Finite Group Databasebeta

← #37545183

Subgroups — #37545183

Order 23168.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#26041902C212yesmaximal, complemented
2#26041572C212yesmaximal, complemented
3#26041558C212yesmaximal, complemented
4#26041527C212yesmaximal, complemented
5#26041515C211yesmaximal, characteristic
6#26041503C212yesmaximal, complemented
7#26041497C212yesmaximal, complemented
8#26041305C211yesmaximal, characteristic, complemented
9#467×C181C211yesmaximal, characteristic, complemented
10C2×#15668436C2212yes—
11C2×D1448C2211yescharacteristic
12C2×#15668433C2211yescharacteristic
13#19153815C2212yes—
14#19153811—22no—
15#19153805C2212yes—
16#19153802—22no—
17#19153798—22no—
18#19153796—22no—
19#19153790C2212yes—
20Q8×Dic181C2212yes—
21#19153778C2212yes—
22#19153775C2211yescharacteristic
23#19153750C2211yescharacteristic
24#19153749C2212yes—
25#19153745C2212yes—
26#19153745C2212yes—
27#19153740C2212yes—
28#19153738C2212yes—
29#19153736C2212yes—
30C2×Q16×C181C2211yescharacteristic
31C22×C8×C181C2211yescharacteristic
32C22⋊Q8×C181C2212yes—
33#107×C181C2211yescharacteristic
34#103×C181C2212yes—
35C2×Dic362—22no—
36C2×D724C2312yes—
37C2×C22⋊D181—22no—
38C2×C4×D181—22no—
39#15668436—42no—
40D1448—41no—
41#15668433—41no—
42C2×#13089678C2312yes—
43C4⋊Dic181—22no—
44C4⋊Dic181D411yescharacteristic
45C4⋊Dic181D411yescharacteristic
46C4⋊Dic181C2312yes—
47#15668428—22no—
48C22⋊Dic181D412yes—
49#15668426—22no—
50#15668426—22no—
51#15668426—22no—
52#15668426—22no—
53C4×Dic181—22no—
54C4×Dic181C2312yes—
55C2×Q8×C181C2312yes—
56C22×C4×C181C2311yescharacteristic
57Q16×C181—41no—
58C2×C8×C181—41no—
59C2×C8×C181C2311yescharacteristic
60C2×C8×C181C2311yescharacteristic
61C4⋊C4×C181—22no—
62C4⋊C4×C181C2312yes—
63C22⋊C4×C181—22no—
64C22×D181—22no—
65Dic362—82no—
66D724—22no—
67D724—22no—
68C22⋊D181—82no—
69C2×Dic181—22no—
70C2×Dic181C2×D412yes—
71C2×Dic181C2×D412yes—
72C4×D181—82no—
73#13089678—22no—
74C23×C181C2×D411yescharacteristic
75Q8×C181—22no—
76Q8×C181—22no—
77C2×C4×C181—21no—
78C2×C4×C181—21no—
79C2×C4×C181—22no—
80C2×C4×C181—22no—
81C2×C4×C181C2×D411yescharacteristic
82C2×C4×C181C2411yescharacteristic
83C8×C181—21no—
84C8×C181—21no—
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—42no—
94C4×C181—42no—
95C4×C181H(5,𝔽2)11yescharacteristic
96C4×C1812−1+411yescharacteristic
97Dic181—22no—
98Dic181—22no—
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#467D18111yescharacteristic, complemented
112#466—1811no—
113#464—1812no—
114#463—1812no—
115#424—1811no—
116C2×Q16C2×D18111yescharacteristic
117C2×SD16—1812no—
118C2×D8—1811no—
119C2×M16—1811no—
120C22×C8C2×D18111yescharacteristic
121C4⋊Q8—1812no—
122#126—3622no—
123#124—3622no—
124#124—1812no—
125#123—3622no—
126C22⋊Q8—3622no—
127C22⋊Q8C2×D18112yes—
128#121—1812no—
129C4×Q8—1812no—
130C4×D4—1812no—
131#117—1811no—
132#107—1811no—
133#107—1812no—
134#107C2×D18111yescharacteristic
135C4⋊C8—1812no—
136C4⋊C8—1812no—
137#103—1812no—
138#103C2×D18112yes—
139#102—1812no—
140#102—1812no—
141C2×Q8—3622no—
142C2×Q8C22×D18112yes—
143C2×D4—3622no—
144C2×D4—1812no—
145C22×C4—3622no—
146C22×C4C22×D18111yescharacteristic
147Q16—41no—
148SD16—7242no—
149D8—7241no—
150M16—7241no—
151C2×C8—41no—
152C2×C8—1812no—
153C2×C8C22×D18111yescharacteristic
154C2×C8C22×D18111yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—3622no—
157C4⋊C4—3622no—
158C4⋊C4—1812no—
159C4⋊C4—1811no—
160C4⋊C4—1811no—
161C4⋊C4C22×D18112yes—
162C22⋊C4—22no—
163C22⋊C4—3622no—
164C22⋊C4—3622no—
165C22⋊C4—3622no—
166C22⋊C4—3622no—
167C22⋊C4—1812no—
168C42—3622no—
169C42—1812no—
170C23—3622no—
171C23D4×D18111yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—14482no—
175D4—3622no—
176D4—3622no—
177D4—14482no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—3622no—
183C2×C4—14482no—
184C2×C4—1812no—
185C2×C4—1812no—
186C2×C4D4×D18111yescharacteristic
187C2×C4C23×D18111yescharacteristic
188C8—21no—
189C8—21no—
190C8—3622no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—7242no—
195C22—7242no—
196C22—7242no—
197C22C2×D4×D18111yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—3622no—
202C4—3622no—
203C4—7242no—
204C4#1915386111yescharacteristic
205C4#1915386311yescharacteristic
206C2—41no—
207C2—14482no—
208C2#2604185611yescharacteristic
209C2#2604208111yescharacteristic
210C2#2604209211yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#467×C181C2
socleC22×C181C22×D4
prime radical#26041515C2
hypercenter#467D181
hypocenterC181#3262
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #26041902C2semidirect
2. #26041572C2semidirect
3. #26041558C2semidirect
4. #26041527C2semidirect
6. #26041503C2semidirect
7. #26041497C2semidirect
8. #26041305C2semidirect
9. #467×C181C2semidirect
105. C181#3262semidirect
111. #467D181semidirect

Maximal subgroups

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

subgroupindex
#260419022
#260415722
#260415582
#260415272
#260415152
#260415032
#260414972
#260413052
#467×C1812
#3262181