Finite Group Databasebeta

← #33000249

Subgroups — #33000249

Order 18688.

232 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 71 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#22612186C211yesmaximal, characteristic
2Q8×#11924003C211yesmaximal, characteristic, complemented
3#22609301C211yesmaximal, characteristic, complemented
4#22608004C211yesmaximal, characteristic, complemented
5#22607979C211yesmaximal, characteristic, complemented
6#22607977C211yesmaximal, characteristic
7C73×#1668C211yesmaximal, characteristic
8C2×#14556535—21no—
9C2×#14556535—21no—
10#17988967C2211yescharacteristic
11#17988872—21no—
12C4⋊#11924003—21no—
13C4⋊#11924003C2211yescharacteristic
14C4×#11924003—21no—
15C4×#11924003C2211yescharacteristic
16#17988340C2211yescharacteristic
17#17988334—21no—
18#493×C73C2211yescharacteristic
19#460×C73—21no—
20C8×Q8×C73C2211yescharacteristic
21#335×C73C2211yescharacteristic
22#331×C73—21no—
23C2×Dic292—21no—
24C2×Dic292—21no—
25#14556591—21no—
26#14556535—42no—
27#14556535—42no—
28C2×#11924003—21no—
29C2×#11924003D411yescharacteristic
30C2×#11924003D411yescharacteristic
31#14556522—41no—
32C4⋊#10596831—21no—
33#14556515—41no—
34C2×SD16×C73—41no—
35C2×D8×C73—21no—
36C2×D8×C73—21no—
37#127×C73—21no—
38C4×Q8×C73D411yescharacteristic
39C4⋊C8×C73—21no—
40C4⋊C8×C73—21no—
41C4⋊C8×C73D411yescharacteristic
42#102×C73—41no—
43C4×C8×C73—21no—
44C4×C8×C73C2311yescharacteristic
45C2×Dic146—41no—
46Dic292—41no—
47Dic292—41no—
48Dic292—81no—
49C2×#10596831—41no—
50C4⋊Dic73—41no—
51C4⋊Dic73—41no—
52#11924003—21no—
53#11924003—41no—
54#11924003SD1612yescomplemented
55C2×Q8×C73D811yescharacteristic
56C2×D4×C73—41no—
57C2×D4×C73—41no—
58C2×D4×C73—41no—
59SD16×C73—82no—
60D8×C73—41no—
61D8×C73—41no—
62D8×C73—81no—
63C2×C8×C73—21no—
64C2×C8×C73—41no—
65C2×C8×C73C2×D411yescharacteristic
66C2×C8×C73#4111yescharacteristic
67C4⋊C4×C73—21no—
68C4⋊C4×C73D811yescharacteristic
69C42×C73—21no—
70C42×C73C2×D411yescharacteristic
71C23×C73—81no—
72Q8×C73—21no—
73Q8×C73SD3212yescomplemented
74D4×C73—81no—
75D4×C73—81no—
76D4×C73—81no—
77D4×C73—81no—
78D4×C73—161no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73#12111yescharacteristic
82C2×C4×C73C2×D811yescharacteristic
83C2×C4×C73#13511yescharacteristic
84C2×Dic73—81no—
85Dic146—81no—
86Dic146—81no—
87C8×C73—21no—
88C8×C73—41no—
89C8×C73—81no—
90C8×C73C2×SD1611yescharacteristic
91C8×C73AGL(1,ℤ/8)11yescharacteristic
92#10596831—81no—
93C22×C73—161no—
94C22×C73—161no—
95C22×C73—161no—
96C22×C73#46611yescharacteristic
97C4×C73—22no—
98C4×C73—21no—
99C4×C73—41no—
100C4×C73#46011yescharacteristic
101C4×C73#47211yescharacteristic
102C4×C73C2×SD3211yescharacteristic
103C4×C73#50811yescharacteristic
104Dic73—161no—
105#12441—731nomaximal, Sylow
106C2×C73—321no—
107C2×C73#166811yescharacteristic
108C2×C73#221711yescharacteristic
109C2×C73#222311yescharacteristic
110#2248—731no—
111Q8×C16—731no—
112#1670—731no—
113#1668D7311yescharacteristic
114#1372—731no—
115#1362—731no—
116#1361—731no—
117C73#1244111yescharacteristic, complemented, Sylow
118C2×SD32—1461no—
119C2×SD32—1461no—
120#494—731no—
121#493C2×D7311yescharacteristic
122#475—1461no—
123#460—21no—
124C8×Q8C2×D7311yescharacteristic
125C4⋊C16—731no—
126C4⋊C16—1461no—
127C4×C16—731no—
128C4×C16—1461no—
129#335—731no—
130#335C2×D7311yescharacteristic
131#332—1461no—
132#331—21no—
133C2×Q16—1461no—
134C2×Q16—1461no—
135C2×SD16—41no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—1461no—
139#127—21no—
140C4×Q8C22⋊D7311yescharacteristic
141SD32—2922no—
142SD32—2922no—
143C2×C16—731no—
144C2×C16—731no—
145C2×C16—1461no—
146#106—2921no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—1461no—
150C4⋊C8C22⋊D7311yescharacteristic
151#103—2921no—
152#102—41no—
153C4×C8—21no—
154C4×C8C22×D7311yescharacteristic
155C2×Q8—2921no—
156C2×Q8#1192401311yescharacteristic
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—2921no—
161Q16—2921no—
162Q16—5841no—
163SD16—82no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—2921no—
170C2×C8D4×D7311yescharacteristic
171C2×C8#1192402711yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—2921no—
174C4⋊C4—2921no—
175C4⋊C4#1192401311yescharacteristic
176C42—21no—
177C42C2×C22⋊D7311yescharacteristic
178C16—2921no—
179C16—732no—
180C16—1461no—
181C23—81no—
182Q8—21no—
183Q8—5841no—
184Q8—5841no—
185Q8#1455653512yescomplemented
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—5841no—
194C2×C4#1455656311yescharacteristic
195C2×C4C2×#1192401311yescharacteristic
196C2×C4#1455661111yescharacteristic
197C8—21no—
198C8—41no—
199C8—81no—
200C8—5841no—
201C8SD16×D7311yescharacteristic
202C8#1455661811yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1798882111yescharacteristic
207C4—22no—
208C4—21no—
209C4—41no—
210C4—11681no—
211C4#1798878511yescharacteristic
212C4#1798885511yescharacteristic
213C4C2×#1455653511yescharacteristic
214C4#1798901711yescharacteristic
215C2—321no—
216C2#2260928911yescharacteristic
217C2#2261202511yescharacteristic
218C2#2261205911yescharacteristic
subgroupG/H
centerC22#17988821
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1668C2
socleC22×C73#466
prime radical#493×C73C22
hypercenter#1668D73
hypocenterC73#12441
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1668

Complements

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

NH ≅ G/Ntype
2. Q8×#11924003C2semidirect
3. #22609301C2semidirect
4. #22608004C2semidirect
5. #22607979C2semidirect
54. #11924003SD16semidirect
73. Q8×C73SD32semidirect
117. C73#12441semidirect
185. Q8#14556535semidirect

Maximal subgroups

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

subgroupindex
#226121862
Q8×#119240032
#226093012
#226080042
#226079792
#226079772
C73×#16682
#1244173