Finite Group Databasebeta

← #28340344

Subgroups — #28340344

Order 13696.

660 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812094C212yesmaximal, complemented
2C2×(C22⋊D107)⋊C4C212yesmaximal, complemented
3#20292543C212yesmaximal, complemented
4#20292536C211yesmaximal, characteristic
5#20292523C212yesmaximal, complemented
6#20292519C212yesmaximal, complemented
7#20292499C211yesmaximal, characteristic, complemented
8#20292498C212yesmaximal, complemented
9#399×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C22×C22⋊D107C2212yes—
12C23×Dic107C2211yescharacteristic
13C22×C4×D107C2212yes—
14#16812094—28no—
15(C22⋊D107)⋊C4—28no—
16C2×C4⋊Dic107C2212yes—
17C2×Dic107⋊C4C2212yes—
18C2×Dic107⋊C4C2212yes—
19C2×C22⋊Dic107C2211yescharacteristic
20C2×(C2×D107)⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C2×C4×Dic107C2212yes—
25C107⋊#95C2212yes—
26C107⋊#95C2212yes—
27C107⋊#95C2211yescharacteristic
28C2×C4⋊C4×C107C2212yes—
29C2×C4⋊C4×C107C2211yescharacteristic
30C2×C22⋊C4×C107C2212yes—
31C2×C22⋊C4×C107C2211yescharacteristic
32#95×C107C2211yescharacteristic
33C23×D107C2312yes—
34C2×D428—24no—
35C2×D428—41no—
36C2×C22⋊D107—24no—
37C2×C22⋊D107—24no—
38C2×C22⋊D107—24no—
39C22×Dic107—24no—
40C22×Dic107—41no—
41C22×Dic107C2312yes—
42C22×Dic107C2312yes—
43C22×Dic107C2312yes—
44C2×C4×D107—24no—
45C2×C4×D107—24no—
46C2×C4×D107—42no—
47C4⋊Dic107—24no—
48Dic107⋊C4—24no—
49Dic107⋊C4—24no—
50C22⋊Dic107—22no—
51(C2×D107)⋊C4—24no—
52(C2×D107)⋊C4—24no—
53(C2×D107)⋊C4—24no—
54(C2×D107)⋊C4D418yescomplemented
55C4×Dic107—24no—
56C24×C107C2311yescharacteristic
57C22×C4×C107C2311yescharacteristic
58C22×C4×C107C2312yes—
59C22×C4×C107C2311yescharacteristic
60C22×C4×C107C2312yes—
61C4⋊C4×C107—24no—
62C4⋊C4×C107—41no—
63C22⋊C4×C107—24no—
64C22⋊C4×C107—41no—
65C23×C107—22no—
66C23×C107—22no—
67C23×C107—21no—
68C23×C107—21no—
69C23×C107#4112yes—
70C23×C107C2411yescharacteristic
71C22×D107—22no—
72C22×D107—22no—
73C22×D107—22no—
74C22×D107—22no—
75C22×D107—22no—
76C22×D107—22no—
77C22×D107C2×D414yes—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—21no—
85C2×C4×C107—21no—
86C2×C4×C107—22no—
87C2×C4×C107C2×D414yes—
88C22⋊D107—216no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—44no—
97C2×Dic107C2×D414yes—
98C2×Dic107#4114yes—
99D428—44no—
100C4×D107—48no—
101C22×C107—24no—
102C22×C107—22no—
103C22×C107—41no—
104C22×C107—42no—
105C22×C107—41no—
106C22×C107C22×D412yes—
107C22×C107C2×#4112yes—
108C22×C107C2×#4111yescharacteristic
109C22×C107H(5,𝔽2)11yescharacteristic
110C22×C1072−1+411yescharacteristic
111C2×D107—24no—
112C2×D107—24no—
113C2×D107—24no—
114C2×D107—42no—
115C2×D107—42no—
116C2×D107—42no—
117C2×D107—42no—
118C4×C107—24no—
119C4×C107—42no—
120C4×C107—41no—
121C4×C107—41no—
122Dic107—24no—
123Dic107—24no—
124Dic107—42no—
125C2×C107—42no—
126C2×C107#52512yes—
127C2×C107#53911yescharacteristic
128C2×C107D4211yescharacteristic
129C2×C107#54712yes—
130C2×C107#55911yescharacteristic
131D107—44no—
132#2455—1071nomaximal, Sylow
133C107#245511yescharacteristic, complemented, Sylow
