Finite Group Databasebeta

← #37533917

Subgroups — #37533917

Order 23168.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#19153774C211yesmaximal, characteristic
2C2×#19153747C211yesmaximal, characteristic, complemented
3C2×#19153728C211yesmaximal, characteristic
4#26041136C211yesmaximal, characteristic
5#26041042C211yesmaximal, characteristic, complemented
6#26041041C211yesmaximal, characteristic, complemented
7C2×#107×C181C211yesmaximal, characteristic, complemented
8C22×C4×D181C2211yescharacteristic
9C22×#13089678C2211yescharacteristic
10#19153774—22no—
11#19153774—22no—
12C2×C4⋊Dic181C2211yescharacteristic
13C2×C4⋊Dic181C2211yescharacteristic
14C2×#15668426—21no—
15C2×#15668426—21no—
16C2×C4×Dic181—21no—
17#19153747—22no—
18#19153728C414yescomplemented
19#19153723—21no—
20C22×C8×C181C2211yescharacteristic
21C2×C4⋊C4×C181C2211yescharacteristic
22C2×C4⋊C4×C181C2211yescharacteristic
23#107×C181—22no—
24C23×D181D411yescharacteristic
25C22×Dic181—21no—
26C22×Dic181—21no—
27C22×Dic181D411yescharacteristic
28C2×C4×D181—21no—
29C2×C4×D181—21no—
30C2×C4×D181—21no—
31C2×C4×D181—21no—
32C2×C4×D181C2×C412yes—
33C2×C4×D181Q812yes—
34C2×#13089678—21no—
35C2×#13089678—21no—
36C2×#13089678C2×C412yes—
37C4⋊Dic181—21no—
38C4⋊Dic181—21no—
39C4⋊Dic181—21no—
40C4⋊Dic181D412yes—
41#15668426—24no—
42#15668426—42no—
43C4×Dic181—24no—
44C22×C4×C181—21no—
45C22×C4×C181—21no—
46C22×C4×C181C2311yescharacteristic
47C2×C8×C181—21no—
48C2×C8×C181—21no—
49C2×C8×C181C2×C412yes—
50C4⋊C4×C181—21no—
51C4⋊C4×C181—21no—
52C4⋊C4×C181—21no—
53C4⋊C4×C181D412yes—
54C22×D181—21no—
55C22×D181—21no—
56C22×D181—21no—
57C22×D181—21no—
58C22×D181—21no—
59C22×D181—21no—
60C22×D181C4⋊C412yes—
61C2×Dic181—21no—
62C2×Dic181—21no—
63C2×Dic181—22no—
64C2×Dic181—41no—
65C2×Dic181—41no—
66C2×Dic181—41no—
67C2×Dic181—41no—
68C2×Dic181—41no—
69C2×Dic181C4⋊C412yes—
70C4×D181—22no—
71C4×D181—22no—
72C4×D181—22no—
73C4×D181—22no—
74#13089678—22no—
75C23×C181C2×D411yescharacteristic
76C2×C4×C181—22no—
77C2×C4×C181—41no—
78C2×C4×C181—41no—
79C2×C4×C181—41no—
80C2×C4×C181—41no—
81C2×C4×C181—41no—
82C2×C4×C181C22×C411yescharacteristic
83C2×C4×C181C2×D411yescharacteristic
84C2×C4×C181C2×D411yescharacteristic
85C2×C4×C181C2×Q811yescharacteristic
86C2×C4×C181#4111yescharacteristic
87C2×C4×C181#4111yescharacteristic
88C8×C181—22no—
89C22×C181C2×C4⋊C411yescharacteristic
90C22×C181#12011yescharacteristic
91C22×C181#12311yescharacteristic
92C22×C181#13511yescharacteristic
93C22×C181#13511yescharacteristic
94C22×C181AGL(1,ℤ/8)11yescharacteristic
95C22×C181#13711yescharacteristic
96C2×D181—22no—
97C2×D181—22no—
98C2×D181—22no—
99C2×D181—41no—
100C2×D181—41no—
101C2×D181—41no—
102C2×D181—41no—
103C4×C181—42no—
104C4×C181—81no—
105C4×C181C4×D411yescharacteristic
106C4×C181C4×D411yescharacteristic
107C4×C181C22⋊Q811yescharacteristic
108C4×C181C22⋊Q811yescharacteristic
109Dic181—22no—
110Dic181—42no—
111Dic181—81no—
112C2×C181#38511yescharacteristic
113C2×C181#42711yescharacteristic
114C2×C181#42811yescharacteristic
115C2×C181#44911yescharacteristic
116C2×C181#45211yescharacteristic
117C2×C181#48211yescharacteristic
118C2×C181#48511yescharacteristic
