Finite Group Databasebeta

← #33062253

Subgroups — #33062253

Order 18688.

1068 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 179 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#22620388C211yesmaximal, characteristic, complemented
2#22620022C212yesmaximal, complemented
3#22609076C214yesmaximal, complemented
4#22609014C214yesmaximal, complemented
5#22608657C211yesmaximal, characteristic, complemented
6#22608374C212yesmaximal, complemented
7C2×#555×C73C211yesmaximal, characteristic, complemented
8C2×#14556622—22no—
9#17989137C2211yescharacteristic
10#17988979—22no—
11#17988966C2212yescomplemented
12#17988961C2212yescomplemented
13#17988905—22no—
14#17988877C2214yescomplemented
15#17988782C2214yescomplemented
16#17988721—22no—
17#17988697—24no—
18#17988644C2212yes—
19C4×#11924012C2211yescharacteristic
20#17988336C2214yescomplemented
21D4⋊#10596831C2218yescomplemented
22#555×C73—22no—
23#555×C73—24no—
24#555×C73C2214yes—
25C2×C4⋊Q8×C73C2211yescharacteristic
26C2×#124×C73—22no—
27C2×#121×C73—22no—
28C2×C4×D4×C73C2212yes—
29C2×#117×C73—22no—
30C2×#11924028—21no—
31#14556622—48no—
32C2×#11924018—24no—
33C2×#11924015—24no—
34C2×#11924013—24no—
35C2×#11924012—21no—
36#14556591C2311yescharacteristic
37#14556588C2311yescharacteristic
38#14556579—21no—
39#14556565—22no—
40C4×D292—22no—
41#14556522—24no—
42C4⋊#10596831C2314yes—
43#14556512—24no—
44#14556511—24no—
45C22⋊#10596831—22no—
46#14556502—22no—
47#14556502—21no—
48C4×#10596831C2312yes—
49C22×Q8×C73D412yes—
50C22×D4×C73—22no—
51C22×D4×C73D412yes—
52C23×C4×C73—22no—
53C4⋊Q8×C73—22no—
54C4⋊Q8×C73D412yes—
55C4⋊Q8×C73C2312yes—
56#124×C73—28no—
57#124×C73—28no—
58#121×C73—28no—
59#121×C73—28no—
60C4×D4×C73—22no—
61C4×D4×C73—22no—
62C4×D4×C73D414yes—
63C4×D4×C73C2314yes—
64#117×C73—28no—
65#117×C73—28no—
66C2×C4⋊C4×C73—22no—
67C2×C4⋊C4×C73D412yes—
68C2×C22⋊C4×C73—24no—
69C2×C22⋊C4×C73—22no—
70C2×C42×C73—22no—
71C2×C42×C73C2311yescharacteristic
72#11924028—44no—
73C2×Dic146—21no—
74C2×D292—22no—
75C2×D292—21no—
76C2×C22⋊D73—41no—
77C2×C4×D73—41no—
78#11924018—48no—
79#11924015—48no—
80#11924013—48no—
81#11924012—24no—
82C2×#10596831—22no—
83C2×#10596831—24no—
84C4⋊Dic73—22no—
85#11924005—42no—
86C24×C73—22no—
87C2×Q8×C73—22no—
88C2×Q8×C73—44no—
89C2×Q8×C73C2×D414yes—
90C2×D4×C73—28no—
91C2×D4×C73—22no—
92C2×D4×C73—44no—
93C2×D4×C73—84no—
94C2×D4×C73C2×D414yes—
95C22×C4×C73—22no—
96C22×C4×C73—22no—
97C22×C4×C73—24no—
98C22×C4×C73—24no—
99C22×C4×C73—48no—
100C22×C4×C73C2×D411yescharacteristic
101C22×C4×C73C2×D412yes—
102C4⋊C4×C73—24no—
103C4⋊C4×C73—22no—
104C4⋊C4×C73—24no—
105C4⋊C4×C73C2×D414yes—
106C22⋊C4×C73—216no—
107C22⋊C4×C73—24no—
108C22⋊C4×C73—24no—
109C42×C73—28no—
110C42×C73C2×D411yescharacteristic
111C42×C73C2×D412yes—
112C42×C73C2411yescharacteristic
113C23×C73—24no—
114C23×C73—44no—
115C23×C73—44no—
116C23×C73—44no—
117C23×C73#12011yescharacteristic
118C22×D73—41no—
119Q8×C73—44no—
120Q8×C73—82no—
121D4×C73—24no—
122D4×C73—24no—
123D4×C73—44no—
124D4×C73—88no—
125C2×C4×C73—22no—
126C2×C4×C73—24no—
127C2×C4×C73—24no—
128C2×C4×C73—24no—
129C2×C4×C73—48no—
130C2×C4×C73—44no—
131C2×C4×C73—44no—
132C2×C4×C73—48no—
133C2×C4×C73#12011yescharacteristic
134C2×C4×C73#12012yes—
135C2×C4×C73C22×D411yescharacteristic
136C2×C4×C73C22×D412yes—
137C22⋊D73—82no—
138C2×Dic73—41no—
139D292—42no—
140D292—44no—
141C4×D73—82no—
142Dic146—42no—
143#10596831—24no—
144#10596831—44no—
145C22×C73—22no—
146C22×C73—44no—
147C22×C73—44no—
148C22×C73—44no—
149C22×C73—88no—
150C22×C73#45312yes—
151C22×C73C2×#12011yescharacteristic
152C2×D73—82no—
153C2×D73—81no—
154C4×C73—22no—
155C4×C73—21no—
156C4×C73—21no—
157C4×C73—44no—
158C4×C73—44no—
159C4×C73#57514yes—
160Dic73—81no—
161#18547—731nomaximal, Sylow
162C2×C73—22no—
