Finite Group Databasebeta

← #28346290

Subgroups — #28346290

Order 13696.

398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 133 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#20293258C211yesmaximal, characteristic, complemented
2#20293101C211yesmaximal, characteristic
3#20292930C212yesmaximal, complemented
4#20292926C212yesmaximal, complemented
5#20292916C212yesmaximal, complemented
6#20292906C212yesmaximal, complemented
7#20292663C211yesmaximal, characteristic, complemented
8#20292655C211yesmaximal, characteristic, complemented
9#20292633C211yesmaximal, characteristic
10C2×#16812029C211yesmaximal, characteristic
11#428×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×C107⋊C8C2211yescharacteristic
14#16812108—21no—
15#16812108C2212yescomplemented
16#16812100—21no—
17#16812100C2212yescomplemented
18#16812088—22no—
19C4×D428—21no—
20C4×D428—21no—
21#16812071C2212yes—
22#16812071C2212yes—
23#16812068C2211yescharacteristic
24#16812068C2211yescharacteristic
25C22⋊C4×D107—22no—
26#16812042C2211yescharacteristic, complemented
27#16812042C2211yescharacteristic, complemented
28#16812040C2211yescharacteristic, complemented
29#16812040C2211yescharacteristic, complemented
30Dic107⋊C8C2212yescomplemented
31Dic107⋊C8C2212yescomplemented
32#16812031C2212yes—
33#16812031C2212yes—
34#16812029C2212yes—
35#16812029C2212yes—
36C2×M16×C107C2211yescharacteristic
37#117×C107C2211yescharacteristic
38C2×C4⋊C4×C107C2211yescharacteristic
39#107×C107C2212yescomplemented
40#106×C107C2212yescomplemented
41C23×D107—21no—
42C2×Dic214—22no—
43C2×D428—21no—
44C2×D428—22no—
45C2×D428Q814yescomplemented
46C2×D428C2312yes—
47C2×C4×D107—22no—
48C2×C4×D107—22no—
49C2×C107⋊C8—22no—
50C2×C107⋊C8C2311yescharacteristic
51C2×C107⋊C8C2311yescharacteristic
52C4⋊Dic107—22no—
53C4⋊Dic107D411yescharacteristic
54C4⋊Dic107D411yescharacteristic
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2311yescharacteristic
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59C22⋊Dic107—21no—
60C22⋊Dic107D412yes—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63C4×Dic107—21no—
64C4×Dic107C2312yes—
65C22×C4×C107—21no—
66C22×C4×C107C2311yescharacteristic
67M16×C107—22no—
68C2×C8×C107C2312yes—
69C4⋊C4×C107—21no—
70C4⋊C4×C107C2311yescharacteristic
71C4⋊C4×C107C2311yescharacteristic
72C4⋊C4×C107C2312yes—
73C22⋊C4×C107—21no—
74C42×C107—21no—
75C23×C107C2×D411yescharacteristic
76C22×D107—22no—
77C22×D107—24no—
78C22×D107—81no—
79C22×D107—81no—
80C2×C4×C107—21no—
81C2×C4×C107—21no—
82C2×C4×C107—22no—
83C2×C4×C107—81no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×Q812yes—
86C2×C4×C107#4112yes—
87C2×C4×C107C2411yescharacteristic
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107C2×D412yes—
91C2×Dic107C2×D412yes—
92D428—24no—
93D428—22no—
94D428C2×Q814yes—
95C4×D107—82no—
96C4×D107—82no—
97Dic214—82no—
98C8×C107—22no—
99C107⋊C8—21no—
100C107⋊C8—21no—
101C22×C107—21no—
102C22×C107—21no—
103C22×C107C22⋊Q812yes—
104C22×C107C22×D411yescharacteristic
105C2×D107—24no—
106C2×D107—42no—
107C2×D107—42no—
108C2×D107—84no—
109C4×C107—42no—
110C4×C107—41no—
111C4×C107—41no—
112C4×C107C22⋊Q812yes—
113C4×C107C22×Q811yescharacteristic
114C4×C107C2×#4111yescharacteristic
115Dic107—22no—
116Dic107—22no—
117Dic107—41no—
118Dic107—41no—
119C2×C107—22no—
120C2×C107C2×C22⋊Q811yescharacteristic
121C2×C107#57611yescharacteristic
122C2×C107#57711yescharacteristic
