Finite Group Databasebeta

← #32999667

Subgroups — #32999667

Order 18688.

298 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 75 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#22612156C211yesmaximal, characteristic
2#22611971C211yesmaximal, characteristic
3#22609291C211yesmaximal, characteristic
4#22607986C211yesmaximal, characteristic, complemented
5Q8⋊#11924003C211yesmaximal, characteristic, complemented
6#22607895C211yesmaximal, characteristic, complemented
7C73×#3383C211yesmaximal, characteristic, complemented
8C2×#14556534—21no—
9#17988964C2211yescharacteristic
10#17988787—21no—
11C4⋊#11924003C2211yescharacteristic
12#17988387—21no—
13#17988387—22no—
14C4×#11924003C2211yescharacteristic
15#17988340C2211yescharacteristic
16#17988330—21no—
17#557×C73—21no—
18#500×C73C2211yescharacteristic
19#499×C73—21no—
20#479×C73—22no—
21#475×C73—21no—
22C8×Q8×C73C2211yescharacteristic
23C4×Q16×C73—21no—
24C4×Q16×C73C2211yescharacteristic
25C2×#11924015—41no—
26C2×D584—21no—
27C2×D584—21no—
28#14556588—21no—
29#14556534—42no—
30C2×#11924003—21no—
31C2×#11924003—21no—
32C4⋊#10596831—21no—
33#14556511—41no—
34C2×Q16×C73D411yescharacteristic
35C4⋊Q8×C73—21no—
36C4⋊Q8×C73—41no—
37#125×C73—22no—
38#125×C73—42no—
39C4×Q8×C73—22no—
40C4×Q8×C73—21no—
41C4×Q8×C73—41no—
42C4×Q8×C73D411yescharacteristic
43#107×C73—21no—
44#107×C73D411yescharacteristic
45#107×C73D412yes—
46#106×C73—22no—
47C4⋊C8×C73—21no—
48C4⋊C8×C73D411yescharacteristic
49#103×C73—22no—
50#103×C73—21no—
51C4×C8×C73—21no—
52C4×C8×C73C2311yescharacteristic
53C2×D292—41no—
54C2×D292—41no—
55C2×D292—41no—
56#11924015—82no—
57D584—42no—
58D584—41no—
59D584—41no—
60C2×#10596831—41no—
61#11924003—41no—
62#11924003—41no—
63C2×Q8×C73—21no—
64C2×Q8×C73—41no—
65C2×Q8×C73D811yescharacteristic
66Q16×C73—21no—
67Q16×C73SD1612yescomplemented
68C2×C8×C73—21no—
69C2×C8×C73C2×D411yescharacteristic
70C2×C8×C73C2×D411yescharacteristic
71C4⋊C4×C73—21no—
72C4⋊C4×C73—21no—
73C4⋊C4×C73—22no—
74C4⋊C4×C73—42no—
75C4⋊C4×C73—42no—
76C4⋊C4×C73—42no—
77C4⋊C4×C73—41no—
78C4⋊C4×C73—41no—
79C4⋊C4×C73D811yescharacteristic
80C42×C73—21no—
81C42×C73—42no—
82C42×C73—41no—
83C42×C73C2×D411yescharacteristic
84C22×D73—81no—
85Q8×C73—22no—
86Q8×C73—21no—
87Q8×C73—21no—
88Q8×C73—41no—
89Q8×C73—81no—
90Q8×C73—81no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—41no—
94C2×C4×C73—41no—
95C2×C4×C73—42no—
96C2×C4×C73#12011yescharacteristic
97C2×C4×C73C2×D811yescharacteristic
98C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
99D292—81no—
100D292—81no—
101D292—81no—
102D292—81no—
103D292—161no—
104C8×C73—21no—
105C8×C73—41no—
106C8×C73C2×SD1611yescharacteristic
107C8×C73#13711yescharacteristic
108#10596831—81no—
109C22×C73#44711yescharacteristic
110C2×D73—161no—
111C2×D73—161no—
112C2×D73—161no—
113C4×C73—21no—
114C4×C73—42no—
115C4×C73—41no—
116C4×C73—41no—
117C4×C73—82no—
118C4×C73—81no—
119C4×C73#44811yescharacteristic
120C4×C73#45411yescharacteristic
121C4×C73#50811yescharacteristic
122C4×C73#50911yescharacteristic
123#12353—731nomaximal, Sylow
124C2×C73#162711yescharacteristic
125C2×C73#218611yescharacteristic
