Finite Group Databasebeta

← #28347011

Subgroups — #28347011

Order 13696.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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#20293180C211yesmaximal, characteristic
2C2×#16812100C211yesmaximal, characteristic
3#20293035C212yesmaximal, complemented
4#20293034C212yesmaximal, complemented
5#20293028C212yesmaximal, complemented
6#20293027C212yesmaximal, complemented
7#20292670C211yesmaximal, characteristic
8#20292669C211yesmaximal, characteristic
9#20292667C211yesmaximal, characteristic
10M16×Dic107C211yesmaximal, characteristic
11#428×C107C211yesmaximal, characteristic
12C2×#13133802C2211yescharacteristic
13#16812108C2211yescharacteristic
14#16812108C2211yescharacteristic
15#16812100—21no—
16#16812100—21no—
17#16812100C2212yes—
18#16812100C2212yes—
19#16812100C2211yescharacteristic
20#16812100C2211yescharacteristic
21#16812086—21no—
22#16812086—21no—
23C4×Dic214—21no—
24C4×Dic214—21no—
25#16812068C2211yescharacteristic
26C2×C4⋊Dic107—21no—
27C2×C4⋊Dic107C2211yescharacteristic
28C2×Dic107⋊C4—21no—
29C2×C4×Dic107C2211yescharacteristic
30#16812043C2212yes—
31#16812042C2212yes—
32#16812042C2212yes—
33#16812041C2212yes—
34#16812040C2212yes—
35#16812040C2212yes—
36#16812019C2212yes—
37C8×Dic107C2212yes—
38C2×M16×C107C2211yescharacteristic
39#117×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#107×C107C2212yes—
42#106×C107C2212yes—
43C2×Dic214—21no—
44C2×Dic214—21no—
45C22×Dic107—21no—
46C22×Dic107D411yescharacteristic
47C22×Dic107D411yescharacteristic
48#13133802Q814yescomplemented
49C2×C107⋊C8C2312yes—
50C4⋊Dic107—21no—
51C4⋊Dic107—22no—
52C4⋊Dic107—22no—
53C4⋊Dic107—21no—
54C4⋊Dic107—21no—
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2312yes—
57C4⋊Dic107C2311yescharacteristic
58Dic107⋊C4—22no—
59Dic107⋊C4—22no—
60Dic107⋊C4—21no—
61Dic107⋊C4—21no—
62Dic107⋊C4—21no—
63Dic107⋊C4—21no—
64C22⋊Dic107—21no—
65C4×Dic107—21no—
66C4×Dic107D411yescharacteristic
67C4×Dic107D411yescharacteristic
68C4×Dic107C2311yescharacteristic
69C4×Dic107C2311yescharacteristic
70C22×C4×C107—21no—
71C22×C4×C107C2311yescharacteristic
72M16×C107Q814yescomplemented
73C2×C8×C107C2312yes—
74C4⋊C4×C107—21no—
75C4⋊C4×C107C2311yescharacteristic
76C4⋊C4×C107C2312yes—
77C4⋊C4×C107C2311yescharacteristic
78C22⋊C4×C107—21no—
79C42×C107—21no—
80C23×C107C2×D411yescharacteristic
81C2×C4×C107—21no—
82C2×C4×C107—21no—
83C2×C4×C107—22no—
84C2×C4×C107—42no—
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2×Q812yes—
87C2×C4×C107C2×Q812yes—
88C2×C4×C107C2411yescharacteristic
89C2×Dic107—21no—
90C2×Dic107—21no—
91C2×Dic107—22no—
92C2×Dic107—41no—
93C2×Dic107—41no—
94C2×Dic107—42no—
95C2×Dic107C2×D411yescharacteristic
96C2×Dic107C2×D411yescharacteristic
97C2×Dic107C2×D411yescharacteristic
98C2×Dic107C2×D411yescharacteristic
99Dic214—42no—
100Dic214—42no—
101C8×C107C2×Q814yes—
102C107⋊C8C2×Q814yes—
103C22×C107—21no—
104C22×C107—21no—
105C22×C107C4⋊Q812yes—
106C22×C107C22×D411yescharacteristic
107C4×C107—41no—
108C4×C107—41no—
109C4×C107—42no—
110C4×C107C4⋊Q812yes—
111C4×C107C22×Q811yescharacteristic
112C4×C107C22×Q811yescharacteristic
113Dic107—21no—
114Dic107—21no—
115Dic107—21no—
