Finite Group Databasebeta

← #28366415

Subgroups — #28366415

Order 13952.

786 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×#16812683C211yesmaximal, characteristic, complemented
2C2×#16812669C211yesmaximal, characteristic
3C2×#16812655C211yesmaximal, characteristic, complemented
4C22×#13182006C211yesmaximal, characteristic, complemented
5#20294948C212yesmaximal, complemented
6#20294944C212yesmaximal, complemented
7#20294927C212yesmaximal, complemented
8#20294918C212yesmaximal, complemented
9#20294905C211yesmaximal, characteristic, complemented
10#20294881C211yesmaximal, characteristic, complemented
11#385×C109C211yesmaximal, characteristic, complemented
12C24×D109C2211yescharacteristic
13C22×D436C2212yes—
14C22×C22⋊D109C2211yescharacteristic
15C23×Dic109C2211yescharacteristic
16#16812683—24no—
17#16812669—24no—
18#16812655—24no—
19C2×C4⋊Dic109C2211yescharacteristic, complemented
20C2×C4⋊Dic109C2211yescharacteristic, complemented
21C2×#13182008C2212yescomplemented
22C2×C22⋊Dic109C2211yescharacteristic
23C2×#13182006—24no—
24C2×#13182006C2212yes—
25C2×#13182006C2212yes—
26C2×#13182006C2212yes—
27C2×#13182006C2212yes—
28C2×#13182006C2212yes—
29C2×#13182006C2211yescharacteristic, complemented
30C2×#13182006C2211yescharacteristic, complemented
31C2×#13182006C2212yescomplemented
32C2×C4×Dic109C2212yescomplemented
33#16812606C2212yescomplemented
34C23×C4×C109C2211yescharacteristic
35C2×C4⋊C4×C109C2212yescomplemented
36C2×C22⋊C4×C109C2211yescharacteristic
37C2×C22⋊C4×C109C2211yescharacteristic
38#95×C109C2212yescomplemented
39C23×D109—22no—
40C23×D109—24no—
41C23×D109—22no—
42C23×D109—22no—
43C23×D109—42no—
44C23×D109C2311yescharacteristic
45C23×D109C2312yes—
46C2×D436—24no—
47C2×D436—24no—
48C2×D436—42no—
49C2×C22⋊D109—22no—
50C2×C22⋊D109—22no—
51C2×C22⋊D109—22no—
52C22×Dic109—22no—
53C22×Dic109—22no—
54C22×Dic109D414yescomplemented
55C22×Dic109C2311yescharacteristic
56C22×Dic109C2312yes—
57C22×Dic109C2311yescharacteristic
58C22×Dic109C2311yescharacteristic
59C4⋊Dic109—22no—
60C4⋊Dic109—22no—
61#13182008—24no—
62C22⋊Dic109—22no—
63#13182006—24no—
64#13182006—24no—
65#13182006—24no—
66#13182006—24no—
67#13182006—22no—
68#13182006—22no—
69#13182006—42no—
70#13182006—42no—
71C4×Dic109—24no—
72C24×C109C2311yescharacteristic
73C22×C4×C109—22no—
74C22×C4×C109—42no—
75C22×C4×C109C2311yescharacteristic
76C22×C4×C109C2312yes—
77C22×C4×C109C2311yescharacteristic
78C22×C4×C109C2312yes—
79C4⋊C4×C109—42no—
80C22⋊C4×C109—22no—
81C22⋊C4×C109D414yes—
82C23×C109—22no—
83C23×C109—22no—
84C23×C109C2×D412yes—
85C23×C109C2×D412yes—
86C23×C109#4112yes—
87C23×C109C2411yescharacteristic
88C22×D109—28no—
89C22×D109—22no—
90C22×D109—22no—
91C22×D109—22no—
92C22×D109—22no—
93C22×D109—22no—
94C22×D109—22no—
95C22×D109—21no—
96C22×D109—21no—
97C22×D109—21no—
98C22×D109—22no—
