Finite Group Databasebeta

← #28366390

Subgroups — #28366390

Order 13952.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D109C211yesmaximal, characteristic, complemented
2C2×#16812676C212yesmaximal, complemented
3C2×#16812659C211yesmaximal, characteristic
4#20294947C211yesmaximal, characteristic, complemented
5#20294945C211yesmaximal, characteristic, complemented
6#20294938C211yesmaximal, characteristic, complemented
7#20294930C212yesmaximal, complemented
8#20294902C212yesmaximal
9#20294900C212yesmaximal, complemented
10#20294885C211yesmaximal, characteristic, complemented
11#393×C109C211yesmaximal, characteristic, complemented
12C22×Dic218C2211yescharacteristic
13C22×Dic218C2212yes—
14C2×Q8×D109—28no—
15C2×Q8×D109—24no—
16C22×C22⋊D109C2211yescharacteristic
17C23×Dic109C2211yescharacteristic
18C22×C4×D109C2212yes—
19C22×C4×D109C2211yescharacteristic
20#16812676—28no—
21#16812659—24no—
22C2×C4⋊Dic109C2212yes—
23C2×#13182008C2212yes—
24C2×#13182008C2211yescharacteristic
25C2×#13182008C2212yes—
26C2×C22⋊Dic109C2212yes—
27C2×#13182006C2211yescharacteristic
28C2×#13182006C2212yes—
29C2×C4×Dic109C2211yescharacteristic
30#16812606C2212yes—
31#16812606C2212yes—
32#16812606C2212yes—
33#16812606C2211yescharacteristic
34C22×Q8×C109C2211yescharacteristic
35C2×C22⋊C4×C109C2211yescharacteristic
36C2×C22⋊C4×C109C2212yes—
37#95×C109C2212yes—
38#95×C109C2211yescharacteristic
39C23×D109C2311yescharacteristic
40C2×Dic218—22no—
41C2×Dic218—24no—
42C2×Dic218—24no—
43C2×Dic218—24no—
44C2×Dic218D418yescomplemented
45Q8×D109—232no—
46C2×C22⋊D109—22no—
47C2×C22⋊D109—22no—
48C2×C22⋊D109Q814yescomplemented
49C22×Dic109—22no—
50C22×Dic109—24no—
51C22×Dic109C2312yes—
52C22×Dic109C2311yescharacteristic
53C22×Dic109C2312yes—
54C22×Dic109C2311yescharacteristic
55C22×Dic109C2311yescharacteristic
56C2×C4×D109—24no—
57C2×C4×D109—24no—
58C2×C4×D109—24no—
59C2×C4×D109—24no—
60C2×C4×D109D414yes—
61C4⋊Dic109—24no—
62#13182008—22no—
63#13182008—24no—
64#13182008—24no—
65C22⋊Dic109—24no—
66#13182006—22no—
67#13182006—24no—
68C4×Dic109—22no—
69C24×C109C2311yescharacteristic
70C2×Q8×C109—24no—
71C2×Q8×C109—22no—
72C22×C4×C109C2311yescharacteristic
73C22×C4×C109C2312yes—
74C22×C4×C109C2312yes—
75C22×C4×C109C2311yescharacteristic
76C22⋊C4×C109—22no—
77C22⋊C4×C109—24no—
78C23×C109—22no—
79C23×C109—21no—
80C23×C109—21no—
81C23×C109—22no—
82C23×C109C2×Q812yescomplemented
83C23×C109C2411yescharacteristic
84C22×D109—22no—
85C22×D109—22no—
86C22×D109C2×D412yes—
87C22×D109C2×Q812yes—
88C22×D109#4112yes—
89Q8×C109—28no—
90C2×C4×C109—22no—
91C2×C4×C109—22no—
92C2×C4×C109—22no—
93C2×C4×C109—21no—
94C2×C4×C109—22no—
95C2×C4×C109—22no—
96C2×C4×C109—22no—
97C2×C4×C109—22no—
98C2×C4×C109C2×D414yes—
99C2×C4×C109C2×D412yes—
100C22⋊D109—28no—
101C2×Dic109—21no—
102C2×Dic109—22no—
103C2×Dic109—22no—
104C2×Dic109—21no—
105C2×Dic109—21no—
106C2×Dic109—22no—
107C2×Dic109—22no—
108C2×Dic109—22no—
109C2×Dic109—22no—
110C2×Dic109—44no—
111C2×Dic109C2×D414yes—
112C2×Dic109C2×D412yes—
113C2×Dic109C2×D414yes—
114C2×Dic109C2×Q812yescomplemented
115C4×D109—216no—
116C4×D109—28no—
117Dic218—28no—
118Dic218—216no—
119C22×C109—22no—
120C22×C109—24no—
121C22×C109—41no—
122C22×C109—41no—
123C22×C109—42no—
124C22×C109C22×D411yescharacteristic
125C22×C109C22×D412yes—
126C22×C109C22×Q811yescharacteristic
127C22×C109C2×#4111yescharacteristic
128C22×C1092−1+412yes—
129C2×D109—22no—
130C2×D109—22no—
131C2×D109—24no—
132C2×D109—24no—
133C2×D109C22⋊Q814yescomplemented
134C4×C109—22no—
135C4×C109—24no—
136C4×C109—41no—
137C4×C109—42no—
138Dic109—24no—
139Dic109—22no—
140Dic109—41no—
141Dic109—42no—
142Dic109#12014yes—
143C2×C109—42no—
144C2×C109C2×#12011yescharacteristic
145C2×C109C2×C22⋊Q811yescharacteristic