116Dic107—21no—
117Dic107—41no—
118Dic107—41no—
119Dic107—42no—
120C2×C107—22no—
121C2×C107C2×C4⋊Q811yescharacteristic
122C2×C107#57511yescharacteristic
123C2×C107#57511yescharacteristic
124#3166—1071nomaximal, Sylow
125C107#316611yescharacteristic, complemented, Sylow
126#533—1071no—
127C2×#125—1071no—
128#501—1072no—
129#501—1072no—
130#499—1072no—
131#499—1072no—
132#428—1071no—
133#428—1071no—
134#428—1071no—
135#428D10711yescharacteristic
136C4×M16—1071no—
137C2×M16—1071no—
138C2×M16C2×D10711yescharacteristic
139C4⋊Q8—1071no—
140C4⋊Q8—1071no—
141#125—1072no—
142#125—1072no—
143#125—1071no—
144#125—1071no—
145#125—2141no—
146#125—2141no—
147C22⋊Q8—2141no—
148C22⋊Q8—2141no—
149C4×Q8—2141no—
150C4×Q8—2141no—
151#117—1071no—
152#117C2×D10711yescharacteristic
153C2×C4⋊C4—1071no—
154C2×C4⋊C4—2141no—
155C2×C4⋊C4—2141no—
156C2×C4⋊C4C2×D10711yescharacteristic
157C2×C42—1071no—
158#107—1072no—
159#107—1072no—
160#107—1072no—
161#107C2×D10712yes—
162#106—1072no—
163#106—1072no—
164#106—1072no—
165#106C2×D10712yes—
166#97—1072no—
167C4×C8—1072no—
168C2×Q8—2141no—
169C2×Q8—2141no—
170C22×C4—21no—
171C22×C4—1071no—
172C22×C4—1071no—
173C22×C4—2141no—
174C22×C4C22×D10711yescharacteristic
175M16—1074no—
176M16Dic21414yescomplemented
177C2×C8—1072no—
178C2×C8C22×D10712yes—
179C4⋊C4—21no—
180C4⋊C4—1072no—
181C4⋊C4—1071no—
182C4⋊C4—1071no—
183C4⋊C4—2142no—
184C4⋊C4—2142no—
185C4⋊C4—2142no—
186C4⋊C4—2142no—
187C4⋊C4—2141no—
188C4⋊C4—2141no—
189C4⋊C4—2141no—
190C4⋊C4—2141no—
191C4⋊C4—2141no—
192C4⋊C4—2141no—
193C4⋊C4—2141no—
194C4⋊C4C22×D10711yescharacteristic
195C4⋊C4C22×D10712yes—
196C4⋊C4C22×D10711yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—2141no—
199C42—21no—
200C42—1071no—
201C42—1071no—
202C42—1071no—
203C42—1071no—
204C42—2141no—
205C23D4×D10711yescharacteristic
206Q8—4282no—
207Q8—4282no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—1071no—
213C2×C4—1071no—
214C2×C4—1071no—
215C2×C4—1071no—
216C2×C4—4282no—
217C2×C4—4281no—
218C2×C4—4281no—
219C2×C4—2141no—
220C2×C4—2141no—
221C2×C4—2142no—
222C2×C4D4×D10711yescharacteristic
223C2×C4Q8×D10712yes—
224C2×C4C2×Dic21412yes—
225C2×C4C23×D10711yescharacteristic
226C8—1074no—
227C8C2×Dic21414yes—
228C22—21no—
229C22—21no—
230C22#1681210812yes—
231C22C2×D4×D10711yescharacteristic
232C4—42no—
233C4—41no—
234C4—41no—
235C4—4281no—
236C4—4282no—
237C4—4281no—
238C4—2141no—
239C4—2141no—
240C4—2141no—
241C4—2141no—
242C4#1681210812yes—
243C4C2×Q8×D10711yescharacteristic
244C4C22×Dic21411yescharacteristic
245C2—22no—
246C2C2×#1681210811yescharacteristic
247C2#2029342311yescharacteristic
248C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenter#428D107
hypocenterC107#3166
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #20293035C2semidirect
4. #20293034C2semidirect
5. #20293028C2semidirect
6. #20293027C2semidirect
48. #13133802Q8semidirect
72. M16×C107Q8semidirect
125. C107#3166semidirect
176. M16Dic214semidirect

Maximal subgroups

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

subgroupindex
#202931802
C2×#168121002
#202930352
#202930342
#202930282
#202930272
#202926702
#202926692
#202926672
M16×Dic1072
#428×C1072
#3166107