Finite Group Databasebeta

← #28344554

Subgroups — #28344554

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#20292553C212yesmaximal, complemented
4#20292549C212yesmaximal, complemented
5#20292545C212yesmaximal, complemented
6#20292542C212yesmaximal, complemented
7#20292536C211yesmaximal, characteristic
8#20292506C212yesmaximal, complemented
9#385×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C22×C22⋊D107C2212yes—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×Dic107⋊C4C2212yes—
14C2×Dic107⋊C4C2212yes—
15C2×C22⋊Dic107C2212yes—
16C2×(C2×D107)⋊C4C2212yes—
17C2×(C2×D107)⋊C4C2212yes—
18C2×(C2×D107)⋊C4C2212yes—
19C2×(C2×D107)⋊C4C2212yes—
20C2×(C2×D107)⋊C4C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2211yescharacteristic
24C107⋊#95C2212yes—
25C107⋊#95C2211yescharacteristic
26C23×C4×C107C2211yescharacteristic
27C2×C4⋊C4×C107C2212yes—
28C2×C22⋊C4×C107C2211yescharacteristic
29C2×C22⋊C4×C107C2211yescharacteristic
30#95×C107C2212yes—
31C23×D107C2312yes—
32C2×D428—41no—
33C2×D428—42no—
34C2×C22⋊D107—24no—
35C2×C22⋊D107—42no—
36C2×C22⋊D107—42no—
37C22×Dic107C2312yes—
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C4⋊Dic107—41no—
41Dic107⋊C4—42no—
42Dic107⋊C4—42no—
43C22⋊Dic107—42no—
44(C2×D107)⋊C4—42no—
45(C2×D107)⋊C4—42no—
46(C2×D107)⋊C4—42no—
47(C2×D107)⋊C4—42no—
48(C2×D107)⋊C4—42no—
49C24×C107C2311yescharacteristic
50C22×C4×C107—42no—
51C22×C4×C107—41no—
52C22×C4×C107C2312yes—
53C22×C4×C107C2311yescharacteristic
54C22×C4×C107C2312yes—
55C22×C4×C107C2311yescharacteristic
56C4⋊C4×C107—42no—
57C22⋊C4×C107—22no—
58C22⋊C4×C107—41no—
59C23×C107—21no—
60C23×C107—22no—
61C23×C107—22no—
62C23×C107—21no—
63C23×C107#4112yes—
64C23×C107C2411yescharacteristic
65C22×D107—22no—
66C22×D107—22no—
67C22×D107—22no—
68C22×D107—22no—
69C22×D107—22no—
70C22×D107—22no—
71C22×D107—22no—
72C2×C4×C107—22no—
73C2×C4×C107—21no—
74C2×C4×C107—21no—
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—44no—
83C22⋊D107—48no—
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—
93D428—44no—
94C22×C107—24no—
95C22×C107—22no—
96C22×C107—41no—
97C22×C107—41no—
98C22×C107—42no—
99C22×C107C2×#4111yescharacteristic
100C22×C107H(5,𝔽2)12yes—
101C22×C107H(5,𝔽2)11yescharacteristic
102C22×C107H(5,𝔽2)12yes—
103C22×C1072−1+411yescharacteristic
104C2×D107—42no—
105C2×D107—42no—
106C2×D107—42no—
107C2×D107—42no—
108C2×D107—42no—
109C2×D107—42no—
110C2×D107—42no—
111C4×C107—42no—
112C4×C107—41no—
113C4×C107—41no—
114C4×C107—42no—
115Dic107—42no—
116Dic107—42no—
117Dic107—42no—
118C2×C107—42no—
119C2×C107#53812yes—
120C2×C107#53911yescharacteristic
121C2×C107#56011yescharacteristic
122C2×C107#56011yescharacteristic
123C2×C107#56212yes—
124D107—82no—
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#392—1071no—
134#386—1072no—
135#385D10711yescharacteristic, complemented
136C22×D4—1072no—
137C22×D4—1071no—
138C23×C4C2×D10711yescharacteristic
139C2×C4⋊C4—1072no—
140C2×C4⋊C4—1072no—
141C2×C4⋊C4—1071no—
142C2×C4⋊C4C2×D10712yes—
143C2×C22⋊C4—1072no—
144C2×C22⋊C4—1072no—
145C2×C22⋊C4—1072no—
146C2×C22⋊C4—1072no—
147C2×C22⋊C4—1072no—
148C2×C22⋊C4—1072no—
149C2×C22⋊C4C2×D10711yescharacteristic
150C2×C22⋊C4C2×D10711yescharacteristic
151#95—1072no—
152#95—1072no—
153#95—1071no—
154#95—1072no—
155#95—1071no—
156#95C2×D10712yes—
157C24—1072no—
158C24C22×D10711yescharacteristic
159C2×D4—4281no—
160C2×D4—4282no—
161C2×D4—4282no—
162C2×D4—4282no—
163C2×D4—2144no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—1072no—
167C22×C4—1072no—
168C22×C4—1072no—
169C22×C4C22×D10712yes—
170C22×C4C22×D10711yescharacteristic
171C22×C4C22×D10712yes—
172C22×C4C22×D10711yescharacteristic
173C4⋊C4—42no—
174C4⋊C4—4282no—
175C4⋊C4—4282no—
176C4⋊C4—4281no—
177C22⋊C4—22no—
178C22⋊C4—41no—
179C22⋊C4—4282no—
180C22⋊C4—4282no—
181C22⋊C4—4282no—
182C22⋊C4—4282no—
183C22⋊C4—4282no—
184C22⋊C4—4282no—
185C23—21no—
186C23—22no—
187C23—22no—
188C23—21no—
189C23—2142no—
190C23—2142no—
191C23—2142no—
192C23—2142no—
193C23—2142no—
194C23—2142no—
195C23—2142no—
196C23#1313381812yes—
197C23C23×D10711yescharacteristic
198D4—4284no—
199D4—4288no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—44no—
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—4282no—
226C22—4282no—
227C22—4282no—
228C22—4282no—
229C22—4282no—
230C22—4282no—
231C22—4282no—
232C22C2×#1313381811yescharacteristic
233C22#1681215412yes—
234C22#1681215411yescharacteristic
235C22#1681215512yes—
236C22#1681215711yescharacteristic
237C4—42no—
238C4—42no—
239C4—41no—
240C4—41no—
241C4—4282no—
242C4—4282no—
243C4—4282no—
244C2—42no—
245C2—8562no—
246C2#2029321012yes—
247C2#2029322411yescharacteristic
248C2#2029336611yescharacteristic
249C2#2029336711yescharacteristic
250C2#2029337312yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#385×C107C2
prime radical#20292536C2
hypercenter#385D107
hypocenterC107#2826
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #20292568C2semidirect
2. #20292563C2semidirect
3. #20292553C2semidirect
4. #20292549C2semidirect
5. #20292545C2semidirect
6. #20292542C2semidirect
8. #20292506C2semidirect
9. #385×C107C2semidirect
126. C107#2826semidirect
135. #385D107semidirect

Maximal subgroups

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

subgroupindex
#202925682
#202925632
#202925532
#202925492
#202925452
#202925422
#202925362
#202925062
#385×C1072
#2826107