Finite Group Databasebeta

← #28366413

Subgroups — #28366413

Order 13952.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812680C211yesmaximal, characteristic, complemented
2C2×#16812655C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic109C211yesmaximal, characteristic, complemented
4#20294949C212yesmaximal, complemented
5#20294940C212yesmaximal, complemented
6#20294929C212yesmaximal, complemented
7#20294914C212yesmaximal, complemented
8#20294906C211yesmaximal, characteristic, complemented
9#20294901C211yesmaximal, characteristic, complemented
10#20294882C211yesmaximal, characteristic, complemented
11C2×#120×C109C211yesmaximal, characteristic
12C23×Dic109C2211yescharacteristic
13C23×Dic109C2211yescharacteristic
14#16812680—24no—
15#16812655—24no—
16C2×C4⋊Dic109C2212yescomplemented
17C2×#13182008C2211yescharacteristic, complemented
18C2×#13182008C2212yescomplemented
19C2×#13182008C2211yescharacteristic, complemented
20C2×C22⋊Dic109—24no—
21C2×C22⋊Dic109C2212yes—
22C2×C22⋊Dic109C2212yes—
23C2×C22⋊Dic109C2212yes—
24C2×C22⋊Dic109C2211yescharacteristic
25C2×C22⋊Dic109C2211yescharacteristic
26C2×C22⋊Dic109C2212yes—
27C2×C22⋊Dic109C2211yescharacteristic, complemented
28C2×C22⋊Dic109C2211yescharacteristic, complemented
29C2×C22⋊Dic109C2212yescomplemented
30C2×C4×Dic109C2212yescomplemented
31#16812606C2212yescomplemented
32#16812606C2212yescomplemented
33C25×C109C2211yescharacteristic
34C22×D4×C109C2211yescharacteristic
35C22×D4×C109C2212yes—
36#120×C109—24no—
37C2×C22⋊C4×C109C2212yes—
38C2×C22⋊C4×C109C2211yescharacteristic
39C22×Dic109—22no—
40C22×Dic109—22no—
41C22×Dic109—22no—
42C22×Dic109—42no—
43C22×Dic109D414yescomplemented
44C22×Dic109C2311yescharacteristic
45C22×Dic109C2312yes—
46C22×Dic109C2312yes—
47C22×Dic109C2311yescharacteristic
48C22×Dic109C2311yescharacteristic
49C22×Dic109C2311yescharacteristic
50C4⋊Dic109—42no—
51#13182008—24no—
52#13182008—22no—
53#13182008—22no—
54C22⋊Dic109—22no—
55C22⋊Dic109—22no—
56C22⋊Dic109—24no—
57C22⋊Dic109—24no—
58C22⋊Dic109—24no—
59C22⋊Dic109—22no—
60C22⋊Dic109—22no—
61C22⋊Dic109—42no—
62C22⋊Dic109—42no—
63C4×Dic109—24no—
64C24×C109—22no—
65C24×C109—24no—
66C24×C109—22no—
67C24×C109—22no—
68C24×C109—42no—
69C24×C109C2311yescharacteristic
70C24×C109C2312yes—
71C24×C109C2311yescharacteristic
72C2×D4×C109—22no—
73C2×D4×C109—22no—
74C2×D4×C109—24no—
75C2×D4×C109—22no—
76C2×D4×C109—24no—
77C2×D4×C109—42no—
78C22×C4×C109C2311yescharacteristic
79C22×C4×C109C2312yes—
80C22⋊C4×C109—24no—
81C22⋊C4×C109D414yes—
82C23×C109—22no—
83C23×C109—22no—
84C23×C109—28no—
85C23×C109—22no—
86C23×C109—22no—
87C23×C109—22no—
88C23×C109—22no—
89C23×C109—22no—
90C23×C109—22no—
91C23×C109—21no—
