Finite Group Databasebeta

← #28337526

Subgroups — #28337526

Order 13696.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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#20293425C211yesmaximal, characteristic, complemented
2#20293138C211yesmaximal, characteristic, complemented
3#20292645C211yesmaximal, characteristic, complemented
4#20292644C211yesmaximal, characteristic, complemented
5#20292591C211yesmaximal, characteristic, complemented
6#20292383C211yesmaximal, characteristic, complemented
7#525×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812140—21no—
10#16812139—21no—
11#16812135—21no—
12#16812134—21no—
13#16812129—22no—
14C2×#13133802C2211yescharacteristic
15C2×#13133802C2211yescharacteristic
16#16812092—21no—
17#16812088—21no—
18#16812083—21no—
19#16812078—21no—
20C4×C22⋊D107—22no—
21#16812068C2211yescharacteristic
22#16812037—22no—
23#16812036—22no—
24C22⋊(C107⋊C8)—22no—
25C2×#41×C107C2211yescharacteristic
26C23×C4×C107C2211yescharacteristic
27#123×C107—21no—
28C22⋊Q8×C107—21no—
29#121×C107—21no—
30#120×C107—21no—
31C4×D4×C107—22no—
32#117×C107C2211yescharacteristic
33#13133818—21no—
34#13133818—21no—
35#13133818—21no—
36#13133818—21no—
37C2×Dic214D411yescharacteristic, complemented
38C2×D428D411yescharacteristic, complemented
39C2×C22⋊D107—22no—
40C2×C22⋊D107—21no—
41C2×C4×D107—21no—
42#13133808—22no—
43#13133806—22no—
44#13133805—22no—
45#13133803—22no—
46#13133802—21no—
47#13133802—21no—
48#13133802—21no—
49#13133802—21no—
50C2×C107⋊C8—21no—
51C2×C107⋊C8D412yescomplemented
52C4⋊Dic107—21no—
53Dic107⋊C4—22no—
54C22⋊Dic107—22no—
55C22⋊Dic107—21no—
56(C2×D107)⋊C4—22no—
57C4×Dic107D412yescomplemented
58C24×C107D411yescharacteristic
59#41×C107—21no—
60#41×C107—21no—
61#41×C107—21no—
62#41×C107—21no—
63C2×Q8×C107D411yescharacteristic, complemented
64C2×D4×C107—22no—
65C2×D4×C107—21no—
66C2×D4×C107D411yescharacteristic, complemented
67C22×C4×C107—21no—
68C22×C4×C107—21no—
69C22×C4×C107—21no—
70C22×C4×C107—21no—
71C22×C4×C107—21no—
72C22×C4×C107—21no—
73C22×C4×C107—21no—
74C22×C4×C107D411yescharacteristic
75C22×C4×C107C2311yescharacteristic
76C4⋊C4×C107—22no—
77C4⋊C4×C107—21no—
78C22⋊C4×C107—22no—
79C22⋊C4×C107—22no—
80C22⋊C4×C107—21no—
81C42×C107D412yescomplemented
82C23×C107—21no—
83C23×C107—21no—
84C23×C107—21no—
85C23×C107—21no—
86C23×C107—42no—
87C23×C107C2×D411yescharacteristic
88C22×D107—21no—
89Q8×C107—21no—
90Q8×C107—21no—
91D4×C107—21no—
92D4×C107—21no—
93D4×C107—44no—
94D4×C107—41no—
95D4×C107—41no—
96C2×C4×C107—22no—
97C2×C4×C107—21no—
98C2×C4×C107—21no—
99C2×C4×C107—21no—
100C2×C4×C107—21no—
101C2×C4×C107—41no—
102C2×C4×C107—41no—
103C2×C4×C107—41no—
104C2×C4×C107—41no—
105C2×C4×C107—41no—
106C2×C4×C107—41no—
107C2×C4×C107C2×D411yescharacteristic
108C2×C4×C107C2×D411yescharacteristic
109C2×C4×C107C2×D411yescharacteristic
110C2×C4×C107C2×D411yescharacteristic
111C2×C4×C107C2×D411yescharacteristic
112C2×C4×C107#4111yescharacteristic
113C22⋊D107—44no—
114C22⋊D107—41no—
115C22⋊D107—41no—
116C2×Dic107—22no—
117C2×Dic107—21no—
118D428—21no—
119D428—21no—
120C4×D107—41no—
121C4×D107—41no—
122Dic214—21no—
123Dic214—21no—
124C107⋊C8—22no—
125C107⋊C8—41no—
126C22×C107—22no—
127C22×C107—22no—
128C22×C107—22no—
129C22×C107—41no—
130C22×C107—41no—
131C22×C107—41no—
132C22×C107—41no—
133C22×C107—42no—
134C22×C107—41no—
135C22×C107—41no—
136C22×C107#12011yescharacteristic
137C22×C107#12011yescharacteristic
138C22×C107#12111yescharacteristic
139C2×D107—42no—
140C2×D107—41no—
141C4×C107—42no—
142C4×C107—41no—
143C4×C107—41no—
144C4×C107#12011yescharacteristic
145C4×C107#12111yescharacteristic
146C4×C107#12111yescharacteristic
147C4×C107#12711yescharacteristic
148Dic107—42no—
149Dic107—41no—
