Finite Group Databasebeta

← #28336841

Subgroups — #28336841

Order 13696.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#20293393C211yesmaximal, characteristic, complemented
2#20293173C211yesmaximal, characteristic
3#20292644C211yesmaximal, characteristic, complemented
4#20292600C211yesmaximal, characteristic, complemented
5#20292388C211yesmaximal, characteristic, complemented
6#20292386C211yesmaximal, characteristic, complemented
7#421×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812117—22no—
10#16812115—22no—
11C2×#13133802C2211yescharacteristic
12M16×D107—22no—
13#16812107C411yescharacteristic, complemented
14#16812105C411yescharacteristic, complemented
15#16812098C411yescharacteristic, complemented
16#16812096C411yescharacteristic, complemented
17#16812081—22no—
18C4×C22⋊D107—22no—
19#16812068C2211yescharacteristic
20C2×C4×Dic107C2211yescharacteristic
21#16812036—22no—
22#16812023—22no—
23C2×#41×C107C2211yescharacteristic
24C2×M16×C107C2211yescharacteristic
25#117×C107C2211yescharacteristic
26#104×C107—22no—
27#13133818—21no—
28#13133818—21no—
29#13133818—21no—
30#13133818—21no—
31C2×Dic214C2×C411yescharacteristic
32C2×D428C2×C411yescharacteristic
33C2×C22⋊D107—22no—
34C2×C22⋊D107D412yes—
35C22×Dic107—21no—
36C2×C4×D107D412yes—
37#13133802—21no—
38#13133802D412yescomplemented
39#13133801—24no—
40C2×C107⋊C8—21no—
41C8×D107—24no—
42C4⋊Dic107—21no—
43Dic107⋊C4—22no—
44C22⋊Dic107—22no—
45C22⋊Dic107—21no—
46(C2×D107)⋊C4—22no—
47C4×Dic107—21no—
48C4×Dic107—21no—
49C4×Dic107C2×C411yescharacteristic
50C4×Dic107C2×C411yescharacteristic
51#41×C107—21no—
52#41×C107—21no—
53#41×C107—21no—
54#41×C107—21no—
55C2×Q8×C107C2×C411yescharacteristic
56C2×D4×C107—21no—
57C2×D4×C107C2×C411yescharacteristic
58C22×C4×C107—21no—
59C22×C4×C107C2311yescharacteristic
60M16×C107—21no—
61M16×C107D412yescomplemented
62C2×C8×C107—21no—
63C4⋊C4×C107—21no—
64C22⋊C4×C107—21no—
65C42×C107—21no—
66C23×C107—21no—
67C23×C107C2×D411yescharacteristic
68C22×D107C22⋊C412yes—
69Q8×C107—21no—
70Q8×C107—21no—
71D4×C107—21no—
72D4×C107—21no—
73D4×C107—41no—
74D4×C107—41no—
75C2×C4×C107—21no—
76C2×C4×C107—41no—
77C2×C4×C107—41no—
78C2×C4×C107—41no—
79C2×C4×C107C22×C411yescharacteristic
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107#4111yescharacteristic
84C2×C4×C107#4111yescharacteristic
85C22⋊D107—22no—
86C22⋊D107—22no—
87C22⋊D107—44no—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—42no—
91C2×Dic107—41no—
92C2×Dic107C22⋊C412yes—
93D428—21no—
94D428—21no—
95C4×D107—22no—
96C4×D107—22no—
97Dic214—21no—
98Dic214—21no—
99C8×C107—22no—
100C107⋊C8—22no—
101C22×C107—41no—
102C22×C107—42no—
103C22×C107—41no—
104C22×C107C2×C22⋊C411yescharacteristic
105C22×C107#12111yescharacteristic
106C22×C107#12111yescharacteristic
107C2×D107—22no—
108C2×D107—42no—
109C4×C107—41no—
110C4×C107—81no—
111C4×C107C4×D411yescharacteristic
112C4×C107C4×D411yescharacteristic
113C4×C107#12011yescharacteristic
114C4×C107#12311yescharacteristic
115Dic107—22no—
116Dic107—22no—
117Dic107—41no—
118Dic107—81no—
119C2×C107—21no—
120C2×C107—21no—
121C2×C107—21no—
122C2×C107—81no—
123C2×C107#38611yescharacteristic
124D107—42no—
