Finite Group Databasebeta

← #37545168

Subgroups — #37545168

Order 23168.

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

normal subgroups only

This group has 346 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#26041891C211yesmaximal, characteristic, complemented
2#26041884C211yesmaximal, characteristic
3#26041587C212yesmaximal, complemented
4#26041580C212yesmaximal, complemented
5#26041525C212yesmaximal, complemented
6#26041508C212yesmaximal, complemented
7#26041485C212yesmaximal, complemented
8#26041480C212yesmaximal, complemented
9#424×C181C211yesmaximal, characteristic, complemented
10C2×#15668438C2212yes—
11C2×#15668438C2212yes—
12#19153816C2211yescharacteristic
13#19153813C2211yescharacteristic
14#19153809—22no—
15#19153808C2211yescharacteristic
16#19153808C2211yescharacteristic
17#19153806—22no—
18#19153801—22no—
19#19153795C2212yes—
20#19153790C2212yes—
21#19153789—22no—
22C4×Dic362C2212yes—
23C4×D724C2212yes—
24#19153750C2212yes—
25#19153748C2212yes—
26#19153741C2212yes—
27#19153741C2212yes—
28#19153738C2212yes—
29#19153738C2212yes—
30C2×M16×C181C2211yescharacteristic
31C22×C8×C181C2211yescharacteristic
32#117×C181C2211yescharacteristic
33C4⋊C8×C181C2212yes—
34C4⋊C8×C181C2212yes—
35C2×Dic362C2312yes—
36C2×D724—22no—
37C2×D724C2312yes—
38C2×C22⋊D181—22no—
39C2×C4×D181—22no—
40#15668438—42no—
41#15668438—42no—
42C4⋊Dic181—22no—
43C4⋊Dic181C2312yes—
44C4⋊Dic181C2312yes—
45#15668428—22no—
46#15668428—22no—
47#15668428—22no—
48#15668428—22no—
49C22⋊Dic181—22no—
50#15668426—22no—
51#15668426—22no—
52C4×Dic181—22no—
53C22×C4×C181C2311yescharacteristic
54M16×C181—41no—
55C2×C8×C181—41no—
56C2×C8×C181C2312yes—
57C2×C8×C181C2312yes—
58C4⋊C4×C181D412yes—
59C22⋊C4×C181D412yes—
60C42×C181C2311yescharacteristic
61C42×C181C2311yescharacteristic
62C22×D181—22no—
63Dic362—22no—
64Dic362—22no—
65D724—22no—
66D724—22no—
67D724—82no—
68C22⋊D181—82no—
69C2×Dic181—22no—
70C2×Dic181—22no—
71C2×Dic181—22no—
72C4×D181—82no—
73C23×C181C2×D411yescharacteristic
74C2×C4×C181—21no—
75C2×C4×C181—21no—
76C2×C4×C181C2×D412yes—
77C2×C4×C181C2×D412yes—
78C2×C4×C181C2×D411yescharacteristic
79C2×C4×C181C2411yescharacteristic
80C8×C181—22no—
81C8×C181—22no—
82C22×C181—21no—
83C22×C181—22no—
84C22×C181C22×D411yescharacteristic
85C2×D181—42no—
86C2×D181—42no—
87C2×D181—42no—
88C4×C181—21no—
89C4×C181—22no—
90C4×C181—22no—
91C4×C181H(5,𝔽2)11yescharacteristic
92C4×C1812−1+411yescharacteristic
93Dic181—42no—
94Dic181—42no—
95Dic181—42no—
96C2×C181—41no—
97C2×C181#53711yescharacteristic
98C2×C181#57712yes—
99D181—82no—
100C181#326011yescharacteristic, complemented, Sylow
101#3260—1811nomaximal, Sylow
102#541—1811no—
103#540—1811no—
104#479—1812no—
105#478—1812no—
106#468—1812no—
107#465—1812no—
108#461—1812no—
109#460—1812no—
110#424D18111yescharacteristic, complemented
111C2×SD16—1812no—
112C2×SD16—1812no—
113C2×M16C2×D18111yescharacteristic
114C22×C8C2×D18111yescharacteristic
115C4⋊Q8—1811no—
116#127—1811no—
117#126—3622no—
118#125—3622no—
119#125—1811no—
120#125—1811no—
121#123—3622no—
122C22⋊Q8—1812no—
123#121—3622no—
124#121—1812no—
125C4×Q8—1812no—
126C4×D4—1812no—
127#117C2×D18111yescharacteristic
128#107—1812no—
129#106—1812no—
130C4⋊C8C2×D18112yes—
131C4⋊C8C2×D18112yes—
132#103—1812no—
133#103—1812no—
134#102—1812no—
135#102—1812no—
136C2×Q8—1812no—
137C2×D4—3622no—
138C2×D4—3622no—
139C2×D4—1812no—
140C22×C4—3622no—
141C22×C4C22×D18111yescharacteristic
142SD16—7242no—
143SD16—7242no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D18112yes—
147C2×C8C22×D18112yes—
148C4⋊C4—3622no—
149C4⋊C4—3622no—
150C4⋊C4—3622no—
151C4⋊C4—3622no—
152C4⋊C4—3622no—
153C4⋊C4—1812no—
154C4⋊C4—1812no—
155C4⋊C4D72412yes—
156C22⋊C4—3622no—
157C22⋊C4—3622no—
158C22⋊C4—3622no—
159C22⋊C4D72412yes—
160C42—3622no—
161C42C22×D18111yescharacteristic
162C42C22×D18111yescharacteristic
163C23—3622no—
164C23C2×D72411yescharacteristic
165Q8—3622no—
166Q8—3622no—
167D4—3622no—
168D4—3622no—
169D4—14482no—
170D4—14482no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—3622no—
174C2×C4—3622no—
175C2×C4—3622no—
176C2×C4—14482no—
177C2×C4C2×D72412yes—
178C2×C4C2×D72412yes—
179C2×C4C2×D72411yescharacteristic
180C2×C4C23×D18111yescharacteristic
181C8—22no—
182C8—22no—
183C22—21no—
184C22—22no—
185C22—7242no—
186C22—7242no—
187C22—7242no—
188C22C22×D72411yescharacteristic
189C4—21no—
190C4—22no—
191C4—22no—
192C4—7242no—
193C4—7242no—
194C4—7242no—
195C4#1915386111yescharacteristic
196C4#1915386411yescharacteristic
197C2—41no—
198C2—14482no—
199C2#2604186011yescharacteristic
200C2#2604209112yes—
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#424×C181C2
socleC22×C181C22×D4
prime radical#26041884C2
hypercenter#424D181
hypocenterC181#3260
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #26041891C2semidirect
3. #26041587C2semidirect
4. #26041580C2semidirect
5. #26041525C2semidirect
6. #26041508C2semidirect
7. #26041485C2semidirect
8. #26041480C2semidirect
9. #424×C181C2semidirect
100. C181#3260semidirect
110. #424D181semidirect

Maximal subgroups

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

subgroupindex
#260418912
#260418842
#260415872
#260415802
#260415252
#260415082
#260414852
#260414802
#424×C1812
#3260181