Finite Group Databasebeta

← #37534259

Subgroups — #37534259

Order 23168.

314 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 93 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#26041757C211yesmaximal, characteristic, complemented
2#26041296C212yesmaximal
3C2×#19153735C211yesmaximal, characteristic, complemented
4#26041035C212yesmaximal, complemented
5#536×C181C211yesmaximal, characteristic
6C2×#15668433C2212yes—
7Q8×Dic181—24no—
8#19153775—22no—
9#19153775C2212yes—
10#19153743—24no—
11#19153735—22no—
12C2×#41×C181C2211yescharacteristic
13C22×Q8×C181C2211yescharacteristic
14C4⋊Q8×C181C412yescomplemented
15#124×C181C412yescomplemented
16#123×C181—22no—
17C22⋊Q8×C181—22no—
18#117×C181C2211yescharacteristic
19#15668433—22no—
20#15668433—22no—
21C2×#13089678—22no—
22C4⋊Dic181—24no—
23C4⋊Dic181—22no—
24C22⋊Dic181—22no—
25C4×Dic181—24no—
26C4×Dic181D414yes—
27#41×C181—22no—
28#41×C181—22no—
29C2×Q8×C181—22no—
30C2×Q8×C181—22no—
31C2×Q8×C181C2×C411yescharacteristic
32C2×Q8×C181C2×C412yes—
33C2×Q8×C181D412yes—
34C2×D4×C181—21no—
35C2×D4×C181C2×C411yescharacteristic
36C22×C4×C181—21no—
37C22×C4×C181D412yes—
38C22×C4×C181C2311yescharacteristic
39C4⋊C4×C181—24no—
40C4⋊C4×C181—21no—
41C22⋊C4×C181—21no—
42C22⋊C4×C181—24no—
43C42×C181C2×C412yes—
44C2×Dic181—24no—
45#13089678—42no—
46C23×C181—21no—
47C23×C181C2×D411yescharacteristic
48Q8×C181—24no—
49Q8×C181—24no—
50Q8×C181—22no—
51D4×C181—22no—
52D4×C181—41no—
53D4×C181—41no—
54C2×C4×C181—22no—
55C2×C4×C181—22no—
56C2×C4×C181—22no—
57C2×C4×C181—21no—
58C2×C4×C181—42no—
59C2×C4×C181C22⋊C414yes—
60C2×C4×C181C22×C411yescharacteristic
61C2×C4×C181C2×D412yes—
62C2×C4×C181C2×D411yescharacteristic
63C2×C4×C181#4112yes—
64C22×C181—41no—
65C22×C181—42no—
66C22×C181—41no—
67C22×C181C2×C22⋊C411yescharacteristic
68C22×C181#12011yescharacteristic
69C22×C181#12311yescharacteristic
70C4×C181—24no—
71C4×C181—42no—
72C4×C181—41no—
73C4×C181C4×D412yes—
74C4×C181#12112yes—
75Dic181—44no—
76C2×C181—21no—
77C2×C181—21no—
78C2×C181—21no—
79C2×C181—81no—
80C2×C181#38611yescharacteristic
81C181#191211yescharacteristic, complemented, Sylow
82#1912—1811nomaximal, Sylow
83#536D18111yescharacteristic
84#519—1811no—
85#421—1812no—
86C2×#101—1811no—
87H(3,ℤ/4)—1812no—
88C2×#41C2×D18111yescharacteristic
89C22×Q8C2×D18111yescharacteristic
90C2×M16—1812no—
91C4⋊Q8Dic18112yescomplemented
92#124Dic18112yescomplemented
93#123—22no—
94C22⋊Q8—22no—
95C4×Q8—3624no—
96#117—3622no—
97#117—1812no—
98#117C2×D18111yescharacteristic
99#104—3624no—
100#101—3622no—
101#41—22no—
102#41—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×Dic18111yescharacteristic
106C2×Q8C2×Dic18112yes—
107C2×Q8C22⋊D18112yes—
108C2×D4—21no—
109C2×D4C2×Dic18111yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D18112yes—
112C22×C4C22×D18111yescharacteristic
113M16—3622no—
114M16—3622no—
115C2×C8—3622no—
116C4⋊C4—24no—
117C4⋊C4—21no—
118C4⋊C4—3624no—
119C4⋊C4—3622no—
120C22⋊C4—21no—
121C22⋊C4—24no—
122C22⋊C4—3622no—
123C42—3624no—
124C42—1814no—
125C42C2×Dic18112yes—
126C23—21no—
127C23C2×C22⋊D18111yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—22no—
131D4—22no—
132D4—41no—
133D4—41no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—42no—
139C2×C4—3624no—
140C2×C4C22⋊Dic18114yes—
141C2×C4C22×Dic18111yescharacteristic
142C2×C4D4×D18112yes—
143C2×C4C2×C22⋊D18111yescharacteristic
144C2×C4#1566844812yes—
145C8—7242no—
146C22—41no—
147C22—42no—
148C22—41no—
149C22C2×C22⋊Dic18111yescharacteristic
150C22#1915378411yescharacteristic
151C22#1915379711yescharacteristic
152C4—24no—
153C4—42no—
154C4—41no—
155C4—7244no—
156C4D4×Dic18112yes—
157C4#1915378812yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—81no—
162C2#2604115911yescharacteristic
subgroupG/H
centerC2#26041159
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#536×C181C2
socleC2×C181#386
prime radicalC2×#41×C181C22
cosocleC22×C4×C181C23
hypercenter#536D181
hypocenterC181#1912
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #26041757C2semidirect
3. C2×#19153735C2semidirect
4. #26041035C2semidirect
14. C4⋊Q8×C181C4semidirect
15. #124×C181C4semidirect
81. C181#1912semidirect
91. C4⋊Q8Dic181semidirect
92. #124Dic181semidirect

Maximal subgroups

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

subgroupindex
#260417572
#260412962
C2×#191537352
#260410352
#536×C1812
#1912181