Finite Group Databasebeta

← #28336843

Subgroups — #28336843

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#20293177C211yesmaximal, characteristic
3#20292645C211yesmaximal, characteristic, complemented
4#20292643C211yesmaximal, characteristic, complemented
5#20292600C211yesmaximal, characteristic, complemented
6#20292387C211yesmaximal, characteristic, complemented
7#339×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C2×#13133818C2211yescharacteristic
10#16812117—22no—
11#16812115—22no—
12C2×#13133802C2211yescharacteristic
13M16×D107—22no—
14#16812109C411yescharacteristic, complemented
15#16812106C411yescharacteristic, complemented
16#16812097C411yescharacteristic, complemented
17#16812097C411yescharacteristic, complemented
18#16812083—22no—
19C4×D428—22no—
20#16812068C2211yescharacteristic
21#16812037—22no—
22#16812035—22no—
23#16812023—22no—
24C2×M16×C107C2211yescharacteristic
25#117×C107C2211yescharacteristic
26C2×C42×C107C2211yescharacteristic
27#13133818—21no—
28#13133818—21no—
29#13133818—21no—
30#13133818—21no—
31#13133818—21no—
32#13133818—21no—
33#13133818—21no—
34#13133818—21no—
35C2×Dic214C2×C411yescharacteristic
36C2×Dic214C2×C411yescharacteristic
37C2×D428—22no—
38C2×D428C2×C411yescharacteristic
39C2×D428C2×C411yescharacteristic
40C2×C22⋊D107—21no—
41C2×C22⋊D107D412yes—
42C2×C4×D107—21no—
43C2×C4×D107D412yes—
44#13133802—21no—
45#13133802D412yescomplemented
46#13133801—24no—
47C2×C107⋊C8—21no—
48C8×D107—24no—
49C4⋊Dic107—22no—
50C4⋊Dic107—21no—
51C22⋊Dic107—21no—
52(C2×D107)⋊C4—22no—
53(C2×D107)⋊C4—22no—
54C4×Dic107—21no—
55C22×C4×C107—21no—
56C22×C4×C107C2311yescharacteristic
57M16×C107—21no—
58M16×C107D412yescomplemented
59C2×C8×C107—21no—
60C4⋊C4×C107—21no—
61C22⋊C4×C107—21no—
62C42×C107—21no—
63C42×C107—21no—
64C42×C107C2×C411yescharacteristic
65C42×C107C2×C411yescharacteristic
66C23×C107C2×D411yescharacteristic
67C22×D107—21no—
68C22×D107C22⋊C412yes—
69C2×C4×C107—21no—
70C2×C4×C107—21no—
71C2×C4×C107—42no—
72C2×C4×C107—41no—
73C2×C4×C107C22×C411yescharacteristic
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107#4111yescharacteristic
78C2×C4×C107#4111yescharacteristic
79C22⋊D107—22no—
80C22⋊D107—22no—
81C22⋊D107—41no—
82C22⋊D107—41no—
83C2×Dic107—21no—
84C2×Dic107—41no—
85C2×Dic107C22⋊C412yes—
86D428—21no—
87D428—21no—
88D428—21no—
89D428—21no—
90D428—44no—
91C4×D107—22no—
92C4×D107—22no—
93C4×D107—41no—
94C4×D107—41no—
95Dic214—21no—
96Dic214—21no—
97Dic214—21no—
98Dic214—21no—
99C8×C107—22no—
100C107⋊C8—22no—
101C22×C107—41no—
102C22×C107C2×C22⋊C411yescharacteristic
103C22×C107#12111yescharacteristic
104C22×C107#12111yescharacteristic
105C2×D107—22no—
106C2×D107—42no—
107C2×D107—42no—
108C2×D107—41no—
109C4×C107—22no—
110C4×C107—41no—
111C4×C107—81no—
112C4×C107C4×D411yescharacteristic
113C4×C107C4×D411yescharacteristic
114C4×C107#12011yescharacteristic
115C4×C107#12311yescharacteristic
116Dic107—22no—
117Dic107—41no—
118Dic107—81no—
119C2×C107—21no—
120C2×C107—21no—
