Finite Group Databasebeta

← #28348327

Subgroups — #28348327

Order 13696.

420 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×Dic214C211yesmaximal, characteristic, complemented
2#20293281C211yesmaximal, characteristic
3#20292978C211yesmaximal, characteristic, complemented
4#20292958C212yesmaximal, complemented
5#20292913C211yesmaximal, characteristic, complemented
6#20292856C212yesmaximal, complemented
7#20292833C211yesmaximal, characteristic, complemented
8#20292768C212yesmaximal, complemented
9C4×#13133807C211yesmaximal, characteristic, complemented
10C2×#16812034C212yesmaximal, complemented
11C8×D4×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—21no—
13C22×Dic214C2212yes—
14C2×#13133807C2211yescharacteristic
15C2×#13133807C2212yes—
16#16812109C2211yescharacteristic
17#16812109C2211yescharacteristic
18#16812108—21no—
19#16812093—22no—
20#16812089—21no—
21#16812088C2212yes—
22#16812086—22no—
23#16812083C2212yes—
24C4×Dic214C2211yescharacteristic
25C4×D428C2211yescharacteristic
26D4×Dic107—21no—
27C2×C4⋊Dic107C2212yes—
28#16812041C2212yes—
29#16812041C2211yescharacteristic
30#16812034C2211yescharacteristic
31#16812034C2212yes—
32#16812034C2212yes—
33#16812034C2212yes—
34#16812034C2212yes—
35#16812031C2211yescharacteristic
36C22×C8×C107C2212yes—
37C4×D4×C107C2211yescharacteristic
38C4⋊C8×C107C2211yescharacteristic
39C22⋊C8×C107C2212yes—
40C4×C8×C107C2211yescharacteristic
41#13133817—44no—
42C2×Dic214—24no—
43C2×Dic214—22no—
44C2×Dic214—21no—
45C2×Dic214—42no—
46C2×Dic214C2312yes—
47C2×Dic214C2311yescharacteristic
48C2×D428C2311yescharacteristic
49C2×C22⋊D107—22no—
50C22×Dic107—22no—
51C22×Dic107—22no—
52C2×C4×D107—21no—
53#13133807—22no—
54#13133807—42no—
55C4⋊Dic107—22no—
56C4⋊Dic107—21no—
57C4⋊Dic107—21no—
58C4⋊Dic107—21no—
59C4⋊Dic107C2311yescharacteristic
60C4⋊Dic107C2312yes—
61C4⋊Dic107C2311yescharacteristic
62Dic107⋊C4—21no—
63Dic107⋊C4—22no—
64C22⋊Dic107—22no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—22no—
67C4×Dic107—21no—
68C2×D4×C107D411yescharacteristic
69C22×C4×C107C2312yes—
70C2×C8×C107—42no—
71C2×C8×C107C2311yescharacteristic
72C2×C8×C107C2311yescharacteristic
73C2×C8×C107C2312yes—
74C4⋊C4×C107D411yescharacteristic
75C22⋊C4×C107D412yes—
76C42×C107C2311yescharacteristic
77C23×C107C2×D412yes—
78C22×D107—21no—
79D4×C107SD1614yescomplemented
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D412yes—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2411yescharacteristic
86C22⋊D107—44no—
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—44no—
93C2×Dic107—44no—
94D428—21no—
95D428—21no—
96C4×D107—42no—
97Dic214—21no—
98Dic214—22no—
99Dic214—22no—
100Dic214—44no—
101Dic214—42no—
102Dic214#4112yescomplemented
103C8×C107—21no—
104C8×C107—21no—
105C8×C107—22no—
106C22×C107—22no—
107C22×C107—22no—
108C22×C107C2×SD1614yes—
109C22×C107C22×D411yescharacteristic
110C2×D107—41no—
111C2×D107—41no—
112C2×D107—41no—
113C4×C107—22no—
114C4×C107—21no—
115C4×C107C2×SD1612yes—
116C4×C107C2×#4111yescharacteristic
117C4×C107H(5,𝔽2)11yescharacteristic
118Dic107—22no—
119Dic107—42no—
120Dic107—42no—
121Dic107—41no—
122Dic107—41no—
123C2×C107—24no—
124C2×C107#54611yescharacteristic
125C2×C107C22×SD1611yescharacteristic
126C2×C107#57811yescharacteristic
127D107—81no—
