Finite Group Databasebeta

← #28346417

Subgroups — #28346417

Order 13696.

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

normal subgroups only

This group has 418 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
1#20293194C211yesmaximal, characteristic
2#20293107C211yesmaximal, characteristic, complemented
3#20292950C212yesmaximal, complemented
4#20292949C212yesmaximal, complemented
5#20292944C212yesmaximal, complemented
6#20292939C212yesmaximal, complemented
7#20292669C211yesmaximal, characteristic, complemented
8#20292663C211yesmaximal, characteristic, complemented
9C2×#16812030C211yesmaximal, characteristic, complemented
10#20292595C211yesmaximal, characteristic, complemented
11#418×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×C107⋊C8C2211yescharacteristic
14#16812105—21no—
15#16812096—21no—
16#16812080C2212yes—
17#16812080C2212yes—
18#16812078—22no—
19C4×Dic214—21no—
20C4×D428—21no—
21(C22⋊D107)⋊C4—22no—
22#16812068C2211yescharacteristic
23#16812067C2212yescomplemented
24C2×C4⋊Dic107C2211yescharacteristic
25C4⋊C4×D107C2212yescomplemented
26#16812043C2212yescomplemented
27#16812042C2211yescharacteristic, complemented
28#16812042C2211yescharacteristic, complemented
29#16812041C2212yescomplemented
30#16812040C2211yescharacteristic, complemented
31#16812040C2211yescharacteristic, complemented
32#16812030C2212yes—
33#16812030C2212yes—
34(C2×D107)⋊C8C2212yescomplemented
35(C2×D107)⋊C8C2212yescomplemented
36C22×D4×C107C2211yescharacteristic
37C2×M16×C107C2211yescharacteristic
38#117×C107C2211yescharacteristic
39#102×C107C2212yes—
40#102×C107C2212yes—
41#13133818—22no—
42#13133818—22no—
43C2×Dic214D411yescharacteristic
44C2×D428D411yescharacteristic
45C2×C22⋊D107—22no—
46C2×C22⋊D107—22no—
47C2×C22⋊D107D412yes—
48C22×Dic107—21no—
49C2×C4×D107—22no—
50C2×C4×D107C2312yes—
51C2×C107⋊C8—22no—
52C2×C107⋊C8C2311yescharacteristic
53C2×C107⋊C8C2311yescharacteristic
54C4⋊Dic107—21no—
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2311yescharacteristic
57C4⋊Dic107C2312yes—
58Dic107⋊C4—22no—
59C22⋊Dic107—21no—
60C22⋊Dic107—22no—
61C22⋊Dic107—22no—
62(C2×D107)⋊C4—22no—
63C4×Dic107—21no—
64C4×Dic107—22no—
65C24×C107—21no—
66C2×D4×C107—21no—
67C2×D4×C107—22no—
68C2×D4×C107—22no—
69C2×D4×C107C2312yes—
70C22×C4×C107C2311yescharacteristic
71M16×C107—22no—
72C2×C8×C107C2312yes—
73C4⋊C4×C107C2311yescharacteristic
74C4⋊C4×C107C2311yescharacteristic
75C22⋊C4×C107—21no—
76C42×C107—21no—
77C23×C107—22no—
78C23×C107—42no—
79C23×C107—81no—
80C23×C107—81no—
81C23×C107C2×D411yescharacteristic
82C22×D107C2×D412yes—
83D4×C107—22no—
84D4×C107—22no—
85D4×C107—42no—
86C2×C4×C107—21no—
87C2×C4×C107—21no—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107#4112yes—
90C2×C4×C107#4112yes—
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—24no—
93C22⋊D107—82no—
94C22⋊D107—82no—
95C2×Dic107—22no—
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107—81no—
99C2×Dic107C2×D412yes—
100D428—22no—
101C4×D107—22no—
102C4×D107—22no—
103C4×D107—82no—
104Dic214—22no—
105C8×C107—22no—
106C107⋊C8—21no—
107C107⋊C8—21no—
108C22×C107—21no—
109C22×C107—21no—
110C22×C107—42no—
111C22×C107—42no—
112C22×C107—42no—
113C22×C107—84no—
114C22×C107#12312yes—
115C22×C107C22×D411yescharacteristic
116C2×D107—22no—
117C2×D107—22no—
118C2×D107—22no—
119C4×C107—41no—
120C4×C107—41no—
121C4×C107#12312yes—
122C4×C107C2×#4111yescharacteristic
123C4×C107C2×#4111yescharacteristic
124Dic107—22no—
125Dic107—42no—
126Dic107—41no—
127Dic107—41no—
128C2×C107—22no—
