Finite Group Databasebeta

← #33067254

Subgroups — #33067254

Order 18688.

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

normal subgroups only

This group has 850 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
1#22621137C212yesmaximal, complemented
2#22620317C211yesmaximal, characteristic
3#22609265C218yesmaximal, complemented
4#22608555C211yesmaximal, characteristic
5#22608533C212yesmaximal, complemented
6C2×#563×C73C211yesmaximal, characteristic
7#17989144C2211yescharacteristic
8#17988997C2214yescomplemented
9#17988957C2214yescomplemented
10#17988946C2214yes—
11#17988941—22no—
12#17988922—22no—
13#17988917—22no—
14#17988891—24no—
15#17988644C2212yes—
16#17988596—21no—
17#17988592—21no—
18C4×#11924012C2211yescharacteristic
19#17988329C2218yescomplemented
20#17988327C2214yes—
21#563×C73—24no—
22#563×C73—22no—
23#563×C73C2214yes—
24C2×C4⋊Q8×C73—21no—
25C2×#126×C73—22no—
26C2×#125×C73C2212yes—
27C2×#124×C73C2211yescharacteristic
28C2×#123×C73—21no—
29C2×C22⋊Q8×C73—22no—
30C2×#11924012—21no—
31#14556591C2312yes—
32#14556583—21no—
33#14556570—22no—
34C4×Dic146—22no—
35#14556550—21no—
36#14556524—24no—
37#14556522—24no—
38C4⋊#10596831—22no—
39#14556515—24no—
40#14556514—24no—
41#14556514—24no—
42#14556512—24no—
43C22⋊#10596831—22no—
44#14556502C2314yes—
45#14556502C2312yes—
46C4×#10596831—21no—
47C22×Q8×C73—21no—
48C22×Q8×C73D411yescharacteristic
49C22×D4×C73D411yescharacteristic
50C23×C4×C73—21no—
51C4⋊Q8×C73—24no—
52C4⋊Q8×C73—24no—
53#126×C73—28no—
54#126×C73—28no—
55#125×C73—22no—
56#125×C73—22no—
57#125×C73D414yes—
58#125×C73C2314yes—
59#124×C73—22no—
60#124×C73D412yes—
61#124×C73C2312yes—
62#123×C73—24no—
63#123×C73—24no—
64C22⋊Q8×C73—28no—
65C22⋊Q8×C73—28no—
66C2×C4⋊C4×C73—22no—
67C2×C4⋊C4×C73—22no—
68C2×C4⋊C4×C73—22no—
69C2×C4⋊C4×C73D412yes—
70C2×C4⋊C4×C73D412yes—
71C2×C22⋊C4×C73—22no—
72C2×C22⋊C4×C73—21no—
73C2×C22⋊C4×C73—22no—
74C2×C42×C73—21no—
75C2×C42×C73C2311yescharacteristic
76C2×Dic146—22no—
77#11924012—42no—
78C2×#10596831—22no—
79C2×#10596831—24no—
80C4⋊Dic73—22no—
81C4⋊Dic73—22no—
82#11924007—42no—
83C22⋊Dic73—41no—
84C4×Dic73—41no—
85C24×C73—21no—
86C2×Q8×C73—24no—
87C2×Q8×C73—21no—
88C2×Q8×C73—42no—
89C2×Q8×C73—82no—
90C2×Q8×C73C2×D412yes—
91C2×D4×C73—21no—
92C2×D4×C73—42no—
93C2×D4×C73C2×D412yes—
94C22×C4×C73—22no—
95C22×C4×C73—21no—
96C22×C4×C73—22no—
97C22×C4×C73—24no—
98C22×C4×C73—82no—
99C22×C4×C73C2×D411yescharacteristic
100C22×C4×C73C2×D412yes—
101C4⋊C4×C73—28no—
102C4⋊C4×C73—24no—
103C4⋊C4×C73—24no—
104C4⋊C4×C73—22no—
105C4⋊C4×C73—22no—
106C4⋊C4×C73—24no—
107C4⋊C4×C73—24no—
108C4⋊C4×C73C2×D414yes—
109C4⋊C4×C73C2×D414yes—
110C22⋊C4×C73—28no—
111C22⋊C4×C73—24no—
112C22⋊C4×C73—24no—
113C22⋊C4×C73—24no—
114C42×C73—24no—
115C42×C73C2×D412yes—
116C42×C73C2×D411yescharacteristic
117C42×C73C2411yescharacteristic
118C23×C73—22no—
119C23×C73—44no—
120C23×C73—42no—
121C23×C73#12011yescharacteristic
122Q8×C73—42no—
123Q8×C73—81no—
124Q8×C73—84no—
125D4×C73—42no—
126D4×C73—81no—
127C2×C4×C73—22no—
128C2×C4×C73—24no—
129C2×C4×C73—22no—
130C2×C4×C73—24no—
131C2×C4×C73—42no—
132C2×C4×C73—44no—
133C2×C4×C73—44no—
134C2×C4×C73—42no—
135C2×C4×C73—84no—
136C2×C4×C73#12012yes—
137C2×C4×C73#12011yescharacteristic
138C2×C4×C73C22×D412yes—
139C2×C4×C73C22×D411yescharacteristic
140C2×Dic73—42no—
141Dic146—44no—
142#10596831—44no—
143#10596831—42no—
144C22×C73—22no—
145C22×C73—42no—
146C22×C73—44no—
147C22×C73—84no—
148C22×C73#45512yes—
149C22×C73C2×#12011yescharacteristic
150C4×C73—21no—
151C4×C73—21no—
152C4×C73—22no—
