Finite Group Databasebeta

← #34011299

Subgroups — #34011299

Order 18688.

378 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 88 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#22626290C211yesmaximal, characteristic, complemented
2#22625852C211yesmaximal, characteristic, complemented
3#22624918C214yesmaximal, complemented
4#22620511C211yesmaximal, characteristic
5#17989533—21no—
6#17989508—21no—
7#17989482C2212yes—
8C4×#11924031—21no—
9#17989461—21no—
10#17989461C2212yes—
11#17989138—21no—
12#17988964C411yescharacteristic, complemented
13#17988935—21no—
14#17988824—21no—
15#17988824C2212yes—
16#17988669—21no—
17#17988669C412yescomplemented
18C4×C8×D73C2211yescharacteristic
19#500×C73C411yescharacteristic, complemented
20C2×C4⋊#64408—22no—
21C2×C4⋊#64408—41no—
22C2×C4⋊#64408—41no—
23C2×#11924031—22no—
24C2×#11924029—41no—
25C2×D584—21no—
26C2×D584C2×C412yes—
27C2×C8×D73D411yescharacteristic
28C2×C8×D73C2311yescharacteristic
29#14556588—21no—
30#14556580—41no—
31#14556566—22no—
32#14556566—41no—
33#14556553—22no—
34#14556553—41no—
35C42×D73D411yescharacteristic
36#14556511—22no—
37#14556511—22no—
38C4×#10596831D411yescharacteristic
39C8×Dic73C2×C411yescharacteristic
40C8×Dic73D411yescharacteristic
41C4⋊Q8×C73—21no—
42#107×C73—21no—
43#107×C73C2×C412yes—
44C4×C8×C73C2×C411yescharacteristic
45C22×#64408—42no—
46C4⋊#64408—42no—
47C4⋊#64408—42no—
48C4⋊#64408—82no—
49C4⋊#64408—82no—
50#11924031—42no—
51#11924029—82no—
52#11924029—82no—
53C2×D292—22no—
54C2×D292—42no—
55C2×D292—41no—
56C2×D292—41no—
57C2×C4×D73—42no—
58C2×C4×D73D811yescharacteristic
59C2×C4×D73D811yescharacteristic
60C2×C4×D73C2×D411yescharacteristic
61D584—42no—
62D584—42no—
63C2×#10596831C22⋊C411yescharacteristic
64C2×#10596831C2×D411yescharacteristic
65C8×D73—41no—
66C8×D73—41no—
67#11924005—42no—
68#11924005—42no—
69C4×Dic73C22⋊C411yescharacteristic
70C4×Dic73SD1611yescharacteristic
71C4×Dic73SD1611yescharacteristic
72C2×Q8×C73—41no—
73C2×C8×C73C22⋊C411yescharacteristic
74C2×C8×C73C22×C411yescharacteristic
75C4⋊C4×C73—22no—
76C4⋊C4×C73—41no—
77C42×C73C22⋊C411yescharacteristic
78C22×D73—42no—
79C22×D73C2×D811yescharacteristic
80C2×#64408—82no—
81C2×#64408—82no—
82C2×#64408—82no—
83Q8×C73—82no—
84C2×C4×C73—42no—
85C2×C4×C73#10211yescharacteristic
86C2×C4×C73#10211yescharacteristic
87C2×C4×C73C2×C22⋊C411yescharacteristic
88C2×Dic73#10211yescharacteristic
89C2×Dic73#10211yescharacteristic
90C2×Dic73C2×SD1611yescharacteristic
91D292—42no—
92D292—42no—
93D292—82no—
94D292—82no—
95D292—82no—
96D292—82no—
97C4×D73—22no—
98C4×D73—21no—
99C4×D73—21no—
100C4×D73—84no—
101C4×D73D1612yes—
102C4×D73SD3212yes—
103C8×C73—21no—
104C8×C73—21no—
105#10596831—21no—
106#10596831—21no—
107C22×C73C2×#10211yescharacteristic
108C2×D73—21no—
109C2×D73—21no—
110C2×D73—82no—
111C2×D73—82no—
112C2×D73—82no—
113C2×D73C2×D1611yescharacteristic
114C2×D73C2×SD3211yescharacteristic
115#64408—162no—
116C4×C73—82no—
117C4×C73#35712yes—
118C4×C73#36012yes—
119C4×C73#41011yescharacteristic
120C4×C73#42111yescharacteristic
121Dic73—21no—
122Dic73—21no—
123Dic73#35712yes—
124#11856—731nomaximal, Sylow
125C2×C73#151711yescharacteristic
