Finite Group Databasebeta

← #28349873

Subgroups — #28349873

Order 13696.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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#20293226C216yesmaximal, complemented
2#20293217C216yesmaximal, complemented
3#20293189C213yesmaximal, complemented
4#20293163C216yesmaximal, complemented
5#20293140C216yesmaximal, complemented
6C2×(C22⋊D107)⋊C4C213yesmaximal, complemented
7#543×C107C211yesmaximal, characteristic, complemented
8C2×#13133819C2212yescomplemented
9C22×C22⋊D107C2213yes—
10C23×Dic107C2211yescharacteristic
11C22×C4×D107C2213yescomplemented
12#16812104C22112yescomplemented
13#16812100C2216yescomplemented
14#16812098C2216yescomplemented
15#16812094C22112yescomplemented
16#16812093C22112yescomplemented
17#16812085C2216yescomplemented
18#16812084C2216yescomplemented
19#16812083C2216yes—
20#16812083C2216yes—
21#16812079C22112yescomplemented
22Q8×Dic107C2212yescomplemented
23#16812071C2216yescomplemented
24(C22⋊D107)⋊C4C22112yescomplemented
25(C22⋊D107)⋊C4C22112yescomplemented
26#16812067C2216yescomplemented
27C2×Dic107⋊C4C2213yescomplemented
28C2×(C2×D107)⋊C4C2213yescomplemented
29C2×C4×Dic107C2213yescomplemented
30C22⋊Q8×C107C2216yescomplemented
31C22⋊Q8×C107C2216yescomplemented
32C2×C22⋊C4×C107C2213yes—
33C23×D107C2313yes—
34#13133819—44no—
35C2×D428C2316yes—
36C2×D428C2316yes—
37C2×C22⋊D107—212no—
38C2×C22⋊D107C23112yes—
39C22×Dic107—42no—
40C22×Dic107C2313yes—
41C22×Dic107C2313yes—
42C22×Dic107C2313yes—
43C2×C4×D107—46no—
44C2×C4×D107C2316yes—
45C2×C4×D107C2316yes—
46C4⋊Dic107C2316yes—
47C4⋊Dic107C2316yes—
48Dic107⋊C4C23112yes—
49(C2×D107)⋊C4C23112yes—
50(C2×D107)⋊C4C23112yes—
51(C2×D107)⋊C4C23112yes—
52C4×Dic107C2316yes—
53C4×Dic107C2316yes—
54C24×C107C2311yescharacteristic
55C2×Q8×C107C2312yes—
56C2×Q8×C107C2312yes—
57C22×C4×C107C2313yes—
58C22×C4×C107C2313yes—
59C4⋊C4×C107C2316yes—
60C4⋊C4×C107C2316yes—
61C22⋊C4×C107C23112yes—
62C23×C107—41no—
63C23×C107—41no—
64C23×C107#4114yescomplemented
65C23×C107C2413yes—
66C22×D107—26no—
67C22×D107—43no—
68C22×D107—43no—
69C22×D107C2416yes—
70Q8×C107—42no—
71Q8×C107—42no—
72C2×C4×C107—43no—
73C2×C4×C107—43no—
74C2×C4×C107C2416yes—
75C2×C4×C107C2416yes—
76C22⋊D107—224no—
77C2×Dic107—26no—
78C2×Dic107—26no—
79C2×Dic107—43no—
80C2×Dic107C2413yes—
81C2×Dic107C2416yes—
82C2×Dic107C2411yescharacteristic
83D428—46no—
84D428—46no—
85C4×D107—212no—
86C4×D107—46no—
87C22×C107—23no—
88C22×C107—23no—
89C22×C107—44no—
90C22×C107C2×#4116yes—
91C22×C107C2511yescharacteristic
92C2×D107—26no—
93C2×D107—26no—
94C2×D107—26no—
95C2×D107—412no—
96C4×C107—26no—
97C4×C107—26no—
98Dic107—26no—
99Dic107—23no—
100Dic107H(5,𝔽2)12yes—
101C2×C107—26no—
102C2×C107C22×#4111yescharacteristic
103C2×C107C2×H(5,𝔽2)11yescharacteristic
104C2×C107#58511yescharacteristic
105D107—46no—
106#3459—1071nomaximal, Sylow
107C107#345911yescharacteristic, complemented, Sylow
108#543D10711yescharacteristic, complemented
109#539—1076no—
110#538—1076no—
111#534—1073no—
112#529—1076no—
113#525—1076no—
114C2×C4×D4—1073no—
115C2×#41—1072no—
116C22×D4—1073no—
117C23×C4—1071no—