153C4×C73—22no—
154C4×C73—44no—
155C4×C73—42no—
156C4×C73—44no—
157Dic73—82no—
158#18999—731nomaximal, Sylow
159C2×C73—22no—
160C2×C73—82no—
161C2×C73C2×#45511yescharacteristic
162C2×C73#302012yes—
163C2×#563D7311yescharacteristic
164#3236—732no—
165#3140—731no—
166#1664—738no—
167#1533—731no—
168#1528—732no—
169C73#1899911yescharacteristic, complemented, Sylow
170#563—24no—
171#563—22no—
172#563C2×D7314yes—
173#533—731no—
174C2×C4⋊Q8—21no—
175C2×#126—22no—
176C2×#125C2×D7312yes—
177C2×#124C2×D7311yescharacteristic
178C2×#123—21no—
179C2×C22⋊Q8—22no—
180#501—734no—
181#491—734no—
182#489—734no—
183#488—1462no—
184#484—1462no—
185#483—1462no—
186#479—1464no—
187#432—732no—
188#419—1461no—
189#418—1461no—
190C4×M16—731no—
191#330—738no—
192#329—734no—
193C22×Q8—21no—
194C22×Q8C22⋊D7311yescharacteristic
195C22×D4C22⋊D7311yescharacteristic
196C23×C4—21no—
197C2×M16—1461no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—732no—
201#126—28no—
202#126—28no—
203#125—22no—
204#125—22no—
205#125—1461no—
206#125C22⋊D7314yes—
207#125C22×D7314yes—
208#124—22no—
209#124C22⋊D7312yes—
210#124C22×D7312yes—
211#123—24no—
212#123—24no—
213C22⋊Q8—28no—
214C22⋊Q8—28no—
215C22⋊Q8—1462no—
216C4×Q8—1462no—
217#117—1461no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4C22⋊D7312yes—
222C2×C4⋊C4C22⋊D7312yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C42—21no—
227C2×C42C22×D7311yescharacteristic
228#107—1464no—
229#106—1464no—
230C4⋊C8—1462no—
231#103—1464no—
232#103—1464no—
233#103—1464no—
234#102—1464no—
235C22⋊C8—1462no—
236#97—734no—
237#97—732no—
238C4×C8—1461no—
239C24—21no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8—42no—
243C2×Q8—82no—
244C2×Q8—1462no—
245C2×Q8C2×C22⋊D7312yes—
246C2×D4—21no—
247C2×D4—42no—
248C2×D4C2×C22⋊D7312yes—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—82no—
254C22×C4C2×C22⋊D7311yescharacteristic
255C22×C4C2×C22⋊D7312yes—
256M16—2922no—
257C2×C8—1462no—
258C2×C8—1464no—
259C4⋊C4—28no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—2922no—
267C4⋊C4—1462no—
268C4⋊C4—1462no—
269C4⋊C4C2×C22⋊D7314yes—
270C4⋊C4C2×C22⋊D7314yes—
271C22⋊C4—28no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—2921no—
276C42—24no—
277C42—2921no—
278C42C2×C22⋊D7311yescharacteristic
279C42C2×C22⋊D7312yes—
280C42C23×D7311yescharacteristic
281C23—22no—
282C23—44no—
283C23—42no—
284C23#1455656011yescharacteristic
285Q8—42no—
286Q8—81no—
287Q8—84no—
288Q8—2924no—
289D4—42no—
290D4—81no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4—2922no—
301C2×C4#1455656011yescharacteristic
302C2×C4#1455656012yes—
303C2×C4C22×C22⋊D7312yes—
304C2×C4C22×C22⋊D7311yescharacteristic
305C8—2922no—
306C8—2924no—
307C22—22no—
308C22—42no—
309C22—44no—
310C22—84no—
311C22#1798876512yes—
312C22C2×#1455656011yescharacteristic
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—44no—
318C4—42no—
319C4—44no—
320C4—5842no—
321C2—22no—
322C2—82no—
323C2C2×#1798876511yescharacteristic
324C2#2261929512yes—
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#563×C73C2
socleC22×C73C2×#120
prime radicalC22×D4×C73D4
hypercenterC2×#563D73
hypocenterC73#18999
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. #22621137C2semidirect
3. #22609265C2semidirect
5. #22608533C2semidirect
8. #17988997C22semidirect
9. #17988957C22semidirect
19. #17988329C22semidirect
169. C73#18999semidirect

Maximal subgroups

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

subgroupindex
#226211372
#226203172
#226092652
#226085552
#226085332
C2×#563×C732
#1899973