Finite Group Databasebeta

← #34011302

Subgroups — #34011302

Order 18688.

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

normal subgroups only

This group has 334 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
2#22625852C211yesmaximal, characteristic
3#22624920C214yesmaximal
4#22620518C211yesmaximal, characteristic
5#17989533—21no—
6#17989508—21no—
7#17989482C2212yes—
8C4×#11924031—21no—
9#17989461—21no—
10#17989461C2212yes—
11#17989142—21no—
12#17988967C411yescharacteristic, complemented
13#17988939—21no—
14#17988829—21no—
15#17988829C2212yes—
16#17988681—21no—
17#17988681C412yescomplemented
18C4×C8×D73C2211yescharacteristic
19#500×C73C411yescharacteristic, complemented
20C2×C4⋊#64408—22no—
21C2×C4⋊#64408—41no—
22C2×C4⋊#64408—41no—
23C2×#11924031—22no—
24C2×Dic292—21no—
25C2×Dic292C2×C412yes—
26C2×C8×D73D411yescharacteristic
27C2×C8×D73C2311yescharacteristic
28#14556591—21no—
29#14556582—41no—
30#14556571—22no—
31#14556567—41no—
32Q8×Dic73—41no—
33#14556556—22no—
34#14556549—41no—
35C42×D73D411yescharacteristic
36#14556514—22no—
37#14556514—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—
51C2×Dic146—22no—
52C2×C4×D73D811yescharacteristic
53C2×C4×D73D811yescharacteristic
54C2×C4×D73C2×D411yescharacteristic
55Dic292—42no—
56Dic292—42no—
57C2×#10596831C22⋊C411yescharacteristic
58C2×#10596831C2×D411yescharacteristic
59C8×D73—41no—
60C8×D73—41no—
61C4⋊Dic73—41no—
62C4⋊Dic73—41no—
63C4⋊Dic73—42no—
64#11924007—42no—
65#11924005—42no—
66C4×Dic73—42no—
67C4×Dic73C22⋊C411yescharacteristic
68C4×Dic73SD1611yescharacteristic
69C4×Dic73SD1611yescharacteristic
70C2×Q8×C73—41no—
71C2×C8×C73C22⋊C411yescharacteristic
72C2×C8×C73C22×C411yescharacteristic
73C4⋊C4×C73—22no—
74C4⋊C4×C73—41no—
75C42×C73C22⋊C411yescharacteristic
76C22×D73C2×D811yescharacteristic
77C2×#64408—82no—
78C2×#64408—82no—
79C2×#64408—82no—
80Q8×C73—82no—
81C2×C4×C73—42no—
82C2×C4×C73#10211yescharacteristic
83C2×C4×C73#10211yescharacteristic
84C2×C4×C73C2×C22⋊C411yescharacteristic
85C2×Dic73—42no—
86C2×Dic73#10211yescharacteristic
87C2×Dic73#10211yescharacteristic
88C2×Dic73C2×SD1611yescharacteristic
89C4×D73—22no—
90C4×D73—21no—
91C4×D73—21no—
92C4×D73SD3212yes—
93C4×D73Q3212yes—
94Dic146—42no—
95Dic146—42no—
96C8×C73—21no—
97C8×C73—21no—
98#10596831—21no—
99#10596831—21no—
100C22×C73C2×#10211yescharacteristic
101C2×D73—21no—
102C2×D73—21no—
103C2×D73C2×SD3211yescharacteristic
104C2×D73C2×Q3211yescharacteristic
105#64408—162no—
106C4×C73—82no—
107C4×C73#35812yes—
108C4×C73#36012yes—
109C4×C73#41011yescharacteristic
110C4×C73#42111yescharacteristic
111Dic73—21no—
112Dic73—21no—
113Dic73—82no—
114Dic73#35812yes—
115#11857—731nomaximal, Sylow
116C2×C73#151711yescharacteristic
117C2×C73C2×#35811yescharacteristic
118C2×C73C2×#36011yescharacteristic
119D73—21no—
120D73—21no—
121#3170—731no—
