Finite Group Databasebeta

← #37535406

Subgroups — #37535406

Order 23168.

250 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 73 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#26041263C212yesmaximal, complemented
2#26041078C212yesmaximal, complemented
3#26041076C212yesmaximal, complemented
4#532×C181C211yesmaximal, characteristic
5C2×#15668433—22no—
6#19153735C2214yes—
7#19153733—22no—
8C2×#41×C181C411yescharacteristic
9C2×#41×C181C2211yescharacteristic
10C4⋊Q8×C181C2211yescharacteristic
11#127×C181C411yescharacteristic
12#124×C181C411yescharacteristic
13#124×C181C2211yescharacteristic
14#121×C181—22no—
15C4×D4×C181—22no—
16C2×C42×C181C411yescharacteristic
17#15668433—24no—
18#15668433—42no—
19C2×#13089678—42no—
20#41×C181—41no—
21#41×C181—41no—
22#41×C181—41no—
23#41×C181—41no—
24C2×Q8×C181C2×C411yescharacteristic
25C2×Q8×C181C2311yescharacteristic
26C2×D4×C181—22no—
27C2×D4×C181—21no—
28C2×D4×C181C2×C411yescharacteristic
29C2×D4×C181C2×C411yescharacteristic
30C2×D4×C181D412yes—
31C22×C4×C181—21no—
32C22×C4×C181—21no—
33C22×C4×C181C2×C411yescharacteristic
34C22×C4×C181D412yes—
35C4⋊C4×C181—22no—
36C22⋊C4×C181—22no—
37C22⋊C4×C181—22no—
38C42×C181—21no—
39C42×C181C2×C411yescharacteristic
40C42×C181C2×C411yescharacteristic
41#13089678—44no—
42C23×C181—21no—
43C23×C181C22⋊C412yes—
44C23×C181C22⋊C411yescharacteristic
45Q8×C181—41no—
46Q8×C181—41no—
47D4×C181—41no—
48D4×C181—42no—
49D4×C181—41no—
50D4×C181—44no—
51D4×C181—42no—
52C2×C4×C181—21no—
53C2×C4×C181—21no—
54C2×C4×C181—21no—
55C2×C4×C181—22no—
56C2×C4×C181—22no—
57C2×C4×C181—21no—
58C2×C4×C181—21no—
59C2×C4×C181—42no—
60C2×C4×C181—41no—
61C2×C4×C181—41no—
62C2×C4×C181C22⋊C411yescharacteristic
63C2×C4×C181C22×C411yescharacteristic
64C2×C4×C181C2×D412yes—
65C22×C181—21no—
66C22×C181—22no—
67C22×C181—41no—
68C22×C181—42no—
69C22×C181—42no—
70C22×C181—41no—
71C22×C181C2×C22⋊C411yescharacteristic
72C4×C181—22no—
73C4×C181—21no—
74C4×C181—22no—
75C4×C181—41no—
76C4×C181—41no—
77C4×C181#9911yescharacteristic
78C4×C181#9911yescharacteristic
79C2×C181—21no—
80C2×C181—41no—
81C2×C181—42no—
82C2×C181—81no—
83C2×C181C2×#9911yescharacteristic
84C181#213211yescharacteristic, complemented, Sylow
85#2132—1811nomaximal, Sylow
86#532D18111yescharacteristic
87#413—1812no—
88#356—1812no—
89#355—1812no—
90C2×#41Dic18111yescharacteristic
91C2×#41C2×D18111yescharacteristic
92C2×M16—3622no—
93C4⋊Q8C2×D18111yescharacteristic
94#127Dic18111yescharacteristic
95#124Dic18111yescharacteristic
96#124C2×D18111yescharacteristic
97#121—22no—
98C4×D4—22no—
99C2×C42Dic18111yescharacteristic
100#101—1814no—
101#100—3622no—
102#41—41no—
103#41—41no—
104#41—41no—
105#41—41no—
106C2×Q8C2×Dic18111yescharacteristic
107C2×Q8C22×D18111yescharacteristic
108C2×D4—22no—
109C2×D4—21no—
110C2×D4C2×Dic18111yescharacteristic
111C2×D4C2×Dic18111yescharacteristic
112C2×D4C22⋊D18112yes—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4C2×Dic18111yescharacteristic
116C22×C4C22⋊D18112yes—
117M16—3624no—
118M16—7242no—
119C2×C8—7242no—
120C4⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C42—21no—
124C42C2×Dic18111yescharacteristic
125C42C2×Dic18111yescharacteristic
126C23—21no—
127C23C22⋊Dic18112yes—
128C23C22⋊Dic18111yescharacteristic
129Q8—41no—
130Q8—41no—
131D4—41no—
132D4—42no—
133D4—41no—
134D4—44no—
135D4—42no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4C22⋊Dic18111yescharacteristic
147C2×C4C22×Dic18111yescharacteristic
148C2×C4C2×C22⋊D18112yes—
149C8—7244no—
150C22—21no—
151C22—22no—
152C22—41no—
153C22—42no—
154C22—42no—
155C22—41no—
156C22C2×C22⋊Dic18111yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—41no—
162C4#1915373111yescharacteristic
163C4#1915373111yescharacteristic
164C2—21no—
165C2—41no—
166C2—42no—
167C2—81no—
168C2C2×#1915373111yescharacteristic
subgroupG/H
centerC4#19153731
commutatorC2×C4×C181C22×C4
FrattiniC2×Q8C22×D181
Fitting, prime radical#532×C181C2
socleC2×C181C2×#99
cosocleC2×Q8×C181C23
hypercenter#532D181
hypocenterC181#2132
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #26041263C2semidirect
2. #26041078C2semidirect
3. #26041076C2semidirect
84. C181#2132semidirect

Maximal subgroups

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

subgroupindex
#260412632
#260410782
#260410762
#532×C1812
#2132181