Finite Group Databasebeta

← #28339620

Subgroups — #28339620

Order 13696.

576 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16812083C211yesmaximal, characteristic
2#20292527C212yesmaximal, complemented
3#20292519C212yesmaximal, complemented
4#20292510C212yesmaximal, complemented
5#20292505C212yesmaximal, complemented
6#20292503C212yesmaximal, complemented
7#20292501C212yesmaximal, complemented
8#20292487C211yesmaximal, characteristic, complemented
9#381×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C22×C22⋊D107C2212yes—
12C22×C4×D107C2212yes—
13#16812083C418yescomplemented
14C2×C4⋊Dic107C2212yes—
15C2×C4⋊Dic107C2211yescharacteristic
16C2×Dic107⋊C4C2212yes—
17C2×C22⋊Dic107C2211yescharacteristic
18C2×C22⋊Dic107C2211yescharacteristic
19C2×(C2×D107)⋊C4C2212yes—
20C2×(C2×D107)⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×C4×Dic107C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C107⋊#95C2212yes—
27C23×C4×C107C2211yescharacteristic
28C2×C22⋊C4×C107C2212yes—
29#95×C107C2211yescharacteristic
30#95×C107C2212yes—
31#95×C107C2211yescharacteristic
32C23×D107C2312yes—
33C2×D428—42no—
34C2×D428C2×C414yes—
35C2×C22⋊D107—42no—
36C2×C22⋊D107—42no—
37C2×C22⋊D107C2×C418yescomplemented
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C22×Dic107C2312yes—
41C2×C4×D107—42no—
42C2×C4×D107—42no—
43C2×C4×D107—42no—
44C4⋊Dic107—42no—
45C4⋊Dic107C2×C414yescomplemented
46Dic107⋊C4—42no—
47C22⋊Dic107—41no—
48C22⋊Dic107—41no—
49(C2×D107)⋊C4—42no—
50(C2×D107)⋊C4—42no—
51(C2×D107)⋊C4—42no—
52(C2×D107)⋊C4C2×C418yescomplemented
53C4×Dic107—42no—
54C24×C107C2311yescharacteristic
55C22×C4×C107—42no—
56C22×C4×C107C2×C414yescomplemented
57C22×C4×C107C2312yes—
58C22×C4×C107C2312yes—
59C22×C4×C107C2311yescharacteristic
60C22×C4×C107C2311yescharacteristic
61C22⋊C4×C107—42no—
62C23×C107—22no—
63C23×C107—21no—
64C23×C107—22no—
65C23×C107—21no—
66C23×C107C22×C412yes—
67C23×C107C2411yescharacteristic
68C22×D107—22no—
69C22×D107—22no—
70C22×D107—22no—
71C22×D107—22no—
72C22×D107—22no—
73C22×D107—22no—
74C22×D107C22×C414yes—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—44no—
84C2×C4×C107C22×C412yes—
85C2×C4×C107C22×C412yes—
86C22⋊D107—48no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107C22×C414yes—
96D428—44no—
97C4×D107—48no—
98C22×C107—22no—
99C22×C107—22no—
100C22×C107—22no—
101C22×C107—42no—
102C22×C107—42no—
103C22×C107C23×C411yescharacteristic
104C22×C107H(5,𝔽2)12yes—
105C22×C107H(5,𝔽2)12yes—
106C22×C107H(5,𝔽2)11yescharacteristic
107C22×C1072−1+411yescharacteristic
108C2×D107—24no—
109C2×D107—24no—
110C2×D107—24no—
111C2×D107—42no—
112C2×D107—42no—
113C2×D107—42no—
114C2×D107—42no—
115C4×C107—22no—
116C4×C107—22no—
117C4×C107—42no—
118C4×C107—42no—
119Dic107—24no—
120Dic107—42no—
121Dic107—42no—
122C2×C107—42no—
123C2×C107#51811yescharacteristic
124C2×C107#51811yescharacteristic
125C2×C107#52011yescharacteristic
126C2×C107#56012yes—
127C2×C107#56212yes—
128D107—44no—
129#2377—1071nomaximal, Sylow
130C107#237711yescharacteristic, complemented, Sylow
131C2×#121—1071no—
132#390—1072no—
133#388—1072no—
