Finite Group Databasebeta

← #28376444

Subgroups — #28376444

Order 13952.

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#20295625C216yesmaximal, complemented
2#20295616C216yesmaximal, complemented
3#20295588C213yesmaximal, complemented
4#20295562C216yesmaximal, complemented
5#20295539C216yesmaximal, complemented
6C2×#16812659C213yesmaximal, complemented
7#543×C109C211yesmaximal, characteristic, complemented
8C2×#13182030C2212yescomplemented
9C22×C22⋊D109C2213yes—
10C23×Dic109C2211yescharacteristic
11C22×C4×D109C2213yescomplemented
12#16812694C22112yescomplemented
13#16812690C2216yescomplemented
14#16812688C2216yescomplemented
15#16812684C22112yescomplemented
16#16812683C22112yescomplemented
17#16812675C2216yescomplemented
18#16812674C2216yescomplemented
19#16812673C2216yes—
20#16812673C2216yes—
21#16812669C22112yescomplemented
22Q8×Dic109C2212yescomplemented
23#16812661C2216yescomplemented
24#16812659C22112yescomplemented
25#16812659C22112yescomplemented
26#16812657C2216yescomplemented
27C2×#13182008C2213yescomplemented
28C2×#13182006C2213yescomplemented
29C2×C4×Dic109C2213yescomplemented
30C22⋊Q8×C109C2216yescomplemented
31C22⋊Q8×C109C2216yescomplemented
32C2×C22⋊C4×C109C2213yes—
33C23×D109C2313yes—
34#13182030—44no—
35C2×D436C2316yes—
36C2×D436C2316yes—
37C2×C22⋊D109—212no—
38C2×C22⋊D109C23112yes—
39C22×Dic109—42no—
40C22×Dic109C2313yes—
41C22×Dic109C2313yes—
42C22×Dic109C2313yes—
43C2×C4×D109—46no—
44C2×C4×D109C2316yes—
45C2×C4×D109C2316yes—
46C4⋊Dic109C2316yes—
47C4⋊Dic109C2316yes—
48#13182008C23112yes—
49#13182006C23112yes—
50#13182006C23112yes—
51#13182006C23112yes—
52C4×Dic109C2316yes—
53C4×Dic109C2316yes—
54C24×C109C2311yescharacteristic
55C2×Q8×C109C2312yes—
56C2×Q8×C109C2312yes—
57C22×C4×C109C2313yes—
58C22×C4×C109C2313yes—
59C4⋊C4×C109C2316yes—
60C4⋊C4×C109C2316yes—
61C22⋊C4×C109C23112yes—
62C23×C109—41no—
63C23×C109—41no—
64C23×C109#4114yescomplemented
65C23×C109C2413yes—
66C22×D109—26no—
67C22×D109—43no—
68C22×D109—43no—
69C22×D109C2416yes—
70Q8×C109—42no—
71Q8×C109—42no—
72C2×C4×C109—43no—
73C2×C4×C109—43no—
74C2×C4×C109C2416yes—
75C2×C4×C109C2416yes—
76C22⋊D109—224no—
77C2×Dic109—26no—
78C2×Dic109—26no—
79C2×Dic109—43no—
80C2×Dic109C2413yes—
81C2×Dic109C2416yes—
82C2×Dic109C2411yescharacteristic
83D436—46no—
84D436—46no—
85C4×D109—212no—
86C4×D109—46no—
87C22×C109—23no—
88C22×C109—23no—
89C22×C109—44no—
90C22×C109C2×#4116yes—
91C22×C109C2511yescharacteristic
92C2×D109—26no—
93C2×D109—26no—
94C2×D109—26no—
95C2×D109—412no—
96C4×C109—26no—
97C4×C109—26no—
98Dic109—26no—
99Dic109—23no—
100Dic109H(5,𝔽2)12yes—
101C2×C109—26no—
102C2×C109C22×#4111yescharacteristic
103C2×C109C2×H(5,𝔽2)11yescharacteristic
104C2×C109#58511yescharacteristic
105D109—46no—
106#3459—1091nomaximal, Sylow
107C109#345911yescharacteristic, complemented, Sylow
108#543D10911yescharacteristic, complemented
109#539—1096no—
110#538—1096no—
111#534—1093no—
112#529—1096no—
113#525—1096no—
114C2×C4×D4—1093no—
115C2×#41—1092no—
116C22×D4—1093no—
117C23×C4—1091no—
118C23×C4—1093no—
