Finite Group Databasebeta

← #37534103

Subgroups — #37534103

Order 23168.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#26042045C211yesmaximal, characteristic, complemented
2#26041829C211yesmaximal, characteristic
3#26041297C211yesmaximal, characteristic, complemented
4#26041295C211yesmaximal, characteristic, complemented
5#26041252C211yesmaximal, characteristic, complemented
6#26041039C211yesmaximal, characteristic, complemented
7#339×C181C211yesmaximal, characteristic, complemented
8C2×#15668449C2211yescharacteristic
9C2×#15668449C2211yescharacteristic
10#19153824—22no—
11#19153822—22no—
12C2×#15668433C2211yescharacteristic
13M16×D181—22no—
14#19153816C411yescharacteristic, complemented
15#19153813C411yescharacteristic, complemented
16#19153804C411yescharacteristic, complemented
17#19153804C411yescharacteristic, complemented
18#19153790—22no—
19C4×D724—22no—
20#19153775C2211yescharacteristic
21#19153744—22no—
22#19153742—22no—
23#19153730—22no—
24C2×M16×C181C2211yescharacteristic
25#117×C181C2211yescharacteristic
26C2×C42×C181C2211yescharacteristic
27#15668449—21no—
28#15668449—21no—
29#15668449—21no—
30#15668449—21no—
31#15668449—21no—
32#15668449—21no—
33#15668449—21no—
34#15668449—21no—
35C2×Dic362C2×C411yescharacteristic
36C2×Dic362C2×C411yescharacteristic
37C2×D724—22no—
38C2×D724C2×C411yescharacteristic
39C2×D724C2×C411yescharacteristic
40C2×C22⋊D181—21no—
41C2×C22⋊D181D412yes—
42C2×C4×D181—21no—
43C2×C4×D181D412yes—
44#15668433—21no—
45#15668433D412yescomplemented
46#15668432—24no—
47C2×#13089678—21no—
48C8×D181—24no—
49C4⋊Dic181—22no—
50C4⋊Dic181—21no—
51C22⋊Dic181—21no—
52#15668426—22no—
53#15668426—22no—
54C4×Dic181—21no—
55C22×C4×C181—21no—
56C22×C4×C181C2311yescharacteristic
57M16×C181—21no—
58M16×C181D412yescomplemented
59C2×C8×C181—21no—
60C4⋊C4×C181—21no—
61C22⋊C4×C181—21no—
62C42×C181—21no—
63C42×C181—21no—
64C42×C181C2×C411yescharacteristic
65C42×C181C2×C411yescharacteristic
66C22×D181—21no—
67C22×D181C22⋊C412yes—
68Dic362—21no—
69Dic362—21no—
70Dic362—21no—
71Dic362—21no—
72D724—21no—
73D724—21no—
74D724—21no—
75D724—21no—
76D724—44no—
77C22⋊D181—22no—
78C22⋊D181—22no—
79C22⋊D181—41no—
80C22⋊D181—41no—
81C2×Dic181—21no—
82C2×Dic181—41no—
83C2×Dic181C22⋊C412yes—
84C4×D181—22no—
85C4×D181—22no—
86C4×D181—41no—
87C4×D181—41no—
88#13089678—22no—
89C23×C181C2×D411yescharacteristic
90C2×C4×C181—21no—
91C2×C4×C181—21no—
92C2×C4×C181—42no—
93C2×C4×C181—41no—
94C2×C4×C181C22×C411yescharacteristic
95C2×C4×C181C2×D411yescharacteristic
96C2×C4×C181C2×D411yescharacteristic
97C2×C4×C181C2×D411yescharacteristic
98C2×C4×C181#4111yescharacteristic
99C2×C4×C181#4111yescharacteristic
100C8×C181—22no—
101C22×C181—41no—
102C22×C181C2×C22⋊C411yescharacteristic
103C22×C181#12111yescharacteristic
104C22×C181#12111yescharacteristic
105C2×D181—22no—
106C2×D181—42no—
107C2×D181—42no—
108C2×D181—41no—
109C4×C181—22no—
110C4×C181—41no—
111C4×C181—81no—
112C4×C181C4×D411yescharacteristic
113C4×C181C4×D411yescharacteristic
114C4×C181#12011yescharacteristic
115C4×C181#12311yescharacteristic
116Dic181—22no—
117Dic181—41no—
118Dic181—81no—
119C2×C181—21no—
120C2×C181—21no—
121C2×C181—21no—
122C2×C181#38611yescharacteristic
123D181—42no—
124D181—81no—
