Finite Group Databasebeta

← #28338113

Subgroups — #28338113

Order 13696.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#20293199C211yesmaximal, characteristic
2#20292608C211yesmaximal, characteristic
3#20292599C211yesmaximal, characteristic
4#20292418C211yesmaximal, characteristic, complemented
5#20292417C211yesmaximal, characteristic, complemented
6#20292416C211yesmaximal, characteristic, complemented
7#352×C107C211yesmaximal, characteristic, complemented
8C2×#13133817C2211yescharacteristic
9C22×Dic214C411yescharacteristic, complemented
10M16×D107—21no—
11#16812108—21no—
12#16812095—21no—
13#16812093C411yescharacteristic, complemented
14#16812092C411yescharacteristic, complemented
15#16812090C2211yescharacteristic
16#16812087—21no—
17#16812086—21no—
18#16812065—21no—
19#16812065C411yescharacteristic, complemented
20#16812027—21no—
21#16812026C2211yescharacteristic
22#16812024C2211yescharacteristic
23#16812023—21no—
24#123×C107C2211yescharacteristic
25#100×C107C2211yescharacteristic
26#99×C107C2211yescharacteristic
27#13133817—41no—
28#13133817—41no—
29C2×Dic214—21no—
30C2×Dic214—21no—
31C2×Dic214—41no—
32C2×Dic214—41no—
33C2×Dic214D411yescharacteristic
34C2×C22⋊D107C2×C411yescharacteristic
35C2×C22⋊D107D411yescharacteristic
36C22×Dic107C2×C411yescharacteristic
37C22×Dic107C2×C411yescharacteristic
38C22×Dic107D411yescharacteristic
39C2×C4×D107D411yescharacteristic
40#13133802—21no—
41#13133801—41no—
42C8×D107—41no—
43C4⋊Dic107—21no—
44Dic107⋊C4—21no—
45Dic107⋊C4—21no—
46Dic107⋊C4—21no—
47Dic107⋊C4—41no—
48C22⋊Dic107—21no—
49C22⋊Dic107—21no—
50C22⋊Dic107C2×C411yescharacteristic
51(C2×D107)⋊C4—21no—
52(C2×D107)⋊C4—21no—
53C4×Dic107—21no—
54C4×Dic107—41no—
55C2×D4×C107C2311yescharacteristic
56C22×C4×C107C2×C411yescharacteristic
57M16×C107—21no—
58C4⋊C4×C107—21no—
59C22⋊C4×C107—21no—
60C22⋊C4×C107—21no—
61C22⋊C4×C107C2×C411yescharacteristic
62C23×C107C22×C411yescharacteristic
63C23×C107C2×D411yescharacteristic
64C22×D107C22⋊C411yescharacteristic
65D4×C107—41no—
66C2×C4×C107—21no—
67C2×C4×C107—21no—
68C2×C4×C107—41no—
69C2×C4×C107—41no—
70C2×C4×C107C2×D411yescharacteristic
71C22⋊D107—41no—
72C22⋊D107—41no—
73C2×Dic107—21no—
74C2×Dic107—21no—
75C2×Dic107—21no—
76C2×Dic107—21no—
77C2×Dic107—21no—
78C2×Dic107—21no—
79C2×Dic107—41no—
80C2×Dic107—41no—
81C2×Dic107C22⋊C411yescharacteristic
82C2×Dic107C22⋊C411yescharacteristic
83C2×Dic107C22⋊C411yescharacteristic
84C4×D107—21no—
85C4×D107—21no—
86Dic214—41no—
87Dic214—42no—
88Dic214—41no—
89Dic214—41no—
90C8×C107—41no—
91C107⋊C8—41no—
92C22×C107—21no—
93C22×C107—21no—
94C22×C107—41no—
95C22×C107—41no—
96C22×C107C2×C22⋊C411yescharacteristic
97C2×D107—21no—
98C2×D107—41no—
99C4×C107—21no—
100C4×C107—41no—
101C4×C107—41no—
102C4×C107—81no—
103Dic107—21no—
104Dic107—21no—
105Dic107—41no—
106Dic107—41no—
107Dic107—81no—
108Dic107#9911yescharacteristic
109Dic107#9911yescharacteristic
