Finite Group Databasebeta

← #28346328

Subgroups — #28346328

Order 13696.

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

normal subgroups only

This group has 376 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#20293258C211yesmaximal, characteristic, complemented
2#20293108C211yesmaximal, characteristic
3#20292937C212yesmaximal, complemented
4#20292926C212yesmaximal, complemented
5#20292916C212yesmaximal, complemented
6#20292911C212yesmaximal, complemented
7#20292667C211yesmaximal, characteristic, complemented
8#20292653C211yesmaximal, characteristic, complemented
9#20292626C211yesmaximal, characteristic
10#20292624C211yesmaximal, characteristic
11C2×#106×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×#13133802C2211yescharacteristic
14#16812108—21no—
15#16812108C2212yes—
16#16812100—21no—
17#16812100C2212yes—
18#16812088—22no—
19#16812074C2212yes—
20#16812071C2212yes—
21C4×C22⋊D107—22no—
22#16812068C2211yescharacteristic
23#16812068C2211yescharacteristic
24#16812067—21no—
25C4⋊C4×D107—21no—
26#16812042C2212yes—
27#16812040C2212yes—
28Dic107⋊C8C2212yes—
29Dic107⋊C8C2212yes—
30#16812034C2211yescharacteristic
31#16812034C2211yescharacteristic
32#16812032C2212yes—
33#16812031C2211yescharacteristic
34#16812031C2211yescharacteristic
35#16812029C2212yes—
36C22×C8×C107C2211yescharacteristic
37C2×C4⋊C4×C107C2211yescharacteristic
38C2×C4⋊C4×C107C2211yescharacteristic
39#106×C107C2212yes—
40#106×C107C2212yes—
41#13133818—22no—
42#13133818Q814yescomplemented
43C2×Dic214—22no—
44C2×Dic214C2311yescharacteristic
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—21no—
47C2×C4×D107—21no—
48C2×C4×D107—22no—
49#13133802—22no—
50C2×C107⋊C8C2312yes—
51C4⋊Dic107—22no—
52C4⋊Dic107D411yescharacteristic
53C4⋊Dic107D411yescharacteristic
54C4⋊Dic107C2311yescharacteristic
55C4⋊Dic107C2311yescharacteristic
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59C22⋊Dic107—21no—
60C22⋊Dic107D412yes—
61(C2×D107)⋊C4—22no—
62C4×Dic107—21no—
63C4×Dic107—22no—
64C4×Dic107C2312yes—
65C22×C4×C107—21no—
66C22×C4×C107—21no—
67C22×C4×C107C2311yescharacteristic
68C2×C8×C107—22no—
69C2×C8×C107C2311yescharacteristic
70C2×C8×C107C2311yescharacteristic
71C4⋊C4×C107—21no—
72C4⋊C4×C107—21no—
73C4⋊C4×C107C2312yes—
74C4⋊C4×C107C2312yes—
75C23×C107C2×D411yescharacteristic
76C22×D107—21no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—81no—
80C2×C4×C107—81no—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×Q812yes—
83C2×C4×C107#4112yes—
84C2×C4×C107C2411yescharacteristic
85C22⋊D107—24no—
86C2×Dic107—21no—
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107C2×D412yes—
90C2×Dic107C2×D412yes—
91D428—21no—
92D428C2×Q812yes—
93C4×D107—24no—
94C4×D107—82no—
95Dic214—21no—
96Dic214—82no—
97Dic214C2×Q812yes—
98C8×C107—21no—
99C8×C107—21no—
100C107⋊C8—22no—
101C22×C107—21no—
102C22×C107—21no—
103C22×C107C22⋊Q812yes—
104C22×C107C22×D411yescharacteristic
105C2×D107—22no—
106C2×D107—41no—
107C2×D107—41no—
108C4×C107—42no—
109C4×C107—42no—
110C4×C107C22⋊Q812yes—
111C4×C107C22×Q811yescharacteristic
112C4×C107C2×#4111yescharacteristic
113Dic107—22no—
114Dic107—22no—
115Dic107—22no—
116Dic107—41no—
117Dic107—41no—
118C2×C107—22no—
