Finite Group Databasebeta

← #28344804

Subgroups — #28344804

Order 13696.

924 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 201 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×#16812105C211yesmaximal, characteristic, complemented
2C2×#16812083C211yesmaximal, characteristic, complemented
3C2×#16812081C214yesmaximal, complemented
4C22×C4×Dic107C211yesmaximal, characteristic, complemented
5#20292530C211yesmaximal, characteristic, complemented
6#20292528C211yesmaximal, characteristic, complemented
7#20292527C214yesmaximal, complemented
8#20292496C211yesmaximal, characteristic, complemented
9C2×#121×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C22×C22⋊D107C2214yes—
12C22×C22⋊D107C2212yes—
13C23×Dic107C2212yes—
14#16812105—24no—
15#16812083—24no—
16#16812081—216no—
17C2×C4⋊Dic107C2211yescharacteristic
18C2×Dic107⋊C4C2214yes—
19C2×C22⋊Dic107C2214yes—
20C2×(C2×D107)⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2214yes—
22C2×C4×Dic107—24no—
23C2×C4×Dic107C2212yes—
24C2×C4×Dic107C2211yescharacteristic
25C2×C4×Dic107C2211yescharacteristic
26C22×D4×C107C2211yescharacteristic
27C22×D4×C107C2212yes—
28C23×C4×C107C2211yescharacteristic
29#121×C107—24no—
30C2×C4⋊C4×C107C2211yescharacteristic
31C2×C22⋊C4×C107C2212yes—
32C23×D107C2312yes—
33C2×D428—22no—
34C2×D428—22no—
35C2×D428—22no—
36C2×C22⋊D107—28no—
37C2×C22⋊D107—24no—
38C2×C22⋊D107—28no—
39C2×C22⋊D107—28no—
40C2×C22⋊D107—28no—
41C22×Dic107—24no—
42C22×Dic107—24no—
43C22×Dic107D418yescomplemented
44C22×Dic107C2312yes—
45C22×Dic107C2312yes—
46C22×Dic107C2312yes—
47C4⋊Dic107—22no—
48Dic107⋊C4—28no—
49C22⋊Dic107—28no—
50(C2×D107)⋊C4—24no—
51(C2×D107)⋊C4—28no—
52C4×Dic107—22no—
53C4×Dic107—24no—
54C4×Dic107D414yescomplemented
55C24×C107C2312yes—
56C24×C107C2311yescharacteristic
57C2×D4×C107—28no—
58C2×D4×C107—24no—
59C2×D4×C107—22no—
60C2×D4×C107—22no—
61C2×D4×C107—22no—
62C22×C4×C107—22no—
63C22×C4×C107—22no—
64C22×C4×C107D414yescomplemented
65C22×C4×C107C2312yes—
66C22×C4×C107C2311yescharacteristic
67C22×C4×C107C2311yescharacteristic
68C4⋊C4×C107—22no—
69C22⋊C4×C107—24no—
70C23×C107—21no—
71C23×C107—21no—
72C23×C107—22no—
73C23×C107—22no—
74C23×C107—24no—
75C23×C107—22no—
76C23×C107—22no—
77C23×C107—24no—
78C23×C107C2×D414yes—
79C23×C107C2×D412yes—
80C23×C107C2411yescharacteristic
81C22×D107—24no—
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—24no—
85C22×D107—22no—
86D4×C107—48no—
87D4×C107—44no—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107—28no—
91C2×C4×C107—22no—
92C2×C4×C107—24no—
93C2×C4×C107C2×D412yes—
94C2×C4×C107C2×D412yes—
95C2×C4×C107C2×D412yes—
96C2×C4×C107#4112yes—
97C22⋊D107—48no—
98C22⋊D107—416no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—216no—
102C2×Dic107—22no—
103C2×Dic107—24no—
104C2×Dic107C2×D414yes—
105C2×Dic107C2×D414yes—
106C2×Dic107C2×D414yes—
107C2×Dic107#4114yes—
108D428—44no—
109C22×C107—24no—
110C22×C107—22no—
111C22×C107—24no—
112C22×C107—22no—
113C22×C107—44no—
114C22×C107—42no—
115C22×C107—42no—
116C22×C107—44no—
117C22×C107—42no—
118C22×C107#12714yescomplemented
119C22×C107C22×D411yescharacteristic
120C22×C107C22×D412yes—
121C22×C107C22×D411yescharacteristic
122C22×C107C2×#4112yes—
123C22×C107C2×#4111yescharacteristic
124C2×D107—42no—
125C2×D107—44no—
126C2×D107—42no—
127C2×D107—44no—
128C2×D107—42no—
129C4×C107—22no—
130C4×C107—42no—
131C4×C107#12114yescomplemented
132Dic107—24no—
133Dic107—42no—
134Dic107#12118yescomplemented
135C2×C107—24no—
136C2×C107—82no—
137C2×C107C2×#12112yes—
138C2×C107C2×#12111yescharacteristic
139C2×C107C2×#12711yescharacteristic
140C2×C107#53212yes—
141C2×C107#53211yescharacteristic
142D107—82no—
143#2856—1071nomaximal, Sylow
