Finite Group Databasebeta

← #28344560

Subgroups — #28344560

Order 13696.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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#20292568C212yesmaximal, complemented
2#20292563C211yesmaximal, characteristic, complemented
3#20292555C212yesmaximal, complemented
4#20292550C212yesmaximal, complemented
5#20292541C212yesmaximal, complemented
6#20292541C212yesmaximal, complemented
7#20292507C212yesmaximal, complemented
8#20292499C211yesmaximal, characteristic, complemented
9#392×C107C211yesmaximal, characteristic
10C23×Dic107C2211yescharacteristic
11C2×C4⋊Dic107C2211yescharacteristic
12C2×Dic107⋊C4C2212yes—
13C2×Dic107⋊C4C2212yes—
14C2×Dic107⋊C4C2212yes—
15C2×C22⋊Dic107C2211yescharacteristic
16C2×C22⋊Dic107C2212yes—
17C2×C22⋊Dic107C2212yes—
18C2×C22⋊Dic107C2212yes—
19C2×C22⋊Dic107C2212yes—
20C2×C22⋊Dic107C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2211yescharacteristic
26C22×D4×C107C2212yes—
27C22×D4×C107C2211yescharacteristic
28C2×C22⋊C4×C107C2211yescharacteristic
29C2×C22⋊C4×C107C2212yes—
30#95×C107C2211yescharacteristic
31C22×Dic107—41no—
32C22×Dic107—42no—
33C22×Dic107C2312yes—
34C22×Dic107C2312yes—
35C22×Dic107C2312yes—
36C22×Dic107C2312yes—
37C4⋊Dic107—41no—
38Dic107⋊C4—42no—
39Dic107⋊C4—42no—
40Dic107⋊C4—42no—
41C22⋊Dic107—22no—
42C22⋊Dic107—42no—
43C22⋊Dic107—42no—
44C22⋊Dic107—42no—
45C22⋊Dic107—42no—
46C22⋊Dic107—42no—
47C24×C107C2312yes—
48C24×C107C2311yescharacteristic
49C2×D4×C107—24no—
50C2×D4×C107—42no—
51C2×D4×C107—41no—
52C2×D4×C107—42no—
53C2×D4×C107—42no—
54C22×C4×C107C2311yescharacteristic
55C22×C4×C107C2311yescharacteristic
56C22×C4×C107C2312yes—
57C22⋊C4×C107—42no—
58C22⋊C4×C107—41no—
59C23×C107—21no—
60C23×C107—22no—
61C23×C107—22no—
62C23×C107—21no—
63C23×C107—22no—
64C23×C107—22no—
65C23×C107—22no—
66C23×C107—22no—
67C23×C107—22no—
68C23×C107—22no—
69C23×C107—22no—
70C23×C107#4112yes—
71C23×C107C2411yescharacteristic
72D4×C107—48no—
73D4×C107—44no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—21no—
77C2×C4×C107—22no—
78C2×C4×C107—21no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×Dic107—22no—
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—
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—44no—
94C22×C107—24no—
95C22×C107—22no—
96C22×C107—41no—
97C22×C107—41no—
98C22×C107—42no—
99C22×C107—42no—
100C22×C107—42no—
101C22×C107—42no—
102C22×C107—42no—
103C22×C107—42no—
104C22×C107—42no—
105C22×C107—42no—
106C22×C107C2×#4111yescharacteristic
107C22×C107H(5,𝔽2)12yes—
108C22×C107H(5,𝔽2)12yes—
109C22×C107H(5,𝔽2)11yescharacteristic
110C22×C1072−1+411yescharacteristic
111C4×C107—42no—
112C4×C107—41no—
113C4×C107—41no—
114Dic107—42no—
115Dic107—42no—
116Dic107—42no—
117Dic107—42no—
118C2×C107—42no—
119C2×C107—82no—
120C2×C107#53812yes—
121C2×C107#53911yescharacteristic
122C2×C107#56011yescharacteristic
123C2×C107#56011yescharacteristic
124C2×C107#56212yes—
125#2826—1071nomaximal, Sylow
126C107#282611yescharacteristic, complemented, Sylow
127#400—1072no—
128#399—1071no—
129#397—1072no—
130#396—1072no—
131#394—1072no—
132#394—1072no—
133#392D10711yescharacteristic
134#386—1072no—
135#385—1071no—
136C22×D4C2×D10712yes—
137C22×D4C2×D10711yescharacteristic
138C23×C4—1071no—
139C2×C4⋊C4—1072no—
140C2×C4⋊C4—1072no—
141C2×C4⋊C4—1072no—
142C2×C4⋊C4—1071no—
143C2×C22⋊C4—1072no—
144C2×C22⋊C4—1072no—
145C2×C22⋊C4—1072no—
146C2×C22⋊C4—1072no—
147C2×C22⋊C4—1072no—
148C2×C22⋊C4—1071no—
149C2×C22⋊C4C2×D10711yescharacteristic
150C2×C22⋊C4C2×D10712yes—
151#95—1072no—
152#95—1072no—
153#95—1072no—
154#95—1072no—
155#95—1071no—
156#95C2×D10711yescharacteristic
157C24C22×D10712yes—
158C24C22×D10711yescharacteristic
159C2×D4—24no—
160C2×D4—42no—
161C2×D4—41no—
162C2×D4—42no—
163C2×D4—42no—
164C22×C4—1072no—
165C22×C4—1072no—
166C22×C4—1072no—
167C22×C4—1072no—
168C22×C4—4281no—
169C22×C4—4282no—
170C22×C4C22×D10712yes—
171C22×C4C22×D10711yescharacteristic
172C22×C4C22×D10711yescharacteristic
173C4⋊C4—4282no—
174C4⋊C4—4282no—
175C4⋊C4—4282no—
176C4⋊C4—4281no—
177C22⋊C4—41no—
178C22⋊C4—42no—
179C22⋊C4—4282no—
180C22⋊C4—4282no—
181C22⋊C4—4282no—
182C22⋊C4—4282no—
183C22⋊C4—4282no—
184C22⋊C4—2142no—
185C23—21no—
186C23—22no—
187C23—22no—
188C23—21no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23#1313381712yes—
197C23C23×D10711yescharacteristic
198D4—48no—
199D4—44no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—4284no—
208C2×C4—2142no—
209C2×C4—2142no—
210C2×C4—2142no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4—2142no—
215C2×C4—2142no—
216C2×C4—2142no—
217C2×C4—2142no—
218C2×C4—2142no—
219C2×C4—2142no—
220C22—24no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22C2×#1313381711yescharacteristic
233C22#1681215412yes—
234C22#1681215412yes—
235C22#1681215411yescharacteristic
236C22#1681215611yescharacteristic
237C4—42no—
238C4—41no—
239C4—41no—
240C4—4282no—
241C4—4282no—
242C4—4282no—
243C4—4282no—
244C2—42no—
245C2—82no—
246C2#2029321212yes—
247C2#2029322211yescharacteristic
248C2#2029336411yescharacteristic
249C2#2029336611yescharacteristic
250C2#2029337212yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting, prime radical#392×C107C2
hypercenter#392D107
hypocenterC107#2826
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #20292568C2semidirect
2. #20292563C2semidirect
3. #20292555C2semidirect
4. #20292550C2semidirect
5. #20292541C2semidirect
6. #20292541C2semidirect
7. #20292507C2semidirect
8. #20292499C2semidirect
126. C107#2826semidirect

Maximal subgroups

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

subgroupindex
#202925682
#202925632
#202925552
#202925502
#202925412
#202925072
#202924992
#392×C1072
#2826107