123D107—44no—
124#3081—1071nomaximal, Sylow
125C107#308111yescharacteristic, complemented, Sylow
126#544—1071no—
127#518—1071no—
128#478—1072no—
129#478—1072no—
130#476—1072no—
131#474—1072no—
132#428D10711yescharacteristic, complemented
133#427—1071no—
134#424—1071no—
135#418—1071no—
136C2×#102—1071no—
137C22×D4—1071no—
138C2×M16C2×D10711yescharacteristic
139C22×C8—1071no—
140C4⋊Q8—1072no—
141C4⋊Q8—2141no—
142#125—1072no—
143#125—2141no—
144C22⋊Q8—2142no—
145C4×D4—1072no—
146C4×D4—1072no—
147C4×D4—2141no—
148C4×D4—2141no—
149#117—1071no—
150#117—1071no—
151#117C2×D10711yescharacteristic
152C2×C4⋊C4C2×D10711yescharacteristic
153C2×C22⋊C4—2142no—
154#107—1071no—
155#107—1071no—
156#107C2×D10712yescomplemented
157#106—1071no—
158#106—1071no—
159#106C2×D10712yescomplemented
160C4⋊C8—1072no—
161C4⋊C8—1072no—
162#102—1072no—
163#102—1072no—
164#102—1072no—
165#102—1072no—
166C24—2141no—
167C2×Q8—2142no—
168C2×D4—1074no—
169C2×D4—1072no—
170C2×D4—2141no—
171C2×D4—2142no—
172C22×C4—21no—
173C22×C4—2142no—
174C22×C4—2142no—
175C22×C4C22×D10711yescharacteristic
176M16—22no—
177C2×C8—1071no—
178C2×C8—1071no—
179C2×C8—2142no—
180C2×C8C22×D10712yes—
181C4⋊C4—21no—
182C4⋊C4—1071no—
183C4⋊C4—1071no—
184C4⋊C4—1071no—
185C4⋊C4—1071no—
186C4⋊C4—2142no—
187C4⋊C4—2142no—
188C4⋊C4—2142no—
189C4⋊C4C22×D10712yes—
190C4⋊C4C22×D10711yescharacteristic
191C4⋊C4C22×D10711yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—1072no—
194C22⋊C4—2142no—
195C22⋊C4—2142no—
196C22⋊C4—2141no—
197C42—21no—
198C42—1072no—
199C42—2141no—
200C23—8561no—
201C23—8561no—
202C23—2144no—
203C23—2142no—
204C23D4×D10711yescharacteristic
205Q8—8562no—
206D4—1074no—
207D4—2144no—
208D4—2142no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—81no—
213C2×C4—8562no—
214C2×C4—8562no—
215C2×C4—1072no—
216C2×C4—1072no—
217C2×C4—2141no—
218C2×C4—2141no—
219C2×C4D4×D10711yescharacteristic
220C2×C4Q8×D10712yes—
221C2×C4#1313381812yes—
222C2×C4C23×D10711yescharacteristic
223C8—22no—
224C8—2141no—
225C8—2141no—
226C22—21no—
227C22—21no—
228C22—8564no—
229C22—4282no—
230C22—4282no—
231C22—2144no—
232C22#1681208712yes—
233C22C2×D4×D10711yescharacteristic
234C4—42no—
235C4—41no—
236C4—41no—
237C4—4281no—
238C4—4281no—
239C4—2142no—
240C4—2142no—
241C4#1681208712yes—
242C4C2×Q8×D10711yescharacteristic
243C4C2×#1313381811yescharacteristic
244C2—22no—
245C2—4284no—
246C2C2×#1681208711yescharacteristic
247C2#2029343111yescharacteristic
248C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radicalC22×D428C22
hypercenter#428D107
hypocenterC107#3081
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
1. #20293258C2semidirect
3. #20292930C2semidirect
4. #20292926C2semidirect
5. #20292916C2semidirect
6. #20292906C2semidirect
7. #20292663C2semidirect
8. #20292655C2semidirect
11. #428×C107C2semidirect
15. #16812108C22semidirect
17. #16812100C22semidirect
26. #16812042C22semidirect
27. #16812042C22semidirect
28. #16812040C22semidirect
29. #16812040C22semidirect
30. Dic107⋊C8C22semidirect
31. Dic107⋊C8C22semidirect
39. #107×C107C22semidirect
40. #106×C107C22semidirect
45. C2×D428Q8semidirect
125. C107#3081semidirect
132. #428D107semidirect
156. #107C2×D107semidirect
159. #106C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932582
#202931012
#202929302
#202929262
#202929162
#202929062
#202926632
#202926552
#202926332
C2×#168120292
#428×C1072
#3081107