Finite Group Databasebeta

← #28336993

Subgroups — #28336993

Order 13696.

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

normal subgroups only

This group has 362 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#20293194C211yesmaximal, characteristic
2#20292645C212yesmaximal, complemented
3C2×#16812026C211yesmaximal, characteristic, complemented
4#20292383C212yesmaximal, complemented
5#518×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C2211yescharacteristic
7C2×#13133802C2212yes—
8#16812105C412yescomplemented
9#16812096C412yescomplemented
10#16812080—22no—
11#16812078—22no—
12#16812068C2211yescharacteristic
13#16812037—24no—
14#16812026—22no—
15C22×D4×C107C2211yescharacteristic
16C4×D4×C107—24no—
17#117×C107C2212yes—
18C2×C22⋊C4×C107—22no—
19#13133818—22no—
20#13133818—22no—
21C2×Dic214C2×C411yescharacteristic, complemented
22C2×D428C2×C411yescharacteristic, complemented
23C2×C22⋊D107—21no—
24C2×C22⋊D107—24no—
25C2×C4×D107—21no—
26#13133802—22no—
27#13133802—22no—
28C2×C107⋊C8—22no—
29C4⋊Dic107—21no—
30C22⋊Dic107—24no—
31C22⋊Dic107—21no—
32C4×Dic107C2×C412yescomplemented
33C24×C107D412yes—
34C2×D4×C107—22no—
35C2×D4×C107—22no—
36C2×D4×C107C2×C412yes—
37C2×D4×C107D412yes—
38C22×C4×C107—24no—
39C22×C4×C107C2311yescharacteristic
40C4⋊C4×C107—22no—
41C22⋊C4×C107—24no—
42C22⋊C4×C107—22no—
43C42×C107D414yescomplemented
44C23×C107—22no—
45C23×C107—22no—
46C23×C107—82no—
47C23×C107C22⋊C414yes—
48C23×C107C2×D411yescharacteristic
49C22×D107—21no—
50D4×C107—24no—
51D4×C107—24no—
52C2×C4×C107—24no—
53C2×C4×C107—84no—
54C2×C4×C107C22×C411yescharacteristic
55C2×C4×C107C2×D412yes—
56C2×C4×C107C2×D411yescharacteristic
57C2×C4×C107#4112yes—
58C22⋊D107—41no—
59C22⋊D107—41no—
60C22⋊D107—84no—
61C2×Dic107—22no—
62C2×Dic107—21no—
63D428—22no—
64C4×D107—42no—
65Dic214—22no—
66C107⋊C8—42no—
67C22×C107—24no—
68C22×C107—41no—
69C22×C107—44no—
70C22×C107—44no—
71C22×C107—82no—
72C22×C107C2×C22⋊C411yescharacteristic
73C22×C107#12011yescharacteristic
74C22×C107#12311yescharacteristic
75C2×D107—42no—
76C2×D107—41no—
77C4×C107—44no—
78C4×C107C4×D412yes—
79C4×C107#12112yes—
80Dic107—42no—
81Dic107—41no—
82C2×C107—21no—
83C2×C107—21no—
84C2×C107—21no—
85C2×C107—44no—
86C2×C107#38611yescharacteristic
87D107—81no—
88#1911—1071nomaximal, Sylow
89C107#191111yescharacteristic, complemented, Sylow
90#535—1071no—
91#518D10711yescharacteristic, complemented
92#421—1072no—
93C2×#100—1071no—
94H(3,ℤ/4)—1072no—
95C2×#41—1071no—
96C22×D4C2×D10711yescharacteristic
97C2×M16—1072no—
98#127—1072no—
99#124—1072no—
100#121—2142no—
101#120—2142no—
102C4×D4—24no—
103#117—1071no—
104#117C2×D10712yes—
105C2×C22⋊C4—22no—
106#104—2144no—
107#100—2142no—
108C24C22⋊D10712yes—
109#41—2142no—
110#41—2142no—
111C2×Q8—1071no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—1071no—
115C2×D4—2144no—
116C2×D4—2141no—
117C2×D4C4×D10712yes—
118C2×D4D42812yes—
119C22×C4—24no—
120C22×C4—2141no—
121C22×C4C22×D10711yescharacteristic
122M16—2142no—
123M16—2142no—
124C2×C8—2142no—
125C4⋊C4—22no—
126C4⋊C4—2141no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—2144no—
130C22⋊C4—2141no—
131C42—1072no—
132C42C22⋊D10714yescomplemented
133C23—22no—
134C23—22no—
135C23—82no—
136C23—2141no—
137C23(C2×D107)⋊C414yes—
138C23C2×C22⋊D10711yescharacteristic
139Q8—2142no—
140D4—24no—
141D4—24no—
142D4—8564no—
143D4—4281no—
144D4—4281no—
145D4—2142no—
146C2×C4—24no—
147C2×C4—84no—
148C2×C4—4282no—
149C2×C4—2142no—
150C2×C4—2141no—
151C2×C4C2×C4×D10711yescharacteristic
152C2×C4C2×C22⋊D10712yes—
153C2×C4C2×D42811yescharacteristic
154C2×C4#1313381812yes—
155C8—4282no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—4282no—
162C22—4281no—
163C22C2×(C2×D107)⋊C411yescharacteristic
164C22#1681207811yescharacteristic
165C22#1681209211yescharacteristic
166C4—44no—
167C4—4282no—
168C4—4281no—
169C4C4×C22⋊D10712yes—
170C4#1681208312yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—8561no—
176C2#2029250611yescharacteristic
subgroupG/H
centerC2#20292506
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#518×C107C2
socleC2×C107#386
prime radical#20293194C2
cosocleC22×C4×C107C23
hypercenter#518D107
hypocenterC107#1911
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #20292645C2semidirect
3. C2×#16812026C2semidirect
4. #20292383C2semidirect
5. #518×C107C2semidirect
8. #16812105C4semidirect
9. #16812096C4semidirect
21. C2×Dic214C2×C4semidirect
22. C2×D428C2×C4semidirect
32. C4×Dic107C2×C4semidirect
43. C42×C107D4semidirect
89. C107#1911semidirect
91. #518D107semidirect
132. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202931942
#202926452
C2×#168120262
#202923832
#518×C1072
#1911107