99C22×D109—22no—
100C22×D109—48no—
101C22×D109—48no—
102C22×D109—44no—
103C22×D109—44no—
104C22×D109#4114yes—
105C2×C4×C109—22no—
106C2×C4×C109—22no—
107C2×C4×C109—21no—
108C2×C4×C109—21no—
109C2×C4×C109—22no—
110C2×C4×C109—22no—
111C2×C4×C109—22no—
112C2×C4×C109—22no—
113C2×C4×C109—22no—
114C2×C4×C109—44no—
115C2×C4×C109C2×D414yes—
116C22⋊D109—44no—
117C2×Dic109—28no—
118C2×Dic109—21no—
119C2×Dic109—21no—
120C2×Dic109—21no—
121C2×Dic109—21no—
122C2×Dic109—22no—
123C2×Dic109—21no—
124C2×Dic109—21no—
125C2×Dic109—22no—
126C2×Dic109—21no—
127C2×Dic109—22no—
128C2×Dic109C2×D412yes—
129C2×Dic109C2×D412yes—
130D436—48no—
131C22×C109—22no—
132C22×C109—22no—
133C22×C109—24no—
134C22×C109—24no—
135C22×C109#12114yes—
136C22×C109C22×D411yescharacteristic
137C22×C109C22×D411yescharacteristic
138C22×C109C2×#4112yes—
139C22×C109C2×#4111yescharacteristic
140C22×C109H(5,𝔽2)12yes—
141C2×D109—24no—
142C2×D109—24no—
143C2×D109—24no—
144C2×D109—416no—
145C2×D109—42no—
146C2×D109—42no—
147C2×D109—42no—
148C2×D109—42no—
149C2×D109—41no—
150C2×D109—41no—
151C2×D109—41no—
152C2×D109—42no—
153C2×D109—42no—
154C4×C109—24no—
155C4×C109—41no—
156C4×C109—41no—
157C4×C109—42no—
158Dic109—22no—
159Dic109—22no—
160Dic109—41no—
161Dic109—42no—
162C2×C109—24no—
163C2×C109C2×#12111yescharacteristic
164C2×C109#52511yescharacteristic
165C2×C109#53511yescharacteristic
166C2×C109#54612yes—
167C2×C109#55112yes—
168D109—44no—
169D109—81no—
170#2411—1091nomaximal, Sylow
171C109#241111yescharacteristic, complemented, Sylow
172C2×#123—1091no—
173C2×#120—1091no—
174C2×#117—1091no—
175C22×C22⋊C4—1091no—
176#396—1092no—
177#394—1092no—
178#390—1092no—
179#388—1092no—
180#386—1091no—
181#385D10911yescharacteristic, complemented
182#379—1091no—
183C25—1091no—
184C22×D4—1091no—
185C22×D4—1092no—
186C23×C4—1091no—
187C23×C4C2×D10911yescharacteristic
188#123—2184no—
189#120—2184no—
190#117—2184no—
191C2×C4⋊C4—1092no—
192C2×C4⋊C4—1091no—
193C2×C4⋊C4—1091no—
194C2×C4⋊C4C2×D10912yescomplemented
195C2×C22⋊C4—1092no—
196C2×C22⋊C4—1092no—
197C2×C22⋊C4—1091no—
198C2×C22⋊C4—1092no—
199C2×C22⋊C4—1092no—
200C2×C22⋊C4—1092no—
201C2×C22⋊C4—1091no—
202C2×C22⋊C4—1091no—
203C2×C22⋊C4—1092no—
204C2×C22⋊C4—2184no—
205C2×C22⋊C4C2×D10911yescharacteristic
206C2×C22⋊C4C2×D10911yescharacteristic
207C2×C42—1092no—
208#95—1092no—
209#95C2×D10912yescomplemented
210C24—1092no—
211C24—1091no—
212C24—4362no—
213C24—2184no—
214C24—2182no—
215C24—2182no—
216C24—2182no—
217C24C22×D10911yescharacteristic
218C2×D4—4362no—
219C2×D4—2184no—
220C2×D4—2182no—
221C2×D4—2182no—
222C2×D4—2182no—
223C2×D4—2184no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—1094no—
227C22×C4—1091no—
228C22×C4—1091no—
229C22×C4—1091no—
230C22×C4—1092no—
231C22×C4—2182no—
232C22×C4—2182no—
233C22×C4C22×D10912yes—
234C22×C4C22×D10911yescharacteristic
