Finite Group Databasebeta

← #37544955

Subgroups — #37544955

Order 23168.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C4⋊Q8×D181C211yesmaximal, characteristic, complemented
2#26041816C211yesmaximal, characteristic
3#26041686C212yesmaximal, complemented
4#26041640C212yesmaximal, complemented
5#26041613C212yesmaximal, complemented
6#26041599C212yesmaximal
7#26041564C211yesmaximal, characteristic, complemented
8#26041560C211yesmaximal, characteristic, complemented
9#26041556C211yesmaximal, characteristic, complemented
10#26041335C211yesmaximal, characteristic, complemented
11#475×C181C211yesmaximal, characteristic, complemented
12C2×Q8×D181—21no—
13#19153816C2211yescharacteristic
14#19153815—21no—
15#19153815C2212yes—
16#19153814—21no—
17#19153811—21no—
18#19153807C2211yescharacteristic
19#19153805C2211yescharacteristic
20#19153801—21no—
21#19153792C2211yescharacteristic
22#19153791C2211yescharacteristic
23C4×D724—21no—
24C4×D724C2211yescharacteristic
25#19153773—21no—
26C4⋊C4×D181—21no—
27C4⋊C4×D181C2212yes—
28C42×D181C2211yescharacteristic
29#19153750C2212yes—
30#19153749C2212yes—
31#19153747C2212yes—
32C4⋊#13089678C2211yescharacteristic
33#19153740C2212yes—
34#19153738C2212yes—
35#19153736C2212yes—
36#19153728C2212yes—
37#19153726C2212yes—
38C4⋊Q8×C181C2211yescharacteristic
39C4×Q8×C181C2211yescharacteristic
40#106×C181C2212yes—
41C4⋊C8×C181C2211yescharacteristic
42#103×C181C2212yes—
43C2×Dic362—22no—
44C2×Dic362—21no—
45Q8×D181—44no—
46C2×D724—21no—
47C2×D724C2311yescharacteristic
48C2×C4×D181—22no—
49C2×C4×D181—21no—
50C2×C4×D181D411yescharacteristic
51C2×C4×D181D411yescharacteristic
52C2×C4×D181C2311yescharacteristic
53C2×#13089678C2312yes—
54C4⋊Dic181—21no—
55C4⋊Dic181—21no—
56C4⋊Dic181C2312yes—
57C4⋊Dic181C2311yescharacteristic
58#15668428—22no—
59#15668428—22no—
60#15668428—21no—
61#15668428—21no—
62#15668426—21no—
63#15668426—21no—
64#15668426—21no—
65#15668426—21no—
66#15668426—21no—
67C4×Dic181D411yescharacteristic
68C4×Dic181D411yescharacteristic
69C4×Dic181C2311yescharacteristic
70C2×Q8×C181—21no—
71C2×Q8×C181C2311yescharacteristic
72C2×C8×C181C2312yes—
73C4⋊C4×C181—21no—
74C4⋊C4×C181—21no—
75C4⋊C4×C181C2312yes—
76C4⋊C4×C181C2311yescharacteristic
77C42×C181—21no—
78C42×C181C2311yescharacteristic
79C22×D181—21no—
80C22×D181C2×D411yescharacteristic
81Dic362—44no—
82Dic362—42no—
83D724—21no—
84D724—21no—
85D724—42no—
86C2×Dic181—21no—
87C2×Dic181—22no—
88C2×Dic181C2×D411yescharacteristic
89C2×Dic181C2×D411yescharacteristic
90C2×Dic181C2×D411yescharacteristic
91C4×D181—22no—
92C4×D181—22no—
93C4×D181—21no—
94C4×D181—21no—
95C4×D181—44no—
96C4×D181—42no—
97#13089678—22no—
98Q8×C181—21no—
99Q8×C181—21no—
100Q8×C181—42no—
101C2×C4×C181—21no—
102C2×C4×C181—21no—
103C2×C4×C181—22no—
104C2×C4×C181C2×D411yescharacteristic
105C2×C4×C181C2×D411yescharacteristic
106C2×C4×C181C2411yescharacteristic
107C8×C181—22no—
108C22×C181C22×D411yescharacteristic
109C2×D181—21no—
110C2×D181—21no—
111C2×D181—41no—
112C2×D181—41no—
113C2×D181—41no—
114C2×D181#13712yes—
115C4×C181—21no—
116C4×C181—41no—
117C4×C181—42no—
118C4×C181—41no—
119C4×C181AGL(1,ℤ/8)12yes—
120C4×C1812−1+411yescharacteristic
121C4×C1812−1+411yescharacteristic
122Dic181—21no—
123Dic181—21no—
124Dic181—21no—
125Dic181—41no—
126Dic181—42no—
127C2×C181#53611yescharacteristic
