Finite Group Databasebeta

← #28337287

Subgroups — #28337287

Order 13696.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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#20293173C211yesmaximal, characteristic
2#20292650C211yesmaximal, characteristic, complemented
3C2×#16812037C211yesmaximal, characteristic, complemented
4C2×#16812036C211yesmaximal, characteristic, complemented
5C2×#16812035C211yesmaximal, characteristic, complemented
6C42×Dic107C211yesmaximal, characteristic, complemented
7C2×#104×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C2×#13133802C2211yescharacteristic
10#16812107C411yescharacteristic, complemented
11#16812105C411yescharacteristic, complemented
12#16812098C411yescharacteristic, complemented
13#16812096C411yescharacteristic, complemented
14#16812081—22no—
15C4×C22⋊D107—22no—
16C2×C4×Dic107—22no—
17C2×C4×Dic107C2211yescharacteristic
18C2×C4×Dic107C2211yescharacteristic
19Dic107⋊C8—22no—
20#16812037—22no—
21#16812036—22no—
22#16812035—22no—
23C2×#41×C107C2211yescharacteristic
24C2×M16×C107C2211yescharacteristic
25C2×C42×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—21no—
34C2×C22⋊D107—22no—
35C22×Dic107—21no—
36C22×Dic107D412yes—
37C2×C4×D107—21no—
38#13133802—21no—
39#13133802—21no—
40C2×C107⋊C8—21no—
41Dic107⋊C4—22no—
42C22⋊Dic107—22no—
43(C2×D107)⋊C4—22no—
44C4×Dic107—28no—
45C4×Dic107—21no—
46C4×Dic107C2×C411yescharacteristic
47C4×Dic107C2×C411yescharacteristic
48C4×Dic107D412yes—
49C4×Dic107D412yescomplemented
50#41×C107—21no—
51#41×C107—21no—
52#41×C107—21no—
53#41×C107—21no—
54C2×Q8×C107C2×C411yescharacteristic
55C2×D4×C107—21no—
56C2×D4×C107C2×C411yescharacteristic
57C22×C4×C107—21no—
58C22×C4×C107—21no—
59C22×C4×C107C2311yescharacteristic
60M16×C107—21no—
61M16×C107—21no—
62C2×C8×C107—21no—
63C42×C107—21no—
64C42×C107D412yescomplemented
65C23×C107—21no—
66C23×C107C2×D411yescharacteristic
67C22×D107—21no—
68Q8×C107—21no—
69Q8×C107—21no—
70D4×C107—21no—
71D4×C107—21no—
72D4×C107—41no—
73D4×C107—41no—
74C2×C4×C107—24no—
75C2×C4×C107—22no—
76C2×C4×C107—21no—
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—41no—
86C22⋊D107—41no—
87C22⋊D107—44no—
88C2×Dic107—24no—
89C2×Dic107—22no—
90C2×Dic107—24no—
91C2×Dic107—21no—
92C2×Dic107C22⋊C412yes—
93C2×Dic107C22⋊C412yes—
94D428—21no—
95D428—21no—
96C4×D107—41no—
97C4×D107—41no—
98Dic214—21no—
99Dic214—21no—
100C8×C107—41no—
101C107⋊C8—41no—
102C22×C107—22no—
103C22×C107—41no—
104C22×C107—42no—
105C22×C107C2×C22⋊C411yescharacteristic
106C22×C107#12111yescharacteristic
107C22×C107#12111yescharacteristic
108C2×D107—41no—
109C2×D107—42no—
110C4×C107—24no—
111C4×C107—41no—
112C4×C107C4×D411yescharacteristic
113C4×C107C4×D411yescharacteristic
114C4×C107#12411yescharacteristic
115C4×C107#12711yescharacteristic
116Dic107—24no—
117Dic107—22no—
118Dic107—41no—
119Dic107#10414yescomplemented
120C2×C107—21no—
121C2×C107—21no—
122C2×C107—81no—
123C2×C107#39011yescharacteristic
124C2×C107C2×#10412yes—
125D107—81no—
126#1962—1071nomaximal, Sylow
127C107#196211yescharacteristic, complemented, Sylow
128#532—1071no—
129#423—1071no—
130C2×#104—1071no—
