Finite Group Databasebeta

← #28340181

Subgroups — #28340181

Order 13696.

520 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D107C211yesmaximal, characteristic
2#20292565C212yesmaximal, complemented
3#20292560C212yesmaximal, complemented
4#20292526C211yesmaximal, characteristic, complemented
5#20292525C211yesmaximal, characteristic, complemented
6#20292498C212yesmaximal
7#20292492C211yesmaximal, characteristic, complemented
8#20292484C212yesmaximal
9#20292484C211yesmaximal, characteristic
10C4×(C2×D107)⋊C4C211yesmaximal, characteristic
11C2×#125×C107C211yesmaximal, characteristic, complemented
12C22×C4×D107C2212yes—
13C22×C4×D107C2211yescharacteristic
14C2×C4⋊Dic107C2211yescharacteristic
15C2×C4⋊Dic107C2211yescharacteristic
16C2×C4⋊Dic107C2212yes—
17C2×Dic107⋊C4C2212yes—
18C2×Dic107⋊C4C2212yes—
19C4⋊C4×D107—24no—
20C2×(C2×D107)⋊C4C2211yescharacteristic
21C2×(C2×D107)⋊C4C2211yescharacteristic
22C2×(C2×D107)⋊C4C2212yes—
23C2×C4×Dic107C2211yescharacteristic
24C107⋊#95C2211yescharacteristic
25C107⋊#95C2211yescharacteristic
26C107⋊#95C2212yes—
27C107⋊#95C2212yes—
28C107⋊#95C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31#125×C107—24no—
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2212yes—
34C2×C4⋊C4×C107C2212yes—
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C42×C107C2211yescharacteristic
37C23×D107C2311yescharacteristic
38C22×Dic107C2311yescharacteristic
39C22×Dic107C2311yescharacteristic
40C22×Dic107C2312yes—
41C22×Dic107C2312yes—
42C22×Dic107C2311yescharacteristic
43C2×C4×D107—22no—
44C2×C4×D107—22no—
45C2×C4×D107—24no—
46C2×C4×D107—24no—
47C2×C4×D107—42no—
48C2×C4×D107Q814yes—
49C4⋊Dic107—22no—
50C4⋊Dic107—22no—
51C4⋊Dic107—42no—
52Dic107⋊C4—24no—
53Dic107⋊C4—42no—
54(C2×D107)⋊C4—22no—
55(C2×D107)⋊C4—24no—
56(C2×D107)⋊C4—22no—
57C4×Dic107—22no—
58C22×C4×C107C2311yescharacteristic
59C22×C4×C107C2311yescharacteristic
60C22×C4×C107C2312yes—
61C22×C4×C107C2312yes—
62C22×C4×C107C2311yescharacteristic
63C4⋊C4×C107—24no—
64C4⋊C4×C107—24no—
65C4⋊C4×C107—22no—
66C4⋊C4×C107D414yes—
67C42×C107—22no—
68C23×C107C2411yescharacteristic
69C22×D107—21no—
70C22×D107—21no—
71C22×D107—21no—
72C22×D107—21no—
73C22×D107C2×Q812yes—
74C22×D107#4112yes—
75C22×D107#4112yes—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—21no—
79C2×C4×C107—21no—
80C2×C4×C107—22no—
81C2×C4×C107—21no—
82C2×C4×C107—21no—
83C2×C4×C107—22no—
84C2×C4×C107—21no—
85C2×C4×C107—21no—
86C2×C4×C107—22no—
87C2×C4×C107C2×D414yes—
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2×Q812yes—
90C2×C4×C107#4112yes—
91C2×Dic107—21no—
92C2×Dic107—21no—
93C2×Dic107—22no—
94C2×Dic107—21no—
95C2×Dic107—21no—
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—21no—
104C2×Dic107—21no—
105C2×Dic107C2×Q812yes—
106C4×D107—28no—
107C4×D107—48no—
108C22×C107C22×D411yescharacteristic
109C22×C107C22×Q811yescharacteristic
110C22×C107C2×#4111yescharacteristic
111C22×C107C2×#4111yescharacteristic
112C22×C107C2×#4111yescharacteristic
113C22×C107H(5,𝔽2)11yescharacteristic
114C22×C1072−1+411yescharacteristic
115C2×D107—22no—
116C2×D107—22no—
117C2×D107—22no—
118C2×D107—22no—
119C2×D107—22no—
120C2×D107—22no—
121C2×D107#12514yescomplemented
122C4×C107—24no—
123C4×C107—41no—
124C4×C107—41no—
125C4×C107—42no—
126C4×C107C22⋊Q814yes—
127Dic107—22no—
128Dic107—42no—
129Dic107—41no—
130Dic107—41no—
131Dic107—42no—
132C2×C107C2×C22⋊Q811yescharacteristic
133C2×C107C2×#12511yescharacteristic
134C2×C107#54411yescharacteristic
135C2×C107#54611yescharacteristic
