Finite Group Databasebeta

← #28336743

Subgroups — #28336743

Order 13696.

374 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 83 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#20293100C211yesmaximal, characteristic
2C2×#16812023C211yesmaximal, characteristic
3#20292504C211yesmaximal, characteristic
4#20292502C211yesmaximal, characteristic, complemented
5#20292499C211yesmaximal, characteristic, complemented
6#20292393C211yesmaximal, characteristic, complemented
7#342×C107C211yesmaximal, characteristic, complemented
8C22×C22⋊D107C2211yescharacteristic
9C23×Dic107C2211yescharacteristic
10C4×C22⋊D107—21no—
11C4×C22⋊D107—21no—
12(C22⋊D107)⋊C4—21no—
13(C22⋊D107)⋊C4—21no—
14#16812065—21no—
15C2×C4⋊Dic107—21no—
16C2×Dic107⋊C4—21no—
17C2×C22⋊Dic107C2211yescharacteristic
18C2×C22⋊Dic107C2211yescharacteristic
19C2×(C2×D107)⋊C4—21no—
20C22⋊C4×D107—21no—
21#16812023C414yescomplemented
22C107⋊#95—21no—
23C107⋊#95—21no—
24C107⋊#95—21no—
25C2×C22⋊C4×C107C2211yescharacteristic
26C2×C22⋊C4×C107C2211yescharacteristic
27C2×C22⋊C4×C107C2211yescharacteristic
28C23×D107D411yescharacteristic
29C2×C22⋊D107—21no—
30C2×C22⋊D107—21no—
31C2×C22⋊D107—21no—
32C2×C22⋊D107—21no—
33C2×C22⋊D107C2×C412yes—
34C2×C22⋊D107Q812yes—
35C22×Dic107—21no—
36C22×Dic107—21no—
37C22×Dic107—21no—
38C22×Dic107—21no—
39C22×Dic107—42no—
40C22×Dic107D411yescharacteristic
41C22×Dic107D411yescharacteristic
42C22×Dic107D411yescharacteristic
43C2×C4×D107—21no—
44C2×C4×D107—21no—
45C4⋊Dic107—42no—
46Dic107⋊C4—21no—
47Dic107⋊C4—21no—
48Dic107⋊C4—42no—
49C22⋊Dic107—21no—
50C22⋊Dic107—21no—
51C22⋊Dic107—21no—
52C22⋊Dic107C2×C412yescomplemented
53(C2×D107)⋊C4—21no—
54(C2×D107)⋊C4—21no—
55(C2×D107)⋊C4—42no—
56C4×Dic107—21no—
57C4×Dic107—21no—
58C24×C107C2311yescharacteristic
59C22×C4×C107—21no—
60C22×C4×C107—21no—
61C22×C4×C107D411yescharacteristic
62C22×C4×C107D411yescharacteristic
63C22⋊C4×C107—21no—
64C22⋊C4×C107—21no—
65C22⋊C4×C107—21no—
66C22⋊C4×C107—21no—
67C22⋊C4×C107—21no—
68C22⋊C4×C107C2×C412yescomplemented
69C23×C107—41no—
70C23×C107—41no—
71C23×C107C22×C411yescharacteristic
72C23×C107C2×D411yescharacteristic
73C23×C107C2×D411yescharacteristic
74C23×C107C2×D411yescharacteristic
75C23×C107C2×Q811yescharacteristic
76C23×C107#4111yescharacteristic
77C23×C107#4111yescharacteristic
78C22×D107—21no—
79C22×D107—21no—
80C22×D107—41no—
81C22×D107—41no—
82C22×D107C4⋊C412yescomplemented
83C2×C4×C107—22no—
84C2×C4×C107—21no—
85C2×C4×C107—21no—
86C2×C4×C107—41no—
87C2×C4×C107—41no—
88C2×C4×C107—41no—
89C2×C4×C107—41no—
90C2×C4×C107—41no—
91C2×C4×C107—41no—
92C2×C4×C107—41no—
93C22⋊D107—22no—
94C22⋊D107—22no—
95C22⋊D107—22no—
96C22⋊D107—22no—
97C2×Dic107—21no—
98C2×Dic107—21no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—21no—
104C2×Dic107—21no—
105C2×Dic107—41no—
106C2×Dic107—41no—
107C2×Dic107—41no—
108C2×Dic107—41no—
109C2×Dic107—41no—
110C2×Dic107—41no—
111C2×Dic107—41no—
112C2×Dic107—41no—
113C2×Dic107C4⋊C412yescomplemented
114C4×D107—81no—
115C4×D107—81no—
116C22×C107—21no—
117C22×C107—21no—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107—21no—
121C22×C107—21no—
122C22×C107—41no—
123C22×C107—41no—
124C22×C107—41no—
125C22×C107—41no—
126C22×C107C2×C4⋊C411yescharacteristic
127C22×C107C4×D411yescharacteristic
128C22×C107C4×D411yescharacteristic
129C22×C107#12011yescharacteristic
130C22×C107C22⋊Q811yescharacteristic
131C22×C107C22⋊Q811yescharacteristic
132C22×C107#12311yescharacteristic
133C2×D107—22no—
134C2×D107—41no—
135C2×D107—41no—
136C2×D107—42no—
137C2×D107—81no—
138C4×C107—42no—
139C4×C107—41no—
140C4×C107—41no—
141C4×C107—81no—
142Dic107—22no—
143Dic107—42no—
144Dic107—41no—
145Dic107—42no—
146Dic107—41no—
147C2×C107—21no—
148C2×C107—21no—
149C2×C107—21no—
150C2×C107—21no—
151C2×C107—21no—
