristic
60C4×C107—21no—
61C4×C107—22no—
62C4×C107—41no—
63Dic107—42no—
64Dic107—42no—
65C2×C107—81no—
66C2×C107—161no—
67C2×C107#45311yescharacteristic
68C2×C107#45412yes—
69#1657—1071nomaximal, Sylow
70C107#165711yescharacteristic, complemented, Sylow
71#563—1071no—
72#496D10711yescharacteristic
73#490—1072no—
74#330—1072no—
75#329—1071no—
76C2×SD16—21no—
77C2×D8—21no—
78C4⋊Q8—2141no—
79#127C2×D10711yescharacteristic
80#126—2142no—
81#125—1072no—
82#124C2×D10711yescharacteristic
83#123—2141no—
84C22⋊Q8—2142no—
85#102—2142no—
86#102—2142no—
87#97—1072no—
88#97C2×D10711yescharacteristic
89C2×Q8—2141no—
90C2×Q8C22⋊D10711yescharacteristic
91C2×D4—22no—
92C2×D4—21no—
93C2×D4C22⋊D10711yescharacteristic
94C22×C4—2141no—
95SD16—42no—
96D8—42no—
97C2×C8—21no—
98C2×C8—2142no—
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—41no—
111Q8—41no—
112Q8—8561no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—44no—
117C2×C4—21no—
118C2×C4—8561no—
119C2×C4—2142no—
120C2×C4—2142no—
121C2×C4D4×D10712yes—
122C2×C4C2×C22⋊D10711yescharacteristic
123C8—22no—
124C8—4282no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22#1681207711yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—4282no—
134C4—4282no—
135C2—81no—
136C2—161no—
137C2#2029279411yescharacteristic
138C2#2029279812yes—
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting, prime radical#496×C107C2
socleC22×C107#120
hypercenter#496D107
hypocenterC107#1657
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #20293377C2semidirect
2. #20292991C2semidirect
3. #20292368C2semidirect
4. #20292367C2semidirect
70. C107#1657semidirect

Maximal subgroups

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

subgroupindex
#202933772
#202929912
#202923682
#202923672
#496×C1072
#1657107