Finite Group Databasebeta

← #28339133

Subgroups — #28339133

Order 13696.

632 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic107C211yesmaximal, characteristic
2C22×C4⋊Dic107C211yesmaximal, characteristic, complemented
3#20292526C212yesmaximal, complemented
4#20292520C212yesmaximal, complemented
5#20292513C212yesmaximal, complemented
6#20292507C211yesmaximal, characteristic
7#20292502C211yesmaximal, characteristic, complemented
8#20292502C212yesmaximal, complemented
9#20292490C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic107C211yesmaximal, characteristic, complemented
11C2×#123×C107C211yesmaximal, characteristic
12C23×Dic107C2211yescharacteristic
13C23×Dic107C2211yescharacteristic
14C23×Dic107C2211yescharacteristic
15D4×Dic107—24no—
16C2×C4⋊Dic107—24no—
17C2×C4⋊Dic107C2212yes—
18C2×C4⋊Dic107C2212yes—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×C4⋊Dic107C2212yes—
21C2×C22⋊Dic107C2211yescharacteristic
22C2×C22⋊Dic107C2211yescharacteristic
23C2×C22⋊Dic107C2211yescharacteristic
24C2×C22⋊Dic107C2212yes—
25C2×C4×Dic107C2211yescharacteristic
26C2×C4×Dic107C2212yes—
27C107⋊#95C2212yes—
28C107⋊#95C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C22×D4×C107C2211yescharacteristic
33C23×C4×C107C2211yescharacteristic
34#123×C107C418yescomplemented
35C2×C4⋊C4×C107C2212yes—
36C2×C22⋊C4×C107C2211yescharacteristic
37C2×C22⋊C4×C107C2212yes—
38C22×Dic107—22no—
39C22×Dic107—22no—
40C22×Dic107—42no—
41C22×Dic107—42no—
42C22×Dic107—42no—
43C22×Dic107D414yes—
44C22×Dic107C2311yescharacteristic
45C22×Dic107C2312yes—
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2312yes—
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2311yescharacteristic
50C4⋊Dic107—24no—
51C4⋊Dic107—24no—
52C4⋊Dic107—22no—
53C4⋊Dic107—42no—
54C22⋊Dic107—22no—
55C22⋊Dic107—22no—
56C22⋊Dic107—22no—
57C22⋊Dic107—42no—
58C4×Dic107—22no—
59C4×Dic107—42no—
60C24×C107C2311yescharacteristic
61C24×C107C2311yescharacteristic
62C2×D4×C107—22no—
63C2×D4×C107—22no—
64C2×D4×C107C2×C414yes—
65C22×C4×C107—22no—
66C22×C4×C107—22no—
67C22×C4×C107C2×C414yescomplemented
68C22×C4×C107C2312yes—
69C22×C4×C107C2311yescharacteristic
70C22×C4×C107C2312yes—
71C4⋊C4×C107C2×C418yescomplemented
72C22⋊C4×C107C2×C414yescomplemented
73C22⋊C4×C107C2×C418yescomplemented
74C23×C107—22no—
75C23×C107—22no—
76C23×C107—21no—
77C23×C107—21no—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107C22×C412yes—
81C23×C107C22×C412yes—
82C23×C107C2×D412yes—
83C23×C107#4112yes—
84C23×C107C2411yescharacteristic
85D4×C107—28no—
86C2×C4×C107—28no—
87C2×C4×C107—22no—
88C2×C4×C107—21no—
89C2×C4×C107—22no—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107C22×C414yes—
94C2×C4×C107C22×C414yes—
95C2×C4×C107C22×C412yes—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—21no—
99C2×Dic107—21no—
100C2×Dic107—22no—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107—22no—
108C2×Dic107—44no—
109C2×Dic107—44no—
110C2×Dic107—44no—
111C2×Dic107C2×D412yes—
112C2×Dic107C2×D412yes—
113C22×C107—24no—
114C22×C107—22no—
115C22×C107—22no—
116C22×C107—24no—
117C22×C107—22no—
118C22×C107—24no—
119C22×C107—41no—
120C22×C107—41no—
121C22×C107—42no—
122C22×C107C4×D414yes—
123C22×C107C23×C411yescharacteristic
124C22×C107C22×D411yescharacteristic
125C22×C107C2×#4111yescharacteristic
126C22×C107H(5,𝔽2)12yes—
127C22×C1072−1+412yes—
128C4×C107—24no—
129C4×C107—22no—
130C4×C107—24no—
131Dic107—22no—
132Dic107—22no—
133Dic107—41no—
134Dic107—42no—
135Dic107—41no—
136Dic107—42no—
137C2×C107—24no—
138C2×C107—42no—
139C2×C107C2×C4×D411yescharacteristic
140C2×C107#52012yes—
141C2×C107#53411yescharacteristic
142C2×C107#53611yescharacteristic
143C2×C107#53912yes—
144#2322—1071nomaximal, Sylow
