Finite Group Databasebeta

← #37545067

Subgroups — #37545067

Order 23168.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#26041810C212yesmaximal, complemented
2#26041585C212yesmaximal, complemented
3#26041577C212yesmaximal, complemented
4#26041520C211yesmaximal, characteristic, complemented
5#26041506C212yesmaximal, complemented
6#26041500C212yesmaximal, complemented
7#26041492C212yesmaximal, complemented
8C2×#19153745C211yesmaximal, characteristic, complemented
9#466×C181C211yesmaximal, characteristic
10C2×Dic724C2211yescharacteristic
11C2×#15668437C2212yes—
12C22×#13089678C2211yescharacteristic
13#19153809—22no—
14#19153807C2212yes—
15#19153803C2212yes—
16#19153797—22no—
17#19153795C2212yes—
18#19153781C2212yes—
19D4×Dic181—22no—
20D4×Dic181C2212yes—
21#19153772—22no—
22C2×C4×Dic181C2211yescharacteristic
23#19153750C2211yescharacteristic
24#19153747C2212yes—
25#19153745C2212yes—
26#19153745C2212yes—
27#19153741C2212yes—
28#19153739C2212yes—
29#19153737C2212yes—
30C2×D8×C181C2211yescharacteristic
31C22×C8×C181C2211yescharacteristic
32#121×C181C2212yes—
33#107×C181C2211yescharacteristic
34#102×C181C2212yes—
35C2×Dic362C2312yes—
36C22×Dic181—22no—
37C22×Dic181D411yescharacteristic
38C22×Dic181D411yescharacteristic
39Dic724—41no—
40#15668437—42no—
41C2×#13089678—41no—
42C2×#13089678C2312yes—
43C4⋊Dic181—22no—
44C4⋊Dic181C2312yes—
45#15668428—22no—
46#15668428—22no—
47#15668428—22no—
48C22⋊Dic181—22no—
49C22⋊Dic181—22no—
50C22⋊Dic181—22no—
51C22⋊Dic181—22no—
52C4×Dic181—22no—
53C4×Dic181D412yes—
54C4×Dic181C2312yes—
55C2×D4×C181—22no—
56C2×D4×C181C2312yes—
57C22×C4×C181C2311yescharacteristic
58D8×C181—41no—
59C2×C8×C181—41no—
60C2×C8×C181C2311yescharacteristic
61C2×C8×C181C2311yescharacteristic
62C4⋊C4×C181C2312yes—
63C22⋊C4×C181—22no—
64Dic362—22no—
65Dic362—22no—
66C2×Dic181—22no—
67C2×Dic181—22no—
68C2×Dic181—22no—
69C2×Dic181—41no—
70C2×Dic181—44no—
71C2×Dic181C2×D412yes—
72C2×Dic181C2×D412yes—
73#13089678—22no—
74C23×C181—22no—
75C23×C181C2×D411yescharacteristic
76D4×C181—22no—
77D4×C181—22no—
78D4×C181—44no—
79C2×C4×C181—21no—
80C2×C4×C181—21no—
81C2×C4×C181—22no—
82C2×C4×C181C2×D411yescharacteristic
83C2×C4×C181C2411yescharacteristic
84C8×C181—21no—
85C8×C181—21no—
86C22×C181—21no—
87C22×C181—21no—
88C22×C181—42no—
89C22×C181—42no—
90C22×C181—42no—
91C22×C181#13512yes—
92C22×C181C22×D411yescharacteristic
93C4×C181—21no—
94C4×C181—42no—
95C4×C181H(5,𝔽2)11yescharacteristic
96C4×C1812−1+411yescharacteristic
97Dic181—22no—
98Dic181—22no—
99Dic181—42no—
100Dic181—42no—
101C2×C181—22no—
102C2×C181—82no—
103C2×C181#53711yescharacteristic
104C2×C181C2×#13511yescharacteristic
105C2×C181#57511yescharacteristic
106C181#325011yescharacteristic, complemented, Sylow
107#3250—1811nomaximal, Sylow
108#529—1812no—
109#479—1812no—
110#478—1812no—
111#467—1811no—
112#466D18111yescharacteristic
113#465—1812no—
114#464—1812no—
115#463—1812no—
116C2×C4⋊C8—1811no—
117C2×Q16—1811no—
118C2×SD16—1812no—
119C2×D8C2×D18111yescharacteristic
120C22×C8—1811no—
121C22×C8C2×D18111yescharacteristic
122#126—3622no—
123#125—1812no—
124#124—1812no—
125#123—3622no—
126C22⋊Q8—1812no—
127#121C2×D18112yes—
128C4×Q8—1812no—
129C4×D4—3622no—
130C4×D4—1812no—
131#117—3622no—
132C2×C42—1811no—
133#107—1811no—
134#107C2×D18111yescharacteristic
135#106—1812no—
136C4⋊C8—1812no—
137C4⋊C8—1812no—
138#103—1812no—
139#103—1812no—
140#102—1812no—
141#102C2×D18112yes—
142C2×Q8—1812no—
143C2×D4—22no—
144C2×D4C22×D18112yes—
145C22×C4—3622no—
146C22×C4—1811no—
147C22×C4—1811no—
148C22×C4C22×D18111yescharacteristic
149Q16—7241no—
150SD16—7242no—
151D8—41no—
152C2×C8—41no—
153C2×C8—1812no—
154C2×C8—7241no—
155C2×C8C22×D18111yescharacteristic
156C2×C8C22×D18111yescharacteristic
157C4⋊C4—3622no—
158C4⋊C4—3622no—
159C4⋊C4—3622no—
160C4⋊C4—3622no—
161C4⋊C4—1812no—
162C4⋊C4C22×D18112yes—
163C22⋊C4—22no—
164C22⋊C4—3622no—
165C22⋊C4—3622no—
166C22⋊C4—3622no—
167C22⋊C4—3622no—
168C42—3622no—
169C42—1812no—
170C42—1812no—
171C23—22no—
172C23D4×D18111yescharacteristic
173Q8—3622no—
174Q8—3622no—
175D4—22no—
176D4—22no—
177D4—44no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—3622no—
182C2×C4—3622no—
183C2×C4—3622no—
184C2×C4—1812no—
185C2×C4—1812no—
186C2×C4—7241no—
187C2×C4—7244no—
188C2×C4D4×D18111yescharacteristic
189C2×C4C23×D18111yescharacteristic
190C8—21no—
191C8—21no—
192C8—3622no—
193C22—21no—
194C22—21no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22#1915383512yes—
199C22C2×D4×D18111yescharacteristic
200C4—21no—
201C4—42no—
202C4—3622no—
203C4—3622no—
204C4—7242no—
205C4—7242no—
206C4#1915386111yescharacteristic
207C4#1915386311yescharacteristic
208C2—22no—
209C2—82no—
210C2#2604185611yescharacteristic
211C2C2×#1915383511yescharacteristic
212C2#2604207611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting, prime radical#466×C181C2
socleC22×C181C22×D4
hypercenter#466D181
hypocenterC181#3250
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #26041810C2semidirect
2. #26041585C2semidirect
3. #26041577C2semidirect
4. #26041520C2semidirect
5. #26041506C2semidirect
6. #26041500C2semidirect
7. #26041492C2semidirect
8. C2×#19153745C2semidirect
106. C181#3250semidirect

Maximal subgroups

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

subgroupindex
#260418102
#260415852
#260415772
#260415202
#260415062
#260415002
#260414922
C2×#191537452
#466×C1812
#3250181