Finite Group Databasebeta

← #28344536

Subgroups — #28344536

Order 13696.

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

normal subgroups only

This group has 456 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
1C2×#16812104C212yesmaximal, complemented
2#20292561C212yesmaximal, complemented
3#20292551C212yesmaximal, complemented
4#20292549C212yesmaximal, complemented
5#20292549C212yesmaximal, complemented
6#20292545C212yesmaximal
7#20292544C211yesmaximal, characteristic
8#20292496C211yesmaximal, characteristic, complemented
9#398×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11#16812104—28no—
12C2×C4⋊Dic107C2212yes—
13C2×C4⋊Dic107C2211yescharacteristic
14C2×Dic107⋊C4C2212yes—
15C2×Dic107⋊C4C2212yes—
16C2×Dic107⋊C4C2212yes—
17C2×Dic107⋊C4C2212yes—
18C2×(C2×D107)⋊C4C2212yes—
19C2×(C2×D107)⋊C4C2212yes—
20C2×(C2×D107)⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C2×C4×Dic107C2211yescharacteristic
25C107⋊#95C2211yescharacteristic
26C107⋊#95C2212yes—
27C2×C4⋊C4×C107C2212yes—
28C2×C4⋊C4×C107C2211yescharacteristic
29C2×C4⋊C4×C107C2211yescharacteristic
30#95×C107C2212yes—
31#95×C107C2211yescharacteristic
32C23×D107C2312yes—
33C2×D428—41no—
34C2×D428—42no—
35C22×Dic107C2312yes—
36C22×Dic107C2312yes—
37C22×Dic107C2312yes—
38C4⋊Dic107—24no—
39C4⋊Dic107—41no—
40Dic107⋊C4—24no—
41Dic107⋊C4—42no—
42Dic107⋊C4—42no—
43Dic107⋊C4—42no—
44(C2×D107)⋊C4—24no—
45(C2×D107)⋊C4—24no—
46(C2×D107)⋊C4—24no—
47(C2×D107)⋊C4—42no—
48(C2×D107)⋊C4—42no—
49(C2×D107)⋊C4—42no—
50C4×Dic107D414yescomplemented
51C22×C4×C107C2312yes—
52C22×C4×C107C2311yescharacteristic
53C22×C4×C107C2312yes—
54C22×C4×C107C2311yescharacteristic
55C22×C4×C107C2311yescharacteristic
56C4⋊C4×C107—24no—
57C4⋊C4×C107—41no—
58C4⋊C4×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—22no—
70C2×C4×C107—22no—
71C2×C4×C107—21no—
72C2×C4×C107—22no—
73C2×C4×C107—21no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107C2×D412yes—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107C2×D414yes—
89D428—44no—
90C22×C107C22×D411yescharacteristic
91C22×C107H(5,𝔽2)11yescharacteristic
92C22×C107H(5,𝔽2)12yes—
93C22×C1072−1+411yescharacteristic
94C22×C1072−1+412yes—
95C2×D107—42no—
96C2×D107—42no—
97C2×D107—42no—
98C2×D107—42no—
99C2×D107—42no—
100C2×D107—42no—
101C2×D107—42no—
102C4×C107—22no—
103C4×C107—42no—
104C4×C107—42no—
105C4×C107—41no—
106C4×C107—41no—
107Dic107—24no—
108Dic107—42no—
109Dic107—42no—
110C2×C107#53711yescharacteristic
111C2×C107#53712yes—
112C2×C107#56011yescharacteristic
113C2×C107#56311yescharacteristic
114C2×C107#56312yes—
115D107—82no—
116#2823—1071nomaximal, Sylow
117C107#282311yescharacteristic, complemented, Sylow
118C2×#126—1072no—
119#398—1072no—
120#398D10711yescharacteristic, complemented
121#396—1072no—
122#396—1072no—
123#396—1072no—
124#394—1071no—
125#394—1072no—
126#384—1071no—
127C22×D4—1071no—
128#126—2148no—
129C2×C4⋊C4—1072no—
130C2×C4⋊C4—1072no—
131C2×C4⋊C4—1072no—
132C2×C4⋊C4—1071no—
133C2×C4⋊C4—1072no—
134C2×C4⋊C4—1072no—
135C2×C4⋊C4C2×D10712yes—
136C2×C4⋊C4C2×D10711yescharacteristic
137C2×C4⋊C4C2×D10711yescharacteristic
138C2×C22⋊C4—1072no—
139C2×C22⋊C4—1072no—
140C2×C22⋊C4—1072no—
141C2×C22⋊C4—1072no—
142C2×C22⋊C4—1072no—
143C2×C22⋊C4—1072no—
144C2×C42—1071no—
145#95—1071no—
146#95—1072no—
147#95C2×D10712yes—
148#95C2×D10711yescharacteristic
149C24—1072no—
150C2×D4—4281no—
151C2×D4—4282no—
152C22×C4—1072no—
153C22×C4—1072no—
154C22×C4—1072no—
155C22×C4C22×D10711yescharacteristic
156C22×C4C22×D10712yes—
157C22×C4C22×D10712yes—
158C22×C4C22×D10711yescharacteristic
159C22×C4C22×D10711yescharacteristic
160C4⋊C4—24no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—4282no—
164C4⋊C4—4282no—
165C4⋊C4—4282no—
166C4⋊C4—4281no—
167C4⋊C4—2144no—
168C4⋊C4—2144no—
169C22⋊C4—4282no—
170C22⋊C4—4282no—
171C22⋊C4—4282no—
172C22⋊C4—2144no—
173C22⋊C4—2144no—
174C22⋊C4—2144no—
175C42—1074no—
176C23—2142no—
177C23—2142no—
178C23—2142no—
179C23—2142no—
180C23—2142no—
181C23—2142no—
182C23—2142no—
183C23C23×D10711yescharacteristic
184D4—4284no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—1074no—
198C2×C4—2142no—
199C2×C4—2142no—
200C2×C4—2142no—
201C2×C4—2142no—
202C2×C4—2142no—
203C2×C4—2142no—
204C2×C4—2142no—
205C2×C4—2142no—
206C2×C4D4×D10712yes—
207C22—4282no—
208C22—4282no—
209C22—4282no—
210C22—4282no—
211C22—4282no—
212C22—4282no—
213C22—4282no—
214C22C2×D4×D10711yescharacteristic
215C22#1681215411yescharacteristic
216C22#1681215512yes—
217C22#1681215611yescharacteristic
218C22#1681215712yes—
219C4—22no—
220C4—42no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—4282no—
225C4—4282no—
226C4—2144no—
227C2—8562no—
228C2#2029320411yescharacteristic
229C2#2029320712yes—
230C2#2029336711yescharacteristic
231C2#2029337611yescharacteristic
232C2#2029338112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#398×C107C2
prime radicalC22×D428C22
hypercenter#398D107
hypocenterC107#2823
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#16812104C2semidirect
2. #20292561C2semidirect
3. #20292551C2semidirect
4. #20292549C2semidirect
5. #20292549C2semidirect
8. #20292496C2semidirect
9. #398×C107C2semidirect
50. C4×Dic107D4semidirect
117. C107#2823semidirect
120. #398D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121042
#202925612
#202925512
#202925492
#202925452
#202925442
#202924962
#398×C1072
#2823107