146C2×C109#53611yescharacteristic
147C2×C109#54812yes—
148C2×C109D4×Q812yes—
149D109—24no—
150#2409—1091nomaximal, Sylow
151C109#240911yescharacteristic, complemented, Sylow
152C23×Q8—1091no—
153C2×C22⋊Q8—1092no—
154C2×C4×D4—1091no—
155#395—1091no—
156#395—1091no—
157#393—1091no—
158#393D10911yescharacteristic, complemented
159#391—1092no—
160#386—1092no—
161#385—1092no—
162#380—1091no—
163C22×Q8—1092no—
164C22×Q8—1091no—
165C22×Q8—2184no—
166C22×Q8—2188no—
167C22×Q8C2×D10911yescharacteristic
168C22×D4—1091no—
169C23×C4—1091no—
170C23×C4—1091no—
171C23×C4—1092no—
172C22⋊Q8—2188no—
173C4×D4—2184no—
174C2×C4⋊C4—1091no—
175C2×C4⋊C4—1092no—
176C2×C4⋊C4—1092no—
177C2×C4⋊C4—1092no—
178C2×C22⋊C4—1092no—
179C2×C22⋊C4—1091no—
180C2×C22⋊C4—1092no—
181C2×C22⋊C4C2×D10911yescharacteristic
182C2×C22⋊C4C2×D10912yes—
183C2×C42—1091no—
184#95—1091no—
185#95—1092no—
186#95—1092no—
187#95—1092no—
188#95C2×D10911yescharacteristic
189#95C2×D10912yes—
190C24—1091no—
191C24C22×D10911yescharacteristic
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—1098no—
195C2×Q8—2184no—
196C2×Q8—2184no—
197C2×Q8—2184no—
198C2×Q8—21832no—
199C2×Q8—2182no—
200C2×D4—1094no—
201C2×D4—2182no—
202C2×D4—2182no—
203C22×C4—1091no—
204C22×C4—1091no—
205C22×C4—1092no—
206C22×C4—1094no—
207C22×C4—1092no—
208C22×C4—1091no—
209C22×C4—2184no—
210C22×C4—2184no—
211C22×C4—2184no—
212C22×C4—2182no—
213C22×C4—2184no—
214C22×C4—2184no—
215C22×C4C22×D10911yescharacteristic
216C22×C4C22×D10912yes—
217C22×C4C22×D10911yescharacteristic
218C22×C4C22×D10912yes—
219C4⋊C4—2184no—
220C4⋊C4—2184no—
221C4⋊C4—2182no—
222C4⋊C4—2184no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—2184no—
226C22⋊C4—2182no—
227C22⋊C4—2184no—
228C42—2182no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—1092no—
234C23—1092no—
235C23—1092no—
236C23—2182no—
237C23—2182no—
238C23Q8×D10912yescomplemented
239C23C23×D10911yescharacteristic
240Q8—28no—
241Q8—21816no—
242Q8—2188no—
243D4—2188no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—1092no—
253C2×C4—1094no—
254C2×C4—1094no—
255C2×C4—1092no—
256C2×C4—4364no—
257C2×C4—2188no—
258C2×C4—21816no—
259C2×C4—2182no—
260C2×C4—2181no—
261C2×C4—2182no—
262C2×C4—2182no—
263C2×C4—2182no—
264C2×C4—2181no—
265C2×C4—2182no—
266C2×C4—2182no—
267C2×C4—2181no—
268C2×C4D4×D10912yes—
269C2×C4D4×D10914yes—
270C22—24no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—1094no—
276C22—2184no—
277C22—2184no—
278C22—2182no—
279C22—2182no—
280C22C2×D4×D10912yes—
281C22C2×D4×D10911yescharacteristic
282C22C2×Q8×D10911yescharacteristic
283C22#41×D10911yescharacteristic
284C22#1681274612yes—
285C4—22no—
286C4—24no—
287C4—42no—
288C4—41no—
289C4—1094no—
290C4—4362no—
291C4—4361no—
292C4—2184no—
293C4—2182no—
294C2—42no—
295C2—2184no—
296C2#120×D10911yescharacteristic
297C2C22⋊Q8×D10911yescharacteristic
298C2#2029559611yescharacteristic
299C2#2029568412yes—
300C2#2029568912yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#393×C109C2
prime radicalC22×C22⋊D109C22
hypercenter#393D109
hypocenterC109#2409
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D109C2semidirect
2. C2×#16812676C2semidirect
4. #20294947C2semidirect
5. #20294945C2semidirect
6. #20294938C2semidirect
7. #20294930C2semidirect
9. #20294900C2semidirect
10. #20294885C2semidirect
11. #393×C109C2semidirect
44. C2×Dic218D4semidirect
48. C2×C22⋊D109Q8semidirect
82. C23×C109C2×Q8semidirect
114. C2×Dic109C2×Q8semidirect
133. C2×D109C22⋊Q8semidirect
151. C109#2409semidirect
158. #393D109semidirect
238. C23Q8×D109semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1092
C2×#168126762
C2×#168126592
#202949472
#202949452
#202949382
#202949302
#202949022
#202949002
#202948852
#393×C1092
#2409109