118C23×C4—1073no—
119#126—10712no—
120#125—1076no—
121#124—1076no—
122#123—10712no—
123#123—10712no—
124C22⋊Q8—1076no—
125C22⋊Q8C2×D10716yescomplemented
126C22⋊Q8C2×D10716yescomplemented
127#121—1076no—
128#121—1076no—
129#121—1076no—
130#120—10712no—
131C4×Q8—1072no—
132C4×D4—1076no—
133C4×D4—10712no—
134C4×D4—10712no—
135#117—1076no—
136C2×C4⋊C4—1073no—
137C2×C22⋊C4—1073no—
138C2×C22⋊C4C2×D10713yes—
139C2×C42—1073no—
140C24—1073no—
141C24C22×D10711yescharacteristic
142#41—4284no—
143C2×Q8C22×D10712yes—
144C2×Q8C22×D10712yes—
145C2×D4—10712no—
146C2×D4—1076no—
147C2×D4—1076no—
148C2×D4—21412no—
149C22×C4—1073no—
150C22×C4—1073no—
151C22×C4—1073no—
152C22×C4—1076no—
153C22×C4—1076no—
154C22×C4—4286no—
155C22×C4—4282no—
156C22×C4C22×D10713yes—
157C22×C4C22×D10713yes—
158C4⋊C4—10712no—
159C4⋊C4—1076no—
160C4⋊C4—1076no—
161C4⋊C4C22×D10716yes—
162C4⋊C4C22×D10716yes—
163C22⋊C4—10712no—
164C22⋊C4—10712no—
165C22⋊C4—10712no—
166C22⋊C4C22×D107112yes—
167C42—1076no—
168C42—1076no—
169C23—41no—
170C23—41no—
171C23—1076no—
172C23—4283no—
173C23—4283no—
174C23—2146no—
175C23#1313381914yescomplemented
176C23C23×D10713yes—
177Q8—42no—
178Q8—42no—
179D4—4286no—
180D4—4286no—
181D4—21424no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—1071no—
185C2×C4—1076no—
186C2×C4—1073no—
187C2×C4—4286no—
188C2×C4—4283no—
189C2×C4—21412no—
190C2×C4—2146no—
191C2×C4—2146no—
192C2×C4C23×D10716yes—
193C2×C4C23×D10716yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—42812no—
198C22—2146no—
199C22—2146no—
200C22—2146no—
201C22C2×#1313381916yes—
202C22C24×D10711yescharacteristic
203C4—26no—
204C4—26no—
205C4—1072no—
206C4—2146no—
207C4—2143no—
208C2—26no—
209C2—4286no—
210C2C22×#1313381911yescharacteristic
211C2H(5,𝔽2)×D10711yescharacteristic
212C2#2029346211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D107
commutator, socle, cosocleC22×C107C25
Fitting#543×C107C2
hypercenter#543D107
hypocenterC107#3459
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #20293226C2semidirect
2. #20293217C2semidirect
3. #20293189C2semidirect
4. #20293163C2semidirect
5. #20293140C2semidirect
6. C2×(C22⋊D107)⋊C4C2semidirect
7. #543×C107C2semidirect
8. C2×#13133819C22semidirect
11. C22×C4×D107C22semidirect
12. #16812104C22semidirect
13. #16812100C22semidirect
14. #16812098C22semidirect
15. #16812094C22semidirect
16. #16812093C22semidirect
17. #16812085C22semidirect
18. #16812084C22semidirect
21. #16812079C22semidirect
22. Q8×Dic107C22semidirect
23. #16812071C22semidirect
24. (C22⋊D107)⋊C4C22semidirect
25. (C22⋊D107)⋊C4C22semidirect
26. #16812067C22semidirect
27. C2×Dic107⋊C4C22semidirect
28. C2×(C2×D107)⋊C4C22semidirect
29. C2×C4×Dic107C22semidirect
30. C22⋊Q8×C107C22semidirect
31. C22⋊Q8×C107C22semidirect
64. C23×C107#41semidirect
107. C107#3459semidirect
108. #543D107semidirect
125. C22⋊Q8C2×D107semidirect
126. C22⋊Q8C2×D107semidirect
175. C23#13133819semidirect

Maximal subgroups

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

subgroupindex
#202932262
#202932172
#202931892
#202931632
#202931402
C2×(C22⋊D107)⋊C42
#543×C1072
#3459107