129C2×C107—82no—
130C2×C107C2×#12311yescharacteristic
131C2×C107#57611yescharacteristic
132C2×C107#57711yescharacteristic
133D107—42no—
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#418D10711yescharacteristic, complemented
145C2×#102—1071no—
146#408—1071no—
147C2×#41—1071no—
148C22×D4C2×D10711yescharacteristic
149C2×M16C2×D10711yescharacteristic
150C22×C8—1071no—
151#127—2141no—
152#124—2141no—
153#121—1072no—
154#121—1072no—
155#120—2142no—
156C4×Q8—2141no—
157C4×D4—2141no—
158C4×D4—2142no—
159#117—1071no—
160#117—1072no—
161#117C2×D10711yescharacteristic
162C2×C4⋊C4—1071no—
163C2×C4⋊C4—1072no—
164#107—1072no—
165#107—1071no—
166#107—1071no—
167#106—1072no—
168#106—1071no—
169#106—1071no—
170#102—1072no—
171#102—1072no—
172#102C2×D10712yes—
173#102C2×D10712yes—
174C22⋊C8—1072no—
175C22⋊C8—1072no—
176C24—21no—
177#41—2142no—
178#41—2142no—
179C2×Q8—1071no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—1072no—
184C2×D4—1071no—
185C2×D4—2142no—
186C2×D4—2142no—
187C2×D4C22×D10712yes—
188C22×C4—1072no—
189C22×C4—2142no—
190C22×C4—2141no—
191C22×C4C22×D10711yescharacteristic
192M16—22no—
193C2×C8—1071no—
194C2×C8—1071no—
195C2×C8—2142no—
196C2×C8C22×D10712yes—
197C4⋊C4—1072no—
198C4⋊C4—1071no—
199C4⋊C4—1071no—
200C4⋊C4—2142no—
201C4⋊C4—2141no—
202C4⋊C4C22×D10711yescharacteristic
203C4⋊C4C22×D10711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—2142no—
206C22⋊C4—2142no—
207C22⋊C4—2142no—
208C22⋊C4—2141no—
209C42—21no—
210C42—2142no—
211C42—2141no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—1072no—
217C23D4×D10711yescharacteristic
218Q8—2142no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—8562no—
223D4—8562no—
224D4—2144no—
225D4—2142no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—8562no—
229C2×C4—8561no—
230C2×C4—1072no—
231C2×C4—2141no—
232C2×C4—2141no—
233C2×C4—2142no—
234C2×C4—2142no—
235C2×C4—2142no—
236C2×C4D4×D10711yescharacteristic
237C2×C4#1313381712yes—
238C2×C4#1313381812yes—
239C2×C4C23×D10711yescharacteristic
240C8—22no—
241C8—2141no—
242C8—2141no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—84no—
249C22—2142no—
250C22—2142no—
251C22—2142no—
252C22#1681209112yes—
253C22C2×D4×D10711yescharacteristic
254C4—41no—
255C4—41no—
256C4—4282no—
257C4—4281no—
258C4—4281no—
259C4—2142no—
260C4#1681209112yes—
261C4C2×#1313381711yescharacteristic
262C4C2×#1313381811yescharacteristic
263C2—22no—
264C2—82no—
265C2—4282no—
266C2C2×#1681209111yescharacteristic
267C2#2029343111yescharacteristic
268C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#418×C107C2
socleC22×C107C22×D4
prime radical#20293194C2
hypercenter#418D107
hypocenterC107#3096
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #20293107C2semidirect
3. #20292950C2semidirect
4. #20292949C2semidirect
5. #20292944C2semidirect
6. #20292939C2semidirect
7. #20292669C2semidirect
8. #20292663C2semidirect
9. C2×#16812030C2semidirect
10. #20292595C2semidirect
11. #418×C107C2semidirect
23. #16812067C22semidirect
25. C4⋊C4×D107C22semidirect
26. #16812043C22semidirect
27. #16812042C22semidirect
28. #16812042C22semidirect
29. #16812041C22semidirect
30. #16812040C22semidirect
31. #16812040C22semidirect
34. (C2×D107)⋊C8C22semidirect
35. (C2×D107)⋊C8C22semidirect
135. C107#3096semidirect
144. #418D107semidirect

Maximal subgroups

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

subgroupindex
#202931942
#202931072
#202929502
#202929492
#202929442
#202929392
#202926692
#202926632
C2×#168120302
#202925952
#418×C1072
#3096107