119D181—42no—
120C181#185411yescharacteristic, complemented, Sylow
121#1854—1811nomaximal, Sylow
122C2×#117—1811no—
123C2×#107D18111yescharacteristic, complemented
124C2×#106—1811no—
125C2×C22⋊C8—1811no—
126#380—1811no—
127#340—1811no—
128#340—1811no—
129C23×C4—1811no—
130C22×C8—1811no—
131C22×C8C2×D18111yescharacteristic
132#117—3622no—
133#117—3622no—
134C2×C4⋊C4—1811no—
135C2×C4⋊C4—1811no—
136C2×C4⋊C4C2×D18111yescharacteristic
137C2×C4⋊C4C2×D18111yescharacteristic
138C2×C22⋊C4—3621no—
139C2×C22⋊C4—3621no—
140C2×C42—3621no—
141#107—22no—
142#106—3622no—
143C22⋊C8—1814no—
144#95—3621no—
145C24—1811no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—3621no—
149C22×C4—3621no—
150C22×C4—3621no—
151C22×C4—3621no—
152C22×C4—3621no—
153C22×C4—3621no—
154C22×C4—1811no—
155C22×C4—1812no—
156C22×C4—1812no—
157C22×C4C22×D18111yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—3621no—
161C2×C8—3621no—
162C2×C8—1812no—
163C2×C8C4×D18112yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—3621no—
168C4⋊C4—3621no—
169C4⋊C4—3621no—
170C4⋊C4—1812no—
171C4⋊C4C22⋊D18112yes—
172C22⋊C4—3624no—
173C22⋊C4—7242no—
174C42—3624no—
175C23—3621no—
176C23—3621no—
177C23—3621no—
178C23—3621no—
179C23—3621no—
180C23—3621no—
181C23—1812no—
182C23D4×D18111yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—3621no—
190C2×C4—3621no—
191C2×C4—3622no—
192C2×C4—3622no—
193C2×C4—3622no—
194C2×C4—3622no—
195C2×C4—3622no—
196C2×C4—1812no—
197C2×C4—7241no—
198C2×C4—7241no—
199C2×C4—7241no—
200C2×C4—7241no—
201C2×C4—7241no—
202C2×C4C2×C4×D18111yescharacteristic
203C2×C4D4×D18111yescharacteristic
204C2×C4C2×C22⋊D18111yescharacteristic
205C2×C4Q8×D18111yescharacteristic
206C2×C4#1566844911yescharacteristic
207C2×C4#1566845011yescharacteristic
208C8—22no—
209C8—3622no—
210C22—3622no—
211C22—3622no—
212C22—3622no—
213C22—7241no—
214C22—7241no—
215C22—7241no—
216C22—7241no—
217C22C4⋊C4×D18111yescharacteristic
218C22#1915378411yescharacteristic
219C22#1915380111yescharacteristic
220C22#1915383511yescharacteristic
221C22#1915383911yescharacteristic
222C22#1915384011yescharacteristic
223C22#1915384811yescharacteristic
224C4—42no—
225C4—81no—
226C4—3622no—
227C4—14481no—
228C4—7242no—
229C4C4×C22⋊D18111yescharacteristic
230C4#1915377811yescharacteristic
231C4#1915379211yescharacteristic
232C4#1915379411yescharacteristic
233C2—7242no—
234C2#2604115011yescharacteristic
235C2#2604131611yescharacteristic
236C2#2604132011yescharacteristic
237C2#2604142111yescharacteristic
238C2#2604144011yescharacteristic
239C2#2604160611yescharacteristic
240C2#2604162211yescharacteristic
subgroupG/H
centerC23D4×D181
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
FittingC2×#107×C181C2
socleC23×C181C2×D4
prime radicalC23×D181D4
cosocleC22×C4×C181C23
hypercenterC2×#107D181
hypocenterC181#1854
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#19153747C2semidirect
5. #26041042C2semidirect
6. #26041041C2semidirect
7. C2×#107×C181C2semidirect
18. #19153728C4semidirect
120. C181#1854semidirect
123. C2×#107D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191537742
C2×#191537472
C2×#191537282
#260411362
#260410422
#260410412
C2×#107×C1812
#1854181