131C2×#104—1071no—
132C2×#104—1071no—
133C2×#104D10711yescharacteristic, complemented
134C43—1071no—
135C2×#41—1071no—
136C2×#41C2×D10711yescharacteristic
137C2×M16—1071no—
138C2×M16C2×D10711yescharacteristic
139C4⋊Q8—1071no—
140#127—1071no—
141#124—1071no—
142#124—1071no—
143#121—2142no—
144C4×D4—2142no—
145C2×C42—1071no—
146C2×C42—1071no—
147C2×C42—2142no—
148C2×C42C2×D10711yescharacteristic
149C4⋊C8—2142no—
150#104—22no—
151#104—2142no—
152#104—2142no—
153#104—2142no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—2141no—
159#41—2141no—
160#41—2141no—
161#41—2141no—
162C2×Q8—1071no—
163C2×Q8C4×D10711yescharacteristic
164C2×D4—21no—
165C2×D4—1071no—
166C2×D4—2142no—
167C2×D4—2141no—
168C2×D4C4×D10711yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1072no—
172C22×C4—2141no—
173C22×C4—2141no—
174C22×C4C22×D10711yescharacteristic
175M16—21no—
176M16—21no—
177M16—2141no—
178M16—2141no—
179C2×C8—21no—
180C2×C8—2141no—
181C4⋊C4—2142no—
182C22⋊C4—2142no—
183C22⋊C4—2142no—
184C42—21no—
185C42—1071no—
186C42—1071no—
187C42—1072no—
188C42—1072no—
189C42—2148no—
190C42—2141no—
191C42C22⋊D10712yescomplemented
192C23—21no—
193C23—2141no—
194C23D4×D10711yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—2141no—
198Q8—2141no—
199D4—21no—
200D4—21no—
201D4—41no—
202D4—41no—
203D4—4284no—
204D4—4281no—
205D4—4281no—
206D4—2141no—
207D4—2141no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—1072no—
214C2×C4—1072no—
215C2×C4—4281no—
216C2×C4—4281no—
217C2×C4—2144no—
218C2×C4—2141no—
219C2×C4—2144no—
220C2×C4—2142no—
221C2×C4C2×C4×D10711yescharacteristic
222C2×C4D4×D10711yescharacteristic
223C2×C4D4×D10711yescharacteristic
224C2×C4C2×C22⋊D10711yescharacteristic
225C2×C4#1313381711yescharacteristic
226C2×C4#1313381811yescharacteristic
227C8—41no—
228C8—4281no—
229C22—22no—
230C22—42no—
231C22—41no—
232C22—4282no—
233C22—4281no—
234C22C22⋊C4×D10711yescharacteristic
235C22#1681208111yescharacteristic
236C22#1681208211yescharacteristic
237C4—24no—
238C4—41no—
239C4—1074no—
240C4—4281no—
241C4—2144no—
242C4—2142no—
243C4(C22⋊D107)⋊C411yescharacteristic
244C4C4×C22⋊D10711yescharacteristic
245C4#1681209511yescharacteristic
246C4#1681210511yescharacteristic
247C2—21no—
248C2—21no—
249C2—81no—
250C2—8561no—
251C2#2029252711yescharacteristic
252C2#104×D10712yes—
subgroupG/H
centerC2×C4D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#104×C107C2
socleC22×C107C2×C22⋊C4
prime radical#20293173C2
cosocleC22×C4×C107C23
hypercenterC2×#104D107
hypocenterC107#1962
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #20292650C2semidirect
3. C2×#16812037C2semidirect
4. C2×#16812036C2semidirect
5. C2×#16812035C2semidirect
6. C42×Dic107C2semidirect
7. C2×#104×C107C2semidirect
10. #16812107C4semidirect
11. #16812105C4semidirect
12. #16812098C4semidirect
13. #16812096C4semidirect
49. C4×Dic107D4semidirect
64. C42×C107D4semidirect
119. Dic107#104semidirect
127. C107#1962semidirect
133. C2×#104D107semidirect
191. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202931732
#202926502
C2×#168120372
C2×#168120362
C2×#168120352
C42×Dic1072
C2×#104×C1072
#1962107