110C2×C107—21no—
111C2×C107—41no—
112C2×C107—41no—
113C2×C107C2×#9911yescharacteristic
114D107—41no—
115#2123—1071nomaximal, Sylow
116C107#212311yescharacteristic, complemented, Sylow
117#536—1071no—
118#413—1071no—
119#410—1071no—
120#352—1071no—
121#352—1071no—
122#352—1071no—
123#352D10711yescharacteristic, complemented
124C2×#41—1071no—
125C22×Q8—1071no—
126C2×M16—2141no—
127C4⋊Q8—2141no—
128#124—2141no—
129#123—1071no—
130#123—1071no—
131#123—1071no—
132#123C2×D10711yescharacteristic
133C22⋊Q8—2141no—
134C22⋊Q8—2141no—
135#117—1071no—
136#117—2141no—
137#101—2141no—
138#100—1071no—
139#100C2×D10711yescharacteristic
140#99—1071no—
141#99—2141no—
142#99C2×D10711yescharacteristic
143#41—4281no—
144#41—4281no—
145C2×Q8—1071no—
146C2×Q8—4281no—
147C2×Q8—4281no—
148C2×Q8—2141no—
149C2×Q8—2141no—
150C2×D4—1071no—
151C2×D4—1071no—
152C2×D4C22×D10711yescharacteristic
153C22×C4—1071no—
154C22×C4—1071no—
155C22×C4—1071no—
156C22×C4—1071no—
157C22×C4C4×D10711yescharacteristic
158M16—21no—
159M16—4281no—
160M16—2141no—
161C2×C8—4281no—
162C4⋊C4—21no—
163C4⋊C4—4281no—
164C4⋊C4—2141no—
165C4⋊C4—2141no—
166C4⋊C4—2141no—
167C4⋊C4—2141no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—1071no—
171C22⋊C4—2141no—
172C22⋊C4—2141no—
173C22⋊C4—2141no—
174C22⋊C4—2141no—
175C22⋊C4C4×D10711yescharacteristic
176C42—4281no—
177C42—2141no—
178C23—1071no—
179C23C2×C4×D10711yescharacteristic
180C23D4×D10711yescharacteristic
181Q8—4281no—
182Q8—4281no—
183Q8—4281no—
184Q8—4282no—
185D4—41no—
186D4—4281no—
187D4—4281no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—1071no—
193C2×C4—1071no—
194C2×C4—1071no—
195C2×C4—4281no—
196C2×C4—4281no—
197C2×C4—2141no—
198C2×C4—2141no—
199C2×C4—2141no—
200C2×C4—2141no—
201C2×C4—2141no—
202C2×C4—2141no—
203C2×C4—2141no—
204C2×C4—2141no—
205C2×C4D4×D10711yescharacteristic
206C8—41no—
207C8—4281no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—4281no—
213C22—2141no—
214C22C22⋊C4×D10711yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—8561no—
220C4—1071no—
221C4—1071no—
222C4—4281no—
223C4—4281no—
224C4—2141no—
225C4—2141no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—4281no—
230C2#99×D10711yescharacteristic
subgroupG/H
centerC2#99×D107
commutatorC23×C107C22×C4
FrattiniC2×D4C22×D107
Fitting#352×C107C2
socleC2×C107C2×#99
prime radicalC2×#13133817C22
cosocleC2×D4×C107C23
hypercenter#352D107
hypocenterC107#2123
LCSG ▷ C23×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #20292418C2semidirect
5. #20292417C2semidirect
6. #20292416C2semidirect
7. #352×C107C2semidirect
9. C22×Dic214C4semidirect
13. #16812093C4semidirect
14. #16812092C4semidirect
19. #16812065C4semidirect
116. C107#2123semidirect
123. #352D107semidirect

Maximal subgroups

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

subgroupindex
#202931992
#202926082
#202925992
#202924182
#202924172
#202924162
#352×C1072
#2123107