Finite Group Databasebeta

← #28349552

Subgroups — #28349552

Order 13696.

380 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 113 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#20293359C212yesmaximal, complemented
2#20293013C212yesmaximal, complemented
3#20292851C212yesmaximal, complemented
4#20292839C212yesmaximal, complemented
5#20292820C212yesmaximal, complemented
6#20292757C211yesmaximal, characteristic, complemented
7#20292728C212yesmaximal, complemented
8C4×D856C211yesmaximal, characteristic, complemented
9#495×C107C211yesmaximal, characteristic, complemented
10C2×#13133805C2212yes—
11C2×D856C2211yescharacteristic
12C2×#13133803C2212yes—
13#16812105C2212yes—
14#16812098C2212yes—
15#16812095—22no—
16#16812093—22no—
17#16812091—22no—
18Q8×Dic107C2212yes—
19C4×Dic214C2211yescharacteristic
20C4×D428C2212yes—
21D4×Dic107C2212yes—
22(C22⋊D107)⋊C4—22no—
23#16812043C2211yescharacteristic
24C4⋊(C107⋊C8)C2211yescharacteristic
25Dic107⋊C8C2212yes—
26#16812033C2212yes—
27#16812031C2212yes—
28#16812030C2212yes—
29#16812019C2212yes—
30C2×Q16×C107C2211yescharacteristic
31C2×SD16×C107C2212yes—
32C2×D8×C107C2211yescharacteristic
33#124×C107C2212yes—
34C4×C8×C107C2211yescharacteristic
35C2×Dic214D411yescharacteristic
36C2×D428C2312yes—
37C2×C22⋊D107—22no—
38C2×C22⋊D107—22no—
39C22×Dic107—22no—
40C2×C4×D107—22no—
41#13133805—42no—
42D856—22no—
43#13133803—42no—
44C2×C107⋊C8C2312yes—
45C4⋊Dic107—22no—
46C4⋊Dic107D411yescharacteristic
47C4⋊Dic107C2312yes—
48Dic107⋊C4D412yes—
49C22⋊Dic107—22no—
50(C2×D107)⋊C4—22no—
51(C2×D107)⋊C4—22no—
52(C2×D107)⋊C4—22no—
53C4×Dic107—22no—
54C4×Dic107C2312yes—
55C2×Q8×C107C2312yes—
56C2×D4×C107C2312yes—
57Q16×C107—41no—
58SD16×C107—24no—
59D8×C107—41no—
60C2×C8×C107C2311yescharacteristic
61C2×C8×C107C2311yescharacteristic
62C22⋊C4×C107—22no—
63C22⋊C4×C107—22no—
64C42×C107C2311yescharacteristic
65C23×C107—22no—
66C22×D107—22no—
67Q8×C107—22no—
68Q8×C107—22no—
69D4×C107—22no—
70D4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107C2411yescharacteristic
75C22⋊D107—44no—
76C22⋊D107—82no—
77C2×Dic107—22no—
78C2×Dic107—82no—
79C2×Dic107C2×D412yes—
80C2×Dic107C2×D412yes—
81D428—22no—
82D428—22no—
83C4×D107—82no—
84Dic214—41no—
85C8×C107—21no—
86C8×C107#4112yes—
87C107⋊C8—22no—
88C22×C107—42no—
89C22×C107—42no—
90C22×C107—42no—
91C22×C107C22×D411yescharacteristic
92C2×D107—42no—
93C2×D107—42no—
94C2×D107—42no—
95C4×C107—21no—
96C4×C107—21no—
97C4×C107—42no—
98C4×C107C2×#4111yescharacteristic
99C4×C107H(5,𝔽2)11yescharacteristic
100Dic107—22no—
101Dic107—22no—
102Dic107—42no—
103C2×C107—82no—
104C2×C107#55011yescharacteristic
105C2×C107#57511yescharacteristic
106C2×C107#57611yescharacteristic
107D107—82no—
108#3403—1071nomaximal, Sylow
109C107#340311yescharacteristic, complemented, Sylow
110#559—1072no—
111#496—1072no—
112#495D10711yescharacteristic, complemented
113#464—1072no—
114#463—1072no—
115#459—1072no—
116#446—1071no—
117#440—1072no—
118C4×D8—1071no—
119C2×Q16C2×D10711yescharacteristic
120C2×SD16—1072no—
121C2×SD16C2×D10712yes—
122C2×D8—1072no—
123C2×D8—1071no—
124C2×D8C2×D10711yescharacteristic
125#127—1072no—
126#124—1072no—
127#124—2142no—
128#124C2×D10712yes—
129#123—2142no—
130#123—2142no—
131C4×Q8—1071no—
132C4×Q8—1072no—
133C4×D4—1072no—
134C4×D4—1072no—
135C4×D4—2142no—
136#107—1071no—
137C4⋊C8—1072no—
138C4⋊C8—1071no—
139#103—1072no—
140#102—1072no—
141#102—1072no—
142#97—1072no—
143C4×C8C2×D10711yescharacteristic
144C2×Q8—1071no—
145C2×Q8C22×D10712yes—
146C2×D4—1072no—
147C2×D4—2142no—
148C2×D4—2142no—
149C2×D4C22×D10712yes—
150C22×C4—2142no—
151C22×C4—2142no—
152Q16—41no—
153SD16—24no—
154SD16—4282no—
155D8—41no—
156D8—4282no—
157D8—2142no—
158C2×C8—1072no—
159C2×C8C22×D10711yescharacteristic
160C2×C8C22×D10711yescharacteristic
161C4⋊C4—1072no—
162C4⋊C4—1071no—
163C4⋊C4—1072no—
164C4⋊C4—2142no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—2142no—
168C22⋊C4—2142no—
169C22⋊C4—2142no—
170C22⋊C4—2142no—
171C42—1072no—
172C42—2142no—
173C42C22×D10711yescharacteristic
174C23—22no—
175C23—2142no—
176Q8—22no—
177Q8—22no—
178Q8—4281no—
179D4—22no—
180D4—22no—
181D4—8562no—
182D4—4284no—
183D4—2142no—
184D4—2142no—
185C2×C4—22no—
186C2×C4—8562no—
187C2×C4—8562no—
188C2×C4—1072no—
189C2×C4—1072no—
190C2×C4—2142no—
191C2×C4D4×D10711yescharacteristic
192C2×C4D4×D10711yescharacteristic
193C2×C4C23×D10711yescharacteristic
194C8—21no—
195C8—2142no—
196C8#1313381712yes—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—4282no—
201C22—4282no—
202C22—4282no—
203C22C2×D4×D10711yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—4282no—
208C4—2142no—
209C4—2142no—
210C4C2×#1313381711yescharacteristic
211C4#1681215411yescharacteristic
212C2—82no—
213C2—8562no—
214C2#2029329711yescharacteristic
215C2#2029342411yescharacteristic
216C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#495×C107C2
socleC22×C107C22×D4
hypercenter#495D107
hypocenterC107#3403
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #20293359C2semidirect
2. #20293013C2semidirect
3. #20292851C2semidirect
4. #20292839C2semidirect
5. #20292820C2semidirect
6. #20292757C2semidirect
7. #20292728C2semidirect
8. C4×D856C2semidirect
9. #495×C107C2semidirect
109. C107#3403semidirect
112. #495D107semidirect

Maximal subgroups

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

subgroupindex
#202933592
#202930132
#202928512
#202928392
#202928202
#202927572
#202927282
C4×D8562
#495×C1072
#3403107