Finite Group Databasebeta

← #37545122

Subgroups — #37545122

Order 23168.

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

normal subgroups only

This group has 354 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#26041813C212yesmaximal, complemented
2#26041587C212yesmaximal, complemented
3#26041580C212yesmaximal, complemented
4#26041537C211yesmaximal, characteristic, complemented
5#26041532C211yesmaximal, characteristic
6#26041525C212yesmaximal, complemented
7#26041502C212yesmaximal, complemented
8#26041495C212yesmaximal, complemented
9#423×C181C211yesmaximal, characteristic, complemented
10C2×Dic724C2211yescharacteristic
11C2×#15668438C2212yes—
12C2×D1448C2211yescharacteristic
13#19153810—22no—
14#19153808C2212yes—
15#19153804C2212yes—
16#19153799—22no—
17#19153795C2212yes—
18#19153790C2212yes—
19C4×Dic362C2212yes—
20C4×D724C2212yes—
21C4×C22⋊D181—22no—
22#19153773—22no—
23#19153750C2212yes—
24#19153748C2211yescharacteristic
25#19153748C2211yescharacteristic
26#19153741C2212yes—
27#19153741C2212yes—
28#19153738C2212yes—
29#19153738C2212yes—
30C2×M16×C181C2211yescharacteristic
31C2×M16×C181C2211yescharacteristic
32C2×C42×C181C2211yescharacteristic
33C4⋊C8×C181C2212yes—
34C4⋊C8×C181C2212yes—
35C2×Dic362C2312yes—
36C2×D724C2312yes—
37C2×C22⋊D181—22no—
38C2×C4×D181—22no—
39Dic724—41no—
40#15668438—42no—
41D1448—41no—
42C4⋊Dic181C2312yes—
43C4⋊Dic181C2312yes—
44#15668428—22no—
45#15668428—22no—
46#15668428—22no—
47#15668428—22no—
48C22⋊Dic181—22no—
49#15668426—22no—
50#15668426—22no—
51#15668426—22no—
52#15668426—22no—
53C4×Dic181—22no—
54C22×C4×C181D412yes—
55C22×C4×C181C2311yescharacteristic
56M16×C181—41no—
57M16×C181—41no—
58C2×C8×C181C2311yescharacteristic
59C2×C8×C181C2312yes—
60C2×C8×C181C2311yescharacteristic
61C42×C181D411yescharacteristic
62C42×C181D411yescharacteristic
63C42×C181C2312yes—
64C22×D181—22no—
65Dic362—22no—
66Dic362—22no—
67D724—22no—
68D724—22no—
69C22⋊D181—44no—
70C2×Dic181—22no—
71C2×Dic181—22no—
72C2×Dic181—22no—
73C4×D181—44no—
74C23×C181C2×D411yescharacteristic
75C2×C4×C181—21no—
76C2×C4×C181—21no—
77C2×C4×C181—42no—
78C2×C4×C181C2×D411yescharacteristic
79C2×C4×C181C2×D412yes—
80C2×C4×C181C2×D412yes—
81C2×C4×C181C2411yescharacteristic
82C8×C181—21no—
83C8×C181—22no—
84C8×C181—21no—
85C22×C181—21no—
86C22×C181—22no—
87C22×C181C22×D411yescharacteristic
88C2×D181—42no—
89C2×D181—42no—
90C2×D181—42no—
91C4×C181—21no—
92C4×C181—22no—
93C4×C181—22no—
94C4×C181H(5,𝔽2)11yescharacteristic
95C4×C1812−1+411yescharacteristic
96Dic181—42no—
97Dic181—42no—
98Dic181—42no—
99C2×C181—41no—
100C2×C181#53711yescharacteristic
101C2×C181#57512yes—
102D181—82no—
103C181#325611yescharacteristic, complemented, Sylow
104#3256—1811nomaximal, Sylow
105#529—1812no—
106#479—1812no—
107#478—1812no—
108#470—1811no—
109#469—1811no—
110#468—1812no—
111#464—1812no—
112#463—1812no—
113#423D18111yescharacteristic, complemented
114C2×Q16—1811no—
115C2×SD16—1812no—
116C2×D8—1811no—
117C2×M16C2×D18111yescharacteristic
118C2×M16C2×D18111yescharacteristic
119#126—3622no—
120#125—1812no—
121#124—1812no—
122#123—3622no—
123C22⋊Q8—1812no—
124#121—1812no—
125C4×Q8—1812no—
126C4×D4—3622no—
127C4×D4—1812no—
128#117—3622no—
129C2×C42C2×D18111yescharacteristic
130#107—1812no—
131#106—1811no—
132#106—1811no—
133C4⋊C8C2×D18112yes—
134C4⋊C8C2×D18112yes—
135#103—1812no—
136#103—1812no—
137#102—1812no—
138#102—1812no—
139C2×Q8—1812no—
140C2×D4—3622no—
141C2×D4—1812no—
142C22×C4—3622no—
143C22×C4D72412yes—
144C22×C4C22×D18111yescharacteristic
145Q16—7241no—
146SD16—7242no—
147D8—7241no—
148M16—41no—
149M16—41no—
150C2×C8C22×D18111yescharacteristic
151C2×C8C22×D18112yes—
152C2×C8C22×D18111yescharacteristic
153C4⋊C4—3622no—
154C4⋊C4—3622no—
155C4⋊C4—3622no—
156C4⋊C4—3622no—
157C4⋊C4—1812no—
158C4⋊C4—1812no—
159C22⋊C4—3622no—
160C22⋊C4—3622no—
161C22⋊C4—3622no—
162C22⋊C4—3622no—
163C22⋊C4—3622no—
164C42—3622no—
165C42D72411yescharacteristic
166C42D72411yescharacteristic
167C42C22×D18112yes—
168C23—3622no—
169C23C2×D72411yescharacteristic
170Q8—3622no—
171Q8—3622no—
172D4—3622no—
173D4—3622no—
174D4—7244no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—3622no—
179C2×C4—3622no—
180C2×C4—3622no—
181C2×C4—7244no—
182C2×C4C2×D72412yes—
183C2×C4C2×D72411yescharacteristic
184C2×C4C2×D72412yes—
185C2×C4C23×D18111yescharacteristic
186C8—21no—
187C8—22no—
188C8—21no—
189C22—21no—
190C22—22no—
191C22—7242no—
192C22—7242no—
193C22—7242no—
194C22C22×D72411yescharacteristic
195C4—21no—
196C4—22no—
197C4—22no—
198C4—7242no—
199C4—7242no—
200C4—7242no—
201C4#1915386111yescharacteristic
202C4#1915386411yescharacteristic
203C2—41no—
204C2—14482no—
205C2#2604186011yescharacteristic
206C2#2604207912yes—
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#423×C181C2
socleC22×C181C22×D4
prime radical#26041532C2
hypercenter#423D181
hypocenterC181#3256
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #26041813C2semidirect
2. #26041587C2semidirect
3. #26041580C2semidirect
4. #26041537C2semidirect
6. #26041525C2semidirect
7. #26041502C2semidirect
8. #26041495C2semidirect
9. #423×C181C2semidirect
103. C181#3256semidirect
113. #423D181semidirect

Maximal subgroups

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

subgroupindex
#260418132
#260415872
#260415802
#260415372
#260415322
#260415252
#260415022
#260414952
#423×C1812
#3256181