134C2×#123—1072no—
135C2×C4×D4—1072no—
136#399D10711yescharacteristic, complemented
137#394—1072no—
138#392—1071no—
139#389—1072no—
140#388—1072no—
141#385—1072no—
142#385—1071no—
143C22×D4—1072no—
144C22×D4—1071no—
145C23×C4—1071no—
146C23×C4—1072no—
147#123—2148no—
148C4×D4—2148no—
149C2×C4⋊C4—1072no—
150C2×C4⋊C4—1072no—
151C2×C4⋊C4—1072no—
152C2×C4⋊C4C2×D10712yes—
153C2×C4⋊C4C2×D10711yescharacteristic
154C2×C22⋊C4—1072no—
155C2×C22⋊C4—1071no—
156C2×C22⋊C4—1072no—
157C2×C22⋊C4—1072no—
158C2×C22⋊C4—1072no—
159C2×C22⋊C4C2×D10712yes—
160C2×C22⋊C4C2×D10711yescharacteristic
161C2×C42—1072no—
162#95—1072no—
163#95—1072no—
164#95—1071no—
165#95C2×D10711yescharacteristic
166C24—1072no—
167C24C22×D10711yescharacteristic
168C2×D4—4281no—
169C2×D4—2144no—
170C2×D4—2144no—
171C2×D4—2144no—
172C2×D4—2144no—
173C22×C4—1072no—
174C22×C4—1072no—
175C22×C4—1072no—
176C22×C4—4282no—
177C22×C4—4281no—
178C22×C4—2144no—
179C22×C4—2144no—
180C22×C4—2144no—
181C22×C4C22×D10711yescharacteristic
182C22×C4C22×D10712yes—
183C22×C4C22×D10711yescharacteristic
184C22×C4C22×D10712yes—
185C4⋊C4—24no—
186C4⋊C4—41no—
187C4⋊C4—2144no—
188C4⋊C4—2144no—
189C4⋊C4—2144no—
190C22⋊C4—24no—
191C22⋊C4—41no—
192C22⋊C4—1078no—
193C22⋊C4—2144no—
194C22⋊C4—2144no—
195C22⋊C4—2142no—
196C22⋊C4—2144no—
197C42—2144no—
198C23—22no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—1074no—
203C23—2142no—
204C23—2142no—
205C23—2142no—
206C23—2142no—
207C23—2142no—
208C23—2142no—
209C23#1313381912yes—
210C23C23×D10711yescharacteristic
211D4—4284no—
212D4—21416no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—1074no—
223C2×C4—1074no—
224C2×C4—4288no—
225C2×C4—4284no—
226C2×C4—2142no—
227C2×C4—2142no—
228C2×C4—2142no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4—2142no—
232C2×C4—2142no—
233C2×C4D4×D10714yes—
234C22—24no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—41no—
239C22—4282no—
240C22—4282no—
241C22—4282no—
242C22—4282no—
243C22—2144no—
244C22—2144no—
245C22—2144no—
246C22C2×D4×D10712yes—
247C22#41×D10712yes—
248C22C2×#1313381911yescharacteristic
249C22#1681215411yescharacteristic
250C22#1681215611yescharacteristic
251C4—24no—
252C4—42no—
253C4—41no—
254C4—41no—
255C4—4282no—
256C4—2144no—
257C4—2144no—
258C2—42no—
259C2—4284no—
260C2#2029314012yes—
261C2#2029322611yescharacteristic
262C2#2029326211yescharacteristic
263C2#2029327512yes—
264C2#2029336111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#399×C107C2
prime radical#20292536C2
hypercenter#399D107
hypocenterC107#2455
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16812094C2semidirect
2. C2×(C22⋊D107)⋊C4C2semidirect
3. #20292543C2semidirect
5. #20292523C2semidirect
6. #20292519C2semidirect
7. #20292499C2semidirect
8. #20292498C2semidirect
9. #399×C107C2semidirect
54. (C2×D107)⋊C4D4semidirect
133. C107#2455semidirect
136. #399D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120942
C2×(C22⋊D107)⋊C42
#202925432
#202925362
#202925232
#202925192
#202924992
#202924982
#399×C1072
#2455107