Finite Group Databasebeta

← #28348323

Subgroups — #28348323

Order 13696.

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

normal subgroups only

This group has 420 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
1Q8×D428C211yesmaximal, characteristic, complemented
2#20293280C211yesmaximal, characteristic
3#20292979C211yesmaximal, characteristic, complemented
4#20292959C212yesmaximal, complemented
5#20292857C212yesmaximal, complemented
6#20292833C211yesmaximal, characteristic, complemented
7#20292767C212yesmaximal, complemented
8C4×#13133805C211yesmaximal, characteristic, complemented
9#20292692C211yesmaximal, characteristic, complemented
10#103×D107C212yesmaximal, complemented
11#475×C107C211yesmaximal, characteristic, complemented
12C2×#13133819—21no—
13C2×Q8×D107C2212yes—
14C2×#13133807C2212yes—
15C2×#13133805C2211yescharacteristic
16C2×C8×D107C2212yes—
17#16812109—21no—
18#16812109C2211yescharacteristic
19#16812094—22no—
20#16812089—22no—
21#16812089C2212yes—
22#16812085—21no—
23#16812084C2212yes—
24C4×D428—21no—
25C4×D428C2211yescharacteristic
26C4×D428C2211yescharacteristic
27C4⋊C4×D107C2212yes—
28#16812040C2211yescharacteristic
29#16812034C2212yes—
30#16812033C2211yescharacteristic
31#16812032C2212yes—
32#16812032C2212yes—
33#16812029C2211yescharacteristic
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C4⋊Q8×C107C2211yescharacteristic
37C4×Q8×C107C2211yescharacteristic
38#106×C107C2212yes—
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41#13133819—44no—
42C2×Dic214—22no—
43C2×Dic214C2312yes—
44Q8×D107—24no—
45Q8×D107—42no—
46C2×D428—22no—
47C2×D428D411yescharacteristic
48C2×D428C2311yescharacteristic
49C2×C4×D107—22no—
50C2×C4×D107—22no—
51C2×C4×D107—21no—
52C2×C4×D107C2312yes—
53#13133807—42no—
54#13133805—22no—
55C2×C107⋊C8C2311yescharacteristic
56C2×C107⋊C8C2311yescharacteristic
57C8×D107—42no—
58C4⋊Dic107—21no—
59C4⋊Dic107—21no—
60C4⋊Dic107—22no—
61C4⋊Dic107D411yescharacteristic
62C4⋊Dic107C2311yescharacteristic
63Dic107⋊C4—22no—
64(C2×D107)⋊C4—22no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4D412yes—
68C2×Q8×C107—21no—
69C2×Q8×C107C2311yescharacteristic
70C2×C8×C107C2312yes—
71C4⋊C4×C107—21no—
72C4⋊C4×C107—21no—
73C4⋊C4×C107C2312yes—
74C4⋊C4×C107C2311yescharacteristic
75C42×C107—21no—
76C42×C107C2311yescharacteristic
77C22×D107—21no—
78C22×D107C2×D412yes—
79Q8×C107—21no—
80Q8×C107—42no—
81Q8×C107#4112yescomplemented
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—22no—
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2411yescharacteristic
88C2×Dic107—21no—
89C2×Dic107—22no—
90C2×Dic107C2×D412yes—
91D428—21no—
92D428—21no—
93D428—44no—
94D428SD1614yescomplemented
95C4×D107—22no—
96C4×D107—22no—
97C4×D107—44no—
98C4×D107—44no—
99C4×D107—42no—
100Dic214—22no—
101Dic214—22no—
102Dic214—44no—
103C8×C107—22no—
104C107⋊C8—21no—
105C107⋊C8—21no—
106C22×C107C22×D411yescharacteristic
107C2×D107—22no—
108C2×D107—22no—
109C2×D107—41no—
110C2×D107—41no—
111C2×D107—41no—
112C2×D107C2×SD1614yes—
113C4×C107—22no—
114C4×C107—21no—
115C4×C107—42no—
116C4×C107—41no—
117C4×C107C2×SD1612yes—
118C4×C107C2×#4111yescharacteristic
119C4×C107H(5,𝔽2)11yescharacteristic
120Dic107—22no—
121Dic107—42no—
122Dic107—41no—
123C2×C107#54611yescharacteristic
124C2×C107C22×SD1611yescharacteristic
125C2×C107#57811yescharacteristic