125C181#188411yescharacteristic, complemented, Sylow
126#1884—1811nomaximal, Sylow
127#568—1811no—
128#532—1811no—
129#421—1811no—
130#421—1811no—
131#410—1811no—
132#339—1811no—
133#339D18111yescharacteristic, complemented
134C2×#41—1811no—
135C2×#41—1811no—
136#131—3622no—
137#131—3622no—
138C2×M16—3622no—
139C2×M16—1811no—
140C2×M16C2×D18111yescharacteristic
141C4⋊Q8—1811no—
142#127—1811no—
143#124—1811no—
144#124—1811no—
145#121—3622no—
146C4×D4—3622no—
147#117—1811no—
148#117C2×D18111yescharacteristic
149C2×C42C2×D18111yescharacteristic
150#104—3622no—
151#104—3622no—
152#99—3622no—
153#41—3621no—
154#41—3621no—
155#41—3621no—
156#41—3621no—
157#41—3621no—
158#41—3621no—
159#41—3621no—
160#41—3621no—
161C2×Q8—1811no—
162C2×Q8—1811no—
163C2×D4—3622no—
164C2×D4—3621no—
165C2×D4—1812no—
166C2×D4—1811no—
167C2×D4—1811no—
168C22×C4—21no—
169C22×C4—3621no—
170C22×C4—1812no—
171C22×C4C22×D18111yescharacteristic
172M16—21no—
173M16—3624no—
174M16—3621no—
175M16—1812no—
176M16D72412yescomplemented
177C2×C8—21no—
178C2×C8—3624no—
179C2×C8—3621no—
180C4⋊C4—21no—
181C4⋊C4—3622no—
182C4⋊C4—3621no—
183C22⋊C4—21no—
184C22⋊C4—3622no—
185C22⋊C4—3622no—
186C22⋊C4—3621no—
187C42—21no—
188C42—21no—
189C42—3621no—
190C42C4×D18111yescharacteristic
191C42C4×D18111yescharacteristic
192C23—3621no—
193C23—1812no—
194C23D4×D18111yescharacteristic
195Q8—3621no—
196Q8—3621no—
197Q8—3621no—
198Q8—3621no—
199D4—3622no—
200D4—3622no—
201D4—3621no—
202D4—3621no—
203D4—3621no—
204D4—3621no—
205D4—7244no—
206D4—7241no—
207D4—7241no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—3622no—
213C2×C4—3622no—
214C2×C4—3621no—
215C2×C4—1812no—
216C2×C4—7241no—
217C2×C4—7241no—
218C2×C4—7241no—
219C2×C4C2×C4×D18111yescharacteristic
220C2×C4D4×D18111yescharacteristic
221C2×C4D4×D18111yescharacteristic
222C2×C4C2×D72411yescharacteristic
223C2×C4#1566844911yescharacteristic
224C2×C4#1566845011yescharacteristic
225C8—22no—
226C8—3622no—
227C22—41no—
228C22—3622no—
229C22—7242no—
230C22—7242no—
231C22—7241no—
232C22C22⋊C4×D18111yescharacteristic
233C22#1915378911yescharacteristic
234C22#1915379111yescharacteristic
235C4—22no—
236C4—41no—
237C4—81no—
238C4—3622no—
239C4—14481no—
240C4—7241no—
241C4#1915377811yescharacteristic
242C4C4×D72411yescharacteristic
243C4#1915378611yescharacteristic
244C4#1915380111yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—14481no—
249C2—7242no—
250C2#2604115711yescharacteristic
subgroupG/H
centerC4#19153786
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#339×C181C2
socleC2×C181#386
prime radical#26041829C2
cosocleC22×C4×C181C23
hypercenter#339D181
hypocenterC181#1884
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #26042045C2semidirect
3. #26041297C2semidirect
4. #26041295C2semidirect
5. #26041252C2semidirect
6. #26041039C2semidirect
7. #339×C181C2semidirect
14. #19153816C4semidirect
15. #19153813C4semidirect
16. #19153804C4semidirect
17. #19153804C4semidirect
45. #15668433D4semidirect
58. M16×C181D4semidirect
125. C181#1884semidirect
133. #339D181semidirect
176. M16D724semidirect

Maximal subgroups

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

subgroupindex
#260420452
#260418292
#260412972
#260412952
#260412522
#260410392
#339×C1812
#1884181