Finite Group Databasebeta

← #32999365

Subgroups — #32999365

Order 18688.

274 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 59 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#22612148C212yesmaximal
2#22611506C211yesmaximal, characteristic
3#22607919C212yesmaximal
4#22607882C211yesmaximal, characteristic
5C73×#3255C211yesmaximal, characteristic
6#17988636C2211yescharacteristic
7#17988409—22no—
8#17988403—22no—
9#17988369C2212yes—
10#17988359—21no—
11#17988347—21no—
12#17988344C2211yescharacteristic
13#533×C73—21no—
14#475×C73—22no—
15#470×C73—21no—
16#470×C73C2212yes—
17#462×C73—22no—
18#423×C73C2211yescharacteristic
19C2×#11924012—21no—
20#14556550—21no—
21#14556531—22no—
22#14556527—22no—
23C2×Q16×C73—21no—
24C2×Q16×C73D412yes—
25C2×M16×C73D411yescharacteristic
26C2×M16×C73C2311yescharacteristic
27C4⋊Q8×C73—21no—
28C4⋊Q8×C73—21no—
29#125×C73—41no—
30C22⋊Q8×C73—22no—
31C22⋊Q8×C73—41no—
32C4×Q8×C73—22no—
33C4×Q8×C73—41no—
34#117×C73—41no—
35C2×C42×C73D411yescharacteristic
36#106×C73—21no—
37#106×C73D412yes—
38C4⋊C8×C73—22no—
39#103×C73—22no—
40#103×C73—22no—
41#11924012—41no—
42#11924012—41no—
43C2×#10596831—41no—
44C4⋊Dic73—41no—
45C22⋊Dic73—41no—
46C4×Dic73—41no—
47#11924003—42no—
48C2×Q8×C73—22no—
49C2×Q8×C73—41no—
50C22×C4×C73—21no—
51C22×C4×C73C2×D411yescharacteristic
52Q16×C73—42no—
53Q16×C73—81no—
54M16×C73—41no—
55M16×C73—41no—
56C2×C8×C73—21no—
57C2×C8×C73C2×D412yes—
58C4⋊C4×C73—22no—
59C4⋊C4×C73—41no—
60C4⋊C4×C73—41no—
61C4⋊C4×C73—42no—
62C4⋊C4×C73—41no—
63C4⋊C4×C73—42no—
64C22⋊C4×C73—42no—
65C42×C73—21no—
66C42×C73—42no—
67C42×C73SD1611yescharacteristic
68C42×C73SD1611yescharacteristic
69C23×C73AGL(1,ℤ/8)11yescharacteristic
70Q8×C73—42no—
71Q8×C73—42no—
72Q8×C73—82no—
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—21no—
77C2×C4×C73—42no—
78C2×C4×C73—42no—
79C2×C4×C73—42no—
80C2×C4×C73#12011yescharacteristic
81C2×C4×C73C2×SD1611yescharacteristic
82C2×Dic73—81no—
83C8×C73—22no—
84C8×C73—41no—
85#10596831—81no—
86C22×C73—21no—
87C22×C73—41no—
88C22×C73#45011yescharacteristic
89C4×C73—22no—
90C4×C73—21no—
91C4×C73—41no—
92C4×C73—82no—
93C4×C73—82no—
94C4×C73#45411yescharacteristic
95C4×C73#45811yescharacteristic
96Dic73—161no—
97#12308—731nomaximal, Sylow
98C2×C73—21no—
99C2×C73—41no—
100C2×C73#161611yescharacteristic
101#3255D7311yescharacteristic
102#2239—732no—
103#2102—731no—
104#1340—732no—
105#1327—731no—
106C73#1230811yescharacteristic, complemented, Sylow
107#533—21no—
108#475—22no—
109#470—21no—
110#470C2×D7312yes—
111#462—22no—
112#430—731no—
113#423C2×D7311yescharacteristic
114#365—1462no—
115#362—1462no—
116#349—732no—
117#344—1461no—
118#339—1461no—
119H(3,ℤ/4)—731no—
120C2×Q16—21no—
121C2×Q16C22⋊D7312yes—
122C2×M16—1461no—
123C2×M16C22⋊D7311yescharacteristic
124C2×M16C22×D7311yescharacteristic
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127#125—41no—
128C22⋊Q8—22no—
129C22⋊Q8—41no—
130C4×Q8—22no—
131C4×Q8—41no—
132#117—41no—
133#117—1461no—
134C2×C42C22⋊D7311yescharacteristic
135M32—1462no—
136#108—1462no—
137#106—21no—
138#106C22⋊D7312yes—
139C4⋊C8—22no—
140#103—22no—
141#103—22no—
142C2×Q8—22no—
143C2×Q8—41no—
144C22×C4—21no—
145C22×C4C2×C22⋊D7311yescharacteristic
146Q16—42no—
147Q16—81no—
148M16—41no—
149M16—41no—
150M16—2921no—
151M16—2921no—
152C2×C8—21no—
153C2×C8—2921no—
154C2×C8C2×C22⋊D7312yes—
155C4⋊C4—22no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—2921no—
162C22⋊C4—42no—
163C22⋊C4—2921no—
164C42—21no—
165C42—42no—
166C42—2921no—
167C42#1192401611yescharacteristic
168C42#1192401611yescharacteristic
169C16—2922no—
170C23#1455661711yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—82no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—5841no—
182C2×C4#1455656011yescharacteristic
183C2×C4C2×#1192401611yescharacteristic
184C8—22no—
185C8—41no—
186C8—5841no—
187C22—21no—
188C22—41no—
189C22#1798873811yescharacteristic
190C4—22no—
191C4—21no—
192C4—41no—
193C4—82no—
194C4—82no—
195C4—11681no—
196C4#1798876011yescharacteristic
197C4#1798877711yescharacteristic
198C2—21no—
199C2—41no—
200C2#2260896811yescharacteristic
subgroupG/H
centerC2#22608968
commutator, cosocleC2×M16×C73C23
FrattiniC2×M16C22×D73
FittingC73×#3255C2
socleC2×C73#1616
prime radicalC23×C73AGL(1,ℤ/8)
hypercenter#3255D73
hypocenterC73#12308
LCSG ▷ C2×M16×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3255

Complements

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

NH ≅ G/Ntype
106. C73#12308semidirect

Maximal subgroups

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

subgroupindex
#226121482
#226115062
#226079192
#226078822
C73×#32552
#1230873