119C2×C107C2×C22⋊Q811yescharacteristic
120C2×C107#57711yescharacteristic
121C2×C107#57711yescharacteristic
122D107—42no—
123#3084—1071nomaximal, Sylow
124C107#308411yescharacteristic, complemented, Sylow
125#544—1071no—
126#520—1071no—
127#479—1072no—
128#478—1072no—
129#476—1072no—
130#475—1072no—
131#428—1071no—
132C2×#106D10711yescharacteristic, complemented
133#424—1071no—
134#417—1071no—
135#416—1071no—
136C2×#41—1071no—
137C2×M16—1071no—
138C22×C8C2×D10711yescharacteristic
139C4⋊Q8—1072no—
140C4⋊Q8—2141no—
141#125—1072no—
142#125—2141no—
143C22⋊Q8—2142no—
144C4×Q8—1072no—
145C4×D4—1072no—
146C4×D4—2142no—
147#117—1071no—
148#117—1071no—
149#117—2141no—
150C2×C4⋊C4—2141no—
151C2×C4⋊C4C2×D10711yescharacteristic
152C2×C4⋊C4C2×D10711yescharacteristic
153#107—1072no—
154#106—1072no—
155#106C2×D10712yes—
156#106C2×D10712yes—
157C4⋊C8—1072no—
158C4⋊C8—1072no—
159#103—1071no—
160#103—1071no—
161#103—1072no—
162#102—1071no—
163#102—1071no—
164#102—1072no—
165#41—1074no—
166#41—2142no—
167C2×Q8—1071no—
168C2×Q8—2142no—
169C2×D4—1071no—
170C2×D4—2141no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—2142no—
174C22×C4—2141no—
175C22×C4C22×D10711yescharacteristic
176M16—2142no—
177C2×C8—22no—
178C2×C8—1072no—
179C2×C8C22×D10711yescharacteristic
180C2×C8C22×D10711yescharacteristic
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—1071no—
184C4⋊C4—1071no—
185C4⋊C4—1071no—
186C4⋊C4—1071no—
187C4⋊C4—2142no—
188C4⋊C4—2142no—
189C4⋊C4—2142no—
190C4⋊C4—2142no—
191C4⋊C4C22×D10712yes—
192C4⋊C4C22×D10712yes—
193C22⋊C4—1072no—
194C22⋊C4—2142no—
195C22⋊C4—2141no—
196C42—1072no—
197C42—2142no—
198C42—2141no—
199C23—2141no—
200C23D4×D10711yescharacteristic
201Q8—8562no—
202Q8—1072no—
203Q8—2141no—
204D4—1072no—
205D4—2144no—
206D4—2141no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—8562no—
212C2×C4—1072no—
213C2×C4—1072no—
214C2×C4—2141no—
215C2×C4—2141no—
216C2×C4—2141no—
217C2×C4—2144no—
218C2×C4D4×D10711yescharacteristic
219C2×C4Q8×D10712yes—
220C2×C4#1313381812yes—
221C2×C4C23×D10711yescharacteristic
222C8—21no—
223C8—21no—
224C8—2142no—
225C22—21no—
226C22—21no—
227C22—4281no—
228C22—4281no—
229C22—2142no—
230C22#1681208712yes—
231C22C2×D4×D10711yescharacteristic
232C4—42no—
233C4—42no—
234C4—4281no—
235C4—4281no—
236C4—2142no—
237C4—2142no—
238C4—2142no—
239C4#1681208712yes—
240C4C2×Q8×D10711yescharacteristic
241C4C2×#1313381811yescharacteristic
242C2—22no—
243C2—4282no—
244C2C2×#1681208711yescharacteristic
245C2#2029343311yescharacteristic
246C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#106×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenterC2×#106D107
hypocenterC107#3084
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
1. #20293258C2semidirect
3. #20292937C2semidirect
4. #20292926C2semidirect
5. #20292916C2semidirect
6. #20292911C2semidirect
7. #20292667C2semidirect
8. #20292653C2semidirect
11. C2×#106×C107C2semidirect
42. #13133818Q8semidirect
124. C107#3084semidirect
132. C2×#106D107semidirect

Maximal subgroups

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

subgroupindex
#202932582
#202931082
#202929372
#202929262
#202929162
#202929112
#202926672
#202926532
#202926262
#202926242
C2×#106×C1072
#3084107