Finite Group Databasebeta

← #32999392

Subgroups — #32999392

Order 18688.

356 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 67 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#22612012C212yesmaximal, complemented
2#22609547C211yesmaximal, characteristic, complemented
3#22607970C211yesmaximal, characteristic, complemented
4#22607904C212yesmaximal, complemented
5C73×#3293C211yesmaximal, characteristic
6#17989027—22no—
7#17988990—22no—
8#17988967C2211yescharacteristic
9#17988408C2212yes—
10#17988402—22no—
11C8×#10596831C2211yescharacteristic
12#578×C73—22no—
13#494×C73C2211yescharacteristic
14#473×C73—22no—
15#456×C73—22no—
16#443×C73C2212yes—
17C2×Dic292—22no—
18#14556591—21no—
19#14556537—42no—
20#14556535—42no—
21#14556531—22no—
22#14556524—44no—
23C4×#10596831—21no—
242−1+4×C73—22no—
25#137×C73—24no—
26#137×C73—44no—
27#135×C73—42no—
28#135×C73D412yes—
29C2×Q16×C73—22no—
30C2×Q16×C73—42no—
31#131×C73—22no—
32C4⋊Q8×C73D412yes—
33#108×C73D412yes—
34#104×C73—22no—
35#101×C73—22no—
36C4×C8×C73C2311yescharacteristic
37C2×Dic146—41no—
38Dic292—42no—
39Dic292—42no—
40C2×#10596831—22no—
41C4⋊Dic73—42no—
42#11924003—42no—
43#41×C73—22no—
44#41×C73—44no—
45#41×C73—82no—
46C2×Q8×C73—22no—
47C2×Q8×C73—44no—
48Q16×C73—44no—
49Q16×C73—42no—
50Q16×C73—42no—
51Q16×C73D812yes—
52SD16×C73—22no—
53SD16×C73—44no—
54D8×C73D812yes—
55M16×C73—22no—
56M16×C73—42no—
57C2×C8×C73—42no—
58C2×C8×C73C2×D412yes—
59C4⋊C4×C73—42no—
60C42×C73C2×D411yescharacteristic
61Q8×C73—22no—
62Q8×C73—44no—
63Q8×C73—82no—
64D4×C73—22no—
65D4×C73—42no—
66D4×C73—82no—
67C2×C4×C73—21no—
68C2×C4×C73—42no—
69C2×C4×C73—42no—
70C2×C4×C73—84no—
71C2×C4×C73#12011yescharacteristic
72C2×Dic73—81no—
73Dic146—82no—
74C8×C73—42no—
75C8×C73C2×D812yes—
76C8×C73AGL(1,ℤ/8)12yes—
77#10596831—24no—
78C22×C73—42no—
79C22×C73#45311yescharacteristic
80C4×C73—22no—
81C4×C73—42no—
82C4×C73—82no—
83C4×C73#44712yes—
84Dic73—161no—
85#12313—731nomaximal, Sylow
86C2×C73—21no—
87C2×C73—82no—
88C2×C73#161911yescharacteristic
89#3293D7311yescharacteristic
90#2214—732no—
91#1713—731no—
92#1358—731no—
93#1335—732no—
94C73#1231311yescharacteristic, complemented, Sylow
95#578—22no—
96#510—1462no—
97#500—1462no—
98#494—731no—
99#494C2×D7311yescharacteristic
100#473—22no—
101#456—22no—
102#443C2×D7312yes—
103#364—732no—
104#362—1462no—
105C82—731no—
1062−1+4—22no—
107#137—24no—
108#137—44no—
109#135—42no—
110#135C22⋊D7312yes—
111C2×Q16—22no—
112C2×Q16—42no—
113C2×Q16—1462no—
114#131—22no—
115C4⋊Q8—1461no—
116C4⋊Q8C22⋊D7312yes—
117Q32—2922no—
118SD32—2922no—
119M32—1462no—
120#108C22⋊D7312yes—
121#107—2924no—
122#104—22no—
123#101—22no—
124C4×C8—1461no—
125C4×C8C22×D7311yescharacteristic
126#41—22no—
127#41—44no—
128#41—82no—
129C2×Q8—22no—
130C2×Q8—44no—
131C2×Q8—2921no—
132Q16—44no—
133Q16—42no—
134Q16—42no—
135Q16—2922no—
136Q16—2922no—
137Q16#1192401312yes—
138SD16—22no—
139SD16—44no—
140D8#1192401312yes—
141M16—22no—
142M16—42no—
143C2×C8—42no—
144C2×C8—1462no—
145C2×C8C2×C22⋊D7312yes—
146C4⋊C4—42no—
147C4⋊C4—2922no—
148C42C2×C22⋊D7311yescharacteristic
149C16—2922no—
150Q8—22no—
151Q8—44no—
152Q8—82no—
153Q8—5842no—
154D4—22no—
155D4—42no—
156D4—82no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—84no—
161C2×C4—5841no—
162C2×C4#1455656011yescharacteristic
163C8—42no—
164C8—1464no—
165C8C2×#1192401312yes—
166C8#1455661712yes—
167C22—42no—
168C22#1798875511yescharacteristic
169C4—22no—
170C4—42no—
171C4—82no—
172C4—11681no—
173C4#1798872112yes—
174C2—21no—
175C2—82no—
176C2#2260898511yescharacteristic
subgroupG/H
centerC2#22608985
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
Fitting, prime radicalC73×#3293C2
socleC2×C73#1619
hypercenter#3293D73
hypocenterC73#12313
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3293

Complements

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

NH ≅ G/Ntype
1. #22612012C2semidirect
2. #22609547C2semidirect
3. #22607970C2semidirect
4. #22607904C2semidirect
94. C73#12313semidirect

Maximal subgroups

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

subgroupindex
#226120122
#226095472
#226079702
#226079042
C73×#32932
#1231373