Finite Group Databasebeta

← #28337617

Subgroups — #28337617

Order 13696.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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#20292632C212yesmaximal
3#20292396C211yesmaximal, characteristic, complemented
4#20292387C212yesmaximal, complemented
5#390×C107C211yesmaximal, characteristic
6C2×#13133802C2212yes—
7#16812108—22no—
8#16812100—22no—
9#16812088—22no—
10#16812068C2212yes—
11#16812030—22no—
12#16812030—22no—
13C22×D4×C107C2211yescharacteristic
14C2×C4⋊C4×C107C2211yescharacteristic
15C2×C22⋊C4×C107—22no—
16C2×C42×C107C2211yescharacteristic
17C2×Dic214—22no—
18#13133802—22no—
19#13133802—22no—
20C2×C107⋊C8—22no—
21C4⋊Dic107—22no—
22C4⋊Dic107D412yes—
23C4⋊Dic107D412yes—
24Dic107⋊C4—24no—
25C22⋊Dic107—22no—
26C4×Dic107—22no—
27C24×C107—21no—
28C2×D4×C107—22no—
29C2×D4×C107—22no—
30C2×D4×C107—21no—
31C2×D4×C107Q812yescomplemented
32C22×C4×C107—21no—
33C22×C4×C107D412yes—
34C22×C4×C107C2311yescharacteristic
35C4⋊C4×C107—22no—
36C22⋊C4×C107—44no—
37C42×C107—22no—
38C23×C107—22no—
39C23×C107—42no—
40C23×C107—42no—
41C23×C107—41no—
42C23×C107—41no—
43C23×C107C2×D411yescharacteristic
44D4×C107—24no—
45D4×C107—42no—
46C2×C4×C107—22no—
47C2×C4×C107—22no—
48C2×C4×C107—41no—
49C2×C4×C107—41no—
50C2×C4×C107—42no—
51C2×C4×C107C2×D412yes—
52C2×C4×C107C2×Q811yescharacteristic
53C2×C4×C107#4112yes—
54C2×C4×C107#4111yescharacteristic
55C2×Dic107—22no—
56C2×Dic107—22no—
57Dic214—82no—
58C107⋊C8—42no—
59C22×C107—21no—
60C22×C107—21no—
61C22×C107—44no—
62C22×C107—42no—
63C22×C107—82no—
64C22×C107—81no—
65C22×C107—81no—
66C22×C107#12011yescharacteristic
67C22×C107C22⋊Q811yescharacteristic
68C22×C107#12411yescharacteristic
69C4×C107—42no—
70C4×C107—42no—
71C4×C107C22⋊Q812yes—
72C4×C107#12412yes—
73Dic107—42no—
74Dic107—42no—
75C2×C107—21no—
76C2×C107—21no—
77C2×C107—82no—
78C2×C107#39311yescharacteristic
79C2×C107#45311yescharacteristic
80C2×C107#45411yescharacteristic
81#2027—1071nomaximal, Sylow
82C107#202711yescharacteristic, complemented, Sylow
83#544—1071no—
84#418—1072no—
85#390D10711yescharacteristic
86#343—1071no—
87#339—1072no—
88C22×D4C2×D10711yescharacteristic
89C2×M16—1072no—
90C4⋊Q8—2142no—
91#125—2142no—
92C22⋊Q8—2142no—
93#117—1072no—
94C2×C4⋊C4C2×D10711yescharacteristic
95C2×C22⋊C4—22no—
96C2×C42C2×D10711yescharacteristic
97#102—2142no—
98#102—2142no—
99C24—21no—
100C2×Q8—2142no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4Dic21412yescomplemented
105C22×C4—21no—
106C22×C4C22⋊D10712yes—
107C22×C4C22×D10711yescharacteristic
108M16—2142no—
109M16—2142no—
110C2×C8—2142no—
111C4⋊C4—22no—
112C4⋊C4—1072no—
113C4⋊C4—1072no—
114C4⋊C4—2142no—
115C4⋊C4—2144no—
116C22⋊C4—44no—
117C22⋊C4—2142no—
118C42—22no—
119C42—2142no—
120C23—22no—
121C23—42no—
122C23—42no—
123C23—41no—
124C23—41no—
125C23C2×C22⋊D10711yescharacteristic
126Q8—8562no—
127D4—24no—
128D4—42no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—2142no—
135C2×C4—2142no—
136C2×C4D4×D10712yes—
137C2×C4C2×Dic21411yescharacteristic
138C2×C4#1313381712yes—
139C2×C4#1313381811yescharacteristic
140C8—4282no—
141C22—21no—
142C22—21no—
143C22—44no—
144C22—42no—
145C22—82no—
146C22—81no—
147C22—81no—
148C22#1681207711yescharacteristic
149C22#1681208811yescharacteristic
150C22#1681209611yescharacteristic
151C4—42no—
152C4—42no—
153C4—4282no—
154C4—4282no—
155C4#1681208612yes—
156C4#1681209512yes—
157C2—21no—
158C2—21no—
159C2—82no—
160C2#2029253911yescharacteristic
161C2#2029279411yescharacteristic
162C2#2029279911yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#390×C107C2
socleC22×C107#120
prime radicalC22×D4×C107C22
hypercenter#390D107
hypocenterC107#2027
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #20293258C2semidirect
3. #20292396C2semidirect
4. #20292387C2semidirect
31. C2×D4×C107Q8semidirect
82. C107#2027semidirect
104. C2×D4Dic214semidirect

Maximal subgroups

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

subgroupindex
#202932582
#202926322
#202923962
#202923872
#390×C1072
#2027107