Finite Group Databasebeta

← #28346288

Subgroups — #28346288

Order 13696.

398 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 141 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#20293258C211yesmaximal, characteristic, complemented
2#20293102C211yesmaximal, characteristic
3#20292929C212yesmaximal, complemented
4#20292925C212yesmaximal, complemented
5#20292915C212yesmaximal, complemented
6#20292905C212yesmaximal, complemented
7#20292667C211yesmaximal, characteristic, complemented
8#20292665C211yesmaximal, characteristic, complemented
9#20292653C211yesmaximal, characteristic, complemented
10#20292632C211yesmaximal, characteristic
11C2×#102×C107C211yesmaximal, characteristic
12C2×#13133802C2211yescharacteristic
13#16812108—21no—
14#16812108C2212yescomplemented
15#16812100—21no—
16#16812100C2212yescomplemented
17#16812088—22no—
18D4×Dic107—21no—
19D4×Dic107—21no—
20D4×Dic107C2212yes—
21D4×Dic107C2212yes—
22#16812068C2211yescharacteristic
23#16812068C2211yescharacteristic
24#16812068C2211yescharacteristic
25C2×C22⋊Dic107—22no—
26#16812043C2211yescharacteristic, complemented
27#16812043C2211yescharacteristic, complemented
28#16812042C2212yescomplemented
29#16812041C2211yescharacteristic, complemented
30#16812041C2211yescharacteristic, complemented
31#16812040C2212yescomplemented
32Dic107⋊C8C2212yescomplemented
33Dic107⋊C8C2212yescomplemented
34#16812030C2212yes—
35#16812030C2212yes—
36C22×D4×C107C2211yescharacteristic
37C22×C8×C107C2211yescharacteristic
38C2×C4⋊C4×C107C2211yescharacteristic
39#102×C107C2212yes—
40#102×C107C2212yes—
41C2×Dic214—22no—
42C22×Dic107—22no—
43C22×Dic107—22no—
44#13133802—22no—
45C2×C107⋊C8C2312yes—
46C4⋊Dic107—22no—
47C4⋊Dic107D411yescharacteristic
48C4⋊Dic107D411yescharacteristic
49C4⋊Dic107C2311yescharacteristic
50C4⋊Dic107C2311yescharacteristic
51C4⋊Dic107C2311yescharacteristic
52C4⋊Dic107C2311yescharacteristic
53Dic107⋊C4—22no—
54Dic107⋊C4—22no—
55C22⋊Dic107—21no—
56C22⋊Dic107—21no—
57C22⋊Dic107—22no—
58C22⋊Dic107—22no—
59C22⋊Dic107D412yes—
60C4×Dic107—21no—
61C4×Dic107—21no—
62C4×Dic107C2312yes—
63C24×C107—21no—
64C2×D4×C107—21no—
65C2×D4×C107—22no—
66C2×D4×C107Q814yescomplemented
67C2×D4×C107C2312yes—
68C22×C4×C107—21no—
69C22×C4×C107C2311yescharacteristic
70C2×C8×C107—22no—
71C2×C8×C107C2311yescharacteristic
72C2×C8×C107C2311yescharacteristic
73C4⋊C4×C107—21no—
74C4⋊C4×C107C2312yes—
75C23×C107—22no—
76C23×C107—24no—
77C23×C107—81no—
78C23×C107—81no—
79C23×C107C2×D411yescharacteristic
80D4×C107—24no—
81D4×C107—22no—
82D4×C107C2×Q814yes—
83C2×C4×C107—22no—
84C2×C4×C107—81no—
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2×Q812yes—
87C2×C4×C107#4112yes—
88C2×C4×C107C2411yescharacteristic
89C2×Dic107—21no—
90C2×Dic107—21no—
91C2×Dic107—21no—
92C2×Dic107—21no—
93C2×Dic107—82no—
94C2×Dic107—82no—
95C2×Dic107C2×D412yes—
96C2×Dic107C2×D412yes—
97Dic214—82no—
98C8×C107—21no—
99C8×C107—21no—
100C107⋊C8—22no—
101C22×C107—21no—
102C22×C107—21no—
103C22×C107—24no—
104C22×C107—42no—
105C22×C107—42no—
106C22×C107—84no—
107C22×C107C22⋊Q812yes—
108C22×C107C22×D411yescharacteristic
109C4×C107—42no—
110C4×C107C22⋊Q812yes—
111C4×C107C22×Q811yescharacteristic
112C4×C107C2×#4111yescharacteristic
113Dic107—22no—
114Dic107—22no—
115Dic107—41no—
116Dic107—41no—
117Dic107—41no—
118Dic107—41no—
119C2×C107—22no—
120C2×C107—44no—
121C2×C107C2×C22⋊Q811yescharacteristic
122C2×C107#57611yescharacteristic
