Finite Group Databasebeta

← #33000252

Subgroups — #33000252

Order 18688.

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#22612187C211yesmaximal, characteristic
2C16×Dic146C211yesmaximal, characteristic, complemented
3#22609302C211yesmaximal, characteristic, complemented
4#22609292C211yesmaximal, characteristic
5#22607979C211yesmaximal, characteristic, complemented
6#22607976C211yesmaximal, characteristic
7C73×#1372C211yesmaximal, characteristic, complemented
8C2×#14556536—21no—
9C2×#14556536—21no—
10#17988967C2211yescharacteristic
11#17988964C2211yescharacteristic
12#17988873—21no—
13#17988788—21no—
14C8×Dic146C2211yescharacteristic
15C4⋊#11924003C2211yescharacteristic
16#17988405—21no—
17C16×Dic73—21no—
18#17988332—21no—
19#17988330—21no—
20C4×C16×C73C2211yescharacteristic
21#335×C73C2211yescharacteristic
22#335×C73C2211yescharacteristic
23C2×Dic292—21no—
24C2×Dic292—21no—
25C2×#11924017—41no—
26C2×D584—21no—
27C2×D584—21no—
28#14556591—21no—
29#14556588—21no—
30C4×Dic146D411yescharacteristic
31#14556536—42no—
32#14556536—42no—
33C2×#11924003—21no—
34#14556523—41no—
35C4⋊#10596831D411yescharacteristic
36#14556520—21no—
37#14556516—41no—
38#14556513—41no—
39C8×Dic73—21no—
40C2×C16×C73D411yescharacteristic
41C2×C16×C73D411yescharacteristic
42C4⋊C8×C73—21no—
43C4⋊C8×C73—21no—
44C4×C8×C73C2311yescharacteristic
45C2×Dic146—41no—
46C2×Dic146D811yescharacteristic
47C2×D292—41no—
48C2×D292—41no—
49C2×D292—41no—
50Dic292—41no—
51Dic292—41no—
52Dic292—81no—
53#11924017—82no—
54D584—41no—
55D584—41no—
56D584—81no—
57C2×#10596831—21no—
58C4⋊Dic73—41no—
59C4⋊Dic73—41no—
60C4⋊Dic73D811yescharacteristic
61#11924007—21no—
62C4×Dic73—21no—
63#11924003—41no—
64C2×C8×C73—41no—
65C2×C8×C73—41no—
66C2×C8×C73C2×D411yescharacteristic
67C2×C8×C73#4111yescharacteristic
68C42×C73C2×D411yescharacteristic
69C16×C73—21no—
70C16×C73SD1612yescomplemented
71C22×D73—81no—
72C2×C4×C73#12111yescharacteristic
73C2×C4×C73C2×D811yescharacteristic
74C2×C4×C73#13511yescharacteristic
75C2×Dic73—21no—
76C2×Dic73—21no—
77C2×Dic73—81no—
78D292—81no—
79D292—81no—
80D292—81no—
81D292—81no—
82D292—161no—
83Dic146—21no—
84Dic146—81no—
85Dic146—81no—
86Dic146SD3212yescomplemented
87C8×C73—21no—
88C8×C73—81no—
89C8×C73—81no—
90C8×C73C2×SD1611yescharacteristic
91C8×C73AGL(1,ℤ/8)11yescharacteristic
92#10596831—41no—
93C22×C73#46611yescharacteristic
94C2×D73—161no—
95C2×D73—161no—
96C2×D73—161no—
97C4×C73—21no—
98C4×C73#46011yescharacteristic
99C4×C73#47211yescharacteristic
100C4×C73C2×SD3211yescharacteristic
101C4×C73#50811yescharacteristic
102Dic73—22no—
103Dic73—41no—
104Dic73—161no—
105#12441—731nomaximal, Sylow
106C2×C73#166811yescharacteristic
107C2×C73#221711yescharacteristic
108C2×C73#222311yescharacteristic
109D73—321no—
110#2248—731no—
111Q8×C16—731no—
112#1670—731no—
113#1668—731no—
114#1372D7311yescharacteristic, complemented
