Finite Group Databasebeta

← #28339468

Subgroups — #28339468

Order 13696.

620 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812089C211yesmaximal, characteristic
2C2×#16812074C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D107C211yesmaximal, characteristic
4#20292533C211yesmaximal, characteristic, complemented
5#20292526C211yesmaximal, characteristic, complemented
6#20292525C212yesmaximal, complemented
7#20292512C212yesmaximal, complemented
8#20292501C212yesmaximal
9C4×(C2×D107)⋊C4C212yesmaximal
10C4×(C2×D107)⋊C4C211yesmaximal, characteristic
11C4×C4⋊C4×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×C4×D107C2212yes—
14C22×C4×D107C2211yescharacteristic
15#16812089C418yescomplemented
16#16812074—24no—
17C2×C4⋊Dic107C2211yescharacteristic
18C2×C4⋊Dic107C2212yes—
19C2×C4⋊Dic107C2212yes—
20C2×Dic107⋊C4C2212yes—
21C2×Dic107⋊C4C2212yes—
22C4⋊C4×D107—24no—
23C2×(C2×D107)⋊C4C2211yescharacteristic
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2211yescharacteristic
26C2×C4×Dic107C2212yes—
27C2×C4×Dic107C2211yescharacteristic
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2211yescharacteristic
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C4⋊C4×C107C2211yescharacteristic
35C2×C42×C107C2211yescharacteristic
36C2×C42×C107C2212yes—
37#95×C107C2212yes—
38C23×D107C2311yescharacteristic
39C2×Dic214—22no—
40C2×Dic214—22no—
41C2×Dic214C2×C414yescomplemented
42C22×Dic107C2311yescharacteristic
43C22×Dic107C2312yes—
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×D107C2×C414yes—
53C4⋊Dic107—22no—
54C4⋊Dic107—42no—
55C4⋊Dic107C2×C418yescomplemented
56Dic107⋊C4—24no—
57Dic107⋊C4—24no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4C2×C418yes—
60(C2×D107)⋊C4Q814yescomplemented
61C4×Dic107—24no—
62C4×Dic107—22no—
63C22×C4×C107C2311yescharacteristic
64C22×C4×C107C2312yes—
65C22×C4×C107C2312yes—
66C22×C4×C107C2311yescharacteristic
67C22×C4×C107C2311yescharacteristic
68C4⋊C4×C107C2×C414yescomplemented
69C4⋊C4×C107D414yes—
70C42×C107—22no—
71C42×C107—42no—
72C23×C107C2411yescharacteristic
73C22×D107—21no—
74C22×D107—21no—
75C22×D107—21no—
76C22×D107—21no—
77C22×D107C22×C412yes—
78C22×D107C2×Q812yes—
79C22×D107#4112yes—
80C2×C4×C107—21no—
81C2×C4×C107—22no—
82C2×C4×C107—21no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—21no—
87C2×C4×C107—21no—
88C2×C4×C107—21no—
89C2×C4×C107C22×C414yes—
90C2×C4×C107C22×C412yes—
91C2×C4×C107C2×D414yes—
92C2×C4×C107C2×D412yes—
93C2×C4×C107C2×Q812yes—
94C2×C4×C107#4112yes—
95C2×Dic107—21no—
96C2×Dic107—22no—
97C2×Dic107—21no—
98C2×Dic107—21no—
99C2×Dic107—22no—
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—22no—
103C2×Dic107—21no—
104C2×Dic107—22no—
105C2×Dic107—21no—
106C2×Dic107—22no—
107C2×Dic107C22×C414yes—
108C2×Dic107C22×C412yes—
109C2×Dic107C2×Q812yes—
110C4×D107—28no—
111C4×D107—48no—
112Dic214—28no—
113C22×C107C23×C411yescharacteristic
114C22×C107C22×D411yescharacteristic
115C22×C107C22×Q811yescharacteristic
116C22×C107C2×#4111yescharacteristic
117C22×C107C2×#4111yescharacteristic
118C22×C107H(5,𝔽2)11yescharacteristic
119C22×C1072−1+411yescharacteristic
120C2×D107—22no—
121C2×D107—22no—
122C2×D107—22no—
123C2×D107—22no—
124C2×D107—22no—
125C2×D107—22no—
126C2×D107C4×Q814yescomplemented
127C4×C107—24no—
128C4×C107—24no—
129C4×C107—22no—
130C4×C107—41no—
131C4×C107C4×D414yes—
132Dic107—24no—
133Dic107—22no—
134Dic107—22no—
135Dic107—42no—
136Dic107—41no—
137C2×C107C2×C4×D411yescharacteristic
138C2×C107C2×C4×Q811yescharacteristic
139C2×C107#52011yescharacteristic
140C2×C107#54611yescharacteristic
141C2×C107D4×Q811yescharacteristic
142C2×C107#55211yescharacteristic
143C2×C107#55411yescharacteristic
144D107—24no—
