Finite Group Databasebeta

← #28348203

Subgroups — #28348203

Order 13696.

482 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16812132C211yesmaximal, characteristic, complemented
2C2×Q16×D107C211yesmaximal, characteristic, complemented
3#20293289C212yesmaximal, complemented
4#20293011C211yesmaximal, characteristic, complemented
5#20292869C211yesmaximal, characteristic, complemented
6#20292864C211yesmaximal, characteristic, complemented
7#20292849C212yesmaximal, complemented
8#20292792C212yesmaximal, complemented
9#20292767C212yesmaximal, complemented
10C8×D428C211yesmaximal, characteristic, complemented
11C4×Q16×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×Q8×D107C2212yes—
14#16812132—24no—
15C2×Dic428C2211yescharacteristic
16C2×#13133808C2212yes—
17Q16×D107—24no—
18C2×#13133807C2212yes—
19C2×#13133805C2212yes—
20C2×D856C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic
23#16812097—22no—
24#16812097C2212yes—
25#16812089—22no—
26#16812089C2212yes—
27#16812084—22no—
28#16812084C2212yes—
29C4×D428—22no—
30C4×D428C2211yescharacteristic
31C4⋊(C107⋊C8)C2211yescharacteristic
32#16812032C2212yes—
33#16812029C2212yes—
34(C2×D107)⋊C8C2211yescharacteristic
35(C2×D107)⋊C8C2211yescharacteristic
36C2×Q16×C107C2211yescharacteristic
37C4×Q8×C107C2212yes—
38#107×C107C2211yescharacteristic
39#103×C107C2212yes—
40C4×C8×C107C2211yescharacteristic
41#13133819—24no—
42#13133819—42no—
43C2×Dic214—22no—
44C2×Dic214C2312yes—
45Q8×D107—24no—
46Q8×D107—42no—
47C2×D428—22no—
48C2×D428—22no—
49C2×D428D411yescharacteristic
50C2×D428C2312yes—
51C2×C4×D107—22no—
52C2×C4×D107—22no—
53C2×C4×D107C2311yescharacteristic
54C2×C4×D107C2311yescharacteristic
55Dic428—22no—
56#13133808—24no—
57#13133807—42no—
58#13133805—24no—
59D856—22no—
60C2×C107⋊C8C2311yescharacteristic
61C2×C107⋊C8C2311yescharacteristic
62C8×D107—22no—
63C8×D107—22no—
64C4⋊Dic107—22no—
65C4⋊Dic107D411yescharacteristic
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4D411yescharacteristic
71(C2×D107)⋊C4D411yescharacteristic
72C2×Q8×C107C2312yes—
73Q16×C107D414yescomplemented
74C2×C8×C107C2311yescharacteristic
75C2×C8×C107C2311yescharacteristic
76C4⋊C4×C107—22no—
77C4⋊C4×C107C2312yes—
78C42×C107—22no—
79C42×C107C2311yescharacteristic
80C22×D107—22no—
81C22×D107C2×D411yescharacteristic
82C22×D107C2×D411yescharacteristic
83Q8×C107—22no—
84Q8×C107C2×D414yes—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2411yescharacteristic
90C2×Dic107—22no—
91C2×Dic107C2×D411yescharacteristic
92C2×Dic107C2×D411yescharacteristic
93D428—22no—
94D428—22no—
95D428—22no—
96D428—44no—
97D428—44no—
98C4×D107—21no—
99C4×D107—21no—
100C4×D107—21no—
101C4×D107—21no—
102C4×D107—44no—
103C4×D107—44no—
104Dic214—22no—
105Dic214—22no—
106Dic214—44no—
107C8×C107—21no—
108C8×C107C2×D412yes—
109C107⋊C8—21no—
110C107⋊C8—21no—
111C22×C107C22×D411yescharacteristic
112C2×D107—21no—
113C2×D107—21no—
114C2×D107—21no—
115C2×D107—21no—
116C2×D107—21no—
117C2×D107—42no—
118C2×D107—42no—
119C2×D107—42no—
120C2×D107#13512yescomplemented
121C4×C107—24no—
122C4×C107—21no—
123C4×C107—21no—
124C4×C107—42no—
125C4×C107C22×D411yescharacteristic
126C4×C107H(5,𝔽2)11yescharacteristic
127Dic107—21no—
128Dic107—21no—
129Dic107—42no—
130C2×C107D4211yescharacteristic
131C2×C107C2×#13511yescharacteristic
132C2×C107#57811yescharacteristic
133D107—22no—
134D107—41no—
135D107—82no—