126C2×C73#219311yescharacteristic
127D73—321no—
128#3383D7311yescharacteristic, complemented
129#2240—731no—
130#2207—731no—
131#1668—731no—
132#1364—731no—
133Q8⋊C16—731no—
134#1332—731no—
135C73#1235311yescharacteristic, complemented, Sylow
136#557—21no—
137C2×SD32—1461no—
138#500C2×D7311yescharacteristic
139#499—21no—
140#493—731no—
141#479—22no—
142#475—21no—
143#460—1461no—
144C8×Q8C2×D7311yescharacteristic
145C4×Q16—21no—
146C4×Q16C2×D7311yescharacteristic
147C4⋊C16—731no—
148#357—1461no—
149#357—1462no—
150C4×C16—731no—
151#335—731no—
152#331—1461no—
153C2×Q16C22⋊D7311yescharacteristic
154C2×SD16—2921no—
155C2×D8—1461no—
156C2×D8—1461no—
157C4⋊Q8—21no—
158C4⋊Q8—41no—
159#127—1461no—
160#125—22no—
161#125—42no—
162C4×Q8—22no—
163C4×Q8—21no—
164C4×Q8—41no—
165C4×Q8C22⋊D7311yescharacteristic
166SD32—2922no—
167C2×C16—1461no—
168C2×C16—1461no—
169#107—21no—
170#107C22⋊D7311yescharacteristic
171#107C22⋊D7312yes—
172#106—22no—
173C4⋊C8—21no—
174C4⋊C8—1461no—
175C4⋊C8C22⋊D7311yescharacteristic
176#103—22no—
177#103—21no—
178#102—2921no—
179C4×C8—21no—
180C4×C8C22×D7311yescharacteristic
181C2×Q8—21no—
182C2×Q8—41no—
183C2×Q8#1192401311yescharacteristic
184C2×D4—2921no—
185C2×D4—2921no—
186C2×D4—2921no—
187Q16—21no—
188Q16#1192401512yescomplemented
189SD16—5842no—
190D8—2922no—
191D8—2921no—
192D8—2921no—
193C2×C8—21no—
194C2×C8—2921no—
195C2×C8C2×C22⋊D7311yescharacteristic
196C2×C8C2×C22⋊D7311yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4#1192401311yescharacteristic
206C42—21no—
207C42—42no—
208C42—41no—
209C42C2×C22⋊D7311yescharacteristic
210C16—2921no—
211C16—2921no—
212C23—5841no—
213Q8—22no—
214Q8—21no—
215Q8—21no—
216Q8—41no—
217Q8—81no—
218Q8—81no—
219D4—5841no—
220D4—5841no—
221D4—5841no—
222D4—5841no—
223D4—11681no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4#1455656011yescharacteristic
230C2×C4C2×#1192401311yescharacteristic
231C2×C4#1455661711yescharacteristic
232C8—21no—
233C8—41no—
234C8—5841no—
235C8C2×#1192401511yescharacteristic
236C8#1455662211yescharacteristic
237C22—11681no—
238C22—11681no—
239C22—11681no—
240C22#1798872111yescharacteristic
241C4—21no—
242C4—42no—
243C4—41no—
244C4—41no—
245C4—82no—
246C4—81no—
247C4#1798872611yescharacteristic
248C4#1798876011yescharacteristic
249C4#1798901711yescharacteristic
250C4#1798902211yescharacteristic
251C2—23361no—
252C2#2260903411yescharacteristic
253C2#2261185411yescharacteristic
254C2#2261189511yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#3383C2
socleC22×C73#447
prime radical#17988964C22
hypercenter#3383D73
hypocenterC73#12353
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3383

Complements

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

NH ≅ G/Ntype
4. #22607986C2semidirect
5. Q8⋊#11924003C2semidirect
6. #22607895C2semidirect
7. C73×#3383C2semidirect
67. Q16×C73SD16semidirect
128. #3383D73semidirect
135. C73#12353semidirect
188. Q16#11924015semidirect

Maximal subgroups

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

subgroupindex
#226121562
#226119712
#226092912
#226079862
Q8⋊#119240032
#226078952
C73×#33832
#1235373