Finite Group Databasebeta

← #37543373

Subgroups — #37543373

Order 23168.

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

normal subgroups only

This group has 460 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
2C2×#19153823C211yesmaximal, characteristic
3#26041850C211yesmaximal, characteristic
4#26041535C214yesmaximal, complemented
5#26041518C214yesmaximal, complemented
6#26041315C211yesmaximal, characteristic, complemented
7#26041288C211yesmaximal, characteristic, complemented
8#26041287C211yesmaximal, characteristic, complemented
9#536×C181C211yesmaximal, characteristic
10C22×Dic362C2211yescharacteristic
11C2×#15668439—24no—
12C2×#15668439C2212yes—
13C2×#15668439C2212yes—
14#19153823—24no—
15C2×#15668433C2212yes—
16C2×#15668433C2211yescharacteristic
17C22×#13089678C2212yes—
18C22×#13089678C2211yescharacteristic
19#19153814—21no—
20#19153803—21no—
21#19153797—22no—
22#19153795C2214yes—
23#19153775C2211yescharacteristic
24#19153749C2212yes—
25#19153747C2212yes—
26#19153740C2214yes—
27#19153739C2214yes—
28C2×#41×C181C2211yescharacteristic
29C22×Q8×C181C2211yescharacteristic
30C4⋊Q8×C181—21no—
31#124×C181—21no—
32#123×C181—22no—
33C22⋊Q8×C181C2214yes—
34#117×C181C2211yescharacteristic
35C2×Dic362—21no—
36C2×Dic362—22no—
37C2×Dic362—22no—
38C2×Dic362C2312yes—
39C22×Dic181—21no—
40#15668439—24no—
41#15668439—24no—
42#15668433—24no—
43#15668433—22no—
44C2×#13089678—24no—
45C2×#13089678D414yes—
46C2×#13089678C2312yes—
47C2×#13089678C2311yescharacteristic
48C2×#13089678C2311yescharacteristic
49C4⋊Dic181C2312yes—
50#15668428—24no—
51C22⋊Dic181—21no—
52C22⋊Dic181—24no—
53C4×Dic181—21no—
54#41×C181—22no—
55#41×C181—22no—
56C2×Q8×C181—21no—
57C2×Q8×C181—22no—
58C2×Q8×C181—22no—
59C2×Q8×C181D411yescharacteristic
60C2×Q8×C181C2312yes—
61C2×D4×C181D412yes—
62C2×D4×C181D411yescharacteristic
63C22×C4×C181—21no—
64C22×C4×C181C2312yes—
65C22×C4×C181C2311yescharacteristic
66C4⋊C4×C181—24no—
67C4⋊C4×C181C2312yes—
68C22⋊C4×C181—21no—
69C22⋊C4×C181—24no—
70C42×C181—21no—
71Dic362—22no—
72Dic362—22no—
73Dic362—42no—
74C2×Dic181—22no—
75C2×Dic181—22no—
76C2×Dic181—42no—
77#13089678—22no—
78#13089678C2×D412yes—
79#13089678C2×D412yes—
80C23×C181C2×D412yes—
81C23×C181C2×D411yescharacteristic
82Q8×C181—22no—
83Q8×C181—22no—
84Q8×C181—22no—
85Q8×C181—42no—
86D4×C181—24no—
87D4×C181—22no—
88C2×C4×C181—22no—
89C2×C4×C181—22no—
90C2×C4×C181—22no—
91C2×C4×C181—22no—
92C2×C4×C181—42no—
93C2×C4×C181C2×D412yes—
94C2×C4×C181C2×D412yes—
95C2×C4×C181C2×D411yescharacteristic
96C2×C4×C181#4112yes—
97C2×C4×C181C2411yescharacteristic
98C22×C181—22no—
99C22×C181—22no—
100C22×C181—24no—
101C22×C181#12112yes—
102C22×C181C22×D411yescharacteristic
103C4×C181—22no—
104C4×C181—42no—
105C4×C181—42no—
106C4×C181#12112yes—
107C4×C181C22×D411yescharacteristic
108C4×C181C2×#4111yescharacteristic
109Dic181—42no—