136#3285—1071nomaximal, Sylow
137C107#328511yescharacteristic, complemented, Sylow
138C2×#135—1071no—
139C22×Q16—1071no—
140#548—1072no—
141#495—1071no—
142#467—1071no—
143#466—1071no—
144#464—1072no—
145#452—1072no—
146#448—1072no—
147C4×Q16D10711yescharacteristic, complemented
148C8×D4—1071no—
149C2×#41—1072no—
150C22×Q8—1072no—
151#135—2144no—
152C2×Q16—1072no—
153C2×Q16—1071no—
154C2×Q16—2144no—
155C2×Q16C2×D10711yescharacteristic
156C2×SD16—1072no—
157C2×SD16—1072no—
158C2×D8—1071no—
159C22×C8—1071no—
160C22×C8—1071no—
161#124—1072no—
162#124—2142no—
163C22⋊Q8—1072no—
164C22⋊Q8—2142no—
165#121—1072no—
166#121—2142no—
167C4×Q8C2×D10712yes—
168C4×D4—1071no—
169C4×D4—2142no—
170#107C2×D10711yescharacteristic
171C4⋊C8—1071no—
172#103—1072no—
173#103C2×D10712yes—
174#102—1072no—
175C22⋊C8—1071no—
176C22⋊C8—1071no—
177C4×C8C2×D10711yescharacteristic
178#41—4282no—
179#41—2144no—
180C2×Q8—1072no—
181C2×Q8—4282no—
182C2×Q8—2142no—
183C2×Q8—2144no—
184C2×Q8C22×D10712yes—
185C2×D4—1071no—
186C2×D4—1072no—
187C2×D4—2142no—
188C2×D4—2142no—
189C22×C4—1071no—
190C22×C4—1071no—
191C22×C4—2142no—
192C22×C4—2142no—
193Q16—2144no—
194Q16—2142no—
195Q16D42814yescomplemented
196SD16—4282no—
197SD16—2144no—
198D8—2142no—
199C2×C8—1071no—
200C2×C8—1071no—
201C2×C8—2142no—
202C2×C8—2142no—
203C2×C8C22×D10711yescharacteristic
204C2×C8C22×D10711yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—1071no—
207C4⋊C4—2142no—
208C4⋊C4C22×D10712yes—
209C22⋊C4—1071no—
210C22⋊C4—1071no—
211C22⋊C4—2142no—
212C22⋊C4—2142no—
213C22⋊C4—2142no—
214C22⋊C4—2142no—
215C42—22no—
216C42C22×D10711yescharacteristic
217C23—1071no—
218C23—1071no—
219C23—2142no—
220Q8—22no—
221Q8—4284no—
222Q8—2142no—
223Q8—2142no—
224Q8C2×D42814yes—
225D4—4284no—
226D4—4284no—
227D4—2142no—
228D4—2142no—
229D4—2142no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—1071no—
233C2×C4—1071no—
234C2×C4—4284no—
235C2×C4—4284no—
236C2×C4—2142no—
237C2×C4—2141no—
238C2×C4—2141no—
239C2×C4—2141no—
240C2×C4—2141no—
241C2×C4D4×D10711yescharacteristic
242C2×C4D4×D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—21no—
245C8—2141no—
246C8—2141no—
247C8C2×D42812yes—
248C22—1072no—
249C22—4282no—
250C22—4282no—
251C22—4282no—
252C22—2141no—
253C22—2141no—
254C22—2141no—
255C22—2141no—
256C22—2141no—
257C22C2×D4×D10711yescharacteristic
258C4—21no—
259C4—21no—
260C4—24no—
261C4—42no—
262C4—4282no—
263C4—2141no—
264C4—2141no—
265C4C22×D42811yescharacteristic
266C4#1681215511yescharacteristic
267C2—8562no—
268C2—4281no—
269C2—2142no—
270C2D4×D42811yescharacteristic
271C2#135×D10711yescharacteristic
272C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×Q16×C107C2
socleC22×C107C22×D4
hypercenterC4×Q16D107
hypocenterC107#3285
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#16812132C2semidirect
2. C2×Q16×D107C2semidirect
3. #20293289C2semidirect
4. #20293011C2semidirect
5. #20292869C2semidirect
6. #20292864C2semidirect
7. #20292849C2semidirect
8. #20292792C2semidirect
9. #20292767C2semidirect
10. C8×D428C2semidirect
11. C4×Q16×C107C2semidirect
73. Q16×C107D4semidirect
120. C2×D107#135semidirect
137. C107#3285semidirect
147. C4×Q16D107semidirect
195. Q16D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121322
C2×Q16×D1072
#202932892
#202930112
#202928692
#202928642
#202928492
#202927922
#202927672
C8×D4282
C4×Q16×C1072
#3285107