144C107#285611yescharacteristic, complemented, Sylow
145C2×#127—1071no—
146C2×#121—1074no—
147C2×#121—1071no—
148C2×#121D10711yescharacteristic, complemented
149C22×C42—1071no—
150#390—1071no—
151#390—1071no—
152#390—1074no—
153#384—1071no—
154C22×D4—1072no—
155C22×D4—1074no—
156C22×D4—1071no—
157C22×D4C2×D10712yes—
158C22×D4C2×D10711yescharacteristic
159C23×C4—1072no—
160C23×C4C2×D10711yescharacteristic
161#127—2144no—
162#121—24no—
163#121—21416no—
164#121—2144no—
165C2×C4⋊C4—1074no—
166C2×C4⋊C4—1071no—
167C2×C4⋊C4C2×D10711yescharacteristic
168C2×C22⋊C4—1074no—
169C2×C22⋊C4—1074no—
170C2×C22⋊C4—1072no—
171C2×C22⋊C4C2×D10712yes—
172C2×C42—1071no—
173C2×C42—1072no—
174C2×C42—1071no—
175C2×C42—2144no—
176C24—1072no—
177C24C22×D10712yes—
178C24C22×D10711yescharacteristic
179C2×D4—28no—
180C2×D4—24no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—2148no—
185C2×D4—2148no—
186C2×D4—2148no—
187C2×D4—2148no—
188C2×D4—2144no—
189C2×D4—2142no—
190C2×D4—2142no—
191C2×D4—2142no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—1072no—
195C22×C4—1078no—
196C22×C4—1072no—
197C22×C4—1072no—
198C22×C4—2144no—
199C22×C4—2144no—
200C22×C4C22⋊D10714yescomplemented
201C22×C4C22×D10712yes—
202C22×C4C22×D10711yescharacteristic
203C22×C4C22×D10711yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—2148no—
206C4⋊C4—2142no—
207C22⋊C4—24no—
208C22⋊C4—2148no—
209C22⋊C4—2148no—
210C22⋊C4—2144no—
211C42—1074no—
212C42—2144no—
213C42—2142no—
214C23—21no—
215C23—22no—
216C23—21no—
217C23—24no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—22no—
222C23—2144no—
223C23—2142no—
224C23—2142no—
225C23—2144no—
226C23—2142no—
227C23D4×D10714yes—
228C23C2×C22⋊D10712yes—
229C23C23×D10711yescharacteristic
230D4—48no—
231D4—44no—
232D4—42816no—
233D4—4288no—
234D4—4284no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—28no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—1074no—
241C2×C4—1074no—
242C2×C4—1074no—
243C2×C4—1074no—
244C2×C4—21416no—
245C2×C4—2144no—
246C2×C4—2142no—
247C2×C4—2142no—
248C2×C4—2142no—
249C2×C4D4×D10712yes—
250C2×C4C2×C22⋊D10712yes—
251C2×C4C2×C22⋊D10712yes—
252C2×C4#1313381712yes—
253C22—24no—
254C22—22no—
255C22—24no—
256C22—22no—
257C22—42no—
258C22—44no—
259C22—42no—
260C22—44no—
261C22—42no—
262C22—4282no—
263C22—4284no—
264C22—4282no—
265C22—4284no—
266C22—4282no—
267C22#1681210514yescomplemented
268C22C2×D4×D10712yes—
269C22C2×D4×D10711yescharacteristic
270C22C22×C22⋊D10711yescharacteristic
271C22#41×D10712yes—
272C22C2×#1313381711yescharacteristic
273C4—22no—
274C4—42no—
275C4—1078no—
276C4—4282no—
277C4—2144no—
278C4#1681208114yescomplemented
279C2—24no—
280C2—82no—
281C2—8562no—
282C2#121×D10712yes—
283C2C2×#1681208111yescharacteristic
284C2C2×#1681210511yescharacteristic
285C2#2029317212yes—
286C2#2029317311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#121×C107C2
hypercenterC2×#121D107
hypocenterC107#2856
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16812105C2semidirect
2. C2×#16812083C2semidirect
3. C2×#16812081C2semidirect
4. C22×C4×Dic107C2semidirect
5. #20292530C2semidirect
6. #20292528C2semidirect
7. #20292527C2semidirect
8. #20292496C2semidirect
9. C2×#121×C107C2semidirect
43. C22×Dic107D4semidirect
54. C4×Dic107D4semidirect
64. C22×C4×C107D4semidirect
118. C22×C107#127semidirect
131. C4×C107#121semidirect
134. Dic107#121semidirect
144. C107#2856semidirect
148. C2×#121D107semidirect
200. C22×C4C22⋊D107semidirect
267. C22#16812105semidirect
278. C4#16812081semidirect

Maximal subgroups

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

subgroupindex
C2×#168121052
C2×#168120832
C2×#168120812
C22×C4×Dic1072
#202925302
#202925282
#202925272
#202924962
C2×#121×C1072
#2856107