145#2360—1071nomaximal, Sylow
146C107#236011yescharacteristic, complemented, Sylow
147C2×C22⋊Q8—1071no—
148C2×C4×Q8—1071no—
149C22×C4⋊C4—1071no—
150#391—1071no—
151#389—1071no—
152#389—1072no—
153#387—1072no—
154#385—1072no—
155C4×C4⋊C4D10711yescharacteristic, complemented
156C4×C22⋊C4—1071no—
157C4×C22⋊C4—1072no—
158C22×Q8—1071no—
159C23×C4—1071no—
160C23×C4—1072no—
161C22⋊Q8—1078no—
162C4×Q8—2144no—
163C2×C4⋊C4—1072no—
164C2×C4⋊C4—1072no—
165C2×C4⋊C4—1072no—
166C2×C4⋊C4—1071no—
167C2×C4⋊C4—1072no—
168C2×C4⋊C4—2144no—
169C2×C4⋊C4C2×D10711yescharacteristic
170C2×C4⋊C4C2×D10711yescharacteristic
171C2×C22⋊C4—1072no—
172C2×C22⋊C4—1071no—
173C2×C22⋊C4—1071no—
174C2×C42—1072no—
175C2×C42—1071no—
176C2×C42C2×D10711yescharacteristic
177C2×C42C2×D10712yes—
178#95—1071no—
179#95—1072no—
180#95—1071no—
181#95—1072no—
182#95—1072no—
183#95C2×D10712yes—
184C24—1071no—
185C2×Q8—1074no—
186C2×Q8—2142no—
187C2×Q8—2142no—
188C22×C4—1071no—
189C22×C4—1072no—
190C22×C4—1071no—
191C22×C4—1074no—
192C22×C4—1072no—
193C22×C4—1071no—
194C22×C4—4282no—
195C22×C4—2144no—
196C22×C4—2144no—
197C22×C4—2142no—
198C22×C4—2142no—
199C22×C4C22×D10711yescharacteristic
200C22×C4C22×D10712yes—
201C22×C4C22×D10712yes—
202C22×C4C22×D10711yescharacteristic
203C22×C4C22×D10711yescharacteristic
204C4⋊C4—1078no—
205C4⋊C4—4282no—
206C4⋊C4—2144no—
207C4⋊C4—2142no—
208C4⋊C4—2144no—
209C4⋊C4C4×D10714yescomplemented
210C4⋊C4D42814yes—
211C22⋊C4—1078no—
212C22⋊C4—1074no—
213C22⋊C4—2142no—
214C42—22no—
215C42—42no—
216C42—2142no—
217C42—2144no—
218C23—1072no—
219C23—1072no—
220C23—1072no—
221C23—2141no—
222C23—2141no—
223C23—2141no—
224C23—2141no—
225C23C23×D10711yescharacteristic
226Q8—2148no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—1074no—
237C2×C4—1072no—
238C2×C4—1072no—
239C2×C4—4288no—
240C2×C4—2141no—
241C2×C4—2142no—
242C2×C4—2141no—
243C2×C4—2142no—
244C2×C4—2141no—
245C2×C4—2142no—
246C2×C4—2141no—
247C2×C4—2142no—
248C2×C4—2141no—
249C2×C4—2141no—
250C2×C4—2142no—
251C2×C4—2141no—
252C2×C4—2148no—
253C2×C4C2×C4×D10714yes—
254C2×C4C2×C4×D10712yes—
255C2×C4C2×D42814yes—
256C2×C4C2×D42812yes—
257C2×C4Q8×D10712yes—
258C2×C4#1313381812yes—
259C22—1074no—
260C22—2142no—
261C22—2142no—
262C22—2142no—
263C22—2142no—
264C22—2142no—
265C22—2142no—
266C22C22×C4×D10711yescharacteristic
267C22C22×D42811yescharacteristic
268C22C2×Q8×D10711yescharacteristic
269C22#41×D10711yescharacteristic
270C22C2×#1313381811yescharacteristic
271C22#1681215411yescharacteristic
272C22#1681215611yescharacteristic
273C4—24no—
274C4—22no—
275C4—24no—
276C4—41no—
277C4—4282no—
278C4—4281no—
279C4—2142no—
280C4—2142no—
281C4—2144no—
282C4C4×D42814yes—
283C2—2144no—
284C2C2×C4×D42811yescharacteristic
285C2C4×Q8×D10711yescharacteristic
286C2#2029310611yescharacteristic
287C2#2029327211yescharacteristic
288C2Q8×D42811yescharacteristic
289C2#2029331611yescharacteristic
290C2#2029333211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
FittingC4×C4⋊C4×C107C2
socle, cosocleC23×C107C24
prime radicalC23×D107C23
hypercenterC4×C4⋊C4D107
hypocenterC107#2360
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16812074C2semidirect
4. #20292533C2semidirect
5. #20292526C2semidirect
6. #20292525C2semidirect
7. #20292512C2semidirect
11. C4×C4⋊C4×C107C2semidirect
15. #16812089C4semidirect
41. C2×Dic214C2×C4semidirect
55. C4⋊Dic107C2×C4semidirect
60. (C2×D107)⋊C4Q8semidirect
68. C4⋊C4×C107C2×C4semidirect
126. C2×D107C4×Q8semidirect
146. C107#2360semidirect
155. C4×C4⋊C4D107semidirect
209. C4⋊C4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120892
C2×#168120742
C2×C4⋊C4×D1072
#202925332
#202925262
#202925252
#202925122
#202925012
C4×(C2×D107)⋊C42
C4×C4⋊C4×C1072
#2360107