Finite Group Databasebeta

← #37546757

Subgroups — #37546757

Order 23168.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#26042004C211yesmaximal, characteristic, complemented
2#26042003C211yesmaximal, characteristic
3#26041672C212yesmaximal
4#26041503C212yesmaximal
5#26041487C212yesmaximal, complemented
6#26041486C211yesmaximal, characteristic, complemented
7#26041481C211yesmaximal, characteristic
8#26041409C211yesmaximal, characteristic, complemented
9#26041386C212yesmaximal, complemented
10C4×#15668438C211yesmaximal, characteristic
11#494×C181C211yesmaximal, characteristic, complemented
12C2×#15668439C2212yes—
13C2×#15668438C2211yescharacteristic
14C2×#15668436C2212yes—
15#19153814C2212yes—
16#19153811—22no—
17#19153807—21no—
18#19153806—22no—
19#19153805C2212yes—
20#19153796—21no—
21Q8×Dic181C2212yes—
22Q8×Dic181C2212yes—
23C4×Dic362C2211yescharacteristic
24C4×Dic362C2211yescharacteristic
25#19153781—21no—
26C4×D724C2211yescharacteristic
27#19153774—21no—
28#19153748C2211yescharacteristic
29C4⋊#13089678C2211yescharacteristic
30#19153745C2212yes—
31#19153741C2211yescharacteristic
32#19153740C2212yes—
33#19153740C2212yes—
34#19153738C2211yescharacteristic
35#19153726C2212yes—
36C2×Q16×C181C2212yes—
37C2×Q16×C181C2212yes—
38C4⋊Q8×C181C2211yescharacteristic
39C4⋊Q8×C181C2211yescharacteristic
40C4×C8×C181C2211yescharacteristic
41C2×Dic362D411yescharacteristic
42C2×Dic362C2311yescharacteristic
43C2×D724C2311yescharacteristic
44C2×C4×D181—21no—
45#15668439—42no—
46#15668438—22no—
47#15668436—42no—
48C2×#13089678C2312yes—
49C4⋊Dic181—22no—
50C4⋊Dic181—22no—
51C4⋊Dic181D411yescharacteristic
52C4⋊Dic181C2311yescharacteristic
53C4⋊Dic181C2311yescharacteristic
54#15668428—22no—
55#15668428—22no—
56#15668428—21no—
57#15668428D412yes—
58#15668426—22no—
59#15668426—22no—
60#15668426—21no—
61C4×Dic181—22no—
62C4×Dic181—22no—
63C4×Dic181—21no—
64C4×Dic181C2312yes—
65C2×Q8×C181C2312yes—
66C2×Q8×C181C2312yes—
67Q16×C181—24no—
68Q16×C181—42no—
69C2×C8×C181C2311yescharacteristic
70C2×C8×C181C2311yescharacteristic
71C4⋊C4×C181—21no—
72C4⋊C4×C181—21no—
73C4⋊C4×C181—21no—
74C4⋊C4×C181—21no—
75C42×C181C2311yescharacteristic
76C22×D181—21no—
77Dic362—22no—
78Dic362—21no—
79Dic362—21no—
80D724—21no—
81D724—21no—
82C2×Dic181—21no—
83C2×Dic181—21no—
84C2×Dic181—21no—
85C2×Dic181C2×D412yes—
86C2×Dic181C2×D412yes—
87C4×D181—42no—
88#13089678—22no—
89Q8×C181—22no—
90Q8×C181—22no—
91Q8×C181—22no—
92Q8×C181—22no—
93C2×C4×C181—22no—
94C2×C4×C181—22no—
95C2×C4×C181C2×D411yescharacteristic
96C2×C4×C181C2×D411yescharacteristic
97C2×C4×C181C2411yescharacteristic
98C8×C181—21no—
99C8×C181#4112yes—
100C22×C181C22×D411yescharacteristic
101C2×D181—41no—
102C2×D181—41no—
103C2×D181—41no—
104C4×C181—21no—
105C4×C181—42no—
106C4×C181—42no—
107C4×C181#13712yes—
108C4×C181C2×#4111yescharacteristic
109C4×C181H(5,𝔽2)11yescharacteristic
110Dic181—22no—
111Dic181—22no—
112Dic181—41no—
113Dic181—41no—
114Dic181—41no—
115C2×C181#55011yescharacteristic
