Finite Group Databasebeta

← #28346413

Subgroups — #28346413

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#20293110C211yesmaximal, characteristic, complemented
2#20292948C212yesmaximal, complemented
3#20292948C212yesmaximal, complemented
4#20292943C212yesmaximal, complemented
5#20292938C212yesmaximal, complemented
6#20292670C211yesmaximal, characteristic, complemented
7#20292663C211yesmaximal, characteristic, complemented
8#20292632C211yesmaximal, characteristic, complemented
9C2×#16812030C211yesmaximal, characteristic, complemented
10#20292594C211yesmaximal, characteristic, complemented
11#535×C107C211yesmaximal, characteristic
12C2×#13133802C2211yescharacteristic
13C22×C107⋊C8C2211yescharacteristic
14Q8×Dic107—21no—
15D4×Dic107—22no—
16D4×Dic107—21no—
17#16812068C2211yescharacteristic
18#16812068C2212yescomplemented
19C2×C4⋊Dic107C2211yescharacteristic
20C2×C4⋊Dic107C2212yescomplemented
21#16812042C2211yescharacteristic, complemented
22#16812042C2211yescharacteristic, complemented
23#16812042C2212yescomplemented
24#16812040C2211yescharacteristic, complemented
25#16812040C2211yescharacteristic, complemented
26#16812040C2212yescomplemented
27#16812030C2212yes—
28#16812030C2212yes—
29#16812030C2212yes—
30#16812030C2212yes—
31C22⋊(C107⋊C8)C2212yescomplemented
32C22⋊(C107⋊C8)C2212yescomplemented
33C2×#41×C107C2211yescharacteristic
34C22×D4×C107C2211yescharacteristic
35#127×C107—21no—
36#124×C107—21no—
37#121×C107C2212yes—
38#121×C107C2212yes—
39#120×C107—22no—
40#117×C107C2211yescharacteristic
41C22×Dic107—21no—
42C22×Dic107—22no—
43#13133802—22no—
44C2×C107⋊C8—22no—
45C2×C107⋊C8C2311yescharacteristic
46C2×C107⋊C8C2312yes—
47C2×C107⋊C8C2311yescharacteristic
48C4⋊Dic107—21no—
49C4⋊Dic107—22no—
50C4⋊Dic107C2312yes—
51C4⋊Dic107C2311yescharacteristic
52C4⋊Dic107C2311yescharacteristic
53C22⋊Dic107—21no—
54C22⋊Dic107—22no—
55C4×Dic107—21no—
56C4×Dic107—22no—
57C24×C107—21no—
58#41×C107—22no—
59#41×C107—22no—
60C2×Q8×C107D411yescharacteristic
61C2×D4×C107—22no—
62C2×D4×C107—22no—
63C2×D4×C107—21no—
64C2×D4×C107—22no—
65C2×D4×C107—22no—
66C2×D4×C107D412yes—
67C2×D4×C107D411yescharacteristic
68C2×D4×C107C2312yes—
69C22×C4×C107C2312yes—
70C22×C4×C107C2311yescharacteristic
71C4⋊C4×C107C2311yescharacteristic
72C4⋊C4×C107C2311yescharacteristic
73C22⋊C4×C107—21no—
74C22⋊C4×C107—22no—
75C22⋊C4×C107—22no—
76C42×C107—21no—
77C23×C107—22no—
78C23×C107—42no—
79C23×C107—81no—
80C23×C107—81no—
81C23×C107C2×D412yes—
82C23×C107C2×D411yescharacteristic
83Q8×C107—22no—
84D4×C107—24no—
85D4×C107—22no—
86D4×C107—22no—
87D4×C107—22no—
88D4×C107—42no—
89D4×C107—82no—
90D4×C107—82no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—21no—
94C2×C4×C107—21no—
95C2×C4×C107C2×D412yes—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107#4112yes—
98C2×C4×C107#4112yes—
99C2×C4×C107C2411yescharacteristic
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—22no—
103C2×Dic107—81no—
104C2×Dic107—82no—
105C107⋊C8—21no—
106C107⋊C8—21no—
107C107⋊C8—22no—
108C22×C107—21no—
109C22×C107—21no—
110C22×C107—22no—
111C22×C107—22no—
112C22×C107—22no—
113C22×C107—42no—
114C22×C107—42no—
115C22×C107—42no—
116C22×C107—84no—
117C22×C107#12312yes—
118C22×C107C22×D411yescharacteristic
119C4×C107—22no—
120C4×C107—41no—
121C4×C107—41no—
122C4×C107#12312yes—
