Finite Group Databasebeta

← #28340166

Subgroups — #28340166

Order 13696.

608 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812108C211yesmaximal, characteristic, complemented
2C2×#16812087C212yesmaximal
3C2×C4⋊C4×D107C211yesmaximal, characteristic
4#20292558C212yesmaximal, complemented
5#20292547C212yesmaximal, complemented
6#20292525C211yesmaximal, characteristic, complemented
7#20292523C211yesmaximal, characteristic, complemented
8#20292498C212yesmaximal
9#20292484C211yesmaximal, characteristic
10C4×(C2×D107)⋊C4C211yesmaximal, characteristic
11#384×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2212yes—
13C22×C4×D107C2212yes—
14C22×C4×D107C2211yescharacteristic
15#16812108—24no—
16#16812087—28no—
17C2×C4⋊Dic107C2211yescharacteristic
18C2×C4⋊Dic107C2211yescharacteristic
19C2×Dic107⋊C4C2212yes—
20C2×Dic107⋊C4C2212yes—
21C2×Dic107⋊C4C2212yes—
22C2×Dic107⋊C4C2212yes—
23C4⋊C4×D107—24no—
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×C4×Dic107C2211yescharacteristic
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2211yescharacteristic
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C2×C4⋊C4×C107C2212yes—
34C2×C4⋊C4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C4⋊C4×C107C2212yes—
37C2×C42×C107C2211yescharacteristic
38C23×D107C2311yescharacteristic
39C2×Dic214—24no—
40C2×Dic214—24no—
41C2×Dic214—42no—
42C22×Dic107C2312yes—
43C22×Dic107C2311yescharacteristic
44C22×Dic107C2312yes—
45C22×Dic107C2311yescharacteristic
46C22×Dic107C2311yescharacteristic
47C2×C4×D107—22no—
48C2×C4×D107—22no—
49C2×C4×D107—24no—
50C2×C4×D107—24no—
51C2×C4×D107—42no—
52C2×C4×D107Q814yescomplemented
53C4⋊Dic107—22no—
54C4⋊Dic107D414yes—
55Dic107⋊C4—24no—
56Dic107⋊C4—24no—
57Dic107⋊C4—24no—
58Dic107⋊C4—24no—
59(C2×D107)⋊C4—24no—
60(C2×D107)⋊C4D414yes—
61(C2×D107)⋊C4Q814yescomplemented
62C4×Dic107—22no—
63C22×C4×C107C2312yes—
64C22×C4×C107C2312yes—
65C22×C4×C107C2311yescharacteristic
66C22×C4×C107C2311yescharacteristic
67C22×C4×C107C2311yescharacteristic
68C4⋊C4×C107—22no—
69C4⋊C4×C107—24no—
70C4⋊C4×C107—22no—
71C4⋊C4×C107—42no—
72C42×C107—22no—
73C23×C107C2411yescharacteristic
74C22×D107—21no—
75C22×D107—21no—
76C22×D107—21no—
77C22×D107—21no—
78C22×D107C2×D412yes—
79C22×D107C2×Q812yes—
80C22×D107C2×Q812yes—
81C2×C4×C107—21no—
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—21no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107C2×D412yes—
92C2×C4×C107C2×D412yes—
93C2×C4×C107C2×Q812yes—
94C2×C4×C107C2×Q812yes—
95C2×C4×C107#4112yes—
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107—21no—
99C2×Dic107—22no—
100C2×Dic107—21no—
101C2×Dic107—22no—
102C2×Dic107—21no—
103C2×Dic107—21no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107C2×D412yes—
108C2×Dic107C2×D414yes—
109C2×Dic107C2×Q812yes—
110C2×Dic107C2×Q812yes—
111C4×D107—28no—
112C4×D107—48no—
113Dic214—48no—
114C22×C107C22×D411yescharacteristic
115C22×C107C22×D411yescharacteristic
116C22×C107C22×Q811yescharacteristic
117C22×C107C22×Q811yescharacteristic
118C22×C107C2×#4111yescharacteristic
119C22×C107H(5,𝔽2)11yescharacteristic
120C22×C1072−1+411yescharacteristic
121C2×D107—22no—
122C2×D107—22no—
123C2×D107—22no—
124C2×D107—22no—
125C2×D107—22no—
126C2×D107—22no—
127C2×D107C4⋊Q814yescomplemented
128C4×C107—22no—
129C4×C107—22no—
130C4×C107—42no—
131C4×C107—42no—
132C4×C107C22⋊Q814yescomplemented
133Dic107—22no—
134Dic107—22no—
135Dic107—24no—
136Dic107—22no—
137Dic107—42no—
138C2×C107C2×C22⋊Q811yescharacteristic
139C2×C107C2×C4⋊Q811yescharacteristic
140C2×C107#54411yescharacteristic
141C2×C107D4211yescharacteristic
142C2×C107#54711yescharacteristic
143C2×C107D4×Q811yescharacteristic
144C2×C107#55411yescharacteristic
