Finite Group Databasebeta

← #37544616

Subgroups — #37544616

Order 23168.

378 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 117 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#26041850C212yesmaximal
2#26041621C214yesmaximal, complemented
3#26041537C212yesmaximal, complemented
4#26041520C212yesmaximal, complemented
5#26041437C214yesmaximal, complemented
6#408×C181C211yesmaximal, characteristic
7C22×Dic362C2212yes—
8C2×Dic724C2212yes—
9C2×Dic724C2212yes—
10#19153814—22no—
11#19153803—22no—
12#19153797—22no—
13#19153797—22no—
14#19153795C2214yes—
15#19153795C2214yes—
16#19153775C2212yes—
17#19153750C2212yes—
18#19153748C2212yes—
19#19153741C2214yes—
20#19153741C2214yes—
21C2×#41×C181C2211yescharacteristic
22C2×M16×C181C2211yescharacteristic
23C22×C8×C181C2211yescharacteristic
24C22⋊C8×C181C2214yes—
25C2×Dic362—22no—
26C2×Dic362—42no—
27C2×Dic362—42no—
28C2×Dic362C2314yes—
29C22×Dic181—22no—
30Dic724—42no—
31Dic724—42no—
32C4⋊Dic181C2314yes—
33#15668428—24no—
34#15668428—24no—
35C22⋊Dic181—22no—
36C22⋊Dic181—24no—
37C22⋊Dic181—24no—
38C4×Dic181—22no—
39#41×C181—41no—
40#41×C181—41no—
41C2×Q8×C181D411yescharacteristic
42C2×D4×C181D411yescharacteristic
43C2×D4×C181D412yes—
44C22×C4×C181C2312yes—
45C22×C4×C181C2311yescharacteristic
46M16×C181—41no—
47C2×C8×C181—41no—
48C2×C8×C181C2312yes—
49C2×C8×C181C2312yes—
50Dic362—24no—
51Dic362—24no—
52Dic362—44no—
53C2×Dic181—24no—
54C2×Dic181—24no—
55C2×Dic181—44no—
56C23×C181C2×D411yescharacteristic
57C23×C181C2×D412yes—
58Q8×C181—41no—
59D4×C181—41no—
60D4×C181—42no—
61C2×C4×C181—21no—
62C2×C4×C181—21no—
63C2×C4×C181—22no—
64C2×C4×C181—22no—
65C2×C4×C181C2×D412yes—
66C2×C4×C181C2×D411yescharacteristic
67C2×C4×C181C2411yescharacteristic
68C8×C181—22no—
69C8×C181—22no—
70C22×C181—21no—
71C22×C181—22no—
72C22×C181—22no—
73C22×C181—24no—
74C22×C181C22×D411yescharacteristic
75C4×C181—21no—
76C4×C181—22no—
77C4×C181H(5,𝔽2)11yescharacteristic
78C4×C181H(5,𝔽2)11yescharacteristic
79Dic181—44no—
80Dic181—44no—
81C2×C181—41no—
82C2×C181—42no—
83C2×C181#53411yescharacteristic
84C2×C181#57812yes—
85C181#320211yescharacteristic, complemented, Sylow
86#3202—1811nomaximal, Sylow
87#536—1812no—
88#485—1814no—
89#470—1812no—
90#467—1812no—
91#451—1814no—
92#408D18111yescharacteristic
93C2×#41C2×D18111yescharacteristic
94C22×Q8—1812no—
95C2×Q16—1812no—
96C2×Q16—1812no—
97C2×M16C2×D18111yescharacteristic
98C22×C8C2×D18111yescharacteristic
99C4⋊Q8—3622no—
100#124—3622no—
101#123—3622no—
102#123—3622no—
103C22⋊Q8—1814no—
104C22⋊Q8—1814no—
105#117—1812no—
106#107—1812no—
107#106—1812no—
108#103—1814no—
109#103—1814no—
110C22⋊C8C2×D18114yes—
111#41—41no—
112#41—41no—
113C2×Q8—3622no—
114C2×Q8—1814no—
115C2×Q8—7242no—
116C2×Q8—7242no—
117C2×Q8D72411yescharacteristic
118C2×D4D72411yescharacteristic
119C2×D4D72412yes—
120C22×C4—3622no—
121C22×C4C22×D18112yes—
122C22×C4C22×D18111yescharacteristic
123Q16—7242no—
124Q16—7242no—
125M16—41no—
126C2×C8—41no—
127C2×C8C22×D18112yes—
128C2×C8C22×D18112yes—
129C4⋊C4—3624no—
130C4⋊C4—3624no—
131C4⋊C4—1814no—
132C22⋊C4—3622no—
133C22⋊C4—3624no—
134C22⋊C4—3624no—
135C42—3622no—
136C23C2×D72411yescharacteristic
137C23C2×D72412yes—
138Q8—41no—
139Q8—3624no—
140Q8—3624no—
141Q8—7244no—
142D4—42no—
143D4—41no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—3624no—
149C2×C4—3624no—
150C2×C4—7244no—
151C2×C4C2×D72412yes—
152C2×C4C2×D72411yescharacteristic
153C2×C4C23×D18111yescharacteristic
154C8—22no—
155C8—22no—
156C22—22no—
157C22—24no—
158C22—22no—
159C22—21no—
160C22C22×D72411yescharacteristic
161C4—22no—
162C4—21no—
163C4—7244no—
164C4—7244no—
165C4#1915386111yescharacteristic
166C4#1915386111yescharacteristic
167C2—42no—
168C2—41no—
169C2#2604184111yescharacteristic
170C2#2604209612yes—
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#408×C181C2
socleC22×C181C22×D4
prime radicalC2×#41×C181C22
hypercenter#408D181
hypocenterC181#3202
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #26041621C2semidirect
3. #26041537C2semidirect
4. #26041520C2semidirect
5. #26041437C2semidirect
85. C181#3202semidirect

Maximal subgroups

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

subgroupindex
#260418502
#260416212
#260415372
#260415202
#260414372
#408×C1812
#3202181