125#1884—1071nomaximal, Sylow
126C107#188411yescharacteristic, complemented, Sylow
127#568—1071no—
128#532—1071no—
129#421—1071no—
130#421D10711yescharacteristic, complemented
131#410—1071no—
132#339—1071no—
133#339—1071no—
134C2×#41—1071no—
135C2×#41C2×D10711yescharacteristic
136#131—2142no—
137#131—2142no—
138C2×M16—1071no—
139C2×M16—2142no—
140C2×M16C2×D10711yescharacteristic
141C4⋊Q8—1071no—
142#127—1071no—
143#124—1071no—
144#124—1071no—
145#121—2142no—
146C4×D4—2142no—
147#117—1071no—
148#117C2×D10711yescharacteristic
149C2×C42—1071no—
150#104—22no—
151#104—2142no—
152#99—2142no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—2141no—
158#41—2141no—
159#41—2141no—
160#41—2141no—
161C2×Q8—1071no—
162C2×Q8C4×D10711yescharacteristic
163C2×D4—21no—
164C2×D4—1072no—
165C2×D4—1071no—
166C2×D4—2142no—
167C2×D4C4×D10711yescharacteristic
168C22×C4—21no—
169C22×C4—1072no—
170C22×C4—2141no—
171C22×C4C22×D10711yescharacteristic
172M16—21no—
173M16—1072no—
174M16—2144no—
175M16—2141no—
176M16C22⋊D10712yescomplemented
177C2×C8—21no—
178C2×C8—2144no—
179C2×C8—2141no—
180C4⋊C4—21no—
181C4⋊C4—2142no—
182C4⋊C4—2141no—
183C22⋊C4—21no—
184C22⋊C4—2142no—
185C22⋊C4—2142no—
186C22⋊C4—2141no—
187C42—21no—
188C42—1071no—
189C42—1071no—
190C42—2141no—
191C42—2141no—
192C23—21no—
193C23—1072no—
194C23D4×D10711yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—2141no—
198Q8—2141no—
199D4—21no—
200D4—21no—
201D4—41no—
202D4—41no—
203D4—4284no—
204D4—2142no—
205D4—2142no—
206D4—2141no—
207D4—2141no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—1072no—
213C2×C4—4281no—
214C2×C4—4282no—
215C2×C4—2141no—
216C2×C4—2141no—
217C2×C4—2142no—
218C2×C4—2142no—
219C2×C4C2×C4×D10711yescharacteristic
220C2×C4D4×D10711yescharacteristic
221C2×C4D4×D10711yescharacteristic
222C2×C4C2×C22⋊D10711yescharacteristic
223C2×C4#1313381711yescharacteristic
224C2×C4#1313381811yescharacteristic
225C8—22no—
226C8—2142no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—4282no—
231C22—2142no—
232C22C22⋊C4×D10711yescharacteristic
233C22#1681208011yescharacteristic
234C22#1681208211yescharacteristic
235C4—41no—
236C4—81no—
237C4—8561no—
238C4—4281no—
239C4—2142no—
240C4—2142no—
241C4(C22⋊D107)⋊C411yescharacteristic
242C4C4×C22⋊D10711yescharacteristic
243C4#1681207711yescharacteristic
244C4#1681209111yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—81no—
249C2—4282no—
250C2#2029250311yescharacteristic
subgroupG/H
centerC4#16812077
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#421×C107C2
socleC2×C107#386
prime radical#20293173C2
cosocleC22×C4×C107C23
hypercenter#421D107
hypocenterC107#1884
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #20293393C2semidirect
3. #20292644C2semidirect
4. #20292600C2semidirect
5. #20292388C2semidirect
6. #20292386C2semidirect
7. #421×C107C2semidirect
13. #16812107C4semidirect
14. #16812105C4semidirect
15. #16812098C4semidirect
16. #16812096C4semidirect
38. #13133802D4semidirect
61. M16×C107D4semidirect
126. C107#1884semidirect
130. #421D107semidirect
176. M16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202933932
#202931732
#202926442
#202926002
#202923882
#202923862
#421×C1072
#1884107