Finite Group Databasebeta

← #28349516

Subgroups — #28349516

Order 13696.

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

normal subgroups only

This group has 362 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#20293362C211yesmaximal, characteristic
2#20293356C211yesmaximal, characteristic, complemented
3#20293021C212yesmaximal, complemented
4#20293000C211yesmaximal, characteristic
5#20292850C212yesmaximal, complemented
6#20292838C211yesmaximal, characteristic, complemented
7#20292833C212yesmaximal, complemented
8#20292828C211yesmaximal, characteristic
9#20292729C212yesmaximal, complemented
10C4×Dic428C211yesmaximal, characteristic, complemented
11#446×C107C211yesmaximal, characteristic, complemented
12C2×Dic428C2211yescharacteristic
13C2×Dic428C2212yes—
14C2×#13133807C2212yes—
15C2×#13133807C2212yes—
16C2×#13133807C2212yes—
17#16812109C2211yescharacteristic
18#16812109C2212yes—
19#16812106C2211yescharacteristic
20#16812100—21no—
21#16812100—22no—
22#16812098—21no—
23#16812097C2212yes—
24#16812094—22no—
25Q8×Dic107—21no—
26C4×Dic214C2211yescharacteristic
27C4×Dic214C2211yescharacteristic
28C4×Dic214C2212yes—
29C4×D428C2212yes—
30#16812071—21no—
31#16812043C2211yescharacteristic
32#16812034C2211yescharacteristic
33#16812034C2211yescharacteristic
34#16812034C2212yes—
35#16812031C2212yes—
36C4×Q8×C107C2211yescharacteristic
37C4⋊C8×C107C2211yescharacteristic
38C4⋊C8×C107C2212yes—
39#97×C107C2212yes—
40C4×C8×C107C2211yescharacteristic
41C2×Dic214C2311yescharacteristic
42C2×Dic214C2312yes—
43C2×Dic214C2311yescharacteristic
44C2×D428—21no—
45C2×D428—21no—
46C2×D428C2312yes—
47C2×C4×D107—22no—
48Dic428—22no—
49Dic428—42no—
50#13133807—24no—
51#13133807—42no—
52#13133807—42no—
53C4⋊Dic107—22no—
54C4⋊Dic107—21no—
55C4⋊Dic107—22no—
56C4⋊Dic107—21no—
57C4⋊Dic107C2311yescharacteristic
58C4⋊Dic107C2311yescharacteristic
59Dic107⋊C4—21no—
60Dic107⋊C4—22no—
61Dic107⋊C4—21no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4—22no—
65C4×Dic107—21no—
66C4×Dic107—22no—
67C4×Dic107—21no—
68C2×Q8×C107D411yescharacteristic
69C2×C8×C107C2311yescharacteristic
70C2×C8×C107C2312yes—
71C2×C8×C107C2311yescharacteristic
72C4⋊C4×C107D411yescharacteristic
73C4⋊C4×C107D412yes—
74C42×C107C2312yes—
75C42×C107C2311yescharacteristic
76C22×D107—22no—
77Q8×C107—22no—
78C2×C4×C107C2×D412yes—
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107C2×D412yes—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2411yescharacteristic
83C2×Dic107—21no—
84C2×Dic107—21no—
85C2×Dic107—21no—
86C2×Dic107—22no—
87C2×Dic107—21no—
88D428—22no—
89D428—22no—
90D428—42no—
91D428—81no—
92C4×D107—82no—
93Dic214—22no—
94Dic214—21no—
95Dic214—21no—
96Dic214—21no—
97Dic214—22no—
98Dic214—21no—
99C8×C107—21no—
100C8×C107—22no—
101C8×C107#4112yes—
102C22×C107C22×D411yescharacteristic
103C2×D107—42no—
104C2×D107—42no—
105C2×D107—42no—
106C4×C107—22no—
107C4×C107—22no—
108C4×C107—21no—
109C4×C107#13712yes—
110C4×C107C2×#4111yescharacteristic
111C4×C107H(5,𝔽2)11yescharacteristic
112Dic107—42no—
113Dic107—41no—
114Dic107—41no—
115Dic107—41no—
116Dic107—41no—
117C2×C107#55011yescharacteristic