119#126—10912no—
120#125—1096no—
121#124—1096no—
122#123—10912no—
123#123—10912no—
124C22⋊Q8—1096no—
125C22⋊Q8C2×D10916yescomplemented
126C22⋊Q8C2×D10916yescomplemented
127#121—1096no—
128#121—1096no—
129#121—1096no—
130#120—10912no—
131C4×Q8—1092no—
132C4×D4—1096no—
133C4×D4—10912no—
134C4×D4—10912no—
135#117—1096no—
136C2×C4⋊C4—1093no—
137C2×C22⋊C4—1093no—
138C2×C22⋊C4C2×D10913yes—
139C2×C42—1093no—
140C24—1093no—
141C24C22×D10911yescharacteristic
142#41—4364no—
143C2×Q8C22×D10912yes—
144C2×Q8C22×D10912yes—
145C2×D4—10912no—
146C2×D4—1096no—
147C2×D4—1096no—
148C2×D4—21812no—
149C22×C4—1093no—
150C22×C4—1093no—
151C22×C4—1093no—
152C22×C4—1096no—
153C22×C4—1096no—
154C22×C4—4366no—
155C22×C4—4362no—
156C22×C4C22×D10913yes—
157C22×C4C22×D10913yes—
158C4⋊C4—10912no—
159C4⋊C4—1096no—
160C4⋊C4—1096no—
161C4⋊C4C22×D10916yes—
162C4⋊C4C22×D10916yes—
163C22⋊C4—10912no—
164C22⋊C4—10912no—
165C22⋊C4—10912no—
166C22⋊C4C22×D109112yes—
167C42—1096no—
168C42—1096no—
169C23—41no—
170C23—41no—
171C23—1096no—
172C23—4363no—
173C23—4363no—
174C23—2186no—
175C23#1318203014yescomplemented
176C23C23×D10913yes—
177Q8—42no—
178Q8—42no—
179D4—4366no—
180D4—4366no—
181D4—21824no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—1091no—
185C2×C4—1096no—
186C2×C4—1093no—
187C2×C4—4366no—
188C2×C4—4363no—
189C2×C4—21812no—
190C2×C4—2186no—
191C2×C4—2186no—
192C2×C4C23×D10916yes—
193C2×C4C23×D10916yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—43612no—
198C22—2186no—
199C22—2186no—
200C22—2186no—
201C22C2×#1318203016yes—
202C22C24×D10911yescharacteristic
203C4—26no—
204C4—26no—
205C4—1092no—
206C4—2186no—
207C4—2183no—
208C2—26no—
209C2—4366no—
210C2C22×#1318203011yescharacteristic
211C2H(5,𝔽2)×D10911yescharacteristic
212C2#2029586111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D109
commutator, socle, cosocleC22×C109C25
Fitting#543×C109C2
hypercenter#543D109
hypocenterC109#3459
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #20295625C2semidirect
2. #20295616C2semidirect
3. #20295588C2semidirect
4. #20295562C2semidirect
5. #20295539C2semidirect
6. C2×#16812659C2semidirect
7. #543×C109C2semidirect
8. C2×#13182030C22semidirect
11. C22×C4×D109C22semidirect
12. #16812694C22semidirect
13. #16812690C22semidirect
14. #16812688C22semidirect
15. #16812684C22semidirect
16. #16812683C22semidirect
17. #16812675C22semidirect
18. #16812674C22semidirect
21. #16812669C22semidirect
22. Q8×Dic109C22semidirect
23. #16812661C22semidirect
24. #16812659C22semidirect
25. #16812659C22semidirect
26. #16812657C22semidirect
27. C2×#13182008C22semidirect
28. C2×#13182006C22semidirect
29. C2×C4×Dic109C22semidirect
30. C22⋊Q8×C109C22semidirect
31. C22⋊Q8×C109C22semidirect
64. C23×C109#41semidirect
107. C109#3459semidirect
108. #543D109semidirect
125. C22⋊Q8C2×D109semidirect
126. C22⋊Q8C2×D109semidirect
175. C23#13182030semidirect

Maximal subgroups

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

subgroupindex
#202956252
#202956162
#202955882
#202955622
#202955392
C2×#168126592
#543×C1092
#3459109