110Dic181—42no—
111C2×C181—22no—
112C2×C181—42no—
113C2×C181C2×#12111yescharacteristic
114C2×C181#57812yes—
115C181#305911yescharacteristic, complemented, Sylow
116#3059—1811nomaximal, Sylow
117C22×Q16—1811no—
118C2×#131—1811no—
119#536—1811no—
120#536D18111yescharacteristic
121#470—1814no—
122#467—1814no—
123#427—1811no—
124#419—1811no—
125#419—1811no—
126C2×#41C2×D18111yescharacteristic
127C22×Q8—1811no—
128C22×Q8C2×D18111yescharacteristic
129C2×Q16—3624no—
130C2×Q16—1812no—
131C2×Q16—1812no—
132#131—3624no—
133C2×M16—1811no—
134C2×M16—1812no—
135C22×C8—1812no—
136C22×C8—1811no—
137C4⋊Q8—21no—
138C4⋊Q8—3621no—
139#124—21no—
140#124—3621no—
141#123—22no—
142#123—3622no—
143C22⋊Q8—1814no—
144C22⋊Q8C2×D18114yes—
145#117—1811no—
146#117C2×D18111yescharacteristic
147#107—1812no—
148#106—1812no—
149#103—1814no—
150#103—1814no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—3621no—
157C2×Q8—3622no—
158C2×Q8—3622no—
159C2×Q8—1812no—
160C2×Q8C22⋊D18111yescharacteristic
161C2×Q8C22×D18112yes—
162C2×D4C22⋊D18112yes—
163C2×D4C22⋊D18111yescharacteristic
164C22×C4—21no—
165C22×C4—3621no—
166C22×C4C22×D18112yes—
167C22×C4C22×D18111yescharacteristic
168Q16—3624no—
169Q16—3624no—
170M16—3622no—
171M16—3624no—
172C2×C8—3624no—
173C2×C8—1812no—
174C2×C8—1814no—
175C2×C8—1811no—
176C2×C8—1811no—
177C4⋊C4—24no—
178C4⋊C4—3624no—
179C4⋊C4—1812no—
180C4⋊C4C22×D18112yes—
181C22⋊C4—21no—
182C22⋊C4—24no—
183C22⋊C4—3621no—
184C22⋊C4—3624no—
185C42—21no—
186C42—3621no—
187C23C2×C22⋊D18112yes—
188C23C2×C22⋊D18111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—3622no—
194Q8—3622no—
195Q8—7242no—
196D4—24no—
197D4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—3622no—
204C2×C4—3622no—
205C2×C4—7242no—
206C2×C4D4×D18112yes—
207C2×C4C2×C22⋊D18112yes—
208C2×C4C2×C22⋊D18111yescharacteristic
209C2×C4#1566844812yes—
210C2×C4C23×D18111yescharacteristic
211C8—3622no—
212C8—1812no—
213C8—1812no—
214C22—22no—
215C22—22no—
216C22—24no—
217C22#1915378812yes—
218C22C22×C22⋊D18111yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—7242no—
223C4—7242no—
224C4#1915378812yes—
225C4C2×D4×D18111yescharacteristic
226C4C2×#1566844811yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#1915378811yescharacteristic
230C2#2604209412yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#536×C181C2
socleC22×C181C22×D4
prime radicalC2×#41×C181C22
hypercenter#536D181
hypocenterC181#3059
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#15668439C2semidirect
4. #26041535C2semidirect
5. #26041518C2semidirect
6. #26041315C2semidirect
7. #26041288C2semidirect
8. #26041287C2semidirect
115. C181#3059semidirect

Maximal subgroups

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

subgroupindex
C22×#156684392
C2×#191538232
#260418502
#260415352
#260415182
#260413152
#260412882
#260412872
#536×C1812
#3059181