Finite Group Databasebeta

← #33000248

Subgroups — #33000248

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
1C16×Dic146C211yesmaximal, characteristic, complemented
2#22609298C211yesmaximal, characteristic, complemented
3#22609288C211yesmaximal, characteristic
4#22608005C211yesmaximal, characteristic, complemented
5#22607980C211yesmaximal, characteristic, complemented
6#22607977C211yesmaximal, characteristic
7C73×#2248C211yesmaximal, characteristic
8#17988869—21no—
9#17988784—21no—
10C8×Dic146C2211yescharacteristic
11C4⋊#11924003C2211yescharacteristic
12#17988405—21no—
13C16×Dic73—21no—
14#17988340C2211yescharacteristic
15#17988340C2211yescharacteristic
16#17988333—21no—
17#17988331—21no—
18C2×SD32×C73—21no—
19C2×SD32×C73—21no—
20#494×C73C2211yescharacteristic
21#493×C73C2211yescharacteristic
22C4×C16×C73C2211yescharacteristic
23C2×#11924016—41no—
24C4×Dic146D411yescharacteristic
25C2×#11924003—21no—
26#14556522—41no—
27C4⋊#10596831—21no—
28C4⋊#10596831—21no—
29C4⋊#10596831D411yescharacteristic
30#14556520—21no—
31#14556514—41no—
32#14556512—41no—
33C8×Dic73—21no—
34C2×Q16×C73—21no—
35C2×Q16×C73—21no—
36C2×D8×C73—21no—
37C2×D8×C73—21no—
38C4⋊Q8×C73—21no—
39#127×C73—21no—
40SD32×C73—42no—
41SD32×C73—42no—
42C2×C16×C73D411yescharacteristic
43C2×C16×C73D411yescharacteristic
44C4×C8×C73C2311yescharacteristic
45C2×Dic146D811yescharacteristic
46#11924016—82no—
47C2×#10596831—21no—
48C2×#10596831—41no—
49C2×#10596831—41no—
50C4⋊Dic73D811yescharacteristic
51#11924007—21no—
52C4×Dic73—21no—
53#11924003—41no—
54C2×Q8×C73—41no—
55C2×D4×C73—41no—
56C2×D4×C73—41no—
57C2×D4×C73—41no—
58Q16×C73—41no—
59Q16×C73—41no—
60Q16×C73—81no—
61D8×C73—41no—
62D8×C73—41no—
63D8×C73—81no—
64C2×C8×C73C2×D411yescharacteristic
65C2×C8×C73#4111yescharacteristic
66C4⋊C4×C73—41no—
67C4⋊C4×C73—41no—
68C42×C73C2×D411yescharacteristic
69C16×C73—21no—
70C16×C73SD1612yescomplemented
71C23×C73—81no—
72Q8×C73—81no—
73Q8×C73—81no—
74D4×C73—81no—
75D4×C73—81no—
76D4×C73—81no—
77D4×C73—81no—
78D4×C73—161no—
79C2×C4×C73—81no—
80C2×C4×C73#12111yescharacteristic
81C2×C4×C73C2×D811yescharacteristic
82C2×C4×C73#13511yescharacteristic
83C2×Dic73—21no—
84C2×Dic73—21no—
85Dic146—21no—
86Dic146SD3212yescomplemented
87C8×C73—21no—
88C8×C73C2×SD1611yescharacteristic
89C8×C73AGL(1,ℤ/8)11yescharacteristic
90#10596831—41no—
91#10596831—81no—
92#10596831—81no—
93C22×C73—161no—
94C22×C73—161no—
95C22×C73—161no—
96C22×C73#46611yescharacteristic
97C4×C73—21no—
98C4×C73—161no—
99C4×C73#46011yescharacteristic
100C4×C73#47211yescharacteristic
101C4×C73C2×SD3211yescharacteristic
102C4×C73#50811yescharacteristic
103Dic73—22no—
104Dic73—41no—
105#12441—731nomaximal, Sylow
106C2×C73—321no—
107C2×C73#166811yescharacteristic
108C2×C73#221711yescharacteristic
109C2×C73#222311yescharacteristic
110#2248D7311yescharacteristic
111Q8×C16—731no—
112#1670—731no—
113#1668—731no—
114#1372—731no—
115#1362—731no—
116#1361—731no—
117C73#1244111yescharacteristic, complemented, Sylow
118C2×SD32—21no—
119C2×SD32—21no—
120#494C2×D7311yescharacteristic
121#493C2×D7311yescharacteristic
122#475—1461no—
123#460—1461no—
124C8×Q8—731no—
125C4⋊C16—731no—
126C4⋊C16—1461no—
127C4×C16—1461no—
128C4×C16C2×D7311yescharacteristic
129#335—731no—
130#335—731no—
131#332—1461no—
132#331—1461no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—2921no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—21no—
139#127—21no—
140C4×Q8—731no—
141SD32—42no—
142SD32—42no—
143C2×C16—1461no—
144C2×C16C22⋊D7311yescharacteristic
145C2×C16C22⋊D7311yescharacteristic
146#106—2921no—
147C4⋊C8—731no—
148C4⋊C8—1461no—
149C4⋊C8—1461no—
150C4⋊C8—1461no—
151#103—2921no—
152#102—2921no—
153C4×C8—1461no—
154C4×C8C22×D7311yescharacteristic
155C2×Q8—41no—
156C2×Q8—731no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—5842no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—2921no—
168C2×C8—2921no—
169C2×C8—1461no—
170C2×C8C2×C22⋊D7311yescharacteristic
171C2×C8#1192402711yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—731no—
175C4⋊C4—1461no—
176C42—1461no—
177C42D4×D7311yescharacteristic
178C16—21no—
179C16—2921no—
180C16#1192401612yescomplemented
181C23—81no—
182Q8—81no—
183Q8—81no—
184Q8—732no—
185Q8—1461no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—81no—
192C2×C4—1461no—
193C2×C4—1461no—
194C2×C4#1455656211yescharacteristic
195C2×C4D8×D7311yescharacteristic
196C2×C4#1455661011yescharacteristic
197C8—21no—
198C8—2921no—
199C8—5841no—
200C8—5841no—
201C8C2×#1192401611yescharacteristic
202C8#1455661711yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1798882011yescharacteristic
207C4—21no—
208C4—161no—
209C4—2921no—
210C4—1462no—
211C4#1798878411yescharacteristic
212C4#1798885411yescharacteristic
213C4SD32×D7311yescharacteristic
214C4#1798901811yescharacteristic
215C2—321no—
216C2#2260928811yescharacteristic
217C2#2261202411yescharacteristic
218C2#2261205811yescharacteristic
subgroupG/H
centerC22#17988820
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#2248C2
socleC22×C73#466
prime radical#493×C73C22
hypercenter#2248D73
hypocenterC73#12441
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. C16×Dic146C2semidirect
2. #22609298C2semidirect
4. #22608005C2semidirect
5. #22607980C2semidirect
70. C16×C73SD16semidirect
86. Dic146SD32semidirect
117. C73#12441semidirect
180. C16#11924016semidirect

Maximal subgroups

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

subgroupindex
C16×Dic1462
#226092982
#226092882
#226080052
#226079802
#226079772
C73×#22482
#1244173