150C2×C107—21no—
151C2×C107—21no—
152C2×C107—21no—
153C2×C107—42no—
154C2×C107—81no—
155C2×C107#39211yescharacteristic
156D107—81no—
157#2010—1071nomaximal, Sylow
158C107#201011yescharacteristic, complemented, Sylow
159#575—1071no—
160#525—1071no—
161#525D10711yescharacteristic, complemented
162#421—1071no—
163#421—1071no—
164#407—1071no—
165H(3,ℤ/4)—1071no—
166C2×#41—1071no—
167C2×#41C2×D10711yescharacteristic
168C23×C4C2×D10711yescharacteristic
169#137—2141no—
170#137—2141no—
171AGL(1,ℤ/8)—2141no—
172AGL(1,ℤ/8)—2141no—
173#135—2142no—
174C2×M16—1071no—
175C2×M16—1071no—
176#123—21no—
177#123—2141no—
178C22⋊Q8—21no—
179C22⋊Q8—2141no—
180#121—21no—
181#121—2141no—
182#120—21no—
183#120—2141no—
184C4×D4—22no—
185C4×D4—2142no—
186#117—1071no—
187#117C2×D10711yescharacteristic
188#104—2142no—
189#104—2142no—
190C22⋊C8—2142no—
191C24C22⋊D10711yescharacteristic
192#41—21no—
193#41—21no—
194#41—21no—
195#41—21no—
196#41—2141no—
197#41—2141no—
198#41—2141no—
199#41—2141no—
200C2×Q8—1071no—
201C2×Q8C22⋊D10711yescharacteristic, complemented
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—1071no—
205C2×D4—2142no—
206C2×D4—2141no—
207C2×D4C22⋊D10711yescharacteristic, complemented
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—2141no—
216C22×C4C22⋊D10711yescharacteristic
217C22×C4C22×D10711yescharacteristic
218Q16—2142no—
219SD16—2142no—
220SD16—2142no—
221D8—2142no—
222M16—2141no—
223M16—2141no—
224M16—2141no—
225M16—2141no—
226C2×C8—1072no—
227C2×C8—2141no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—2142no—
231C4⋊C4—2141no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—21no—
235C22⋊C4—2142no—
236C22⋊C4—2142no—
237C22⋊C4—2141no—
238C42—1072no—
239C42C22⋊D10712yescomplemented
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—42no—
245C23—2141no—
246C23C2×C22⋊D10711yescharacteristic
247Q8—21no—
248Q8—21no—
249Q8—2141no—
250Q8—2141no—
251D4—21no—
252D4—21no—
253D4—44no—
254D4—41no—
255D4—41no—
256D4—4284no—
257D4—4281no—
258D4—4281no—
259D4—2141no—
260D4—2141no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—4281no—
273C2×C4—4281no—
274C2×C4—2142no—
275C2×C4—2141no—
276C2×C4D4×D10711yescharacteristic
277C2×C4D4×D10711yescharacteristic
278C2×C4D4×D10711yescharacteristic
279C2×C4C2×C22⋊D10711yescharacteristic
280C2×C4C2×C22⋊D10711yescharacteristic
281C2×C4#1313381711yescharacteristic
282C8—4281no—
283C8—2142no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—4282no—
295C22—4281no—
296C22#1681207711yescharacteristic
297C22#1681207811yescharacteristic
298C22#1681208111yescharacteristic
299C4—42no—
300C4—41no—
301C4—41no—
302C4—4282no—
303C4—4281no—
304C4#1681207711yescharacteristic
305C4#1681208011yescharacteristic
306C4#1681208111yescharacteristic
307C4#1681210511yescharacteristic
308C2—21no—
309C2—21no—
310C2—21no—
311C2—42no—
312C2—81no—
313C2—8561no—
314C2#2029253511yescharacteristic
subgroupG/H
centerC4#16812105
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#525×C107C2
socleC2×C107#392
hypercenter#525D107
hypocenterC107#2010
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #20293425C2semidirect
2. #20293138C2semidirect
3. #20292645C2semidirect
4. #20292644C2semidirect
5. #20292591C2semidirect
6. #20292383C2semidirect
7. #525×C107C2semidirect
37. C2×Dic214D4semidirect
38. C2×D428D4semidirect
51. C2×C107⋊C8D4semidirect
57. C4×Dic107D4semidirect
63. C2×Q8×C107D4semidirect
66. C2×D4×C107D4semidirect
81. C42×C107D4semidirect
158. C107#2010semidirect
161. #525D107semidirect
201. C2×Q8C22⋊D107semidirect
207. C2×D4C22⋊D107semidirect
239. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202934252
#202931382
#202926452
#202926442
#202925912
#202923832
#525×C1072
#2010107