Finite Group Databasebeta

← #28345738

Subgroups — #28345738

Order 13696.

752 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#16812139C211yesmaximal, characteristic, complemented
2C22×#13133806C211yesmaximal, characteristic, complemented
3#20293198C211yesmaximal, characteristic, complemented
4#20292789C214yesmaximal, complemented
5#20292778C214yesmaximal, complemented
6#20292632C211yesmaximal, characteristic, complemented
7#20292623C211yesmaximal, characteristic, complemented
8#20292594C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic, complemented
11#16812139—24no—
12C2×#13133808C2212yescomplemented
13C2×#13133806—24no—
14C2×#13133806C2214yescomplemented
15C2×#13133806C2212yescomplemented
16C2×#13133802C2211yescharacteristic, complemented
17C22×C107⋊C8C2211yescharacteristic, complemented
18#16812107—21no—
19#16812096—21no—
20#16812090—22no—
21#16812088C2214yescomplemented
22#16812068C2211yescharacteristic, complemented
23#16812033C2212yescomplemented
24#16812030C2212yescomplemented
25#16812030C2214yescomplemented
26C22⋊(C107⋊C8)C2214yescomplemented
27H(5,𝔽2)×C107—24no—
28H(5,𝔽2)×C107—24no—
29C2×#41×C107—21no—
30C2×#41×C107C2212yes—
31C2×#41×C107C2211yescharacteristic
32C2×#41×C107C2211yescharacteristic
33C22×D4×C107—22no—
34C22×D4×C107—21no—
35C22×D4×C107C2212yes—
36C22×D4×C107C2211yescharacteristic
37C2×Dic214—22no—
38C2×Dic214—22no—
39C2×Dic214—21no—
40C2×Dic214C2312yes—
41C22×Dic107—21no—
42#13133808—24no—
43#13133806—28no—
44#13133806—24no—
45#13133802—22no—
46C2×C107⋊C8—22no—
47C2×C107⋊C8C2312yes—
48C2×C107⋊C8C2312yes—
49C4⋊Dic107C2312yes—
50Dic107⋊C4—24no—
51C22⋊Dic107—21no—
52C22⋊Dic107—24no—
53C4×Dic107—21no—
54C24×C107—22no—
55C24×C107—21no—
56#41×C107—28no—
57#41×C107—24no—
58#41×C107—24no—
59#41×C107—22no—
60#41×C107—22no—
61#41×C107—22no—
62#41×C107D414yes—
63C2×Q8×C107D411yescharacteristic
64C2×Q8×C107C2311yescharacteristic
65C2×D4×C107—216no—
66C2×D4×C107—28no—
67C2×D4×C107—22no—
68C2×D4×C107—21no—
69C2×D4×C107—24no—
70C2×D4×C107—24no—
71C2×D4×C107—22no—
72C2×D4×C107—21no—
73C2×D4×C107—22no—
74C2×D4×C107D412yes—
75C2×D4×C107D411yescharacteristic
76C2×D4×C107D414yes—
77C2×D4×C107C2311yescharacteristic
78C2×D4×C107C2312yes—
79C22×C4×C107—22no—
80C22×C4×C107—21no—
81C22×C4×C107C2312yes—
82C22×C4×C107C2311yescharacteristic
83C23×C107—28no—
84C23×C107—22no—
85C23×C107—21no—
86C23×C107—24no—
87C23×C107—82no—
88C23×C107—84no—
89C23×C107C2×D412yes—
90C23×C107C2×D411yescharacteristic
91Q8×C107—22no—
92Q8×C107—21no—
93Q8×C107C2×D412yes—
94D4×C107—28no—
95D4×C107—28no—
96D4×C107—24no—
97D4×C107—24no—
98D4×C107—22no—
99D4×C107—24no—
100D4×C107—22no—
101D4×C107—21no—
102D4×C107C2×D412yes—
103D4×C107C2×D414yes—
104C2×C4×C107—28no—
105C2×C4×C107—21no—
106C2×C4×C107—24no—
107C2×C4×C107—22no—
108C2×C4×C107—22no—
109C2×C4×C107C2×D412yes—
110C2×C4×C107C2×D411yescharacteristic
111C2×C4×C107C2×D412yes—
112C2×C4×C107C2×D412yes—
113C2×C4×C107C2411yescharacteristic
114C2×Dic107—22no—
115C2×Dic107—22no—
116C2×Dic107—42no—
117Dic214—22no—
118Dic214—22no—
119Dic214—42no—
120C107⋊C8—22no—
121C107⋊C8—22no—
122C22×C107—22no—
123C22×C107—22no—
124C22×C107—24no—
125C22×C107—22no—
126C22×C107—24no—
127C22×C107—44no—
128C22×C107—42no—
129C22×C107—84no—
130C22×C107—88no—
131C22×C107#12012yes—
132C22×C107C22×D411yescharacteristic
133C4×C107—22no—
134C4×C107—22no—
135C4×C107#12012yes—
136C4×C107C22×D411yescharacteristic
137C4×C107C22×D411yescharacteristic
138Dic107—42no—
139Dic107—42no—
140C2×C107—22no—
141C2×C107—42no—
142C2×C107—44no—
143C2×C107—42no—
144C2×C107C2×#12011yescharacteristic
145C2×C107#57712yes—
