Finite Group Databasebeta

← #28344438

Subgroups — #28344438

Order 13696.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#20292553C212yesmaximal, complemented
2#20292544C212yesmaximal
3#20292540C212yesmaximal, complemented
4#20292533C211yesmaximal, characteristic, complemented
5#20292529C211yesmaximal, characteristic
6#20292519C212yesmaximal, complemented
7#20292517C212yesmaximal, complemented
8#20292513C212yesmaximal, complemented
9#389×C107C211yesmaximal, characteristic, complemented
10C22×Dic214C2211yescharacteristic
11C22×D428C2211yescharacteristic
12C2×C4⋊Dic107C2211yescharacteristic
13C2×C4⋊Dic107C2211yescharacteristic
14C2×(C2×D107)⋊C4C2212yes—
15C2×(C2×D107)⋊C4C2212yes—
16C2×(C2×D107)⋊C4C2212yes—
17C2×(C2×D107)⋊C4C2212yes—
18C2×(C2×D107)⋊C4C2212yes—
19C2×(C2×D107)⋊C4C2212yes—
20C2×C4×Dic107C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C2×C4⋊C4×C107C2212yes—
27C2×C4⋊C4×C107C2211yescharacteristic
28C2×C4⋊C4×C107C2211yescharacteristic
29C2×C42×C107C2211yescharacteristic
30#95×C107C2212yes—
31C23×D107C2312yes—
32C2×Dic214—22no—
33C2×Dic214—41no—
34C2×Dic214—41no—
35C2×D428—22no—
36C2×D428—41no—
37C2×D428—41no—
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C22×Dic107C2312yes—
41C4⋊Dic107—41no—
42C4⋊Dic107—41no—
43(C2×D107)⋊C4—24no—
44(C2×D107)⋊C4—24no—
45(C2×D107)⋊C4—42no—
46(C2×D107)⋊C4—42no—
47(C2×D107)⋊C4—42no—
48(C2×D107)⋊C4—42no—
49C4×Dic107—42no—
50C22×C4×C107C2312yes—
51C22×C4×C107C2312yes—
52C22×C4×C107C2311yescharacteristic
53C22×C4×C107C2311yescharacteristic
54C22×C4×C107C2311yescharacteristic
55C4⋊C4×C107—24no—
56C4⋊C4×C107—22no—
57C4⋊C4×C107—41no—
58C42×C107—41no—
59C23×C107C2411yescharacteristic
60C22×D107—22no—
61C22×D107—22no—
62C22×D107—22no—
63C22×D107—22no—
64C22×D107—22no—
65C22×D107—22no—
66C22×D107—22no—
67C2×C4×C107—21no—
68C2×C4×C107—21no—
69C2×C4×C107—21no—
70C2×C4×C107—21no—
71C2×C4×C107—21no—
72C2×C4×C107—21no—
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107#4112yes—
80C2×C4×C107#4112yes—
81C2×C4×C107#4112yes—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91D428—44no—
92Dic214—44no—
93C22×C107C2×#4111yescharacteristic
94C22×C107C2×#4111yescharacteristic
95C22×C107C2×#4111yescharacteristic
96C22×C107H(5,𝔽2)11yescharacteristic
97C22×C107H(5,𝔽2)11yescharacteristic
98C22×C107H(5,𝔽2)11yescharacteristic
99C22×C1072−1+411yescharacteristic
100C2×D107—42no—
101C2×D107—42no—
102C2×D107—42no—
103C2×D107—42no—
104C2×D107—42no—
105C2×D107—42no—
106C2×D107—42no—
107C4×C107—22no—
108C4×C107—22no—
109C4×C107—22no—
110C4×C107—42no—
111C4×C107—42no—
112Dic107—42no—
113Dic107—42no—
114Dic107—42no—
115C2×C107#53811yescharacteristic
116C2×C107#53811yescharacteristic
117C2×C107#53911yescharacteristic
118C2×C107#55511yescharacteristic
119C2×C107#55611yescharacteristic
120C2×C107#55911yescharacteristic
121C2×C107#56211yescharacteristic
122D107—82no—
123#2808—1071nomaximal, Sylow
124C107#280811yescharacteristic, complemented, Sylow