118C2×C107C2×#13711yescharacteristic
119C2×C107#57711yescharacteristic
120D107—82no—
121#3400—1071nomaximal, Sylow
122C107#340011yescharacteristic, complemented, Sylow
123#559—1071no—
124#557—1071no—
125#497—1072no—
126#492—1071no—
127#464—1072no—
128#462—1071no—
129#461—1072no—
130#460—1071no—
131#446D10711yescharacteristic, complemented
132#440—1072no—
133C4×Q16—1071no—
134C2×Q16—1072no—
135C2×Q16—1071no—
136C2×SD16—1072no—
137C2×SD16—1072no—
138C2×SD16—1072no—
139C4⋊Q8—1072no—
140C4⋊Q8—1071no—
141#127—1071no—
142#125—2141no—
143#125—2142no—
144#124—1072no—
145#124—2141no—
146#123—2142no—
147C4×Q8—1071no—
148C4×Q8—1072no—
149C4×Q8—1071no—
150C4×Q8—2141no—
151C4×Q8C2×D10711yescharacteristic
152C4×D4—1072no—
153C4×D4—2141no—
154#107—1071no—
155C4⋊C8C2×D10711yescharacteristic
156C4⋊C8C2×D10712yes—
157#103—1071no—
158#103—1071no—
159#103—1072no—
160#102—1072no—
161#97C2×D10712yes—
162C4×C8C2×D10711yescharacteristic
163C2×Q8—1071no—
164C2×Q8—1072no—
165C2×Q8—1071no—
166C2×Q8D42811yescharacteristic
167C2×D4—1072no—
168C2×D4—2141no—
169C2×D4—2141no—
170C22×C4—2142no—
171Q16—4282no—
172Q16—2142no—
173SD16—4282no—
174SD16—4282no—
175SD16—2144no—
176C2×C8C22×D10711yescharacteristic
177C2×C8C22×D10712yes—
178C2×C8C22×D10711yescharacteristic
179C4⋊C4—1071no—
180C4⋊C4—1071no—
181C4⋊C4—2141no—
182C4⋊C4—2142no—
183C4⋊C4—2141no—
184C4⋊C4—2142no—
185C4⋊C4—2141no—
186C4⋊C4—2142no—
187C4⋊C4—2141no—
188C4⋊C4D42811yescharacteristic
189C4⋊C4D42812yes—
190C22⋊C4—2142no—
191C22⋊C4—2142no—
192C22⋊C4—2142no—
193C42—2141no—
194C42—2142no—
195C42—2141no—
196C42C22×D10711yescharacteristic
197C42C22×D10712yes—
198C23—2142no—
199Q8—22no—
200Q8—2142no—
201Q8—2141no—
202Q8—2141no—
203Q8—2141no—
204Q8—2142no—
205Q8—2141no—
206D4—8561no—
207D4—4282no—
208D4—2142no—
209D4—2142no—
210C2×C4—8562no—
211C2×C4—2141no—
212C2×C4—2141no—
213C2×C4—2141no—
214C2×C4—2142no—
215C2×C4—2141no—
216C2×C4C2×D42812yes—
217C2×C4C2×D42812yes—
218C2×C4C2×D42811yescharacteristic
219C2×C4C2×D42811yescharacteristic
220C2×C4C23×D10711yescharacteristic
221C8—21no—
222C8—22no—
223C8#1313381912yes—
224C22—4282no—
225C22—4282no—
226C22—4282no—
227C22C22×D42811yescharacteristic
228C4—22no—
229C4—22no—
230C4—21no—
231C4—4281no—
232C4—4281no—
233C4—4281no—
234C4—4282no—
235C4—4281no—
236C4#1681214212yes—
237C4C2×#1313381911yescharacteristic
238C4#1681215511yescharacteristic
239C2—8562no—
240C2#2029329911yescharacteristic
241C2C2×#1681214211yescharacteristic
242C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#446×C107C2
socleC22×C107C22×D4
prime radical#16812106C22
hypercenter#446D107
hypocenterC107#3400
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
2. #20293356C2semidirect
3. #20293021C2semidirect
5. #20292850C2semidirect
6. #20292838C2semidirect
7. #20292833C2semidirect
9. #20292729C2semidirect
10. C4×Dic428C2semidirect
11. #446×C107C2semidirect
122. C107#3400semidirect
131. #446D107semidirect

Maximal subgroups

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

subgroupindex
#202933622
#202933562
#202930212
#202930002
#202928502
#202928382
#202928332
#202928282
#202927292
C4×Dic4282
#446×C1072
#3400107