Finite Group Databasebeta

← #28346420

Subgroups — #28346420

Order 13696.

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

normal subgroups only

This group has 418 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#20293191C211yesmaximal, characteristic
2#20293110C211yesmaximal, characteristic, complemented
3#20292952C212yesmaximal, complemented
4#20292952C212yesmaximal, complemented
5#20292946C212yesmaximal, complemented
6#20292941C212yesmaximal, complemented
7#20292669C211yesmaximal, characteristic, complemented
8#20292665C211yesmaximal, characteristic, complemented
9#20292633C211yesmaximal, characteristic, complemented
10C2×#16812031C211yesmaximal, characteristic, complemented
11#408×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13#16812105—21no—
14#16812098—21no—
15#16812083C2212yes—
16#16812083C2212yes—
17#16812077—22no—
18Q8×Dic107—21no—
19D4×Dic107—22no—
20D4×Dic107—21no—
21#16812068C2211yescharacteristic
22#16812068C2211yescharacteristic
23#16812068C2212yescomplemented
24C2×C4⋊Dic107C2211yescharacteristic
25C2×C4⋊Dic107C2212yescomplemented
26#16812043C2211yescharacteristic, complemented
27#16812043C2211yescharacteristic, complemented
28#16812043C2212yescomplemented
29#16812041C2211yescharacteristic, complemented
30#16812041C2211yescharacteristic, complemented
31#16812041C2212yescomplemented
32#16812031C2212yes—
33#16812031C2212yes—
34#16812031C2212yes—
35#16812031C2212yes—
36C2×#41×C107C2211yescharacteristic
37C2×M16×C107C2211yescharacteristic
38C22×C8×C107C2211yescharacteristic
39C22⋊C8×C107C2212yescomplemented
40C22⋊C8×C107C2212yescomplemented
41C23×D107—21no—
42C2×D428—22no—
43C2×D428—22no—
44C2×D428—21no—
45C2×D428C2312yes—
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48C22×Dic107—21no—
49C22×Dic107—22no—
50C4⋊Dic107—21no—
51C4⋊Dic107—22no—
52C4⋊Dic107C2311yescharacteristic
53C4⋊Dic107C2311yescharacteristic
54C4⋊Dic107C2312yes—
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2311yescharacteristic
57C22⋊Dic107—21no—
58C22⋊Dic107—21no—
59C22⋊Dic107—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62C4×Dic107—21no—
63C4×Dic107—21no—
64C4×Dic107—22no—
65#41×C107—22no—
66#41×C107—22no—
67C2×Q8×C107D411yescharacteristic
68C2×D4×C107D412yes—
69C2×D4×C107D411yescharacteristic
70C22×C4×C107C2312yes—
71C22×C4×C107C2311yescharacteristic
72M16×C107—22no—
73C2×C8×C107—22no—
74C2×C8×C107C2311yescharacteristic
75C2×C8×C107C2312yes—
76C2×C8×C107C2311yescharacteristic
77C23×C107C2×D412yes—
78C23×C107C2×D411yescharacteristic
79C22×D107—22no—
80C22×D107—42no—
81C22×D107—81no—
82C22×D107—81no—
83Q8×C107—22no—
84D4×C107—24no—
85D4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107#4112yes—
91C2×C4×C107#4112yes—
92C2×C4×C107C2411yescharacteristic
93C22⋊D107—82no—
94C22⋊D107—82no—
95C2×Dic107—21no—
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107—21no—
99C2×Dic107—22no—
100C2×Dic107—81no—
101C2×Dic107—82no—
102D428—22no—
103D428—22no—
104D428—42no—
105C8×C107—21no—
106C8×C107—21no—
107C8×C107—22no—
108C22×C107—21no—
109C22×C107—21no—
110C22×C107—22no—
111C22×C107—22no—
112C22×C107—22no—
113C22×C107#12312yes—
114C22×C107C22×D411yescharacteristic
115C2×D107—42no—
116C2×D107—42no—
117C2×D107—42no—
118C2×D107—84no—
119C4×C107—22no—
120C4×C107#12312yes—
121C4×C107C2×#4111yescharacteristic
122C4×C107C2×#4111yescharacteristic
123Dic107—41no—
124Dic107—42no—
125Dic107—41no—
126Dic107—41no—
127Dic107—41no—
128C2×C107—22no—
129C2×C107—42no—
130C2×C107C2×#12311yescharacteristic
131C2×C107#57611yescharacteristic
132C2×C107#57711yescharacteristic
133D107—82no—