125#397—1072no—
126#394—1072no—
127#393—1072no—
128#391—1071no—
129#390—1071no—
130#389D10711yescharacteristic, complemented
131#388—1072no—
132#388—1072no—
133#387—1072no—
134C22×Q8—1071no—
135C22×D4—1071no—
136C2×C4⋊C4—1071no—
137C2×C4⋊C4—1071no—
138C2×C4⋊C4C2×D10712yes—
139C2×C4⋊C4C2×D10711yescharacteristic
140C2×C4⋊C4C2×D10711yescharacteristic
141C2×C22⋊C4—1072no—
142C2×C22⋊C4—1072no—
143C2×C22⋊C4—1072no—
144C2×C22⋊C4—1072no—
145C2×C22⋊C4—1072no—
146C2×C22⋊C4—1072no—
147C2×C42—1072no—
148C2×C42C2×D10711yescharacteristic
149#95—1072no—
150#95—1072no—
151#95—1072no—
152#95—1072no—
153#95—1072no—
154#95C2×D10712yes—
155C24—1072no—
156C2×Q8—4281no—
157C2×Q8—4281no—
158C2×Q8—2142no—
159C2×D4—4281no—
160C2×D4—4281no—
161C2×D4—2142no—
162C22×C4—1072no—
163C22×C4—1072no—
164C22×C4—1072no—
165C22×C4C22×D10712yes—
166C22×C4C22×D10712yes—
167C22×C4C22×D10711yescharacteristic
168C22×C4C22×D10711yescharacteristic
169C22×C4C22×D10711yescharacteristic
170C4⋊C4—24no—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—4281no—
174C4⋊C4—4281no—
175C22⋊C4—4282no—
176C22⋊C4—4282no—
177C22⋊C4—4282no—
178C22⋊C4—4282no—
179C22⋊C4—2144no—
180C22⋊C4—2144no—
181C42—41no—
182C42—4282no—
183C23—2142no—
184C23—2142no—
185C23—2142no—
186C23—2142no—
187C23—2142no—
188C23—2142no—
189C23—2142no—
190C23C23×D10711yescharacteristic
191Q8—4284no—
192D4—4284no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—2142no—
206C2×C4—2142no—
207C2×C4—2142no—
208C2×C4—2142no—
209C2×C4—2142no—
210C2×C4—2142no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4#1313381712yes—
215C2×C4#1313381812yes—
216C2×C4#1313381912yes—
217C22—4282no—
218C22—4282no—
219C22—4282no—
220C22—4282no—
221C22—4282no—
222C22—4282no—
223C22—4282no—
224C22C2×#1313381711yescharacteristic
225C22C2×#1313381811yescharacteristic
226C22C2×#1313381911yescharacteristic
227C22#1681215411yescharacteristic
228C22#1681215511yescharacteristic
229C22#1681215511yescharacteristic
230C22#1681215711yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—42no—
235C4—42no—
236C4—4282no—
237C4—4282no—
238C4—4282no—
239C2—8562no—
240C2#2029321411yescharacteristic
241C2#2029321711yescharacteristic
242C2#2029322411yescharacteristic
243C2#2029334211yescharacteristic
244C2#2029335111yescharacteristic
245C2#2029336211yescharacteristic
246C2#2029337511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#389×C107C2
prime radicalC22×D428C22
hypercenter#389D107
hypocenterC107#2808
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #20292553C2semidirect
3. #20292540C2semidirect
4. #20292533C2semidirect
6. #20292519C2semidirect
7. #20292517C2semidirect
8. #20292513C2semidirect
9. #389×C107C2semidirect
124. C107#2808semidirect
130. #389D107semidirect

Maximal subgroups

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

subgroupindex
#202925532
#202925442
#202925402
#202925332
#202925292
#202925192
#202925172
#202925132
#389×C1072
#2808107