Finite Group Databasebeta

← #36468696

Subgroups — #36468696

Order 22784.

232 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 69 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#26022508C211yesmaximal, characteristic
2C16×Dic178C211yesmaximal, characteristic, complemented
3#26019623C211yesmaximal, characteristic, complemented
4#26019613C211yesmaximal, characteristic
5#26018300C211yesmaximal, characteristic, complemented
6#26018297C211yesmaximal, characteristic
7C89×#1372C211yesmaximal, characteristic, complemented
8C2×#15667957—21no—
9C2×#15667957—21no—
10#19151978C2211yescharacteristic
11#19151975C2211yescharacteristic
12#19151884—21no—
13#19151799—21no—
14C8×Dic178C2211yescharacteristic
15C4⋊#13083549C2211yescharacteristic
16#19151416—21no—
17C16×Dic89—21no—
18#19151343—21no—
19#19151341—21no—
20C4×C16×C89C2211yescharacteristic
21#335×C89C2211yescharacteristic
22#335×C89C2211yescharacteristic
23C2×Dic356—21no—
24C2×Dic356—21no—
25C2×#13083563—41no—
26C2×D712—21no—
27C2×D712—21no—
28#15668012—21no—
29#15668009—21no—
30C4×Dic178D411yescharacteristic
31#15667957—42no—
32#15667957—42no—
33C2×#13083549—21no—
34#15667944—41no—
35C4⋊#10623494D411yescharacteristic
36#15667941—21no—
37#15667937—41no—
38#15667934—41no—
39C8×Dic89—21no—
40C2×C16×C89D411yescharacteristic
41C2×C16×C89D411yescharacteristic
42C4⋊C8×C89—21no—
43C4⋊C8×C89—21no—
44C4×C8×C89C2311yescharacteristic
45C2×Dic178—41no—
46C2×Dic178D811yescharacteristic
47C2×D356—41no—
48C2×D356—41no—
49C2×D356—41no—
50Dic356—41no—
51Dic356—41no—
52Dic356—81no—
53#13083563—82no—
54D712—41no—
55D712—41no—
56D712—81no—
57C2×#10623494—21no—
58C4⋊Dic89—41no—
59C4⋊Dic89—41no—
60C4⋊Dic89D811yescharacteristic
61#13083553—21no—
62C4×Dic89—21no—
63#13083549—41no—
64C2×C8×C89—41no—
65C2×C8×C89—41no—
66C2×C8×C89C2×D411yescharacteristic
67C2×C8×C89#4111yescharacteristic
68C42×C89C2×D411yescharacteristic
69C16×C89—21no—
70C16×C89SD1612yescomplemented
71C22×D89—81no—
72C2×C4×C89#12111yescharacteristic
73C2×C4×C89C2×D811yescharacteristic
74C2×C4×C89#13511yescharacteristic
75C2×Dic89—21no—
76C2×Dic89—21no—
77C2×Dic89—81no—
78D356—81no—
79D356—81no—
80D356—81no—
81D356—81no—
82D356—161no—
83Dic178—21no—
84Dic178—81no—
85Dic178—81no—
86Dic178SD3212yescomplemented
87C8×C89—21no—
88C8×C89—81no—
89C8×C89—81no—
90C8×C89C2×SD1611yescharacteristic
91C8×C89AGL(1,ℤ/8)11yescharacteristic
92#10623494—41no—
93C22×C89#46611yescharacteristic
94C2×D89—161no—
95C2×D89—161no—
96C2×D89—161no—
97C4×C89—21no—
98C4×C89#46011yescharacteristic
99C4×C89#47211yescharacteristic
100C4×C89C2×SD3211yescharacteristic
101C4×C89#50811yescharacteristic
102Dic89—22no—
103Dic89—41no—
104Dic89—161no—
105#12441—891nomaximal, Sylow
106C2×C89#166811yescharacteristic
107C2×C89#221711yescharacteristic
108C2×C89#222311yescharacteristic
109D89—321no—
110#2248—891no—
111Q8×C16—891no—
112#1670—891no—
113#1668—891no—
114#1372D8911yescharacteristic, complemented
