Finite Group Databasebeta

← #33000250

Subgroups — #33000250

Order 18688.

232 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 65 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
2#22611833C211yesmaximal, characteristic, complemented
3#22609302C211yesmaximal, characteristic, complemented
4#22609288C211yesmaximal, characteristic
5#22608004C211yesmaximal, characteristic, complemented
6#22607981C211yesmaximal, characteristic, complemented
7C73×#1361C211yesmaximal, characteristic
8C2×#14556535—21no—
9C2×#14556535—21no—
10#17988967C2211yescharacteristic
11#17988873—21no—
12#17988784—21no—
13C8×Dic146C2211yescharacteristic
14#17988405—21no—
15C4×#11924003C2211yescharacteristic
16C16×Dic73—21no—
17#17988340C2211yescharacteristic
18#17988332—21no—
19#17988331—21no—
20#493×C73C2211yescharacteristic
21C4⋊C16×C73C2211yescharacteristic
22#335×C73C2211yescharacteristic
23C2×Dic292—21no—
24C2×Dic292—21no—
25C2×#11924016—41no—
26#14556591—21no—
27C4×Dic146D411yescharacteristic
28#14556535—42no—
29#14556535—42no—
30C2×#11924003D411yescharacteristic
31C2×#11924003D411yescharacteristic
32#14556523—41no—
33C4⋊#10596831—21no—
34C4⋊#10596831D411yescharacteristic
35#14556520—21no—
36#14556516—41no—
37#14556512—41no—
38C8×Dic73—21no—
39C2×D8×C73—21no—
40C2×D8×C73—21no—
41#127×C73—21no—
42C2×C16×C73—21no—
43C4⋊C8×C73—21no—
44C4×C8×C73C2311yescharacteristic
45C2×Dic146—41no—
46C2×Dic146D811yescharacteristic
47Dic292—41no—
48Dic292—41no—
49Dic292—81no—
50#11924016—82no—
51C2×#10596831—21no—
52C2×#10596831—41no—
53C4⋊Dic73—41no—
54C4⋊Dic73—41no—
55C4⋊Dic73D811yescharacteristic
56#11924007—21no—
57C4×Dic73—21no—
58#11924003—21no—
59#11924003SD1612yescomplemented
60C2×D4×C73—41no—
61C2×D4×C73—41no—
62C2×D4×C73—41no—
63D8×C73—41no—
64D8×C73—41no—
65D8×C73—81no—
66C2×C8×C73—41no—
67C2×C8×C73C2×D411yescharacteristic
68C2×C8×C73#4111yescharacteristic
69C42×C73C2×D411yescharacteristic
70C16×C73—41no—
71C23×C73—81no—
72D4×C73—81no—
73D4×C73—81no—
74D4×C73—81no—
75D4×C73—81no—
76D4×C73—161no—
77C2×C4×C73#12111yescharacteristic
78C2×C4×C73C2×D811yescharacteristic
79C2×C4×C73#13511yescharacteristic
80C2×Dic73—21no—
81C2×Dic73—21no—
82C2×Dic73—81no—
83Dic146—21no—
84Dic146—81no—
85Dic146—81no—
86Dic146SD3212yescomplemented
87C8×C73—21no—
88C8×C73—81no—
89C8×C73C2×SD1611yescharacteristic
90C8×C73AGL(1,ℤ/8)11yescharacteristic
91#10596831—41no—
92#10596831—81no—
93C22×C73—161no—
94C22×C73—161no—
95C22×C73—161no—
96C22×C73#46611yescharacteristic
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—321no—
107C2×C73#166811yescharacteristic
108C2×C73#221711yescharacteristic
109C2×C73#222311yescharacteristic
110#2248—731no—
111Q8×C16—731no—
112#1670—731no—
113#1668—731no—
114#1372—731no—
115#1362—731no—
116#1361D7311yescharacteristic
117C73#1244111yescharacteristic, complemented, Sylow
118C2×SD32—1461no—
119C2×SD32—1461no—
120#494—731no—
121#493C2×D7311yescharacteristic
122#475—1461no—
123#460—1461no—
124C8×Q8—731no—
125C4⋊C16—1461no—
126C4⋊C16C2×D7311yescharacteristic
127C4×C16—731no—
128C4×C16—1461no—
129#335—731no—
130#335C2×D7311yescharacteristic
131#332—1461no—
132#331—1461no—
133C2×Q16—1461no—
134C2×Q16—1461no—
135C2×SD16—2921no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—1461no—
139#127—21no—
140C4×Q8—731no—
141SD32—2922no—
142SD32—2922no—
143C2×C16—21no—
144C2×C16—731no—
145C2×C16—731no—
146#106—2921no—
147C4⋊C8—21no—
148C4⋊C8—731no—
149C4⋊C8—1461no—
150C4⋊C8—1461no—
151#103—2921no—
152#102—2921no—
153C4×C8—1461no—
154C4×C8C22×D7311yescharacteristic
155C2×Q8—2921no—
156C2×Q8—731no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—2921no—
161Q16—2921no—
162Q16—5841no—
163SD16—5842no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—41no—
168C2×C8—2921no—
169C2×C8—1461no—
170C2×C8D4×D7311yescharacteristic
171C2×C8#1192402811yescharacteristic
172C4⋊C4—2921no—
173C4⋊C4—2921no—
174C4⋊C4—731no—
175C4⋊C4—1461no—
176C42—1461no—
177C42D4×D7311yescharacteristic
178C16—41no—
179C16—732no—
180C16—1461no—
181C23—81no—
182Q8—5841no—
183Q8—5841no—
184Q8—732no—
185Q8—1461no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—5841no—
192C2×C4—1461no—
193C2×C4—1461no—
194C2×C4#1455656411yescharacteristic
195C2×C4D8×D7311yescharacteristic
196C2×C4#1455661211yescharacteristic
197C8—21no—
198C8—81no—
199C8—2921no—
200C8—5841no—
201C8SD16×D7311yescharacteristic
202C8#1455661511yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1798882211yescharacteristic
207C4—21no—
208C4—2921no—
209C4—11681no—
210C4—1462no—
211C4#1798878611yescharacteristic
212C4#1798885611yescharacteristic
213C4SD32×D7311yescharacteristic
214C4#1798901611yescharacteristic
215C2—321no—
216C2#2260929011yescharacteristic
217C2#2261202611yescharacteristic
218C2#2261206011yescharacteristic
subgroupG/H
centerC22#17988822
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1361C2
socleC22×C73#466
prime radical#493×C73C22
hypercenter#1361D73
hypocenterC73#12441
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1361

Complements

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

NH ≅ G/Ntype
2. #22611833C2semidirect
3. #22609302C2semidirect
5. #22608004C2semidirect
6. #22607981C2semidirect
59. #11924003SD16semidirect
86. Dic146SD32semidirect
117. C73#12441semidirect

Maximal subgroups

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

subgroupindex
#226121862
#226118332
#226093022
#226092882
#226080042
#226079812
C73×#13612
#1244173