152C2×C107—21no—
153C2×C107#38511yescharacteristic
154C2×C107B(4,𝔽2)11yescharacteristic
155C2×C107#45811yescharacteristic
156D107—42no—
157#1867—1071nomaximal, Sylow
158C107#186711yescharacteristic, complemented, Sylow
159#518—1071no—
160C2×#99—1071no—
161#386—1071no—
162#385—1071no—
163#385—1071no—
164#342—1071no—
165#342D10711yescharacteristic, complemented
166C22×D4—1071no—
167C23×C4—1071no—
168C4×D4—2141no—
169C4×D4—2141no—
170C4×D4—2141no—
171C4×D4—2141no—
172#117—2141no—
173C2×C4⋊C4—2141no—
174C2×C4⋊C4—2141no—
175C2×C22⋊C4—1071no—
176C2×C22⋊C4—1071no—
177C2×C22⋊C4—2141no—
178C2×C22⋊C4—2141no—
179C2×C22⋊C4C2×D10711yescharacteristic
180C2×C22⋊C4C2×D10711yescharacteristic
181C2×C22⋊C4C2×D10711yescharacteristic
182#99—1074no—
183#95—2141no—
184#95—2141no—
185#95—2141no—
186C24—1071no—
187C24C22×D10711yescharacteristic
188C2×D4—1072no—
189C2×D4—1072no—
190C2×D4—2141no—
191C2×D4—2141no—
192C2×D4—2141no—
193C2×D4—2141no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—1071no—
197C22×C4—1071no—
198C22×C4—1071no—
199C22×C4—4282no—
200C22×C4—2141no—
201C22×C4—2141no—
202C22×C4—2141no—
203C22×C4—2141no—
204C22×C4—2141no—
205C22×C4—2141no—
206C22×C4D42811yescharacteristic
207C22×C4D42811yescharacteristic
208C4⋊C4—4282no—
209C4⋊C4—4282no—
210C4⋊C4—2141no—
211C4⋊C4—2141no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—1072no—
218C22⋊C4—4282no—
219C22⋊C4—2141no—
220C22⋊C4—2141no—
221C22⋊C4—2141no—
222C22⋊C4—2141no—
223C22⋊C4—2141no—
224C22⋊C4C4×D10712yescomplemented
225C42—2141no—
226C42—2141no—
227C23—41no—
228C23—41no—
229C23—1072no—
230C23—4281no—
231C23—4281no—
232C23—2141no—
233C23—2141no—
234C23C2×C4×D10711yescharacteristic
235C23D4×D10711yescharacteristic
236C23D4×D10711yescharacteristic
237C23C2×D42811yescharacteristic
238C23Q8×D10711yescharacteristic
239C23#1313381711yescharacteristic
240C23#1313381811yescharacteristic
241D4—2142no—
242D4—2142no—
243D4—2142no—
244D4—2142no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—8561no—
256C2×C4—8561no—
257C2×C4—1072no—
258C2×C4—4281no—
259C2×C4—4281no—
260C2×C4—4281no—
261C2×C4—4281no—
262C2×C4—4281no—
263C2×C4—4281no—
264C2×C4—4281no—
265C2×C4—4281no—
266C2×C4—2142no—
267C2×C4—2142no—
268C2×C4—2142no—
269C2×C4—2142no—
270C2×C4—2141no—
271C2×C4—2141no—
272C2×C4—2141no—
273C2×C4—2141no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—8561no—
285C22—4282no—
286C22—4281no—
287C22—4281no—
288C22—2142no—
289C22C4⋊C4×D10711yescharacteristic
290C22(C22⋊D107)⋊C411yescharacteristic
291C22C4×D42811yescharacteristic
292C22#1681207911yescharacteristic
293C22#1681208711yescharacteristic
294C22#1681208911yescharacteristic
295C22#1681209111yescharacteristic
296C4—42no—
297C4—41no—
298C4—41no—
299C4—81no—
300C4—4282no—
301C4—4281no—
302C4—4282no—
303C4—4281no—
304C4—2142no—
305C2—21no—
306C2—21no—
307C2—21no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2—4282no—
312C2#2029250111yescharacteristic
313C2#2029281511yescharacteristic
314C2#2029281811yescharacteristic
subgroupG/H
centerC22#16812079
commutatorC23×C107C22×C4
FrattiniC24C22×D107
Fitting#342×C107C2
socleC22×C107#120
prime radicalC22×C22⋊D107C22
cosocleC24×C107C23
hypercenter#342D107
hypocenterC107#1867
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
4. #20292502C2semidirect
5. #20292499C2semidirect
6. #20292393C2semidirect
7. #342×C107C2semidirect
21. #16812023C4semidirect
52. C22⋊Dic107C2×C4semidirect
68. C22⋊C4×C107C2×C4semidirect
82. C22×D107C4⋊C4semidirect
113. C2×Dic107C4⋊C4semidirect
158. C107#1867semidirect
165. #342D107semidirect
224. C22⋊C4C4×D107semidirect

Maximal subgroups

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

subgroupindex
#202931002
C2×#168120232
#202925042
#202925022
#202924992
#202923932
#342×C1072
#1867107