92C23×C109—21no—
93C23×C109—21no—
94C23×C109—22no—
95C23×C109—22no—
96C23×C109—48no—
97C23×C109—48no—
98C23×C109—44no—
99C23×C109—44no—
100C23×C109C2×D412yes—
101C23×C109C2×D412yes—
102C23×C109#4114yes—
103C23×C109#4112yes—
104C23×C109C2411yescharacteristic
105D4×C109—48no—
106D4×C109—44no—
107C2×C4×C109—22no—
108C2×C4×C109—22no—
109C2×C4×C109—21no—
110C2×C4×C109—21no—
111C2×C4×C109—21no—
112C2×C4×C109C2×D414yes—
113C2×Dic109—28no—
114C2×Dic109—21no—
115C2×Dic109—21no—
116C2×Dic109—21no—
117C2×Dic109—21no—
118C2×Dic109—22no—
119C2×Dic109—21no—
120C2×Dic109—21no—
121C2×Dic109—22no—
122C2×Dic109—22no—
123C2×Dic109—22no—
124C2×Dic109—22no—
125C2×Dic109—22no—
126C2×Dic109—22no—
127C2×Dic109—22no—
128C2×Dic109—44no—
129C2×Dic109C2×D412yes—
130C2×Dic109C2×D412yes—
131C22×C109—22no—
132C22×C109—22no—
133C22×C109—24no—
134C22×C109—24no—
135C22×C109—24no—
136C22×C109—24no—
137C22×C109—24no—
138C22×C109—416no—
139C22×C109—42no—
140C22×C109—42no—
141C22×C109—42no—
142C22×C109—42no—
143C22×C109—41no—
144C22×C109—41no—
145C22×C109—41no—
146C22×C109—42no—
147C22×C109—42no—
148C22×C109#12114yes—
149C22×C109C22×D411yescharacteristic
150C22×C109C22×D411yescharacteristic
151C22×C109C2×#4111yescharacteristic
152C22×C109C2×#4112yes—
153C22×C109H(5,𝔽2)12yes—
154C4×C109—24no—
155C4×C109—41no—
156Dic109—22no—
157Dic109—22no—
158Dic109—41no—
159Dic109—41no—
160Dic109—42no—
161Dic109—42no—
162C2×C109—24no—
163C2×C109—44no—
164C2×C109—81no—
165C2×C109C2×#12111yescharacteristic
166C2×C109#52511yescharacteristic
167C2×C109#53511yescharacteristic
168C2×C109#54612yes—
169C2×C109#55112yes—
170#2411—1091nomaximal, Sylow
171C109#241111yescharacteristic, complemented, Sylow
172C2×#123—1091no—
173C2×#120D10911yescharacteristic
174C2×#117—1091no—
175C22×C22⋊C4—1091no—
176#396—1092no—
177#394—1092no—
178#390—1092no—
179#388—1092no—
180#386—1091no—
181#385—1091no—
182#379—1091no—
183C25C2×D10911yescharacteristic
184C22×D4C2×D10911yescharacteristic
185C22×D4C2×D10912yes—
186C23×C4—1091no—
187C23×C4—1091no—
188#123—2184no—
189#120—24no—
190#117—2184no—
191C2×C4⋊C4—1092no—
192C2×C4⋊C4—1092no—
193C2×C4⋊C4—1091no—
194C2×C4⋊C4—1091no—
195C2×C22⋊C4—1092no—
196C2×C22⋊C4—1092no—
197C2×C22⋊C4—1092no—
198C2×C22⋊C4—1092no—
199C2×C22⋊C4—1091no—
200C2×C22⋊C4—1091no—
201C2×C22⋊C4—1091no—
202C2×C22⋊C4—1091no—
203C2×C22⋊C4—1092no—
204C2×C22⋊C4—2184no—
205C2×C22⋊C4C2×D10912yes—
206C2×C22⋊C4C2×D10911yescharacteristic
207C2×C42—1092no—
208#95—1092no—
209#95—1092no—
210C24—22no—
211C24—24no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24C22×D10911yescharacteristic
216C24C22×D10912yes—
217C24C22×D10911yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—42no—
224C22×C4—1094no—