123C4×C107C2×#4111yescharacteristic
124C4×C107C2×#4111yescharacteristic
125Dic107—41no—
126Dic107—42no—
127Dic107—41no—
128C2×C107—22no—
129C2×C107—42no—
130C2×C107—82no—
131C2×C107C2×#12311yescharacteristic
132C2×C107#57611yescharacteristic
133C2×C107#57711yescharacteristic
134#3096—1071nomaximal, Sylow
135C107#309611yescharacteristic, complemented, Sylow
136#535D10711yescharacteristic
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#408—1071no—
147C2×#41C2×D10711yescharacteristic
148C22×D4C2×D10711yescharacteristic
149C2×M16—1071no—
150C22×C8—1071no—
151#127—21no—
152#124—21no—
153#121C2×D10712yes—
154#121C2×D10712yes—
155#120—22no—
156C4×Q8—2141no—
157C4×D4—2142no—
158C4×D4—2141no—
159#117—1071no—
160#117—1072no—
161#117C2×D10711yescharacteristic
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⋊C8—1072no—
175C22⋊C8—1072no—
176C24—21no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D10711yescharacteristic
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4C22⋊D10712yes—
186C2×D4C22⋊D10711yescharacteristic
187C2×D4C22×D10712yes—
188C22×C4—2142no—
189C22×C4—2141no—
190C22×C4C22×D10712yes—
191C22×C4C22×D10711yescharacteristic
192M16—2142no—
193C2×C8—1071no—
194C2×C8—1072no—
195C2×C8—1071no—
196C2×C8—2142no—
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—22no—
206C22⋊C4—22no—
207C22⋊C4—2142no—
208C22⋊C4—2141no—
209C42—21no—
210C42—2142no—
211C42—2141no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23C2×C22⋊D10712yes—
217C23C2×C22⋊D10711yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4—22no—
223D4—42no—
224D4—82no—
225D4—82no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—8561no—
231C2×C4—8562no—
232C2×C4—2141no—
233C2×C4—2142no—
234C2×C4—2141no—
235C2×C4C2×C22⋊D10712yes—
236C2×C4C2×C22⋊D10711yescharacteristic
237C2×C4#1313381712yes—
238C2×C4#1313381712yes—
239C2×C4C23×D10711yescharacteristic
240C8—2141no—
241C8—2141no—
242C8—2142no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—84no—
252C22#1681209012yes—
253C22C22×C22⋊D10711yescharacteristic
254C4—22no—
255C4—41no—
256C4—41no—
257C4—4282no—
258C4—4281no—
259C4—4281no—
260C4#1681209012yes—
261C4C2×#1313381711yescharacteristic
262C4C2×#1313381711yescharacteristic
263C2—22no—
264C2—42no—
265C2—82no—
266C2C2×#1681209011yescharacteristic
267C2#2029343011yescharacteristic
268C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radical#535×C107C2
socleC22×C107C22×D4
hypercenter#535D107
hypocenterC107#3096
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #20293110C2semidirect
2. #20292948C2semidirect
3. #20292948C2semidirect
4. #20292943C2semidirect
5. #20292938C2semidirect
6. #20292670C2semidirect
7. #20292663C2semidirect
8. #20292632C2semidirect
9. C2×#16812030C2semidirect
10. #20292594C2semidirect
18. #16812068C22semidirect
20. C2×C4⋊Dic107C22semidirect
21. #16812042C22semidirect
22. #16812042C22semidirect
23. #16812042C22semidirect
24. #16812040C22semidirect
25. #16812040C22semidirect
26. #16812040C22semidirect
31. C22⋊(C107⋊C8)C22semidirect
32. C22⋊(C107⋊C8)C22semidirect
135. C107#3096semidirect

Maximal subgroups

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

subgroupindex
#202931102
#202929482
#202929432
#202929382
#202926702
#202926632
#202926322
C2×#168120302
#202925942
#535×C1072
#3096107