Finite Group Databasebeta

← #28339083

Subgroups — #28339083

Order 13696.

600 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×#16812067C211yesmaximal, characteristic
2C2×#16812066C211yesmaximal, characteristic
3C2×C4⋊C4×D107C211yesmaximal, characteristic
4#20292526C211yesmaximal, characteristic, complemented
5#20292525C211yesmaximal, characteristic, complemented
6#20292523C211yesmaximal, characteristic, complemented
7#20292501C212yesmaximal
8#20292498C212yesmaximal
9#20292496C212yesmaximal, complemented
10#20292491C212yesmaximal, complemented
11#389×C107C211yesmaximal, characteristic, complemented
12C22×C4×D107C2212yes—
13C22×C4×D107C2211yescharacteristic
14#16812067C418yescomplemented
15#16812066—24no—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×C4⋊Dic107C2212yes—
18C2×C4⋊Dic107C2211yescharacteristic
19C2×Dic107⋊C4C2211yescharacteristic
20C2×Dic107⋊C4C2212yes—
21C2×Dic107⋊C4C2211yescharacteristic
22C2×Dic107⋊C4C2212yes—
23C2×Dic107⋊C4C2212yes—
24C4⋊C4×D107—24no—
25C2×(C2×D107)⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×(C2×D107)⋊C4C2212yes—
28C2×C4×Dic107C2211yescharacteristic
29C2×C4×Dic107C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C4⋊C4×C107C2212yes—
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C42×C107C2211yescharacteristic
37#95×C107C2212yes—
38C23×D107C2311yescharacteristic
39C22×Dic107C2311yescharacteristic
40C22×Dic107C2311yescharacteristic
41C22×Dic107C2312yes—
42C22×Dic107C2312yes—
43C22×Dic107C2311yescharacteristic
44C2×C4×D107—24no—
45C2×C4×D107—42no—
46C2×C4×D107—42no—
47C2×C4×D107C2×C414yes—
48C2×C4×D107D414yes—
49C2×C4×D107Q814yes—
50C4⋊Dic107—22no—
51C4⋊Dic107—42no—
52C4⋊Dic107C2×C414yescomplemented
53Dic107⋊C4—24no—
54Dic107⋊C4—22no—
55Dic107⋊C4—22no—
56Dic107⋊C4—42no—
57Dic107⋊C4—42no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4C2×C418yes—
61C4×Dic107—22no—
62C4×Dic107C2×C418yescomplemented
63C22×C4×C107C2311yescharacteristic
64C22×C4×C107C2311yescharacteristic
65C22×C4×C107C2312yes—
66C22×C4×C107C2312yes—
67C22×C4×C107C2311yescharacteristic
68C4⋊C4×C107—22no—
69C4⋊C4×C107—42no—
70C4⋊C4×C107C2×C414yescomplemented
71C42×C107—22no—
72C23×C107C2411yescharacteristic
73C22×D107—21no—
74C22×D107—21no—
75C22×D107—21no—
76C22×D107—21no—
77C22×D107C22×C412yes—
78C22×D107C2×D412yes—
79C22×D107C2×Q812yes—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—22no—
85C2×C4×C107—21no—
86C2×C4×C107—22no—
87C2×C4×C107—21no—
88C2×C4×C107—22no—
89C2×C4×C107—21no—
90C2×C4×C107—21no—
91C2×C4×C107C22×C414yes—
92C2×C4×C107C22×C412yes—
93C2×C4×C107C2×D412yes—
94C2×C4×C107C2×Q812yes—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—21no—
98C2×Dic107—21no—
99C2×Dic107—22no—
100C2×Dic107—21no—
101C2×Dic107—22no—
102C2×Dic107—21no—
103C2×Dic107—22no—
104C2×Dic107—21no—
105C2×Dic107—21no—
106C2×Dic107C22×C414yes—
107C2×Dic107C22×C412yes—
108C2×Dic107C2×D412yes—
109C2×Dic107C2×Q812yes—
110C4×D107—48no—
111C4×D107C4⋊C4116yescomplemented
112C22×C107C23×C411yescharacteristic
113C22×C107C22×D411yescharacteristic
114C22×C107C22×Q811yescharacteristic
115C22×C107H(5,𝔽2)11yescharacteristic
116C22×C107H(5,𝔽2)11yescharacteristic
117C22×C1072−1+411yescharacteristic
118C22×C1072−1+411yescharacteristic
119C2×D107—22no—
120C2×D107—22no—
121C2×D107—22no—
122C2×D107—22no—
123C2×D107—22no—
124C2×D107—22no—
125C2×D107C2×C4⋊C414yes—
126C4×C107—24no—
127C4×C107—42no—
128C4×C107—41no—
129C4×C107—41no—
130C4×C107C2×C4⋊C414yes—
131Dic107—24no—
132Dic107—42no—
133Dic107—41no—
134Dic107—41no—
135Dic107C2×C4⋊C414yes—
136C2×C107C22×C4⋊C411yescharacteristic
137C2×C107#52011yescharacteristic
138C2×C107#52011yescharacteristic
139C2×C107#53511yescharacteristic
140C2×C107#53611yescharacteristic