121C2×C107—21no—
122C2×C107#38611yescharacteristic
123D107—42no—
124D107—81no—
125#1884—1071nomaximal, Sylow
126C107#188411yescharacteristic, complemented, Sylow
127#568—1071no—
128#532—1071no—
129#421—1071no—
130#421—1071no—
131#410—1071no—
132#339—1071no—
133#339D10711yescharacteristic, complemented
134C2×#41—1071no—
135C2×#41—1071no—
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×C42C2×D10711yescharacteristic
150#104—2142no—
151#104—2142no—
152#99—2142no—
153#41—2141no—
154#41—2141no—
155#41—2141no—
156#41—2141no—
157#41—2141no—
158#41—2141no—
159#41—2141no—
160#41—2141no—
161C2×Q8—1071no—
162C2×Q8—1071no—
163C2×D4—1072no—
164C2×D4—1071no—
165C2×D4—1071no—
166C2×D4—2142no—
167C2×D4—2141no—
168C22×C4—21no—
169C22×C4—1072no—
170C22×C4—2141no—
171C22×C4C22×D10711yescharacteristic
172M16—21no—
173M16—1072no—
174M16—2144no—
175M16—2141no—
176M16D42812yescomplemented
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—21no—
189C42—2141no—
190C42C4×D10711yescharacteristic
191C42C4×D10711yescharacteristic
192C23—1072no—
193C23—2141no—
194C23D4×D10711yescharacteristic
195Q8—2141no—
196Q8—2141no—
197Q8—2141no—
198Q8—2141no—
199D4—4284no—
200D4—4281no—
201D4—4281no—
202D4—2142no—
203D4—2142no—
204D4—2141no—
205D4—2141no—
206D4—2141no—
207D4—2141no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—1072no—
213C2×C4—4281no—
214C2×C4—4281no—
215C2×C4—4281no—
216C2×C4—2141no—
217C2×C4—2142no—
218C2×C4—2142no—
219C2×C4C2×C4×D10711yescharacteristic
220C2×C4D4×D10711yescharacteristic
221C2×C4D4×D10711yescharacteristic
222C2×C4C2×D42811yescharacteristic
223C2×C4#1313381811yescharacteristic
224C2×C4#1313381911yescharacteristic
225C8—22no—
226C8—2142no—
227C22—41no—
228C22—4282no—
229C22—4282no—
230C22—4281no—
231C22—2142no—
232C22C22⋊C4×D10711yescharacteristic
233C22#1681208211yescharacteristic
234C22#1681208411yescharacteristic
235C4—22no—
236C4—41no—
237C4—81no—
238C4—8561no—
239C4—4281no—
240C4—2142no—
241C4#1681207111yescharacteristic
242C4C4×D42811yescharacteristic
243C4#1681207911yescharacteristic
244C4#1681209411yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—8561no—
249C2—4282no—
250C2#2029250511yescharacteristic
subgroupG/H
centerC4#16812079
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#339×C107C2
socleC2×C107#386
prime radical#20293177C2
cosocleC22×C4×C107C23
hypercenter#339D107
hypocenterC107#1884
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #20293393C2semidirect
3. #20292645C2semidirect
4. #20292643C2semidirect
5. #20292600C2semidirect
6. #20292387C2semidirect
7. #339×C107C2semidirect
14. #16812109C4semidirect
15. #16812106C4semidirect
16. #16812097C4semidirect
17. #16812097C4semidirect
45. #13133802D4semidirect
58. M16×C107D4semidirect
126. C107#1884semidirect
133. #339D107semidirect
176. M16D428semidirect

Maximal subgroups

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

subgroupindex
#202933932
#202931772
#202926452
#202926432
#202926002
#202923872
#339×C1072
#1884107