Finite Group Databasebeta

← #28337618

Subgroups — #28337618

Order 13696.

282 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 67 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#20293258C211yesmaximal, characteristic, complemented
2#20292633C212yesmaximal
3#20292528C211yesmaximal, characteristic
4#20292388C212yesmaximal, complemented
5#343×C107C211yesmaximal, characteristic, complemented
6C22×D428C2211yescharacteristic
7#16812108—22no—
8#16812100—22no—
9#16812088—22no—
10#16812068C2212yes—
11C2×(C2×D107)⋊C4—22no—
12C2×C4×Dic107C2211yescharacteristic
13#16812031—22no—
14#16812031—22no—
15C2×M16×C107C2212yes—
16C2×C4⋊C4×C107C2211yescharacteristic
17C23×D107—21no—
18C2×Dic214—22no—
19C2×D428—22no—
20C2×D428—22no—
21C2×D428—21no—
22C2×D428Q812yescomplemented
23C22×Dic107—21no—
24C4⋊Dic107—22no—
25C4⋊Dic107D412yes—
26C4⋊Dic107D412yes—
27Dic107⋊C4—24no—
28C22⋊Dic107—22no—
29(C2×D107)⋊C4—44no—
30C4×Dic107—22no—
31C4×Dic107—22no—
32C22×C4×C107D412yes—
33C22×C4×C107C2311yescharacteristic
34M16×C107—22no—
35M16×C107—22no—
36C2×C8×C107—22no—
37C4⋊C4×C107—22no—
38C23×C107C2×D411yescharacteristic
39C22×D107—22no—
40C22×D107—42no—
41C22×D107—42no—
42C22×D107—41no—
43C22×D107—41no—
44C2×C4×C107—22no—
45C2×C4×C107—42no—
46C2×C4×C107C2×D412yes—
47C2×C4×C107C2×Q811yescharacteristic
48C2×C4×C107#4111yescharacteristic
49C2×C4×C107#4112yes—
50C2×Dic107—22no—
51C2×Dic107—22no—
52C2×Dic107—22no—
53C2×Dic107—41no—
54C2×Dic107—41no—
55D428—24no—
56D428—42no—
57Dic214—82no—
58C8×C107—42no—
59C22×C107—21no—
60C22×C107—21no—
61C22×C107#12011yescharacteristic
62C22×C107C22⋊Q811yescharacteristic
63C22×C107#12411yescharacteristic
64C2×D107—44no—
65C2×D107—42no—
66C2×D107—82no—
67C2×D107—81no—
68C2×D107—81no—
69C4×C107—42no—
70C4×C107C22⋊Q812yes—
71C4×C107#12412yes—
72Dic107—42no—
73Dic107—42no—
74Dic107—42no—
75C2×C107—21no—
76C2×C107—21no—
77C2×C107#39311yescharacteristic
78C2×C107#45311yescharacteristic
79C2×C107#45411yescharacteristic
80D107—82no—
81#2027—1071nomaximal, Sylow
82C107#202711yescharacteristic, complemented, Sylow
83#544—1071no—
84#418—1072no—
85#390—1071no—
86#343D10711yescharacteristic, complemented
87#339—1072no—
88C22×D4—1071no—
89C2×M16C2×D10712yes—
90C4⋊Q8—2142no—
91#125—2142no—
92C22⋊Q8—2142no—
93#117—1072no—
94C2×C4⋊C4C2×D10711yescharacteristic
95C2×C22⋊C4—2142no—
96C2×C42—1071no—
97#102—2142no—
98#102—2142no—
99C24—2141no—
100C2×Q8—2142no—
101C2×D4—1072no—
102C2×D4—2142no—
103C2×D4—2141no—
104C2×D4—2142no—
105C22×C4—2141no—
106C22×C4D42812yes—
107C22×C4C22×D10711yescharacteristic
108M16—22no—
109M16—22no—
110C2×C8—22no—
111C4⋊C4—22no—
112C4⋊C4—1072no—
113C4⋊C4—1072no—
114C4⋊C4—2142no—
115C4⋊C4—2144no—
116C22⋊C4—4284no—
117C22⋊C4—2142no—
118C42—2142no—
119C42—2142no—
120C23—4282no—
121C23—4282no—
122C23—4281no—
123C23—4281no—
124C23—2142no—
125C23C2×D42811yescharacteristic
126Q8—8562no—
127D4—4282no—
128D4—2144no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—4281no—
132C2×C4—4281no—
133C2×C4—2142no—
134C2×C4—2142no—
135C2×C4—2142no—
136C2×C4D4×D10712yes—
137C2×C4Q8×D10711yescharacteristic
138C2×C4#1313381711yescharacteristic
139C2×C4#1313381812yes—
140C8—42no—
141C22—21no—
142C22—21no—
143C22—8562no—
144C22—8561no—
145C22—8561no—
146C22—4284no—
147C22—4282no—
148C22#1681207911yescharacteristic
149C22#1681208911yescharacteristic
150C22#1681209711yescharacteristic
151C4—42no—
152C4—4282no—
153C4—4282no—
154C4—4282no—
155C4#1681208712yes—
156C4#1681209512yes—
157C2—21no—
158C2—21no—
159C2—8562no—
160C2#2029254011yescharacteristic
161C2#2029279711yescharacteristic
162C2#2029280211yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#343×C107C2
socleC22×C107#120
prime radicalC22×D428C22
hypercenter#343D107
hypocenterC107#2027
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. #20293258C2semidirect
4. #20292388C2semidirect
5. #343×C107C2semidirect
22. C2×D428Q8semidirect
82. C107#2027semidirect
86. #343D107semidirect

Maximal subgroups

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

subgroupindex
#202932582
#202926332
#202925282
#202923882
#343×C1072
#2027107