115#1362—731no—
116#1361—731no—
117C73#1244111yescharacteristic, complemented, Sylow
118C2×SD32—1461no—
119C2×SD32—1461no—
120#494—731no—
121#493—731no—
122#475—1461no—
123#460—1461no—
124C8×Q8—731no—
125C4⋊C16—731no—
126C4⋊C16—1461no—
127C4×C16—1461no—
128C4×C16C2×D7311yescharacteristic
129#335C2×D7311yescharacteristic
130#335C2×D7311yescharacteristic
131#332—1461no—
132#331—1461no—
133C2×Q16—1461no—
134C2×Q16—1461no—
135C2×SD16—2921no—
136C2×D8—1461no—
137C2×D8—1461no—
138C4⋊Q8—1461no—
139#127—1461no—
140C4×Q8—731no—
141SD32—2922no—
142SD32—2922no—
143C2×C16—1461no—
144C2×C16D29211yescharacteristic
145C2×C16D29211yescharacteristic
146#106—2921no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—731no—
150C4⋊C8—1461no—
151#103—2921no—
152#102—2921no—
153C4×C8—1461no—
154C4×C8C22×D7311yescharacteristic
155C2×Q8—2921no—
156C2×Q8—731no—
157C2×D4—2921no—
158C2×D4—2921no—
159C2×D4—2921no—
160Q16—2921no—
161Q16—2921no—
162Q16—5841no—
163SD16—5842no—
164D8—2921no—
165D8—2921no—
166D8—5841no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—1461no—
170C2×C8C2×D29211yescharacteristic
171C2×C8#1192402911yescharacteristic
172C4⋊C4—2921no—
173C4⋊C4—2921no—
174C4⋊C4—731no—
175C4⋊C4—1461no—
176C42—1461no—
177C42D4×D7311yescharacteristic
178C16—21no—
179C16—2921no—
180C16#1192401712yescomplemented
181C23—5841no—
182Q8—5841no—
183Q8—5841no—
184Q8—732no—
185Q8—1461no—
186D4—5841no—
187D4—5841no—
188D4—5841no—
189D4—5841no—
190D4—11681no—
191C2×C4—5841no—
192C2×C4—1461no—
193C2×C4—1461no—
194C2×C4#1455656611yescharacteristic
195C2×C4D8×D7311yescharacteristic
196C2×C4#1455661411yescharacteristic
197C8—21no—
198C8—81no—
199C8—81no—
200C8—2921no—
201C8C2×#1192401711yescharacteristic
202C8#1455661911yescharacteristic
203C22—11681no—
204C22—11681no—
205C22—11681no—
206C22#1798882411yescharacteristic
207C4—21no—
208C4—2921no—
209C4—11681no—
210C4—1462no—
211C4#1798878811yescharacteristic
212C4#1798885811yescharacteristic
213C4SD32×D7311yescharacteristic
214C4#1798901811yescharacteristic
215C2—23361no—
216C2#2260929211yescharacteristic
217C2#2261202811yescharacteristic
218C2#2261206211yescharacteristic
subgroupG/H
centerC22#17988824
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1372C2
socleC22×C73#466
prime radical#17988964C22
hypercenter#1372D73
hypocenterC73#12441
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1372

Complements

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

NH ≅ G/Ntype
2. C16×Dic146C2semidirect
3. #22609302C2semidirect
5. #22607979C2semidirect
7. C73×#1372C2semidirect
70. C16×C73SD16semidirect
86. Dic146SD32semidirect
114. #1372D73semidirect
117. C73#12441semidirect
180. C16#11924017semidirect

Maximal subgroups

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

subgroupindex
#226121872
C16×Dic1462
#226093022
#226092922
#226079792
#226079762
C73×#13722
#1244173