Finite Group Databasebeta

← #37544921

Subgroups — #37544921

Order 23168.

366 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 121 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#26041836C211yesmaximal, characteristic, complemented
2#26041829C211yesmaximal, characteristic
3#26041688C212yesmaximal, complemented
4#26041644C212yesmaximal, complemented
5#26041604C214yesmaximal, complemented
6#26041570C212yesmaximal, complemented
7#26041560C212yesmaximal, complemented
8#432×C181C211yesmaximal, characteristic, complemented
9C2×#15668449—21no—
10#19153816C2211yescharacteristic
11#19153816C2211yescharacteristic
12#19153816C2211yescharacteristic
13#19153813C2211yescharacteristic
14#19153808C2212yes—
15#19153804—21no—
16#19153795—21no—
17#19153795—21no—
18#19153790—21no—
19#19153790C2212yes—
20C4×Dic362—21no—
21C4×Dic362—21no—
22C4×D724—21no—
23C4×D724C2212yes—
24#19153775C2212yes—
25#19153750C2214yes—
26#19153748C2214yes—
27#19153738C2214yes—
28#19153738C2214yes—
29C2×C42×C181C2211yescharacteristic
30C4⋊C8×C181C2212yes—
31C22⋊C8×C181C2212yes—
32#97×C181C2212yes—
33#15668449—44no—
34C2×Dic362—21no—
35C2×Dic362—21no—
36C2×Dic362—21no—
37C2×D724—21no—
38C2×D724—21no—
39C2×D724C2312yes—
40C2×C22⋊D181—22no—
41C2×C4×D181—22no—
42C4⋊Dic181—21no—
43C4⋊Dic181—21no—
44C4⋊Dic181—21no—
45C4⋊Dic181C2312yes—
46C4⋊Dic181C2312yes—
47C4⋊Dic181C2312yes—
48#15668428—22no—
49#15668428—22no—
50C22⋊Dic181—22no—
51#15668426—22no—
52#15668426—22no—
53C4×Dic181—22no—
54C22×C4×C181D411yescharacteristic
55C22×C4×C181D411yescharacteristic
56C22×C4×C181C2311yescharacteristic
57C2×C8×C181C2314yes—
58C42×C181D411yescharacteristic
59C42×C181D411yescharacteristic
60C42×C181C2311yescharacteristic
61C42×C181C2311yescharacteristic
62C22×D181—22no—
63Dic362—42no—
64Dic362—42no—
65Dic362—42no—
66D724—24no—
67D724—42no—
68D724—42no—
69C22⋊D181—44no—
70C2×Dic181—22no—
71C2×Dic181—22no—
72C2×Dic181—22no—
73C4×D181—44no—
74C23×C181C2×D411yescharacteristic
75C2×C4×C181—22no—
76C2×C4×C181—22no—
77C2×C4×C181—41no—
78C2×C4×C181C2×D411yescharacteristic
79C2×C4×C181C2×D411yescharacteristic
80C2×C4×C181C2×D411yescharacteristic
81C2×C4×C181C2×D411yescharacteristic
82C2×C4×C181C2×D411yescharacteristic
83C2×C4×C181C2411yescharacteristic
84C8×C181—24no—
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—21no—
94C4×C181—21no—
95C4×C181—21no—
96C4×C181AGL(1,ℤ/8)12yes—
97C4×C1812−1+412yes—
98Dic181—42no—
99Dic181—42no—
100Dic181—42no—
101C2×C181—41no—
102C2×C181#53611yescharacteristic
103C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
104C2×C181#57511yescharacteristic
105D181—82no—
106C181#323511yescharacteristic, complemented, Sylow
107#3235—1811nomaximal, Sylow
108#533—1811no—
109#532—1811no—
110#501—1812no—
111#490—1812no—
112#482—1814no—
113#476—1812no—
114#474—1812no—
115#432D18111yescharacteristic, complemented
116C2×#41—3621no—
117C4⋊Q8—1811no—
118C4⋊Q8—1811no—
119C4⋊Q8—1811no—
120#127—1811no—
121#125—1812no—
122#124—3621no—
123C22⋊Q8—3621no—
124C22⋊Q8—3621no—
125#121—3621no—
126#121—1812no—
127C4×Q8—3621no—
128C4×Q8—3621no—
129C4×D4—3621no—
130C4×D4—1812no—
131#117—1812no—
132C2×C42C2×D18111yescharacteristic
133#107—1814no—
134#106—1814no—
135C4⋊C8C2×D18112yes—
136#102—1814no—
137#102—1814no—
138C22⋊C8C2×D18112yes—
139#97C2×D18112yes—
140#41—7244no—
141C2×Q8—3621no—
142C2×Q8—3621no—
143C2×Q8—3621no—
144C2×D4—3622no—
145C2×D4—3621no—
146C2×D4—3621no—
147C2×D4—1812no—
148C22×C4—3622no—
149C22×C4D72411yescharacteristic
150C22×C4D72411yescharacteristic
151C22×C4C22×D18111yescharacteristic
152C2×C8C22×D18114yes—
153C4⋊C4—3621no—
154C4⋊C4—3622no—
155C4⋊C4—3622no—
156C4⋊C4—3621no—
157C4⋊C4—3621no—
158C4⋊C4—1812no—
159C4⋊C4—1812no—
160C4⋊C4—1812no—
161C22⋊C4—3622no—
162C22⋊C4—3622no—
163C22⋊C4—3622no—
164C42—3622no—
165C42D72411yescharacteristic
166C42D72411yescharacteristic
167C42C22×D18111yescharacteristic
168C42C22×D18111yescharacteristic
169C23—3622no—
170C23C2×D72411yescharacteristic
171Q8—7242no—
172Q8—7242no—
173Q8—7242no—
174D4—3624no—
175D4—7242no—
176D4—7242no—
177D4—7244no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—3622no—
182C2×C4—3622no—
183C2×C4—3622no—
184C2×C4—7244no—
185C2×C4C2×D72411yescharacteristic
186C2×C4C2×D72411yescharacteristic
187C2×C4C2×D72411yescharacteristic
188C2×C4C2×D72411yescharacteristic
189C2×C4C2×D72411yescharacteristic
190C2×C4C23×D18111yescharacteristic
191C8—24no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—7242no—
196C22—7242no—
197C22—7242no—
198C22C22×D72411yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—7242no—
204C4—7242no—
205C4—7242no—
206C4#1915384412yes—
207C4#1915386312yes—
208C2—41no—
209C2—14482no—
210C2#2604185511yescharacteristic
211C2C2×#1915384411yescharacteristic
212C2#2604207911yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#432×C181C2
socleC22×C181C22×D4
prime radical#26041829C2
hypercenter#432D181
hypocenterC181#3235
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #26041836C2semidirect
3. #26041688C2semidirect
4. #26041644C2semidirect
5. #26041604C2semidirect
6. #26041570C2semidirect
7. #26041560C2semidirect
8. #432×C181C2semidirect
106. C181#3235semidirect
115. #432D181semidirect

Maximal subgroups

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

subgroupindex
#260418362
#260418292
#260416882
#260416442
#260416042
#260415702
#260415602
#432×C1812
#3235181