134#387—1072no—
135#386—1072no—
136#386—1072no—
137#385—1072no—
138#381D10711yescharacteristic, complemented
139#380—1071no—
140C22×D4—1072no—
141C22×D4—1071no—
142C23×C4—1072no—
143C23×C4C2×D10711yescharacteristic
144#121—1078no—
145C2×C4⋊C4—1071no—
146C2×C4⋊C4—1072no—
147C2×C4⋊C4—1072no—
148C2×C22⋊C4—1072no—
149C2×C22⋊C4—1071no—
150C2×C22⋊C4—1071no—
151C2×C22⋊C4—1072no—
152C2×C22⋊C4—1072no—
153C2×C22⋊C4—1072no—
154C2×C22⋊C4C2×D10712yes—
155C2×C42—1072no—
156#95—1072no—
157#95—1072no—
158#95—1072no—
159#95C2×D10711yescharacteristic
160#95C2×D10712yes—
161#95C2×D10711yescharacteristic
162C24—1072no—
163C24C22×D10711yescharacteristic
164C2×D4—1078no—
165C2×D4—1074no—
166C2×D4—4282no—
167C2×D4—4282no—
168C2×D4—4282no—
169C22×C4—42no—
170C22×C4—1072no—
171C22×C4—1072no—
172C22×C4—1072no—
173C22×C4—4282no—
174C22×C4—4282no—
175C22×C4—4282no—
176C22×C4C4×D10714yescomplemented
177C22×C4C22×D10712yes—
178C22×C4C22×D10712yes—
179C22×C4C22×D10711yescharacteristic
180C22×C4C22×D10711yescharacteristic
181C4⋊C4—1074no—
182C4⋊C4—4282no—
183C4⋊C4—4282no—
184C22⋊C4—42no—
185C22⋊C4—1078no—
186C22⋊C4—4282no—
187C22⋊C4—4282no—
188C22⋊C4—4282no—
189C22⋊C4—4281no—
190C22⋊C4—4281no—
191C42—4282no—
192C23—22no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—1074no—
197C23—2142no—
198C23—2142no—
199C23—2142no—
200C23—2142no—
201C23—2142no—
202C23—2142no—
203C23C2×C4×D10712yes—
204C23C23×D10711yescharacteristic
205D4—4284no—
206D4—4288no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—1074no—
217C2×C4—4288no—
218C2×C4—2142no—
219C2×C4—2142no—
220C2×C4—2142no—
221C2×C4—2142no—
222C2×C4—2142no—
223C2×C4—2142no—
224C2×C4—2142no—
225C2×C4—2142no—
226C2×C4C2×C4×D10712yes—
227C2×C4C2×C4×D10712yes—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—42no—
232C22—42no—
233C22—4282no—
234C22—4282no—
235C22—4282no—
236C22—4282no—
237C22—2144no—
238C22—2144no—
239C22—2144no—
240C22C22×C4×D10711yescharacteristic
241C22#1681215412yes—
242C22#1681215512yes—
243C22#1681215511yescharacteristic
244C22#1681215611yescharacteristic
245C4—22no—
246C4—22no—
247C4—42no—
248C4—42no—
249C4—4282no—
250C4—4282no—
251C4—2144no—
252C2—42no—
253C2—4284no—
254C2#2029310011yescharacteristic
255C2#2029310111yescharacteristic
256C2#2029310711yescharacteristic
257C2#2029336712yes—
258C2#2029337412yes—
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#381×C107C2
socle, cosocleC23×C107C24
prime radicalC2×#16812083C2
hypercenter#381D107
hypocenterC107#2377
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #20292527C2semidirect
3. #20292519C2semidirect
4. #20292510C2semidirect
5. #20292505C2semidirect
6. #20292503C2semidirect
7. #20292501C2semidirect
8. #20292487C2semidirect
9. #381×C107C2semidirect
13. #16812083C4semidirect
37. C2×C22⋊D107C2×C4semidirect
45. C4⋊Dic107C2×C4semidirect
52. (C2×D107)⋊C4C2×C4semidirect
56. C22×C4×C107C2×C4semidirect
130. C107#2377semidirect
138. #381D107semidirect
176. C22×C4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120832
#202925272
#202925192
#202925102
#202925052
#202925032
#202925012
#202924872
#381×C1072
#2377107