145C107#232211yescharacteristic, complemented, Sylow
146C2×#123D10711yescharacteristic
147C2×C4×D4—1071no—
148C22×C4⋊C4—1071no—
149#389—1072no—
150#388—1072no—
151#387—1072no—
152#386—1071no—
153#385—1072no—
154#385—1071no—
155#381—1071no—
156C4×C22⋊C4—1071no—
157C22×D4C2×D10711yescharacteristic
158C23×C4—1071no—
159C23×C4—1071no—
160C23×C4—1071no—
161C23×C4C2×D10711yescharacteristic
162#123Dic10718yescomplemented
163C4×D4—2144no—
164C2×C4⋊C4—1072no—
165C2×C4⋊C4—1072no—
166C2×C4⋊C4—1072no—
167C2×C4⋊C4—1071no—
168C2×C4⋊C4—2144no—
169C2×C4⋊C4C2×D10712yes—
170C2×C22⋊C4—1072no—
171C2×C22⋊C4—1071no—
172C2×C22⋊C4—1071no—
173C2×C22⋊C4—1071no—
174C2×C22⋊C4C2×D10711yescharacteristic
175C2×C22⋊C4C2×D10712yes—
176C2×C42—1072no—
177C2×C42—1071no—
178#95—1072no—
179#95—1072no—
180#95—1072no—
181#95—1072no—
182#95—1072no—
183C24C22×D10711yescharacteristic
184C24C22×D10711yescharacteristic
185C2×D4—22no—
186C2×D4—22no—
187C2×D4C2×Dic10714yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—1074no—
191C22×C4—1071no—
192C22×C4—1071no—
193C22×C4—1071no—
194C22×C4—1072no—
195C22×C4—1071no—
196C22×C4—1072no—
197C22×C4—4282no—
198C22×C4—4282no—
199C22×C4—4282no—
200C22×C4—2142no—
201C22×C4—2142no—
202C22×C4C2×Dic10714yescomplemented
203C22×C4C22×D10712yes—
204C22×C4C22×D10711yescharacteristic
205C22×C4C22×D10712yes—
206C4⋊C4—4282no—
207C4⋊C4—2144no—
208C4⋊C4—2144no—
209C4⋊C4—2142no—
210C4⋊C4C2×Dic10718yescomplemented
211C22⋊C4—4282no—
212C22⋊C4—2142no—
213C22⋊C4—2142no—
214C22⋊C4—2142no—
215C22⋊C4C2×Dic10714yescomplemented
216C22⋊C4C2×Dic10718yescomplemented
217C42—4282no—
218C42—2142no—
219C23—22no—
220C23—22no—
221C23—21no—
222C23—21no—
223C23—22no—
224C23—22no—
225C23C22×Dic10712yes—
226C23C22×Dic10712yes—
227C23D4×D10712yes—
228C23#1313381712yes—
229C23C23×D10711yescharacteristic
230D4—28no—
231C2×C4—28no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—1072no—
239C2×C4—1072no—
240C2×C4—4284no—
241C2×C4—4284no—
242C2×C4—4284no—
243C2×C4—2141no—
244C2×C4—2141no—
245C2×C4—2142no—
246C2×C4—2142no—
247C2×C4—2142no—
248C2×C4—2142no—
249C2×C4—2142no—
250C2×C4—2142no—
251C2×C4—2142no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4C22×Dic10714yes—
256C2×C4C22×Dic10714yes—
257C2×C4C22×Dic10712yes—
258C22—22no—
259C22—22no—
260C22—24no—
261C22—24no—
262C22—22no—
263C22—24no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22D4×Dic10714yes—
268C22C23×Dic10711yescharacteristic
269C22C2×D4×D10711yescharacteristic
270C22C2×#1313381711yescharacteristic
271C22#1681215512yes—
272C22#1681215612yes—
273C4—24no—
274C4—22no—
275C4—24no—
276C4—4282no—
277C4—4281no—
278C4—4281no—
279C4—4282no—
280C4—2142no—
281C4—2142no—
282C2—24no—
283C2—42no—
284C2C2×D4×Dic10711yescharacteristic
285C2#2029311012yes—
286C2#2029318811yescharacteristic
287C2#2029319911yescharacteristic
288C2#2029322112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
FittingC2×#123×C107C2
socle, cosocleC23×C107C24
prime radicalC22×D4×C107C22
hypercenterC2×#123D107
hypocenterC107#2322
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic107C2semidirect
3. #20292526C2semidirect
4. #20292520C2semidirect
5. #20292513C2semidirect
7. #20292502C2semidirect
8. #20292502C2semidirect
9. #20292490C2semidirect
10. C22⋊C4×Dic107C2semidirect
34. #123×C107C4semidirect
67. C22×C4×C107C2×C4semidirect
71. C4⋊C4×C107C2×C4semidirect
72. C22⋊C4×C107C2×C4semidirect
73. C22⋊C4×C107C2×C4semidirect
145. C107#2322semidirect
162. #123Dic107semidirect
202. C22×C4C2×Dic107semidirect
210. C4⋊C4C2×Dic107semidirect
215. C22⋊C4C2×Dic107semidirect
216. C22⋊C4C2×Dic107semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1072
C22×C4⋊Dic1072
#202925262
#202925202
#202925132
#202925072
#202925022
#202924902
C22⋊C4×Dic1072
C2×#123×C1072
#2322107