235C22×C4C22×D10911yescharacteristic
236C22×C4C22×D10912yes—
237C4⋊C4—42no—
238C4⋊C4—2182no—
239C4⋊C4—2182no—
240C4⋊C4—2184no—
241C22⋊C4—22no—
242C22⋊C4—4362no—
243C22⋊C4—4362no—
244C22⋊C4—2184no—
245C22⋊C4—2184no—
246C22⋊C4—2182no—
247C22⋊C4—2182no—
248C22⋊C4—2184no—
249C22⋊C4—2182no—
250C22⋊C4—2184no—
251C22⋊C4D43614yes—
252C42—2184no—
253C23—22no—
254C23—22no—
255C23—1094no—
256C23—4364no—
257C23—4368no—
258C23—4368no—
259C23—4364no—
260C23—2188no—
261C23—2181no—
262C23—2181no—
263C23—2182no—
264C23—2181no—
265C23—2182no—
266C23—2182no—
267C23—2182no—
268C23—2182no—
269C23—2182no—
270C23—2182no—
271C23—2182no—
272C23D4×D10912yes—
273C23C2×D43612yes—
274C23#1318203012yes—
275C23C23×D10911yescharacteristic
276D4—4368no—
277D4—4364no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—1092no—
289C2×C4—1092no—
290C2×C4—2181no—
291C2×C4—2182no—
292C2×C4—2181no—
293C2×C4—2182no—
294C2×C4—2181no—
295C2×C4—2182no—
296C2×C4—2188no—
297C2×C4—2181no—
298C2×C4—2181no—
299C2×C4—2181no—
300C2×C4—2181no—
301C2×C4C2×D43614yes—
302C22—24no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—43616no—
307C22—4362no—
308C22—4361no—
309C22—4361no—
310C22—4361no—
311C22—4362no—
312C22—4362no—
313C22—4362no—
314C22—4362no—
315C22—4362no—
316C22—2184no—
317C22—2184no—
318C22—2184no—
319C22#1681267414yes—
320C22C2×D4×D10911yescharacteristic
321C22C22×D43611yescharacteristic
322C22#41×D10912yes—
323C22C2×#1318203011yescharacteristic
324C22#1681274412yes—
325C4—24no—
326C4—41no—
327C4—42no—
328C4—41no—
329C4—4361no—
330C4—4362no—
331C4—2182no—
332C4—2182no—
333C2—24no—
334C2—8721no—
335C2—4364no—
336C2C2×#1681267411yescharacteristic
337C2#2029553411yescharacteristic
338C2#2029559111yescharacteristic
339C2#2029567112yes—
340C2#2029570312yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#385×C109C2
prime radicalC2×#16812669C2
hypercenter#385D109
hypocenterC109#2411
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#16812683C2semidirect
3. C2×#16812655C2semidirect
4. C22×#13182006C2semidirect
5. #20294948C2semidirect
6. #20294944C2semidirect
7. #20294927C2semidirect
8. #20294918C2semidirect
9. #20294905C2semidirect
10. #20294881C2semidirect
11. #385×C109C2semidirect
19. C2×C4⋊Dic109C22semidirect
20. C2×C4⋊Dic109C22semidirect
21. C2×#13182008C22semidirect
29. C2×#13182006C22semidirect
30. C2×#13182006C22semidirect
31. C2×#13182006C22semidirect
32. C2×C4×Dic109C22semidirect
33. #16812606C22semidirect
35. C2×C4⋊C4×C109C22semidirect
38. #95×C109C22semidirect
54. C22×Dic109D4semidirect
171. C109#2411semidirect
181. #385D109semidirect
194. C2×C4⋊C4C2×D109semidirect
209. #95C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126832
C2×#168126692
C2×#168126552
C22×#131820062
#202949482
#202949442
#202949272
#202949182
#202949052
#202948812
#385×C1092
#2411109