141C2×C107#54411yescharacteristic
142C2×C107#54411yescharacteristic
143D107—24no—
144#2317—1071nomaximal, Sylow
145C107#231711yescharacteristic, complemented, Sylow
146C2×#117—1071no—
147C2×#117—1071no—
148C22×C4⋊C4—1071no—
149#389—1071no—
150#389—1071no—
151#389—1071no—
152#389D10711yescharacteristic, complemented
153#385—1072no—
154#385—1072no—
155#384—1072no—
156#382—1072no—
157C23×C4—1071no—
158C23×C4—1072no—
159#117—1078no—
160#117—2144no—
161C2×C4⋊C4—1071no—
162C2×C4⋊C4—1072no—
163C2×C4⋊C4—1072no—
164C2×C4⋊C4—1072no—
165C2×C4⋊C4—1072no—
166C2×C4⋊C4—1071no—
167C2×C4⋊C4—1071no—
168C2×C4⋊C4—1071no—
169C2×C4⋊C4—2144no—
170C2×C4⋊C4C2×D10711yescharacteristic
171C2×C4⋊C4C2×D10712yes—
172C2×C4⋊C4C2×D10711yescharacteristic
173C2×C22⋊C4—1072no—
174C2×C22⋊C4—1071no—
175C2×C22⋊C4—1071no—
176C2×C42—1072no—
177C2×C42—1071no—
178C2×C42C2×D10711yescharacteristic
179#95—1072no—
180#95—1072no—
181#95—1072no—
182#95C2×D10712yes—
183C24—1071no—
184C22×C4—1071no—
185C22×C4—1074no—
186C22×C4—1074no—
187C22×C4—1074no—
188C22×C4—1072no—
189C22×C4—1071no—
190C22×C4—1072no—
191C22×C4—1071no—
192C22×C4—4282no—
193C22×C4—4282no—
194C22×C4—2144no—
195C22×C4C22×D10711yescharacteristic
196C22×C4C22×D10712yes—
197C22×C4C22×D10711yescharacteristic
198C22×C4C22×D10712yes—
199C22×C4C22×D10711yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—1074no—
203C4⋊C4—4282no—
204C4⋊C4—4282no—
205C4⋊C4—4282no—
206C4⋊C4—2144no—
207C4⋊C4—2142no—
208C4⋊C4—2142no—
209C4⋊C4—2142no—
210C4⋊C4C4×D10714yescomplemented
211C22⋊C4—1078no—
212C22⋊C4—2142no—
213C22⋊C4—2142no—
214C42—22no—
215C42—1078no—
216C42—2142no—
217C23—1072no—
218C23—1072no—
219C23—1072no—
220C23—2141no—
221C23—2141no—
222C23—2141no—
223C23—2141no—
224C23C23×D10711yescharacteristic
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—1072no—
237C2×C4—1072no—
238C2×C4—1072no—
239C2×C4—10716no—
240C2×C4—1074no—
241C2×C4—4288no—
242C2×C4—2141no—
243C2×C4—2141no—
244C2×C4—2142no—
245C2×C4—2141no—
246C2×C4—2142no—
247C2×C4—2141no—
248C2×C4—2142no—
249C2×C4—2141no—
250C2×C4—2141no—
251C2×C4—2142no—
252C2×C4—2142no—
253C2×C4C2×C4×D10714yes—
254C2×C4C2×C4×D10712yes—
255C2×C4D4×D10712yes—
256C2×C4Q8×D10712yes—
257C22—1074no—
258C22—2142no—
259C22—2142no—
260C22—2142no—
261C22—2142no—
262C22—2142no—
263C22—2142no—
264C22C22×C4×D10711yescharacteristic
265C22C2×D4×D10711yescharacteristic
266C22C2×Q8×D10711yescharacteristic
267C22#1681215411yescharacteristic
268C22#1681215511yescharacteristic
269C22#1681215611yescharacteristic
270C22#1681215711yescharacteristic
271C4—24no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—1074no—
276C4—4282no—
277C4—4281no—
278C4—4281no—
279C4—2144no—
280C4C4⋊C4×D10714yes—
281C2—2144no—
282C2C2×C4⋊C4×D10711yescharacteristic
283C2#2029310611yescharacteristic
284C2#2029310911yescharacteristic
285C2#2029319311yescharacteristic
286C2#2029320011yescharacteristic
287C2#2029325411yescharacteristic
288C2#2029325711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#389×C107C2
socle, cosocleC23×C107C24
prime radicalC23×D107C23
hypercenter#389D107
hypocenterC107#2317
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
4. #20292526C2semidirect
5. #20292525C2semidirect
6. #20292523C2semidirect
9. #20292496C2semidirect
10. #20292491C2semidirect
11. #389×C107C2semidirect
14. #16812067C4semidirect
52. C4⋊Dic107C2×C4semidirect
62. C4×Dic107C2×C4semidirect
70. C4⋊C4×C107C2×C4semidirect
111. C4×D107C4⋊C4semidirect
145. C107#2317semidirect
152. #389D107semidirect
210. C4⋊C4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120672
C2×#168120662
C2×C4⋊C4×D1072
#202925262
#202925252
#202925232
#202925012
#202924982
#202924962
#202924912
#389×C1072
#2317107