134#3096—1071nomaximal, Sylow
135C107#309611yescharacteristic, complemented, Sylow
136#535—1071no—
137#520—1071no—
138#482—1072no—
139#482—1072no—
140#481—1072no—
141#480—1072no—
142#428—1071no—
143#427—1071no—
144#418—1071no—
145C2×#102—1071no—
146#408D10711yescharacteristic, complemented
147C2×#41C2×D10711yescharacteristic
148C22×D4—1071no—
149C2×M16C2×D10711yescharacteristic
150C22×C8C2×D10711yescharacteristic
151#127—2141no—
152#124—2141no—
153#121—1072no—
154#121—1072no—
155#120—2142no—
156C4×Q8—2141no—
157C4×D4—2142no—
158C4×D4—2141no—
159#117—1071no—
160#117—1072no—
161#117—1071no—
162C2×C4⋊C4—1071no—
163C2×C4⋊C4—1072no—
164#107—1071no—
165#107—1071no—
166#107—1072no—
167#106—1071no—
168#106—1071no—
169#106—1072no—
170#102—1072no—
171#102—1072no—
172#102—1072no—
173#102—1072no—
174C22⋊C8C2×D10712yescomplemented
175C22⋊C8C2×D10712yescomplemented
176C24—2141no—
177#41—22no—
178#41—22no—
179C2×Q8D42811yescharacteristic
180C2×D4—1072no—
181C2×D4—2142no—
182C2×D4—2142no—
183C2×D4—2141no—
184C2×D4—2142no—
185C2×D4—2142no—
186C2×D4D42812yes—
187C2×D4D42811yescharacteristic
188C22×C4—2142no—
189C22×C4—2141no—
190C22×C4C22×D10712yes—
191C22×C4C22×D10711yescharacteristic
192M16—22no—
193C2×C8—22no—
194C2×C8C22×D10711yescharacteristic
195C2×C8C22×D10712yes—
196C2×C8C22×D10711yescharacteristic
197C4⋊C4—1072no—
198C4⋊C4—1071no—
199C4⋊C4—1071no—
200C4⋊C4—1071no—
201C4⋊C4—1071no—
202C4⋊C4—2142no—
203C4⋊C4—2141no—
204C22⋊C4—2142no—
205C22⋊C4—2141no—
206C22⋊C4—2141no—
207C22⋊C4—2142no—
208C22⋊C4—2142no—
209C42—2142no—
210C42—2141no—
211C42—2141no—
212C23—8561no—
213C23—8561no—
214C23—4282no—
215C23—2142no—
216C23C2×D42812yes—
217C23C2×D42811yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—8562no—
222D4—8562no—
223D4—4282no—
224D4—2142no—
225D4—2142no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—8561no—
229C2×C4—8562no—
230C2×C4—2141no—
231C2×C4—2141no—
232C2×C4—2141no—
233C2×C4—2142no—
234C2×C4—2141no—
235C2×C4C2×D42812yes—
236C2×C4C2×D42811yescharacteristic
237C2×C4#1313381712yes—
238C2×C4#1313381712yes—
239C2×C4C23×D10711yescharacteristic
240C8—21no—
241C8—21no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—8564no—
249C22—4282no—
250C22—4282no—
251C22—4282no—
252C22#1681209312yes—
253C22C22×D42811yescharacteristic
254C4—22no—
255C4—4282no—
256C4—4281no—
257C4—4281no—
258C4—4281no—
259C4—4281no—
260C4#1681209312yes—
261C4C2×#1313381711yescharacteristic
262C4C2×#1313381711yescharacteristic
263C2—22no—
264C2—42no—
265C2—8562no—
266C2C2×#1681209311yescharacteristic
267C2#2029343211yescharacteristic
268C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#408×C107C2
socleC22×C107C22×D4
prime radical#20293191C2
hypercenter#408D107
hypocenterC107#3096
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #20293110C2semidirect
3. #20292952C2semidirect
4. #20292952C2semidirect
5. #20292946C2semidirect
6. #20292941C2semidirect
7. #20292669C2semidirect
8. #20292665C2semidirect
9. #20292633C2semidirect
10. C2×#16812031C2semidirect
11. #408×C107C2semidirect
23. #16812068C22semidirect
25. C2×C4⋊Dic107C22semidirect
26. #16812043C22semidirect
27. #16812043C22semidirect
28. #16812043C22semidirect
29. #16812041C22semidirect
30. #16812041C22semidirect
31. #16812041C22semidirect
39. C22⋊C8×C107C22semidirect
40. C22⋊C8×C107C22semidirect
135. C107#3096semidirect
146. #408D107semidirect
174. C22⋊C8C2×D107semidirect
175. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931912
#202931102
#202929522
#202929462
#202929412
#202926692
#202926652
#202926332
C2×#168120312
#408×C1072
#3096107