126C2×C73C2×#35711yescharacteristic
127C2×C73C2×#36011yescharacteristic
128D73—21no—
129D73—21no—
130D73—162no—
131#3169—731no—
132#2162—731no—
133#1834—731no—
134#1347—734no—
135C73#1185611yescharacteristic, complemented, Sylow
136#532—1461no—
137#500#6440811yescharacteristic, complemented
138#493—731no—
139#486—1461no—
140#466—732no—
141#466—1461no—
142C4×D8—732no—
143C4×D8—1461no—
144C2×C4×C8—731no—
145#384—1461no—
146C4⋊C16—1461no—
147C22⋊C16—732no—
148C4×C16—1461no—
149#340—732no—
150#340—1461no—
151C2×#41—2921no—
152C2×D8—732no—
153C2×D8—1461no—
154C22×C8—731no—
155C22×C8—731no—
156C4⋊Q8—21no—
157#127—1461no—
158#124—2921no—
159#121—2921no—
160#121—1462no—
161C4×D4—2921no—
162C4×D4—1462no—
163C2×C4⋊C4—2921no—
164C2×C4⋊C4—2921no—
165C2×C4⋊C4—1462no—
166C2×C42—731no—
167C2×C16—1462no—
168#107—21no—
169#107C2×#6440812yes—
170#102—1462no—
171#102—1462no—
172C4×C8—731no—
173C4×C8—731no—
174C4×C8—731no—
175C4×C8C2×#6440811yescharacteristic
176#41—5842no—
177#41—5842no—
178C2×Q8—41no—
179C2×D4—2922no—
180C2×D4—2921no—
181C2×D4—2921no—
182C2×D4—1462no—
183C22×C4—2922no—
184C22×C4—2922no—
185C22×C4—731no—
186C22×C4—731no—
187C22×C4—731no—
188D8—2922no—
189D8—2922no—
190C2×C8—2921no—
191C2×C8—2921no—
192C2×C8—731no—
193C2×C8—731no—
194C2×C8C22⋊#6440811yescharacteristic
195C2×C8C22×#6440811yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—41no—
198C4⋊C4—2922no—
199C4⋊C4—2922no—
200C4⋊C4—5842no—
201C4⋊C4—5842no—
202C22⋊C4—2922no—
203C22⋊C4—2922no—
204C42—731no—
205C42—731no—
206C42—731no—
207C42C22⋊#6440811yescharacteristic
208C16—2922no—
209C23—2922no—
210C23—731no—
211Q8—82no—
212D4—2922no—
213D4—2922no—
214D4—5842no—
215D4—5842no—
216D4—5842no—
217D4—5842no—
218C2×C4—42no—
219C2×C4—5844no—
220C2×C4—5842no—
221C2×C4—5842no—
222C2×C4—5842no—
223C2×C4—731no—
224C2×C4—732no—
225C2×C4—732no—
226C2×C4—731no—
227C2×C4—731no—
228C2×C4—1462no—
229C2×C4—1461no—
230C2×C4—1461no—
231C2×C4#1455665311yescharacteristic
232C2×C4#1455665311yescharacteristic
233C2×C4C2×C22⋊#6440811yescharacteristic
234C8—21no—
235C8—21no—
236C8—1461no—
237C8—1461no—
238C22—5842no—
239C22—5842no—
240C22—5842no—
241C22—731no—
242C22—731no—
243C22—1461no—
244C22—1461no—
245C22C2×#1455665311yescharacteristic
246C4—82no—
247C4—732no—
248C4—11682no—
249C4—1461no—
250C4—1461no—
251C4#1798949912yes—
252C4#1798950412yes—
253C4#1798956811yescharacteristic
254C4#1798958611yescharacteristic
255C2—11682no—
256C2—1461no—
257C2—1461no—
258C2#2262536211yescharacteristic
259C2C2×#1798949911yescharacteristic
260C2C2×#1798950411yescharacteristic
subgroupG/H
centerC22C2×#14556653
commutatorC2×C8×C73C22×C4
FrattiniC2×C8C22×#64408
Fitting#500×C73C4
socleC22×C73C2×#102
prime radical#22620511C2
cosocleC2×C8×D73C23
hypercenter#500#64408
hypocenterC73#11856
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #22626290C2semidirect
2. #22625852C2semidirect
3. #22624918C2semidirect
12. #17988964C4semidirect
17. #17988669C4semidirect
19. #500×C73C4semidirect
135. C73#11856semidirect
137. #500#64408semidirect

Maximal subgroups

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

subgroupindex
#226262902
#226258522
#226249182
#226205112
#1185673