122#2162—731no—
123#1834—731no—
124#1348—734no—
125C73#1185711yescharacteristic, complemented, Sylow
126#533—1461no—
127#500#6440811yescharacteristic, complemented
128#494—731no—
129#487—1461no—
130#467—732no—
131#467—1461no—
132C4×Q16—732no—
133C4×Q16—1461no—
134C2×C4×C8—731no—
135#384—1461no—
136C4⋊C16—1461no—
137C22⋊C16—732no—
138C4×C16—1461no—
139#340—732no—
140#340—1461no—
141C2×Q16—732no—
142C2×Q16—1461no—
143C22×C8—731no—
144C22×C8—731no—
145C4⋊Q8—21no—
146C4⋊Q8—1461no—
147#125—2921no—
148C22⋊Q8—2921no—
149C22⋊Q8—1462no—
150C4×Q8—2921no—
151C4×Q8—1462no—
152#117—2921no—
153C2×C4⋊C4—2921no—
154C2×C4⋊C4—2921no—
155C2×C4⋊C4—1462no—
156C2×C42—731no—
157C2×C16—1462no—
158#107—21no—
159#107C2×#6440812yes—
160#103—1462no—
161#103—1462no—
162C4×C8—731no—
163C4×C8—731no—
164C4×C8—731no—
165C4×C8C2×#6440811yescharacteristic
166C2×Q8—41no—
167C2×Q8—1462no—
168C22×C4—2922no—
169C22×C4—731no—
170C22×C4—731no—
171C22×C4—731no—
172Q16—2922no—
173Q16—2922no—
174C2×C8—2921no—
175C2×C8—2921no—
176C2×C8—731no—
177C2×C8—731no—
178C2×C8C22⋊#6440811yescharacteristic
179C2×C8C22×#6440811yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—41no—
182C4⋊C4—2922no—
183C4⋊C4—2922no—
184C4⋊C4—2921no—
185C4⋊C4—2921no—
186C4⋊C4—2922no—
187C4⋊C4—2922no—
188C4⋊C4—5842no—
189C4⋊C4—5842no—
190C22⋊C4—2922no—
191C42—2922no—
192C42—731no—
193C42—731no—
194C42—731no—
195C42C22⋊#6440811yescharacteristic
196C16—2922no—
197C23—731no—
198Q8—82no—
199Q8—2922no—
200Q8—2922no—
201C2×C4—42no—
202C2×C4—2922no—
203C2×C4—5842no—
204C2×C4—5842no—
205C2×C4—5842no—
206C2×C4—731no—
207C2×C4—732no—
208C2×C4—732no—
209C2×C4—731no—
210C2×C4—731no—
211C2×C4—1462no—
212C2×C4—1461no—
213C2×C4—1461no—
214C2×C4#1455665311yescharacteristic
215C2×C4#1455665311yescharacteristic
216C2×C4C2×C22⋊#6440811yescharacteristic
217C8—21no—
218C8—21no—
219C8—1461no—
220C8—1461no—
221C22—731no—
222C22—731no—
223C22—1461no—
224C22—1461no—
225C22C2×#1455665311yescharacteristic
226C4—82no—
227C4—5842no—
228C4—732no—
229C4—11682no—
230C4—1461no—
231C4—1461no—
232C4#1798950012yes—
233C4#1798950312yes—
234C4#1798956811yescharacteristic
235C4#1798958611yescharacteristic
236C2—1461no—
237C2—1461no—
238C2#2262536211yescharacteristic
239C2C2×#1798950011yescharacteristic
240C2C2×#1798950311yescharacteristic
subgroupG/H
centerC22C2×#14556653
commutatorC2×C8×C73C22×C4
FrattiniC2×C8C22×#64408
Fitting#500×C73C4
socleC22×C73C2×#102
prime radicalC22×D73C2×D8
cosocleC2×C8×D73C23
hypercenter#500#64408
hypocenterC73#11857
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
12. #17988967C4semidirect
17. #17988681C4semidirect
19. #500×C73C4semidirect
125. C73#11857semidirect
127. #500#64408semidirect

Maximal subgroups

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

subgroupindex
#226262902
#226258522
#226249202
#226205182
#1185773