123C2×C107#57711yescharacteristic
124#3081—1071nomaximal, Sylow
125C107#308111yescharacteristic, complemented, Sylow
126#544—1071no—
127#518—1071no—
128#478—1072no—
129#478—1072no—
130#476—1072no—
131#474—1072no—
132#428—1071no—
133#427—1071no—
134#424—1071no—
135#418—1071no—
136C2×#102D10711yescharacteristic
137C22×D4C2×D10711yescharacteristic
138C2×M16—1071no—
139C22×C8C2×D10711yescharacteristic
140C4⋊Q8—1072no—
141C4⋊Q8—2141no—
142#125—1072no—
143#125—2141no—
144C22⋊Q8—2142no—
145C4×D4—1072no—
146C4×D4—1072no—
147C4×D4—2141no—
148C4×D4—2141no—
149#117—1071no—
150#117—1071no—
151#117—1071no—
152C2×C4⋊C4C2×D10711yescharacteristic
153C2×C22⋊C4—2142no—
154#107—1071no—
155#107—1071no—
156#107—1072no—
157#106—1071no—
158#106—1071no—
159#106—1072no—
160C4⋊C8—1072no—
161C4⋊C8—1072no—
162#102—1072no—
163#102—1072no—
164#102C2×D10712yes—
165#102C2×D10712yes—
166C24—21no—
167C2×Q8—2142no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4Dic21414yescomplemented
171C2×D4C22×D10712yes—
172C22×C4—21no—
173C22×C4—2142no—
174C22×C4—2142no—
175C22×C4C22×D10711yescharacteristic
176M16—2142no—
177C2×C8—22no—
178C2×C8—1072no—
179C2×C8C22×D10711yescharacteristic
180C2×C8C22×D10711yescharacteristic
181C4⋊C4—21no—
182C4⋊C4—1071no—
183C4⋊C4—1071no—
184C4⋊C4—1071no—
185C4⋊C4—1071no—
186C4⋊C4—1071no—
187C4⋊C4—1071no—
188C4⋊C4—2142no—
189C4⋊C4—2142no—
190C4⋊C4—2142no—
191C4⋊C4C22×D10712yes—
192C22⋊C4—1072no—
193C22⋊C4—2142no—
194C22⋊C4—2142no—
195C22⋊C4—2141no—
196C22⋊C4—2141no—
197C42—1072no—
198C42—2141no—
199C42—2141no—
200C23—22no—
201C23—24no—
202C23—81no—
203C23—81no—
204C23D4×D10711yescharacteristic
205Q8—8562no—
206D4—24no—
207D4—22no—
208D4C2×Dic21414yes—
209C2×C4—22no—
210C2×C4—81no—
211C2×C4—8562no—
212C2×C4—8562no—
213C2×C4—1072no—
214C2×C4—1072no—
215C2×C4—2141no—
216C2×C4—2141no—
217C2×C4—2141no—
218C2×C4—2141no—
219C2×C4D4×D10711yescharacteristic
220C2×C4C2×Dic21412yes—
221C2×C4#1313381712yes—
222C2×C4C23×D10711yescharacteristic
223C8—21no—
224C8—21no—
225C8—2142no—
226C22—21no—
227C22—21no—
228C22—24no—
229C22—42no—
230C22—42no—
231C22—84no—
232C22#1681208612yes—
233C22C2×D4×D10711yescharacteristic
234C4—42no—
235C4—4281no—
236C4—4281no—
237C4—4281no—
238C4—4281no—
239C4—2142no—
240C4—2142no—
241C4#1681208612yes—
242C4C22×Dic21411yescharacteristic
243C4C2×#1313381711yescharacteristic
244C2—22no—
245C2—44no—
246C2C2×#1681208611yescharacteristic
247C2#2029342811yescharacteristic
248C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#102×C107C2
socleC22×C107C22×D4
prime radicalC22×D4×C107C22
hypercenterC2×#102D107
hypocenterC107#3081
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #20293258C2semidirect
3. #20292929C2semidirect
4. #20292925C2semidirect
5. #20292915C2semidirect
6. #20292905C2semidirect
7. #20292667C2semidirect
8. #20292665C2semidirect
9. #20292653C2semidirect
14. #16812108C22semidirect
16. #16812100C22semidirect
26. #16812043C22semidirect
27. #16812043C22semidirect
28. #16812042C22semidirect
29. #16812041C22semidirect
30. #16812041C22semidirect
31. #16812040C22semidirect
32. Dic107⋊C8C22semidirect
33. Dic107⋊C8C22semidirect
66. C2×D4×C107Q8semidirect
125. C107#3081semidirect
170. C2×D4Dic214semidirect

Maximal subgroups

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

subgroupindex
#202932582
#202931022
#202929292
#202929252
#202929152
#202929052
#202926672
#202926652
#202926532
#202926322
C2×#102×C1072
#3081107