Finite Group Databasebeta

← #28336842

Subgroups — #28336842

Order 13696.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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#20293394C211yesmaximal, characteristic, complemented
2#20293173C211yesmaximal, characteristic
3#20292645C212yesmaximal, complemented
4#20292386C212yesmaximal, complemented
5#410×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C2211yescharacteristic
7#16812116—24no—
8C2×#13133802—22no—
9C2×#13133802C2212yes—
10#16812107C411yescharacteristic, complemented
11#16812105C411yescharacteristic, complemented
12#16812098C411yescharacteristic, complemented
13#16812096C411yescharacteristic, complemented
14#16812081—22no—
15C4×C22⋊D107—22no—
16C2×C4×Dic107C2211yescharacteristic
17#16812037—24no—
18C2×#41×C107C2211yescharacteristic
19#117×C107C2212yes—
20#99×C107—22no—
21#13133818—22no—
22#13133818—22no—
23C2×Dic214C2×C411yescharacteristic
24C2×D428C2×C411yescharacteristic
25C2×C22⋊D107—22no—
26C2×C22⋊D107—21no—
27C22×Dic107—21no—
28C2×C4×D107—21no—
29#13133802—24no—
30#13133802—22no—
31#13133802D414yescomplemented
32C2×C107⋊C8—24no—
33C2×C107⋊C8—22no—
34Dic107⋊C4—22no—
35C22⋊Dic107—22no—
36(C2×D107)⋊C4—22no—
37C4×Dic107—21no—
38C4×Dic107C2×C411yescharacteristic
39C4×Dic107C2×C411yescharacteristic
40#41×C107—22no—
41#41×C107—22no—
42C2×Q8×C107C2×C411yescharacteristic
43C2×D4×C107C2×C411yescharacteristic
44C2×D4×C107D412yes—
45C22×C4×C107D412yes—
46C22×C4×C107C2311yescharacteristic
47C4⋊C4×C107—22no—
48C22⋊C4×C107—22no—
49C42×C107—22no—
50C23×C107C22⋊C412yes—
51C23×C107C2×D411yescharacteristic
52C22×D107—21no—
53Q8×C107—22no—
54D4×C107—24no—
55D4×C107—22no—
56C2×C4×C107—22no—
57C2×C4×C107—22no—
58C2×C4×C107—42no—
59C2×C4×C107C22⋊C412yes—
60C2×C4×C107C22×C411yescharacteristic
61C2×C4×C107C2×D412yes—
62C2×C4×C107C2×D411yescharacteristic
63C2×C4×C107#4112yes—
64C22⋊D107—44no—
65C22⋊D107—42no—
66C2×Dic107—21no—
67C2×Dic107—21no—
68C2×Dic107—21no—
69C2×Dic107—42no—
70D428—22no—
71C4×D107—41no—
72C4×D107—41no—
73Dic214—22no—
74C107⋊C8—24no—
75C22×C107—22no—
76C22×C107—41no—
77C22×C107—42no—
78C22×C107C2×C22⋊C411yescharacteristic
79C22×C107#12112yes—
80C2×D107—42no—
81C2×D107—41no—
82C4×C107—22no—
83C4×C107—82no—
84C4×C107C4×D412yes—
85C4×C107#12011yescharacteristic
86C4×C107#12311yescharacteristic
87Dic107—22no—
88Dic107—41no—
89Dic107—41no—
90C2×C107—21no—
91C2×C107—22no—
92C2×C107—42no—
93C2×C107#38611yescharacteristic
94D107—81no—
95#1884—1071nomaximal, Sylow
96C107#188411yescharacteristic, complemented, Sylow
97#568—1071no—
98#532—1071no—
99#421—1072no—
100#410D10711yescharacteristic, complemented
101#339—1072no—
102C2×#41—1071no—
103C2×#41C2×D10711yescharacteristic
104#131—2144no—
105C2×M16—1072no—
106C2×M16—2142no—
107C4⋊Q8—1071no—
108#127—1071no—
109#124—1071no—
110#124—1071no—
111#121—2142no—
112C4×D4—2142no—
113#117C2×D10712yes—
114C2×C42—1071no—
115#104—2144no—
116#99—22no—
117#41—22no—
118#41—22no—
119#41—2142no—
120#41—2142no—
121C2×Q8—1071no—
122C2×Q8C4×D10711yescharacteristic
123C2×D4—1071no—
124C2×D4—2142no—
125C2×D4—2141no—
126C2×D4C4×D10711yescharacteristic
127C2×D4C22⋊D10712yes—
128C22×C4—2141no—
129C22×C4—2141no—
130C22×C4D42812yes—
131C22×C4C22×D10711yescharacteristic
132M16—1074no—
133M16—2144no—
134M16—2142no—
135C2×C8—2144no—
136C2×C8—2142no—
137C4⋊C4—22no—
138C4⋊C4—2142no—
139C22⋊C4—22no—
140C22⋊C4—2142no—
141C22⋊C4—2142no—
142C42—22no—
143C42—1071no—
144C42—1071no—
145C42—2141no—
146C23—2141no—
147C23(C2×D107)⋊C412yes—
148C23C2×C22⋊D10711yescharacteristic
149Q8—22no—
150Q8—2142no—
151D4—24no—
152D4—22no—
153D4—4282no—
154D4—4284no—
155D4—2142no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—4282no—
160C2×C4—4281no—
161C2×C4—4281no—
162C2×C4—2141no—
163C2×C4—2141no—
164C2×C4—2141no—
165C2×C4(C2×D107)⋊C412yes—
166C2×C4C2×C4×D10711yescharacteristic
167C2×C4D4×D10712yes—
168C2×C4C2×D42811yescharacteristic
169C2×C4#1313381712yes—
170C8—2144no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—4282no—
175C22—4281no—
176C22C2×(C2×D107)⋊C411yescharacteristic
177C22#1681208112yes—
178C4—22no—
179C4—82no—
180C4—4281no—
181C4—4281no—
182C4—2142no—
183C4(C22⋊D107)⋊C412yes—
184C4#1681207911yescharacteristic
185C4#1681209311yescharacteristic
186C2—22no—
187C2—21no—
188C2—42no—
189C2—8561no—
190C2#2029250411yescharacteristic
subgroupG/H
centerC4#16812079
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#410×C107C2
socleC2×C107#386
prime radical#20293173C2
cosocleC22×C4×C107C23
hypercenter#410D107
hypocenterC107#1884
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #20293394C2semidirect
3. #20292645C2semidirect
4. #20292386C2semidirect
5. #410×C107C2semidirect
10. #16812107C4semidirect
11. #16812105C4semidirect
12. #16812098C4semidirect
13. #16812096C4semidirect
31. #13133802D4semidirect
96. C107#1884semidirect
100. #410D107semidirect

Maximal subgroups

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

subgroupindex
#202933942
#202931732
#202926452
#202923862
#410×C1072
#1884107