128#3298—1071nomaximal, Sylow
129C107#329811yescharacteristic, complemented, Sylow
130D4×Q8—1071no—
131#547—1071no—
132#487—1071no—
133#483—1072no—
134#475—1071no—
135#465—1072no—
136#461—1071no—
137#448—1072no—
138C4×SD16—1071no—
139C8×D4D10711yescharacteristic, complemented
140C2×#103—1072no—
141C2×#41—2141no—
142C22×Q8—1072no—
143C2×SD16—1072no—
144C2×SD16—1071no—
145C22×C8C2×D10712yes—
146C4⋊Q8—1071no—
147C4⋊Q8—1071no—
148C4⋊Q8—2141no—
149#123—2142no—
150C22⋊Q8—1072no—
151C22⋊Q8—2141no—
152C22⋊Q8—2142no—
153#121—1072no—
154C4×Q8—1071no—
155C4×D4—1071no—
156C4×D4—2141no—
157C4×D4C2×D10711yescharacteristic
158C2×C4⋊C4—1072no—
159#106—1071no—
160#106—1072no—
161C4⋊C8C2×D10711yescharacteristic
162#103—1071no—
163#103—1072no—
164#103—1072no—
165#103—1072no—
166#103—1072no—
167#102—1071no—
168C22⋊C8C2×D10712yes—
169C4×C8C2×D10711yescharacteristic
170#41—4284no—
171C2×Q8—1071no—
172C2×Q8—1072no—
173C2×Q8—4282no—
174C2×Q8—2144no—
175C2×Q8—2141no—
176C2×Q8—2142no—
177C2×D4—1071no—
178C2×D4—2142no—
179C2×D4D42811yescharacteristic
180C22×C4—2142no—
181C22×C4—2142no—
182C22×C4—2141no—
183C22×C4C22×D10712yes—
184SD16—4282no—
185SD16—2142no—
186C2×C8—42no—
187C2×C8C22×D10711yescharacteristic
188C2×C8C22×D10711yescharacteristic
189C2×C8C22×D10712yes—
190C4⋊C4—1071no—
191C4⋊C4—1071no—
192C4⋊C4—1072no—
193C4⋊C4—2142no—
194C4⋊C4—2141no—
195C4⋊C4—2141no—
196C4⋊C4—2142no—
197C4⋊C4—2141no—
198C4⋊C4—2141no—
199C4⋊C4D42811yescharacteristic
200C22⋊C4—2142no—
201C22⋊C4—2141no—
202C22⋊C4—2142no—
203C22⋊C4D42812yes—
204C42—2141no—
205C42C22×D10711yescharacteristic
206C23—2141no—
207C23C2×D42812yes—
208Q8—1072no—
209Q8—4284no—
210Q8—4282no—
211Q8—2142no—
212Q8—2142no—
213Q8—2141no—
214D4—4284no—
215D4—2141no—
216D4—2141no—
217D4#1313380714yescomplemented
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—4284no—
221C2×C4—4284no—
222C2×C4—4282no—
223C2×C4—2141no—
224C2×C4—2141no—
225C2×C4—2142no—
226C2×C4—2142no—
227C2×C4—2141no—
228C2×C4C2×D42811yescharacteristic
229C2×C4C2×D42812yes—
230C2×C4C2×D42811yescharacteristic
231C2×C4C23×D10711yescharacteristic
232C8—21no—
233C8—21no—
234C8—22no—
235C22—22no—
236C22—22no—
237C22—4281no—
238C22—4281no—
239C22—4281no—
240C22C2×#1313380714yes—
241C22C22×D42811yescharacteristic
242C4—21no—
243C4—22no—
244C4—4281no—
245C4—4282no—
246C4—4281no—
247C4—4282no—
248C4—2142no—
249C4C2×#1313380712yes—
250C4#41×D10711yescharacteristic
251C4#1681215411yescharacteristic
252C2—24no—
253C2—8561no—
254C2#2029327211yescharacteristic
255C2C22×#1313380711yescharacteristic
256C2#2029344411yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
prime radical#20293281C2
hypercenterC8×D4D107
hypocenterC107#3298
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. D4×Dic214C2semidirect
3. #20292978C2semidirect
4. #20292958C2semidirect
5. #20292913C2semidirect
6. #20292856C2semidirect
7. #20292833C2semidirect
8. #20292768C2semidirect
9. C4×#13133807C2semidirect
10. C2×#16812034C2semidirect
11. C8×D4×C107C2semidirect
79. D4×C107SD16semidirect
102. Dic214#41semidirect
129. C107#3298semidirect
139. C8×D4D107semidirect
217. D4#13133807semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2142
#202932812
#202929782
#202929582
#202929132
#202928562
#202928332
#202927682
C4×#131338072
C2×#168120342
C8×D4×C1072
#3298107