128C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C181C2×#13711yescharacteristic
130D181—22no—
131D181—81no—
132C181#323911yescharacteristic, complemented, Sylow
133#3239—1811nomaximal, Sylow
134C2×C4⋊Q8—1811no—
135#529—1811no—
136#501—1812no—
137#489—1812no—
138#484—1812no—
139#481—1812no—
140#475—1811no—
141#475D18111yescharacteristic, complemented
142#474—1811no—
143#474—1811no—
144#432—1811no—
145C22×Q8—3621no—
146C4⋊Q8—3621no—
147C4⋊Q8—3621no—
148C4⋊Q8—1812no—
149C4⋊Q8—1811no—
150C4⋊Q8C2×D18111yescharacteristic
151#126—3621no—
152#125—1811no—
153#124—1811no—
154#123—3621no—
155C22⋊Q8—1811no—
156#121—1811no—
157C4×Q8C2×D18111yescharacteristic
158C4×D4—3621no—
159C4×D4—1811no—
160#117—3621no—
161C2×C4⋊C4—3621no—
162C2×C4⋊C4—1812no—
163C2×C42—1811no—
164#107—1812no—
165#107—1812no—
166#106—1812no—
167#106C2×D18112yes—
168C4⋊C8—1811no—
169C4⋊C8C2×D18111yescharacteristic
170#103—1812no—
171#103C2×D18112yes—
172#102—1812no—
173#102—1812no—
174C22⋊C8—1812no—
175#97—1812no—
176C2×Q8—21no—
177C2×Q8—3622no—
178C2×Q8—3621no—
179C2×Q8—7244no—
180C2×Q8C22×D18111yescharacteristic
181C2×D4—3621no—
182C2×D4—1811no—
183C22×C4—3622no—
184C22×C4—3621no—
185C22×C4—1811no—
186C22×C4—1811no—
187C22×C4—1811no—
188C2×C8—1812no—
189C2×C8C22×D18112yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—3622no—
193C4⋊C4—3622no—
194C4⋊C4—3621no—
195C4⋊C4—3621no—
196C4⋊C4—3621no—
197C4⋊C4—3621no—
198C4⋊C4—1812no—
199C4⋊C4—1811no—
200C4⋊C4C22×D18112yes—
201C4⋊C4C22×D18111yescharacteristic
202C22⋊C4—3621no—
203C22⋊C4—3621no—
204C22⋊C4—3621no—
205C22⋊C4—3621no—
206C22⋊C4—3621no—
207C42—21no—
208C42—1811no—
209C42—1811no—
210C42—1811no—
211C42C22×D18111yescharacteristic
212C23—3621no—
213C23—1811no—
214Q8—21no—
215Q8—21no—
216Q8—42no—
217Q8—7244no—
218Q8—7242no—
219D4—3621no—
220D4—3621no—
221D4—7242no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—3622no—
226C2×C4—3622no—
227C2×C4—3621no—
228C2×C4—3621no—
229C2×C4—3621no—
230C2×C4—3622no—
231C2×C4—1811no—
232C2×C4—1811no—
233C2×C4—1811no—
234C2×C4—7244no—
235C2×C4—7242no—
236C2×C4D4×D18111yescharacteristic
237C2×C4D4×D18111yescharacteristic
238C2×C4C23×D18111yescharacteristic
239C8—22no—
240C8—3622no—
241C22—3621no—
242C22—3621no—
243C22—1812no—
244C22—7241no—
245C22—7241no—
246C22—7241no—
247C22C2×D4×D18111yescharacteristic
248C4—21no—
249C4—42no—
250C4—41no—
251C4—41no—
252C4—3621no—
253C4—3621no—
254C4—3621no—
255C4—7242no—
256C4—7241no—
257C4#1915384312yes—
258C4#1915386311yescharacteristic
259C4#1915386411yescharacteristic
260C2—3622no—
261C2—14481no—
262C2#2604185211yescharacteristic
263C2C2×#1915384311yescharacteristic
264C2#137×D18111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#475×C181C2
socleC22×C181C22×D4
prime radical#19153791C22
hypercenter#475D181
hypocenterC181#3239
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D181C2semidirect
3. #26041686C2semidirect
4. #26041640C2semidirect
5. #26041613C2semidirect
7. #26041564C2semidirect
8. #26041560C2semidirect
9. #26041556C2semidirect
10. #26041335C2semidirect
11. #475×C181C2semidirect
132. C181#3239semidirect
141. #475D181semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1812
#260418162
#260416862
#260416402
#260416132
#260415992
#260415642
#260415602
#260415562
#260413352
#475×C1812
#3239181