126D107—24no—
127D107—81no—
128#3298—1071nomaximal, Sylow
129C107#329811yescharacteristic, complemented, Sylow
130D4×Q8—1071no—
131#547—1071no—
132#487—1071no—
133#483—1072no—
134#475D10711yescharacteristic, complemented
135#465—1072no—
136#461—1071no—
137#448—1072no—
138C4×SD16—1071no—
139C8×D4—1071no—
140C2×#103—1072no—
141C2×#41—2141no—
142C22×Q8—1072no—
143C2×SD16—1071no—
144C2×SD16—1072no—
145C22×C8—1072no—
146C4⋊Q8—1071no—
147C4⋊Q8—2141no—
148C4⋊Q8C2×D10711yescharacteristic
149#123—2142no—
150C22⋊Q8—1072no—
151C22⋊Q8—2142no—
152C22⋊Q8—2141no—
153#121—1072no—
154C4×Q8C2×D10711yescharacteristic
155C4×D4—1071no—
156C4×D4—1071no—
157C4×D4—2141no—
158C2×C4⋊C4—1072no—
159#106—1071no—
160#106C2×D10712yes—
161C4⋊C8C2×D10711yescharacteristic
162#103—1071no—
163#103—1072no—
164#103—1072no—
165#103—1072no—
166#103C2×D10712yes—
167#102—1071no—
168C22⋊C8—1072no—
169C4×C8—1071no—
170#41—4284no—
171C2×Q8—21no—
172C2×Q8—1072no—
173C2×Q8—4282no—
174C2×Q8—2142no—
175C2×Q8—2144no—
176C2×Q8C22×D10711yescharacteristic
177C2×D4—1071no—
178C2×D4—1071no—
179C2×D4—2142no—
180C22×C4—1072no—
181C22×C4—2142no—
182C22×C4—2142no—
183C22×C4—2141no—
184SD16—4282no—
185SD16—2142no—
186C2×C8—1071no—
187C2×C8—1071no—
188C2×C8—4282no—
189C2×C8C22×D10712yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—1071no—
193C4⋊C4—1071no—
194C4⋊C4—2142no—
195C4⋊C4—2142no—
196C4⋊C4—2141no—
197C4⋊C4—2141no—
198C4⋊C4C22×D10711yescharacteristic
199C4⋊C4C22×D10712yes—
200C22⋊C4—1072no—
201C22⋊C4—2142no—
202C22⋊C4—2142no—
203C22⋊C4—2141no—
204C42—21no—
205C42C22×D10711yescharacteristic
206C23—1072no—
207C23—2141no—
208Q8—21no—
209Q8—42no—
210Q8—4284no—
211Q8—2142no—
212Q8—2142no—
213Q8#1313381912yescomplemented
214D4—1074no—
215D4—4284no—
216D4—2141no—
217D4—2141no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—1072no—
222C2×C4—4284no—
223C2×C4—4284no—
224C2×C4—4282no—
225C2×C4—2141no—
226C2×C4—2142no—
227C2×C4—2142no—
228C2×C4—2142no—
229C2×C4D4×D10711yescharacteristic
230C2×C4D4×D10711yescharacteristic
231C2×C4C23×D10711yescharacteristic
232C8—22no—
233C8—2141no—
234C8—2141no—
235C22—1074no—
236C22—4281no—
237C22—4281no—
238C22—4281no—
239C22—2142no—
240C22—2142no—
241C22C2×D4×D10711yescharacteristic
242C4—21no—
243C4—22no—
244C4—42no—
245C4—41no—
246C4—4281no—
247C4—4282no—
248C4—2142no—
249C4SD16×D10712yes—
250C4C2×#1313381911yescharacteristic
251C4#1681215511yescharacteristic
252C2—8561no—
253C2—2144no—
254C2#2029326811yescharacteristic
255C2C2×SD16×D10711yescharacteristic
256C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#475×C107C2
socleC22×C107C22×D4
prime radical#20293280C2
hypercenter#475D107
hypocenterC107#3298
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. Q8×D428C2semidirect
3. #20292979C2semidirect
4. #20292959C2semidirect
5. #20292857C2semidirect
6. #20292833C2semidirect
7. #20292767C2semidirect
8. C4×#13133805C2semidirect
9. #20292692C2semidirect
10. #103×D107C2semidirect
11. #475×C107C2semidirect
81. Q8×C107#41semidirect
94. D428SD16semidirect
129. C107#3298semidirect
134. #475D107semidirect
213. Q8#13133819semidirect

Maximal subgroups

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

subgroupindex
Q8×D4282
#202932802
#202929792
#202929592
#202928572
#202928332
#202927672
C4×#131338052
#202926922
#103×D1072
#475×C1072
#3298107