136C2×C107#54811yescharacteristic
137C2×C107#55211yescharacteristic
138C2×C107#55311yescharacteristic
139D107—24no—
140#2442—1071nomaximal, Sylow
141C107#244211yescharacteristic, complemented, Sylow
142C2×#125D10711yescharacteristic, complemented
143C22×C4⋊C4—1071no—
144#400—1072no—
145#398—1072no—
146#389—1071no—
147#389—1071no—
148#385—1072no—
149#382—1071no—
150#380—1072no—
151#380—1071no—
152C4×C22⋊C4—1071no—
153C23×C4—1071no—
154C23×C4—1072no—
155#125—24no—
156C2×C4⋊C4—1072no—
157C2×C4⋊C4—1072no—
158C2×C4⋊C4—1072no—
159C2×C4⋊C4—1071no—
160C2×C4⋊C4—1071no—
161C2×C4⋊C4—2144no—
162C2×C4⋊C4C2×D10711yescharacteristic
163C2×C4⋊C4C2×D10712yes—
164C2×C4⋊C4C2×D10712yes—
165C2×C4⋊C4C2×D10711yescharacteristic
166C2×C22⋊C4—1071no—
167C2×C22⋊C4—1071no—
168C2×C22⋊C4—1072no—
169C2×C42—1071no—
170C2×C42C2×D10711yescharacteristic
171#95—1071no—
172#95—1071no—
173#95—1072no—
174#95—1072no—
175#95—1072no—
176#95—1072no—
177#95—1072no—
178C24—1071no—
179C22×C4—1071no—
180C22×C4—1074no—
181C22×C4—1071no—
182C22×C4—1071no—
183C22×C4—1072no—
184C22×C4—1072no—
185C22×C4—4282no—
186C22×C4—2144no—
187C22×C4—2144no—
188C22×C4—2142no—
189C22×C4—2142no—
190C22×C4C22×D10711yescharacteristic
191C22×C4C22×D10712yes—
192C22×C4C22×D10711yescharacteristic
193C22×C4C22×D10712yes—
194C22×C4C22×D10711yescharacteristic
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—22no—
198C4⋊C4—4282no—
199C4⋊C4—4282no—
200C4⋊C4—2144no—
201C4⋊C4—2142no—
202C4⋊C4—2142no—
203C4⋊C4C22⋊D10714yes—
204C22⋊C4—2142no—
205C22⋊C4—2144no—
206C22⋊C4—2142no—
207C42—22no—
208C42—2142no—
209C23—1072no—
210C23—1072no—
211C23—1072no—
212C23—2141no—
213C23—2141no—
214C23—2141no—
215C23—2141no—
216C23C23×D10711yescharacteristic
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—1072no—
229C2×C4—4288no—
230C2×C4—2142no—
231C2×C4—2141no—
232C2×C4—2141no—
233C2×C4—2142no—
234C2×C4—2142no—
235C2×C4—2141no—
236C2×C4—2141no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4—2141no—
240C2×C4—2142no—
241C2×C4—2141no—
242C2×C4—2141no—
243C2×C4—2141no—
244C2×C4—2148no—
245C2×C4C2×C22⋊D10714yes—
246C2×C4C2×C22⋊D10712yes—
247C2×C4Q8×D10712yes—
248C2×C4#1313381912yes—
249C22—1074no—
250C22—2142no—
251C22—2142no—
252C22—2142no—
253C22—2142no—
254C22—2142no—
255C22—2142no—
256C22C22×C22⋊D10711yescharacteristic
257C22C2×Q8×D10711yescharacteristic
258C22#41×D10711yescharacteristic
259C22#41×D10711yescharacteristic
260C22C2×#1313381911yescharacteristic
261C22#1681215511yescharacteristic
262C22#1681215611yescharacteristic
263C4—24no—
264C4—41no—
265C4—41no—
266C4—42no—
267C4—4282no—
268C4—4281no—
269C4—4281no—
270C4—4282no—
271C4—2142no—
272C4#1681208514yes—
273C2—2144no—
274C2C2×#1681208511yescharacteristic
275C2#125×D10711yescharacteristic
276C2#2029325711yescharacteristic
277C2#2029326511yescharacteristic
278C2#2029328411yescharacteristic
279C2#2029331511yescharacteristic
280C2#2029333011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#125×C107C2
prime radicalC23×D107C23
hypercenterC2×#125D107
hypocenterC107#2442
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #20292565C2semidirect
3. #20292560C2semidirect
4. #20292526C2semidirect
5. #20292525C2semidirect
7. #20292492C2semidirect
11. C2×#125×C107C2semidirect
121. C2×D107#125semidirect
141. C107#2442semidirect
142. C2×#125D107semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1072
#202925652
#202925602
#202925262
#202925252
#202924982
#202924922
#202924842
C4×(C2×D107)⋊C42
C2×#125×C1072
#2442107