225C22×C4—1091no—
226C22×C4—1091no—
227C22×C4—1091no—
228C22×C4—1092no—
229C22×C4—1092no—
230C22×C4—1091no—
231C22×C4—4362no—
232C22×C4—2182no—
233C22×C4—2182no—
234C22×C4—2182no—
235C22×C4C22×D10911yescharacteristic
236C22×C4C22×D10912yes—
237C4⋊C4—4362no—
238C4⋊C4—2182no—
239C4⋊C4—2182no—
240C4⋊C4—2184no—
241C22⋊C4—24no—
242C22⋊C4—4362no—
243C22⋊C4—4362no—
244C22⋊C4—2184no—
245C22⋊C4—2184no—
246C22⋊C4—2182no—
247C22⋊C4—2182no—
248C22⋊C4—2182no—
249C22⋊C4—2182no—
250C22⋊C4—2184no—
251C22⋊C4C22⋊D10914yes—
252C42—2184no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—22no—
267C23—48no—
268C23—48no—
269C23—44no—
270C23—44no—
271C23D4×D10912yes—
272C23C2×C22⋊D10912yes—
273C23#1318202814yes—
274C23#1318202812yes—
275C23C23×D10911yescharacteristic
276D4—48no—
277D4—44no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—1092no—
284C2×C4—1092no—
285C2×C4—4364no—
286C2×C4—2181no—
287C2×C4—2181no—
288C2×C4—2182no—
289C2×C4—2182no—
290C2×C4—2182no—
291C2×C4—2182no—
292C2×C4—2182no—
293C2×C4—2182no—
294C2×C4—2182no—
295C2×C4—2182no—
296C2×C4—2188no—
297C2×C4—2181no—
298C2×C4—2181no—
299C2×C4—2181no—
300C2×C4—2181no—
301C2×C4C2×C22⋊D10914yes—
302C22—24no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—42no—
319C22#1681267014yes—
320C22C2×D4×D10911yescharacteristic
321C22C22×C22⋊D10911yescharacteristic
322C22C2×#1318202811yescharacteristic
323C22C2×#1318202812yes—
324C22#1681274412yes—
325C4—24no—
326C4—41no—
327C4—4361no—
328C4—4362no—
329C4—4361no—
330C4—4362no—
331C4—2182no—
332C4—2182no—
333C2—24no—
334C2—44no—
335C2—81no—
336C2C2×#1681267011yescharacteristic
337C2#2029553211yescharacteristic
338C2#2029559111yescharacteristic
339C2#2029567012yes—
340C2#2029570712yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting, prime radicalC2×#120×C109C2
hypercenterC2×#120D109
hypocenterC109#2411
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#16812680C2semidirect
2. C2×#16812655C2semidirect
3. C22×C22⋊Dic109C2semidirect
4. #20294949C2semidirect
5. #20294940C2semidirect
6. #20294929C2semidirect
7. #20294914C2semidirect
8. #20294906C2semidirect
9. #20294901C2semidirect
10. #20294882C2semidirect
16. C2×C4⋊Dic109C22semidirect
17. C2×#13182008C22semidirect
18. C2×#13182008C22semidirect
19. C2×#13182008C22semidirect
27. C2×C22⋊Dic109C22semidirect
28. C2×C22⋊Dic109C22semidirect
29. C2×C22⋊Dic109C22semidirect
30. C2×C4×Dic109C22semidirect
31. #16812606C22semidirect
32. #16812606C22semidirect
43. C22×Dic109D4semidirect
171. C109#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#168126802
C2×#168126552
C22×C22⋊Dic1092
#202949492
#202949402
#202949292
#202949142
#202949062
#202949012
#202948822
C2×#120×C1092
#2411109