Finite Group Databasebeta

← #28344608

Subgroups — #28344608

Order 13696.

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

normal subgroups only

This group has 408 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#20292556C212yesmaximal, complemented
2#20292554C212yesmaximal, complemented
3#20292549C212yesmaximal, complemented
4#20292547C211yesmaximal, characteristic, complemented
5#20292544C212yesmaximal
6#20292540C212yesmaximal, complemented
7#20292528C211yesmaximal, characteristic
8#20292513C212yesmaximal, complemented
9#398×C107C211yesmaximal, characteristic, complemented
10C22×Dic214C2211yescharacteristic
11C22×D428C2211yescharacteristic
12C2×C4⋊Dic107C2212yes—
13C2×Dic107⋊C4C2212yes—
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×Dic107C2211yescharacteristic
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2211yescharacteristic
26C2×C4⋊C4×C107C2212yes—
27C2×C4⋊C4×C107C2211yescharacteristic
28C2×C4⋊C4×C107C2211yescharacteristic
29#95×C107C2212yes—
30#95×C107C2211yescharacteristic
31C23×D107C2312yes—
32C2×Dic214—41no—
33C2×Dic214—42no—
34C2×D428—22no—
35C2×D428—41no—
36C2×D428—41no—
37C22×Dic107C2312yes—
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C4⋊Dic107—42no—
41Dic107⋊C4—42no—
42(C2×D107)⋊C4—42no—
43(C2×D107)⋊C4—42no—
44(C2×D107)⋊C4—42no—
45(C2×D107)⋊C4—42no—
46(C2×D107)⋊C4—42no—
47(C2×D107)⋊C4—42no—
48C4×Dic107—41no—
49C22×C4×C107C2311yescharacteristic
50C22×C4×C107C2312yes—
51C22×C4×C107C2311yescharacteristic
52C22×C4×C107C2312yes—
53C22×C4×C107C2311yescharacteristic
54C4⋊C4×C107—24no—
55C4⋊C4×C107—41no—
56C4⋊C4×C107—41no—
57C23×C107C2411yescharacteristic
58C22×D107—22no—
59C22×D107—22no—
60C22×D107—22no—
61C22×D107—22no—
62C22×D107—22no—
63C22×D107—22no—
64C22×D107—22no—
65C2×C4×C107—21no—
66C2×C4×C107—21no—
67C2×C4×C107—22no—
68C2×C4×C107—22no—
69C2×C4×C107—21no—
70C2×C4×C107—22no—
71C2×C4×C107—21no—
72C2×C4×C107—22no—
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107#4112yes—
78C2×Dic107—22no—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—22no—
87D428—44no—
88Dic214—44no—
89C22×C107C2×#4111yescharacteristic
90C22×C107H(5,𝔽2)11yescharacteristic
91C22×C107H(5,𝔽2)11yescharacteristic
92C22×C107H(5,𝔽2)12yes—
93C22×C1072−1+412yes—
94C2×D107—42no—
95C2×D107—42no—
96C2×D107—42no—
97C2×D107—42no—
98C2×D107—42no—
99C2×D107—42no—
100C2×D107—42no—
101C4×C107—22no—
102C4×C107—42no—
103C4×C107—42no—
104C4×C107—41no—
105C4×C107—41no—
106Dic107—42no—
107Dic107—42no—
108Dic107—42no—
109C2×C107#53811yescharacteristic
110C2×C107#54212yes—
111C2×C107#56011yescharacteristic
112C2×C107#56212yes—
113C2×C107#56311yescharacteristic
114D107—82no—
115#2830—1071nomaximal, Sylow
116C107#283011yescharacteristic, complemented, Sylow
117#398D10711yescharacteristic, complemented
118#397—1072no—
119#397—1072no—
120#396—1072no—
121#395—1071no—
122#394—1072no—
123#393—1072no—
124#390—1071no—
125#387—1072no—
126C22×Q8—1071no—
127C22×D4—1071no—
128C2×C4⋊C4—1072no—
129C2×C4⋊C4—1072no—
130C2×C4⋊C4C2×D10712yes—
131C2×C4⋊C4C2×D10711yescharacteristic
132C2×C4⋊C4C2×D10711yescharacteristic
133C2×C22⋊C4—1072no—
134C2×C22⋊C4—1072no—
135C2×C22⋊C4—1072no—
136C2×C22⋊C4—1072no—
137C2×C22⋊C4—1072no—
138C2×C22⋊C4—1072no—
139C2×C42—1071no—
140#95—1072no—
141#95—1072no—
142#95—1072no—
143#95—1072no—
144#95—1071no—
145#95C2×D10712yes—
146#95C2×D10711yescharacteristic
147C24—1072no—
148C2×Q8—4281no—
149C2×Q8—4282no—
150C2×D4—4281no—
151C2×D4—4281no—
152C2×D4—2142no—
153C22×C4—1072no—
154C22×C4—1072no—
155C22×C4—1072no—
156C22×C4C22×D10712yes—
157C22×C4C22×D10711yescharacteristic
158C22×C4C22×D10712yes—
159C22×C4C22×D10711yescharacteristic
160C22×C4C22×D10711yescharacteristic
161C4⋊C4—24no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—4282no—
165C4⋊C4—4282no—
166C22⋊C4—4282no—
167C22⋊C4—4282no—
168C22⋊C4—4282no—
169C22⋊C4—4282no—
170C22⋊C4—4282no—
171C22⋊C4—4282no—
172C42—4281no—
173C23—2142no—
174C23—2142no—
175C23—2142no—
176C23—2142no—
177C23—2142no—
178C23—2142no—
179C23—2142no—
180C23C23×D10711yescharacteristic
181Q8—4284no—
182D4—4284no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—2142no—
196C2×C4—2142no—
197C2×C4—2142no—
198C2×C4—2142no—
199C2×C4—2142no—
200C2×C4—2142no—
201C2×C4—2142no—
202C2×C4—2142no—
203C2×C4—2142no—
204C2×C4#1313381712yes—
205C22—4282no—
206C22—4282no—
207C22—4282no—
208C22—4282no—
209C22—4282no—
210C22—4282no—
211C22—4282no—
212C22C2×#1313381711yescharacteristic
213C22#1681215411yescharacteristic
214C22#1681215511yescharacteristic
215C22#1681215512yes—
216C22#1681215712yes—
217C4—22no—
218C4—42no—
219C4—42no—
220C4—41no—
221C4—41no—
222C4—4282no—
223C4—4282no—
224C4—4282no—
225C2—8562no—
226C2#2029321411yescharacteristic
227C2#2029325012yes—
228C2#2029336711yescharacteristic
229C2#2029337512yes—
230C2#2029337611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#398×C107C2
prime radicalC22×D428C22
hypercenter#398D107
hypocenterC107#2830
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. #20292556C2semidirect
2. #20292554C2semidirect
3. #20292549C2semidirect
4. #20292547C2semidirect
6. #20292540C2semidirect
8. #20292513C2semidirect
9. #398×C107C2semidirect
116. C107#2830semidirect
117. #398D107semidirect

Maximal subgroups

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

subgroupindex
#202925562
#202925542
#202925492
#202925472
#202925442
#202925402
#202925282
#202925132
#398×C1072
#2830107