Finite Group Databasebeta

← #37543253

Subgroups — #37543253

Order 23168.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#15668439C211yesmaximal, characteristic, complemented
2#26041850C211yesmaximal, characteristic
3#26041757C211yesmaximal, characteristic, complemented
4#26041499C214yesmaximal, complemented
5#26041487C214yesmaximal, complemented
6#26041307C211yesmaximal, characteristic, complemented
7#26041289C211yesmaximal, characteristic, complemented
8#26041275C211yesmaximal, characteristic
9C2×#137×C181C211yesmaximal, characteristic
10C22×Dic362C2211yescharacteristic
11C2×#15668439—24no—
12C2×#15668439C2214yes—
13C22×#13089678C2211yescharacteristic
14#19153814C2212yes—
15#19153803C2212yes—
16#19153797—22no—
17#19153795—22no—
18Q8×Dic181—22no—
19Q8×Dic181C2214yes—
20#19153775—22no—
21#19153775C2211yescharacteristic
22#19153775C2211yescharacteristic
23#19153745C2214yes—
24#19153741C2214yes—
25#19153740C2212yes—
26#19153737C2212yes—
27C2×#41×C181C2211yescharacteristic
28C22×Q8×C181C2211yescharacteristic
29#137×C181—24no—
30C2×Q16×C181C2212yes—
31C2×SD16×C181C2212yes—
32C2×M16×C181C2211yescharacteristic
33C2×Dic362—21no—
34C2×Dic362—22no—
35C2×Dic362—22no—
36C2×Dic362C2312yes—
37C22×Dic181—21no—
38#15668439—28no—
39C2×#13089678—22no—
40C2×#13089678C2312yes—
41C4⋊Dic181—24no—
42C4⋊Dic181D412yes—
43C4⋊Dic181C2312yes—
44#15668428—24no—
45C22⋊Dic181—24no—
46C22⋊Dic181—21no—
47C22⋊Dic181D412yes—
48C4×Dic181—24no—
49C4×Dic181—21no—
50C4×Dic181C2312yes—
51#41×C181—22no—
52#41×C181—22no—
53C2×Q8×C181—21no—
54C2×Q8×C181—22no—
55C2×Q8×C181D414yes—
56C2×Q8×C181C2311yescharacteristic
57C2×Q8×C181C2312yes—
58C2×D4×C181—21no—
59C2×D4×C181C2311yescharacteristic
60C22×C4×C181—21no—
61C22×C4×C181—21no—
62C22×C4×C181C2311yescharacteristic
63Q16×C181—24no—
64SD16×C181—24no—
65M16×C181—22no—
66C2×C8×C181C2312yes—
67Dic362—22no—
68Dic362—22no—
69Dic362—42no—
70C2×Dic181—22no—
71C2×Dic181—22no—
72C2×Dic181—42no—
73C2×Dic181C2×D414yes—
74#13089678—22no—
75C23×C181—21no—
76C23×C181C2×D411yescharacteristic
77Q8×C181—24no—
78Q8×C181—22no—
79Q8×C181—21no—
80Q8×C181—21no—
81Q8×C181C2×D414yes—
82D4×C181—21no—
83D4×C181—21no—
84D4×C181—42no—
85C2×C4×C181—24no—
86C2×C4×C181—22no—
87C2×C4×C181—21no—
88C2×C4×C181—42no—
89C2×C4×C181C2×D411yescharacteristic
90C2×C4×C181C2×D412yes—
91C2×C4×C181#4112yes—
92C2×C4×C181C2411yescharacteristic
93C8×C181—22no—
94C22×C181—22no—
95C22×C181—41no—
96C22×C181—42no—
97C22×C181#12112yes—
98C22×C181C22×D411yescharacteristic
99C4×C181—24no—
100C4×C181—41no—
101C4×C181#12112yes—
102C4×C181C22×D411yescharacteristic
103C4×C181C2×#4111yescharacteristic
104Dic181—24no—
105Dic181—42no—
106Dic181—42no—
107C2×C181—22no—
108C2×C181—81no—
109C2×C181C2×#12111yescharacteristic
110C2×C181#57812yes—
111C181#304411yescharacteristic, complemented, Sylow
112#3044—1811nomaximal, Sylow
113C2×#137D18111yescharacteristic
114C22×Q16—1811no—
115#536—1811no—
116#519—1811no—
117#464—1814no—
118#462—1814no—
119#424—1811no—
120#419—1811no—
121#416—1811no—
122C2×#41C2×D18111yescharacteristic
123C22×Q8—1811no—
124C22×Q8C2×D18111yescharacteristic
125#137—24no—
126C2×Q16—3624no—
127C2×Q16—1814no—
128C2×Q16C2×D18112yes—
129C2×SD16C2×D18112yes—
130C2×M16C2×D18111yescharacteristic
131C22×C8—1811no—
132C4⋊Q8—1812no—
133#124—1812no—
134#123—3622no—
135C22⋊Q8—3622no—
136C4×Q8—3622no—
137C4×Q8—1814no—
138#117—3622no—
139#117—1811no—
140#117—1811no—
141C4⋊C8—1814no—
142#103—1812no—
143#103—1814no—
144#102—1812no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—3621no—
150C2×Q8—3622no—
151C2×Q8—3622no—
152C2×Q8—1812no—
153C2×Q8C22⋊D18114yes—
154C2×Q8C22×D18111yescharacteristic
155C2×Q8C22×D18112yes—
156C2×D4—21no—
157C2×D4C22×D18111yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—3621no—
161C22×C4C22×D18111yescharacteristic
162Q16—24no—
163Q16—3628no—
164SD16—24no—
165M16—22no—
166C2×C8—3622no—
167C2×C8—1812no—
168C2×C8C22×D18112yes—
169C4⋊C4—3624no—
170C4⋊C4—3624no—
171C4⋊C4—1812no—
172C4⋊C4—1812no—
173C22⋊C4—3621no—
174C22⋊C4—3624no—
175C22⋊C4—1812no—
176C42—3624no—
177C42—3621no—
178C42—1812no—
179C23—21no—
180C23D4×D18111yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—21no—
184Q8—21no—
185Q8—3622no—
186Q8—3622no—
187Q8—7242no—
188Q8C2×C22⋊D18114yes—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—3622no—
197C2×C4—3622no—
198C2×C4—1814no—
199C2×C4—7242no—
200C2×C4D4×D18111yescharacteristic
201C2×C4C2×C22⋊D18112yes—
202C2×C4#1566844812yes—
203C2×C4C23×D18111yescharacteristic
204C8—22no—
205C8—3622no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22#1915378812yes—
210C22C2×D4×D18111yescharacteristic
211C4—24no—
212C4—41no—
213C4—3624no—
214C4—7242no—
215C4—7242no—
216C4#1915378812yes—
217C4C22×C22⋊D18111yescharacteristic
218C4C2×#1566844811yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1915378811yescharacteristic
222C2#2604209512yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#137×C181C2
socleC22×C181C22×D4
prime radicalC2×#41×C181C22
hypercenterC2×#137D181
hypocenterC181#3044
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#15668439C2semidirect
3. #26041757C2semidirect
4. #26041499C2semidirect
5. #26041487C2semidirect
6. #26041307C2semidirect
7. #26041289C2semidirect
111. C181#3044semidirect

Maximal subgroups

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

subgroupindex
C22×#156684392
#260418502
#260417572
#260414992
#260414872
#260413072
#260412892
#260412752
C2×#137×C1812
#3044181