145D107—24no—
146#2440—1071nomaximal, Sylow
147C107#244011yescharacteristic, complemented, Sylow
148C2×C4⋊Q8—1071no—
149C2×C22⋊Q8—1072no—
150C22×C4⋊C4—1071no—
151#398—1072no—
152#395—1072no—
153#389—1071no—
154#389—1071no—
155#385—1072no—
156#384D10711yescharacteristic, complemented
157#380—1071no—
158C4×C22⋊C4—1071no—
159C22×Q8—1072no—
160C23×C4—1071no—
161C23×C4—1072no—
162C4⋊Q8—2144no—
163C22⋊Q8—2148no—
164C2×C4⋊C4—1071no—
165C2×C4⋊C4—1072no—
166C2×C4⋊C4—1072no—
167C2×C4⋊C4—1072no—
168C2×C4⋊C4—1072no—
169C2×C4⋊C4—1071no—
170C2×C4⋊C4—2144no—
171C2×C4⋊C4C2×D10711yescharacteristic
172C2×C4⋊C4C2×D10712yes—
173C2×C4⋊C4C2×D10711yescharacteristic
174C2×C4⋊C4C2×D10712yes—
175C2×C22⋊C4—1071no—
176C2×C22⋊C4—1071no—
177C2×C22⋊C4—1072no—
178C2×C42—1071no—
179C2×C42C2×D10711yescharacteristic
180#95—1071no—
181#95—1071no—
182#95—1072no—
183#95—1072no—
184#95—1072no—
185C24—1071no—
186C2×Q8—4282no—
187C2×Q8—2144no—
188C2×Q8—2144no—
189C22×C4—1071no—
190C22×C4—1071no—
191C22×C4—1071no—
192C22×C4—1072no—
193C22×C4—1074no—
194C22×C4—1072no—
195C22×C4—4282no—
196C22×C4—2144no—
197C22×C4—2144no—
198C22×C4—2142no—
199C22×C4—2142no—
200C22×C4C22×D10712yes—
201C22×C4C22×D10712yes—
202C22×C4C22×D10711yescharacteristic
203C22×C4C22×D10711yescharacteristic
204C22×C4C22×D10711yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—24no—
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—1074no—
210C4⋊C4—2144no—
211C4⋊C4—2144no—
212C4⋊C4—2144no—
213C4⋊C4—2144no—
214C4⋊C4—2142no—
215C22⋊C4—1074no—
216C22⋊C4—1074no—
217C22⋊C4—2144no—
218C42—22no—
219C42—2142no—
220C23—1072no—
221C23—1072no—
222C23—1072no—
223C23—2141no—
224C23—2141no—
225C23—2141no—
226C23—2141no—
227C23C23×D10711yescharacteristic
228Q8—4288no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—1072no—
240C2×C4—1074no—
241C2×C4—1072no—
242C2×C4—1072no—
243C2×C4—4288no—
244C2×C4—2142no—
245C2×C4—2142no—
246C2×C4—2142no—
247C2×C4—2141no—
248C2×C4—2141no—
249C2×C4—2141no—
250C2×C4—2142no—
251C2×C4—2141no—
252C2×C4—2142no—
253C2×C4—2141no—
254C2×C4—2141no—
255C2×C4—2148no—
256C2×C4D4×D10712yes—
257C2×C4D4×D10712yes—
258C2×C4Q8×D10712yes—
259C2×C4Q8×D10712yes—
260C2×C4#1313381812yes—
261C22—1074no—
262C22—2142no—
263C22—2142no—
264C22—2142no—
265C22—2142no—
266C22—2142no—
267C22—2142no—
268C22C2×D4×D10711yescharacteristic
269C22C2×D4×D10711yescharacteristic
270C22C2×Q8×D10711yescharacteristic
271C22C2×Q8×D10711yescharacteristic
272C22C2×#1313381811yescharacteristic
273C22#1681215411yescharacteristic
274C22#1681215711yescharacteristic
275C4—22no—
276C4—22no—
277C4—42no—
278C4—42no—
279C4—4282no—
280C4—2142no—
281C4—2144no—
282C4—2142no—
283C4—2142no—
284C4#1681208714yescomplemented
285C2—2144no—
286C2C2×#1681208711yescharacteristic
287C2C4⋊Q8×D10711yescharacteristic
288C2#2029325411yescharacteristic
289C2#2029326211yescharacteristic
290C2#2029327811yescharacteristic
291C2#2029329211yescharacteristic
292C2#2029333211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#384×C107C2
prime radicalC23×D107C23
hypercenter#384D107
hypocenterC107#2440
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#16812108C2semidirect
4. #20292558C2semidirect
5. #20292547C2semidirect
6. #20292525C2semidirect
7. #20292523C2semidirect
11. #384×C107C2semidirect
52. C2×C4×D107Q8semidirect
61. (C2×D107)⋊C4Q8semidirect
127. C2×D107C4⋊Q8semidirect
132. C4×C107C22⋊Q8semidirect
147. C107#2440semidirect
156. #384D107semidirect
284. C4#16812087semidirect

Maximal subgroups

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

subgroupindex
C2×#168121082
C2×#168120872
C2×C4⋊C4×D1072
#202925582
#202925472
#202925252
#202925232
#202924982
#202924842
C4×(C2×D107)⋊C42
#384×C1072
#2440107