163C2×C73—84no—
164C2×C73#300512yes—
165C2×C73C2×#45311yescharacteristic
166D73—161no—
167C2×#555D7311yescharacteristic, complemented
168#3152—731no—
169#3109—732no—
170#1634—734no—
171#1624—734no—
172#1558—731no—
173#1490—732no—
174C73#1854711yescharacteristic, complemented, Sylow
175C2×#137—1462no—
176#555—22no—
177#555—24no—
178#555C2×D7314yes—
179#532—731no—
180C2×C4⋊Q8C2×D7311yescharacteristic
181C2×#124—22no—
182C2×#121—22no—
183C2×C4×D4C2×D7312yes—
184C2×#117—22no—
185#497—1462no—
186#494—732no—
187#492—732no—
188#481—1462no—
189#476—734no—
190#459—734no—
191#447—1462no—
192#442—1464no—
193#432—732no—
194C4×M16—731no—
195#333—734no—
196D4⋊C8—738no—
197C2×#41—1461no—
198C22×Q8C22⋊D7312yes—
199C22×D4—22no—
200C22×D4C22⋊D7312yes—
201C23×C4—22no—
202#137—2928no—
203C2×Q16—1464no—
204C2×SD16—1464no—
205C2×D8—1464no—
206C2×M16—1461no—
207C4⋊Q8—22no—
208C4⋊Q8—731no—
209C4⋊Q8C22⋊D7312yes—
210C4⋊Q8C22×D7312yes—
211#127—731no—
212#124—28no—
213#124—28no—
214#124—1461no—
215#121—28no—
216#121—28no—
217#121—1462no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—1462no—
221C4×D4C22⋊D7314yes—
222C4×D4C22×D7314yes—
223#117—28no—
224#117—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C22⋊D7312yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C42—22no—
230C2×C42C22×D7311yescharacteristic
231#106—1464no—
232C4⋊C8—734no—
233#102—1464no—
234#102—1464no—
235C22⋊C8—1462no—
236#97—1461no—
237#97—1462no—
238C4×C8—732no—
239C24—22no—
240#41—2924no—
241C2×Q8—22no—
242C2×Q8—44no—
243C2×Q8—1461no—
244C2×Q8C2×C22⋊D7314yes—
245C2×D4—28no—
246C2×D4—22no—
247C2×D4—44no—
248C2×D4—84no—
249C2×D4—2921no—
250C2×D4—1462no—
251C2×D4—1461no—
252C2×D4C2×C22⋊D7314yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—48no—
258C22×C4—2921no—
259C22×C4C2×C22⋊D7312yes—
260C22×C4C2×C22⋊D7311yescharacteristic
261Q16—2928no—
262SD16—2928no—
263D8—2928no—
264M16—1464no—
265C2×C8—1464no—
266C2×C8—1462no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—1462no—
271C4⋊C4C2×C22⋊D7314yes—
272C22⋊C4—216no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—2922no—
276C42—28no—
277C42C2×C22⋊D7311yescharacteristic
278C42C2×C22⋊D7312yes—
279C42C23×D7311yescharacteristic
280C23—24no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—2921no—
285C23#1455656011yescharacteristic
286Q8—44no—
287Q8—82no—
288Q8—2922no—
289D4—24no—
290D4—24no—
291D4—44no—
292D4—88no—
293D4—2922no—
294D4—2924no—
295D4—5842no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—2921no—
305C2×C4—5842no—
306C2×C4#1455656011yescharacteristic
307C2×C4#1455656012yes—
308C2×C4C22×C22⋊D7311yescharacteristic
309C2×C4C22×C22⋊D7312yes—
310C8—2924no—
311C8—1464no—
312C22—22no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—88no—
317C22—5842no—
318C22—5841no—
319C22#1798875512yes—
320C22C2×#1455656011yescharacteristic
321C4—22no—
322C4—21no—
323C4—21no—
324C4—44no—
325C4—44no—
326C4—5841no—
327C4#1798938514yes—
328C2—22no—
329C2—84no—
330C2—11681no—
331C2#2261917312yes—
332C2C2×#1798875511yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#555×C73C2
socleC22×C73C2×#120
hypercenterC2×#555D73
hypocenterC73#18547
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #22620388C2semidirect
2. #22620022C2semidirect
3. #22609076C2semidirect
4. #22609014C2semidirect
5. #22608657C2semidirect
6. #22608374C2semidirect
7. C2×#555×C73C2semidirect
11. #17988966C22semidirect
12. #17988961C22semidirect
14. #17988877C22semidirect
15. #17988782C22semidirect
20. #17988336C22semidirect
21. D4⋊#10596831C22semidirect
167. C2×#555D73semidirect
174. C73#18547semidirect

Maximal subgroups

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

subgroupindex
#226203882
#226200222
#226090762
#226090142
#226086572
#226083742
C2×#555×C732
#1854773