146#3009—1071nomaximal, Sylow
147C107#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)D10711yescharacteristic
149C2×#137—1071no—
150C22×SD16—1071no—
151#536—1071no—
152#452—1074no—
153#450—1074no—
154#418—1071no—
155#416—1071no—
156#408—1071no—
157H(5,𝔽2)—24no—
158H(5,𝔽2)—24no—
159C2×#41—21no—
160C2×#41C2×D10712yes—
161C2×#41C2×D10711yescharacteristic
162C2×#41C2×D10711yescharacteristic
163C22×Q8—1071no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4C2×D10712yes—
167C22×D4C2×D10711yescharacteristic
168#137—2144no—
169C2×Q16—1072no—
170C2×SD16—1074no—
171C2×SD16—1072no—
172C2×SD16—2144no—
173C2×M16—1071no—
174C22×C8—1071no—
175C4⋊Q8—2141no—
176#124—2141no—
177#123—2142no—
178C22⋊Q8—1074no—
179#117—1071no—
180#103—1072no—
181#102—1072no—
182#102—1074no—
183C22⋊C8—1074no—
184C24—22no—
185C24—21no—
186#41—28no—
187#41—24no—
188#41—24no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41C22⋊D10714yes—
193C2×Q8—1072no—
194C2×Q8—2142no—
195C2×Q8—2142no—
196C2×Q8—2141no—
197C2×Q8C22⋊D10711yescharacteristic
198C2×Q8C22×D10711yescharacteristic
199C2×D4—216no—
200C2×D4—28no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4C22⋊D10712yes—
209C2×D4C22⋊D10711yescharacteristic
210C2×D4C22⋊D10714yes—
211C2×D4C22×D10711yescharacteristic
212C2×D4C22×D10712yes—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—2141no—
216C22×C4C22×D10712yes—
217C22×C4C22×D10711yescharacteristic
218Q16—2144no—
219SD16—2148no—
220SD16—2144no—
221M16—2142no—
222C2×C8—1072no—
223C2×C8—1072no—
224C2×C8—2142no—
225C4⋊C4—1072no—
226C4⋊C4—2144no—
227C22⋊C4—2141no—
228C22⋊C4—2144no—
229C42—2141no—
230C23—28no—
231C23—22no—
232C23—21no—
233C23—24no—
234C23—82no—
235C23—84no—
236C23C2×C22⋊D10712yes—
237C23C2×C22⋊D10711yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—4282no—
241Q8—2142no—
242Q8—2142no—
243Q8C2×C22⋊D10712yes—
244D4—28no—
245D4—28no—
246D4—24no—
247D4—24no—
248D4—22no—
249D4—24no—
250D4—22no—
251D4—21no—
252D4C2×C22⋊D10712yes—
253D4C2×C22⋊D10714yes—
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—4282no—
260C2×C4—2142no—
261C2×C4—2142no—
262C2×C4C2×C22⋊D10712yes—
263C2×C4C2×C22⋊D10711yescharacteristic
264C2×C4C2×C22⋊D10712yes—
265C2×C4C2×C22⋊D10712yes—
266C2×C4C23×D10711yescharacteristic
267C8—2142no—
268C8—2142no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—24no—
273C22—22no—
274C22—44no—
275C22—42no—
276C22—84no—
277C22—88no—
278C22#1681207812yes—
279C22C22×C22⋊D10711yescharacteristic
280C4—22no—
281C4—22no—
282C4—4282no—
283C4—4282no—
284C4#1681207812yes—
285C4C22×C22⋊D10711yescharacteristic
286C4C22×C22⋊D10711yescharacteristic
287C2—22no—
288C2—42no—
289C2—44no—
290C2—42no—
291C2C2×#1681207811yescharacteristic
292C2#2029343812yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radicalC2×H(5,𝔽2)×C107C2
socleC22×C107C22×D4
hypercenterC2×H(5,𝔽2)D107
hypocenterC107#3009
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16812139C2semidirect
2. C22×#13133806C2semidirect
3. #20293198C2semidirect
4. #20292789C2semidirect
5. #20292778C2semidirect
6. #20292632C2semidirect
7. #20292623C2semidirect
8. #20292594C2semidirect
10. C22×Dic214C22semidirect
12. C2×#13133808C22semidirect
14. C2×#13133806C22semidirect
15. C2×#13133806C22semidirect
16. C2×#13133802C22semidirect
17. C22×C107⋊C8C22semidirect
21. #16812088C22semidirect
22. #16812068C22semidirect
23. #16812033C22semidirect
24. #16812030C22semidirect
25. #16812030C22semidirect
26. C22⋊(C107⋊C8)C22semidirect
147. C107#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
C22×#131338062
#202931982
#202927892
#202927782
#202926322
#202926232
#202925942
C2×H(5,𝔽2)×C1072
#3009107