Finite Group Databasebeta

← #33000762

Subgroups — #33000762

Order 18688.

284 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 75 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#22612188C211yesmaximal, characteristic
2#22612158C211yesmaximal, characteristic, complemented
3#22612155C211yesmaximal, characteristic
4#22609538C211yesmaximal, characteristic
5#22609525C211yesmaximal, characteristic, complemented
6#22607850C211yesmaximal, characteristic, complemented
7C73×#2247C211yesmaximal, characteristic, complemented
8C2×#14556534—22no—
9#17988992C2211yescharacteristic
10#17988983—21no—
11#17988964C2211yescharacteristic
12#17988939—21no—
13#17988935—21no—
14#17988935—21no—
15#17988391—22no—
16#17988389—22no—
17C4×#11924003C2211yescharacteristic
18C8×#10596831C2211yescharacteristic
19C2×Q32×C73—22no—
20#494×C73C2211yescharacteristic
21#494×C73C2211yescharacteristic
22C4×C16×C73C2211yescharacteristic
23C2×D584—21no—
24C2×D584—21no—
25#14556591—21no—
26#14556588—21no—
27#14556534—44no—
28C2×#11924003D412yes—
29#14556525—21no—
30#14556525—21no—
31#14556522—42no—
32#14556514—42no—
33#14556511—42no—
34#14556511—42no—
35C4×#10596831D411yescharacteristic
36C4×#10596831D411yescharacteristic
37C2×Q16×C73—21no—
38C2×Q16×C73—21no—
39C2×Q16×C73—21no—
40C2×Q16×C73—21no—
41C4⋊Q8×C73—21no—
42C4⋊Q8×C73—21no—
43Q32×C73—44no—
44C2×C16×C73D412yes—
45C4×C8×C73C2311yescharacteristic
46C2×Dic146—41no—
47C2×D292—41no—
48C2×D292—41no—
49C2×D292—41no—
50D584—42no—
51D584—42no—
52C2×#10596831D812yes—
53C2×#10596831D812yes—
54C4⋊Dic73—41no—
55C4⋊Dic73—41no—
56#11924003—22no—
57C2×Q8×C73—41no—
58C2×Q8×C73—41no—
59Q16×C73—42no—
60Q16×C73—42no—
61Q16×C73—42no—
62Q16×C73—42no—
63C2×C8×C73C2×D411yescharacteristic
64C2×C8×C73C2×D411yescharacteristic
65C4⋊C4×C73—41no—
66C4⋊C4×C73—41no—
67C4⋊C4×C73—41no—
68C4⋊C4×C73—41no—
69C42×C73C2×D411yescharacteristic
70C16×C73—22no—
71C22×D73—81no—
72Q8×C73—81no—
73Q8×C73—81no—
74Q8×C73—81no—
75Q8×C73—81no—
76C2×C4×C73—81no—
77C2×C4×C73—81no—
78C2×C4×C73#12711yescharacteristic
79C2×C4×C73C2×D811yescharacteristic
80C2×C4×C73C2×D811yescharacteristic
81C2×Dic73—81no—
82D292—81no—
83D292—81no—
84D292—82no—
85D292—82no—
86Dic146—82no—
87C8×C73AGL(1,ℤ/8)12yes—
88C8×C73AGL(1,ℤ/8)12yes—
89#10596831—22no—
90#10596831—22no—
91C22×C73#49311yescharacteristic
92C2×D73—161no—
93C2×D73—161no—
94C2×D73—161no—
95C4×C73—161no—
96C4×C73—161no—
97C4×C73#49611yescharacteristic
98C4×C73#49611yescharacteristic
99C4×C73#50812yes—
100C4×C73#51012yes—
101Dic73—161no—
102#12530—731nomaximal, Sylow
103C2×C73#172111yescharacteristic
104C2×C73#224911yescharacteristic
105C2×C73#225111yescharacteristic
106D73—321no—
107#2248—731no—
108#2247D7311yescharacteristic, complemented
109#2241—731no—
110#2240—731no—
111#1709—731no—
112#1707—731no—
113#1312—731no—
114C73#1253011yescharacteristic, complemented, Sylow
115C2×Q32—22no—
116C2×SD32—1462no—
117#500—731no—
118#498—1461no—
119#494C2×D7311yescharacteristic
120#494C2×D7311yescharacteristic
121#493—731no—
122#487—1461no—
123#486—1461no—
124#486—1461no—
125#358—1462no—
126#357—1462no—
127C4×C16—731no—
128C4×C16C2×D7311yescharacteristic
129C82—731no—
130C2×Q16—21no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×D8—1461no—
135C2×D8—1461no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—1461no—
139#127—1461no—
140Q32—44no—
141SD32—2924no—
142C2×C16—732no—
143C2×C16C22⋊D7312yes—
144#107—1461no—
145#107—1461no—
146#106—2922no—
147#103—2922no—
148#102—2922no—
149#102—2922no—
150C4×C8—731no—
151C4×C8—731no—
152C4×C8C22×D7311yescharacteristic
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—2921no—
156C2×D4—2921no—
157C2×D4—2921no—
158C2×D4—2921no—
159Q16—42no—
160Q16—42no—
161Q16—42no—
162Q16—42no—
163D8—2922no—
164D8—2922no—
165C2×C8—732no—
166C2×C8—732no—
167C2×C8D4×D7311yescharacteristic
168C2×C8C2×C22⋊D7311yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—2921no—
174C4⋊C4—2921no—
175C42D4×D7311yescharacteristic
176C16—22no—
177C16—1462no—
178C23—5841no—
179Q8—81no—
180Q8—81no—
181Q8—81no—
182Q8—81no—
183Q8—5842no—
184D4—5841no—
185D4—5841no—
186D4—5842no—
187D4—5842no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—5841no—
191C2×C4#1455658711yescharacteristic
192C2×C4D8×D7311yescharacteristic
193C2×C4D8×D7311yescharacteristic
194C8—1462no—
195C8—1462no—
196C8#1455661512yes—
197C8#1455661712yes—
198C22—11681no—
199C22—11681no—
200C22—11681no—
201C22#1798896211yescharacteristic
202C4—161no—
203C4—161no—
204C4—11681no—
205C4#1798897411yescharacteristic
206C4#1798897511yescharacteristic
207C4#1798902012yes—
208C4#1798902912yes—
209C2—23361no—
210C2#2260959611yescharacteristic
211C2#2261219311yescharacteristic
212C2#2261219911yescharacteristic
subgroupG/H
centerC22#17988962
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#2247C2
socleC22×C73#493
prime radical#17988964C22
hypercenter#2247D73
hypocenterC73#12530
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2247

Complements

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

NH ≅ G/Ntype
2. #22612158C2semidirect
5. #22609525C2semidirect
6. #22607850C2semidirect
7. C73×#2247C2semidirect
108. #2247D73semidirect
114. C73#12530semidirect

Maximal subgroups

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

subgroupindex
#226121882
#226121582
#226121552
#226095382
#226095252
#226078502
C73×#22472
#1253073