116C2×C181C2×#13711yescharacteristic
117C2×C181#57811yescharacteristic
118D181—81no—
119C181#339811yescharacteristic, complemented, Sylow
120#3398—1811nomaximal, Sylow
121#557—1811no—
122#556—1811no—
123#497—1812no—
124#494D18111yescharacteristic, complemented
125#464—1812no—
126#462—1812no—
127#462—1811no—
128#461—1811no—
129#446—1811no—
130#441—1812no—
131C4×SD16—1811no—
132C2×Q16—1812no—
133C2×Q16C2×D18112yes—
134C2×Q16C2×D18112yes—
135C2×SD16—1812no—
136C2×SD16—1811no—
137C4⋊Q8—1812no—
138C4⋊Q8C2×D18111yescharacteristic
139C4⋊Q8C2×D18111yescharacteristic
140#126—3622no—
141#125—3622no—
142#125—3621no—
143#124—1812no—
144C22⋊Q8—3621no—
145C4×Q8—3621no—
146C4×Q8—1812no—
147C4×Q8—1812no—
148C4×Q8—1811no—
149C4×Q8—1811no—
150C4×D4—1811no—
151#117—3621no—
152#106—1811no—
153C4⋊C8—1812no—
154C4⋊C8—1811no—
155#103—1812no—
156#103—1812no—
157#103—1811no—
158#102—1811no—
159#97—1812no—
160C4×C8C2×D18111yescharacteristic
161C2×Q8—1811no—
162C2×Q8—1811no—
163C2×Q8C22×D18112yes—
164C2×Q8C22×D18112yes—
165C2×D4—1811no—
166C22×C4—3621no—
167Q16—24no—
168Q16—42no—
169Q16—7242no—
170SD16—3622no—
171SD16—7242no—
172C2×C8—1812no—
173C2×C8C22×D18111yescharacteristic
174C2×C8C22×D18111yescharacteristic
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—3622no—
180C4⋊C4—3622no—
181C4⋊C4—3622no—
182C4⋊C4—3622no—
183C4⋊C4—3621no—
184C4⋊C4—1811no—
185C4⋊C4—1811no—
186C4⋊C4—1811no—
187C4⋊C4—1812no—
188C22⋊C4—3622no—
189C22⋊C4—3622no—
190C22⋊C4—3621no—
191C42—3622no—
192C42—3622no—
193C42—3621no—
194C42—1812no—
195C42C22×D18111yescharacteristic
196C23—3621no—
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—22no—
201Q8—3622no—
202Q8—3621no—
203Q8—3621no—
204D4—3621no—
205D4—3621no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—3621no—
209C2×C4—3621no—
210C2×C4—3621no—
211C2×C4—1812no—
212C2×C4—1812no—
213C2×C4—7242no—
214C2×C4D4×D18111yescharacteristic
215C2×C4D4×D18111yescharacteristic
216C2×C4C23×D18111yescharacteristic
217C8—21no—
218C8—3622no—
219C8#1566844812yes—
220C22—7241no—
221C22—7241no—
222C22—7241no—
223C22C2×D4×D18111yescharacteristic
224C4—21no—
225C4—42no—
226C4—42no—
227C4—3622no—
228C4—3622no—
229C4—7241no—
230C4—7241no—
231C4—7241no—
232C4#1915384812yes—
233C4C2×#1566844811yescharacteristic
234C4#1915386111yescharacteristic
235C2—14481no—
236C2#2604194911yescharacteristic
237C2C2×#1915384811yescharacteristic
238C2#2604209211yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#494×C181C2
socleC22×C181C22×D4
prime radicalC2×D724C23
hypercenter#494D181
hypocenterC181#3398
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #26042004C2semidirect
5. #26041487C2semidirect
6. #26041486C2semidirect
8. #26041409C2semidirect
9. #26041386C2semidirect
11. #494×C181C2semidirect
119. C181#3398semidirect
124. #494D181semidirect

Maximal subgroups

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

subgroupindex
#260420042
#260420032
#260416722
#260415032
#260414872
#260414862
#260414812
#260414092
#260413862
C4×#156684382
#494×C1812
#3398181