115#1362—891no—
116#1361—891no—
117C89#1244111yescharacteristic, complemented, Sylow
118C2×SD32—1781no—
119C2×SD32—1781no—
120#494—891no—
121#493—891no—
122#475—1781no—
123#460—1781no—
124C8×Q8—891no—
125C4⋊C16—891no—
126C4⋊C16—1781no—
127C4×C16—1781no—
128C4×C16C2×D8911yescharacteristic
129#335C2×D8911yescharacteristic
130#335C2×D8911yescharacteristic
131#332—1781no—
132#331—1781no—
133C2×Q16—1781no—
134C2×Q16—1781no—
135C2×SD16—3561no—
136C2×D8—1781no—
137C2×D8—1781no—
138C4⋊Q8—1781no—
139#127—1781no—
140C4×Q8—891no—
141SD32—3562no—
142SD32—3562no—
143C2×C16—1781no—
144C2×C16D35611yescharacteristic
145C2×C16D35611yescharacteristic
146#106—3561no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—891no—
150C4⋊C8—1781no—
151#103—3561no—
152#102—3561no—
153C4×C8—1781no—
154C4×C8C22×D8911yescharacteristic
155C2×Q8—891no—
156C2×Q8—3561no—
157C2×D4—3561no—
158C2×D4—3561no—
159C2×D4—3561no—
160Q16—3561no—
161Q16—3561no—
162Q16—7121no—
163SD16—7122no—
164D8—3561no—
165D8—3561no—
166D8—7121no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—1781no—
170C2×C8C2×D35611yescharacteristic
171C2×C8#1308357511yescharacteristic
172C4⋊C4—891no—
173C4⋊C4—3561no—
174C4⋊C4—3561no—
175C4⋊C4—1781no—
176C42—1781no—
177C42D4×D8911yescharacteristic
178C16—21no—
179C16—3561no—
180C16#1308356312yescomplemented
181C23—7121no—
182Q8—892no—
183Q8—1781no—
184Q8—7121no—
185Q8—7121no—
186D4—14241no—
187D4—7121no—
188D4—7121no—
189D4—7121no—
190D4—7121no—
191C2×C4—1781no—
192C2×C4—1781no—
193C2×C4—7121no—
194C2×C4#1566798711yescharacteristic
195C2×C4D8×D8911yescharacteristic
196C2×C4#1566803511yescharacteristic
197C8—21no—
198C8—81no—
199C8—81no—
200C8—3561no—
201C8C2×#1308356311yescharacteristic
202C8#1566804011yescharacteristic
203C22—14241no—
204C22—14241no—
205C22—14241no—
206C22#1915183511yescharacteristic
207C4—21no—
208C4—3561no—
209C4—14241no—
210C4—1782no—
211C4#1915179911yescharacteristic
212C4#1915186911yescharacteristic
213C4SD32×D8911yescharacteristic
214C4#1915202911yescharacteristic
215C2—28481no—
216C2#2601961311yescharacteristic
217C2#2602234911yescharacteristic
218C2#2602238311yescharacteristic
subgroupG/H
centerC22#19151835
commutator, cosocleC4×C8×C89C23
FrattiniC4×C8C22×D89
FittingC89×#1372C2
socleC22×C89#466
prime radical#19151975C22
hypercenter#1372D89
hypocenterC89#12441
LCSG ▷ C4×C8×C89 ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1372

Complements

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

NH ≅ G/Ntype
2. C16×Dic178C2semidirect
3. #26019623C2semidirect
5. #26018300C2semidirect
7. C89×#1372C2semidirect
70. C16×C89SD16semidirect
86. Dic178SD32semidirect
114. #1372D89semidirect
117. C89#12441semidirect
180. C16#13083563semidirect

Maximal subgroups

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

subgroupindex
#260225082
C16×Dic1782
#260196232
#260196132
#260183002
#260182972
C89×#13722
#1244189