Finite Group Databasebeta

← #37545182

Subgroups — #37545182

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#26041900C212yesmaximal, complemented
2#26041575C212yesmaximal, complemented
3#26041560C212yesmaximal, complemented
4#26041525C212yesmaximal, complemented
5#26041520C211yesmaximal, characteristic, complemented
6#26041515C211yesmaximal, characteristic
7#26041502C212yesmaximal, complemented
8#26041495C212yesmaximal, complemented
9#424×C181C211yesmaximal, characteristic, complemented
10C2×Dic724C2211yescharacteristic
11C2×#15668438C2212yes—
12C2×D1448C2211yescharacteristic
13#19153816C2212yes—
14#19153811—22no—
15#19153804C2212yes—
16#19153802—22no—
17#19153798—22no—
18#19153795C2212yes—
19#19153794—22no—
20#19153790C2212yes—
21C4×Dic362C2212yes—
22C4×D724C2212yes—
23#19153750C2211yescharacteristic
24#19153750C2211yescharacteristic
25#19153750C2212yes—
26#19153741C2212yes—
27#19153741C2212yes—
28#19153738C2212yes—
29#19153738C2212yes—
30C2×M16×C181C2211yescharacteristic
31C22×C8×C181C2211yescharacteristic
32#117×C181C2211yescharacteristic
33C4⋊C8×C181C2212yes—
34C4⋊C8×C181C2212yes—
35C2×Dic362—22no—
36C2×Dic362C2312yes—
37C2×D724C2312yes—
38C2×C22⋊D181—22no—
39C2×C4×D181—22no—
40Dic724—41no—
41#15668438—42no—
42D1448—41no—
43C4⋊Dic181—22no—
44C4⋊Dic181C2312yes—
45C4⋊Dic181C2312yes—
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×C181C2311yescharacteristic
55M16×C181—41no—
56C2×C8×C181—41no—
57C2×C8×C181C2311yescharacteristic
58C2×C8×C181C2311yescharacteristic
59C2×C8×C181C2312yes—
60C4⋊C4×C181D411yescharacteristic
61C4⋊C4×C181D411yescharacteristic
62C22⋊C4×C181D412yes—
63C42×C181C2312yes—
64C22×D181—22no—
65Dic362—22no—
66Dic362—22no—
67Dic362—82no—
68D724—22no—
69D724—22no—
70C22⋊D181—82no—
71C2×Dic181—22no—
72C2×Dic181—22no—
73C2×Dic181—22no—
74C4×D181—82no—
75C23×C181C2×D411yescharacteristic
76C2×C4×C181—21no—
77C2×C4×C181—21no—
78C2×C4×C181C2×D412yes—
79C2×C4×C181C2×D412yes—
80C2×C4×C181C2×D411yescharacteristic
81C2×C4×C181C2411yescharacteristic
82C8×C181—21no—
83C8×C181—21no—
84C8×C181—22no—
85C22×C181—21no—
86C22×C181—21no—
87C22×C181—21no—
88C22×C181C22×D411yescharacteristic
89C2×D181—42no—
90C2×D181—42no—
91C2×D181—42no—
92C4×C181—21no—
93C4×C181—22no—
94C4×C181—22no—
95C4×C181H(5,𝔽2)11yescharacteristic
96C4×C1812−1+411yescharacteristic
97Dic181—42no—
98Dic181—42no—
99Dic181—42no—
100C2×C181—41no—
101C2×C181#53711yescharacteristic
102C2×C181#57611yescharacteristic
103C2×C181#57811yescharacteristic
104D181—82no—
105C181#326211yescharacteristic, complemented, Sylow
106#3262—1811nomaximal, Sylow
107#542—1812no—
108#477—1812no—
109#474—1812no—
110#468—1812no—
111#467—1811no—
112#466—1811no—
113#464—1812no—
114#463—1812no—
115#424D18111yescharacteristic, complemented
116C2×Q16—1811no—
117C2×SD16—1812no—
118C2×D8—1811no—
119C2×M16C2×D18111yescharacteristic
120C22×C8C2×D18111yescharacteristic
121C4⋊Q8—1812no—
122#126—3622no—
123#124—3622no—
124#124—1812no—
125#123—3622no—
126C22⋊Q8—3622no—
127C22⋊Q8—1812no—
128#121—1812no—
129C4×Q8—1812no—
130C4×D4—1812no—
131#117C2×D18111yescharacteristic
132#107—1811no—
133#107—1811no—
134#107—1812no—
135C4⋊C8C2×D18112yes—
136C4⋊C8C2×D18112yes—
137#103—1812no—
138#103—1812no—
139#102—1812no—
140#102—1812no—
141C2×Q8—3622no—
142C2×Q8—1812no—
143C2×D4—3622no—
144C2×D4—1812no—
145C22×C4—3622no—
146C22×C4C22×D18111yescharacteristic
147Q16—7241no—
148SD16—7242no—
149D8—7241no—
150M16—41no—
151C2×C8—41no—
152C2×C8C22×D18111yescharacteristic
153C2×C8C22×D18112yes—
154C2×C8C22×D18111yescharacteristic
155C4⋊C4—3622no—
156C4⋊C4—3622no—
157C4⋊C4—3622no—
158C4⋊C4—1812no—
159C4⋊C4—1812no—
160C4⋊C4D72411yescharacteristic
161C4⋊C4D72411yescharacteristic
162C22⋊C4—3622no—
163C22⋊C4—3622no—
164C22⋊C4—3622no—
165C22⋊C4—3622no—
166C22⋊C4—3622no—
167C22⋊C4D72412yes—
168C42—3622no—
169C42C22×D18112yes—
170C23—3622no—
171C23C2×D72411yescharacteristic
172Q8—3622no—
173Q8—3622no—
174Q8—14482no—
175D4—3622no—
176D4—3622no—
177D4—14482no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—3622no—
181C2×C4—3622no—
182C2×C4—3622no—
183C2×C4—14482no—
184C2×C4C2×D72412yes—
185C2×C4C2×D72412yes—
186C2×C4C2×D72411yescharacteristic
187C2×C4C23×D18111yescharacteristic
188C8—21no—
189C8—21no—
190C8—22no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—7242no—
195C22—7242no—
196C22—7242no—
197C22C22×D72411yescharacteristic
198C4—21no—
199C4—22no—
200C4—22no—
201C4—7242no—
202C4—7242no—
203C4—7242no—
204C4#1915386111yescharacteristic
205C4#1915386411yescharacteristic
206C2—41no—
207C2—14482no—
208C2#2604186011yescharacteristic
209C2#2604208411yescharacteristic
210C2#2604209611yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#424×C181C2
socleC22×C181C22×D4
prime radical#26041515C2
hypercenter#424D181
hypocenterC181#3262
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. #26041900C2semidirect
2. #26041575C2semidirect
3. #26041560C2semidirect
4. #26041525C2semidirect
5. #26041520C2semidirect
7. #26041502C2semidirect
8. #26041495C2semidirect
9. #424×C181C2semidirect
105. C181#3262semidirect
115. #424D181semidirect

Maximal subgroups

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

subgroupindex
#260419002
#260415752
#260415602
#260415252
#260415202
#260415152
#260415022
#260414952
#424×C1812
#3262181