Finite Group Databasebeta

← #37534753

Subgroups — #37534753

Order 23168.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19153849C211yesmaximal, characteristic, complemented
2#26041848C211yesmaximal, characteristic
3C2×#19153804C211yesmaximal, characteristic, complemented
4C2×#19153744C211yesmaximal, characteristic, complemented
5#26041287C211yesmaximal, characteristic, complemented
6C2×#19153732C211yesmaximal, characteristic, complemented
7#339×C181C211yesmaximal, characteristic, complemented
8C2×#15668449C2211yescharacteristic
9C22×Dic362C2211yescharacteristic
10C22×D724C2211yescharacteristic
11#19153849—22no—
12#19153849—22no—
13C2×Dic724—21no—
14C2×#15668438—21no—
15C2×#15668433C2211yescharacteristic
16#19153813—21no—
17#19153804—21no—
18#19153804—22no—
19#19153804—22no—
20#19153791—22no—
21#19153786—22no—
22C2×#15668426—22no—
23#19153744—22no—
24#19153739—22no—
25#19153732—22no—
26C2×M16×C181C2211yescharacteristic
27#117×C181C2211yescharacteristic
28C2×C42×C181C2211yescharacteristic
29C23×D181D412yes—
30#15668449—21no—
31#15668449—21no—
32#15668449—21no—
33#15668449—21no—
34C2×Dic362—21no—
35C2×Dic362—21no—
36C2×Dic362—21no—
37C2×Dic362—21no—
38C2×Dic362—21no—
39C2×Dic362D411yescharacteristic
40C2×Dic362D412yescomplemented
41C2×D724—24no—
42C2×D724—21no—
43C2×D724—21no—
44C2×D724—21no—
45C2×D724—21no—
46C2×D724—21no—
47C2×D724—21no—
48C2×D724D411yescharacteristic
49C2×C22⋊D181—21no—
50C22×Dic181—21no—
51C2×C4×D181—21no—
52Dic724—24no—
53#15668438—24no—
54#15668433—21no—
55#15668433—21no—
56C2×#13089678—21no—
57#15668426—28no—
58#15668426—24no—
59C22×C4×C181—21no—
60C22×C4×C181C2311yescharacteristic
61M16×C181—21no—
62M16×C181D412yescomplemented
63C2×C8×C181—21no—
64C4⋊C4×C181D412yes—
65C22⋊C4×C181—21no—
66C42×C181—21no—
67C42×C181—21no—
68C42×C181D412yescomplemented
69C22×D181—22no—
70C22×D181—22no—
71C22×D181—21no—
72C22×D181—22no—
73C22×D181—44no—
74Dic362—22no—
75Dic362—21no—
76Dic362—22no—
77Dic362—21no—
78Dic362—41no—
79Dic362—41no—
80D724—21no—
81D724—21no—
82D724—41no—
83D724—41no—
84D724—41no—
85D724—41no—
86D724—84no—
87C22⋊D181—41no—
88C22⋊D181—41no—
89C2×Dic181—21no—
90C2×Dic181—22no—
91C2×Dic181—41no—
92C2×Dic181—41no—
93C4×D181—41no—
94C4×D181—41no—
95#13089678—41no—
96C23×C181C2×D411yescharacteristic
97C2×C4×C181—22no—
98C2×C4×C181—22no—
99C2×C4×C181—42no—
100C2×C4×C181C2×D411yescharacteristic
101C2×C4×C181C2×D411yescharacteristic
102C2×C4×C181C2×D411yescharacteristic
103C2×C4×C181C2×D411yescharacteristic
104C2×C4×C181C2×D411yescharacteristic
105C2×C4×C181#4111yescharacteristic
106C8×C181—22no—
107C22×C181—22no—
108C22×C181#12011yescharacteristic
109C22×C181#12011yescharacteristic
110C22×C181#12111yescharacteristic
111C2×D181—44no—
112C2×D181—41no—
113C2×D181—42no—
114C2×D181—42no—
115C2×D181—44no—
116C2×D181—82no—
117C4×C181—42no—
118C4×C181—42no—
119C4×C181#12011yescharacteristic
120C4×C181#12111yescharacteristic
121C4×C181#12111yescharacteristic
122C4×C181#12711yescharacteristic
123Dic181—41no—
124Dic181—42no—
125C2×C181—21no—
126C2×C181—21no—
127C2×C181#39211yescharacteristic
128C2×C181#45512yes—
129D181—81no—
130D181—82no—
131C181#200411yescharacteristic, complemented, Sylow
132#2004—1811nomaximal, Sylow
133C2×#137—1811no—
134#535—1811no—
135C2×#124—1811no—
136C2×#104—1811no—
137#419—1811no—
138C2×#100—1811no—
139#339D18111yescharacteristic, complemented
140C2×#41—1811no—
141C22×Q8—1811no—
142C22×D4—1811no—
143#137—3622no—
144#137—3622no—
145C2×Q16—3621no—
146C2×SD16—3621no—
147C2×M16—1811no—
148C2×M16C2×D18111yescharacteristic
149#127—3621no—
150#124—3622no—
151#124—3621no—
152#124—3622no—
153#121—3622no—
154#120—3622no—
155#117C2×D18111yescharacteristic
156C2×C22⋊C4—3622no—
157C2×C42C2×D18111yescharacteristic
158#104—3622no—
159#103—3622no—
160#100—3622no—
161C24—1812no—
162#41—3621no—
163#41—3621no—
164#41—3621no—
165#41—3621no—
166C2×Q8—3621no—
167C2×Q8—3621no—
168C2×Q8—3621no—
169C2×Q8—3621no—
170C2×Q8—3621no—
171C2×Q8—1811no—
172C2×Q8—1812no—
173C2×D4—3624no—
174C2×D4—3621no—
175C2×D4—3621no—
176C2×D4—3621no—
177C2×D4—3621no—
178C2×D4—3621no—
179C2×D4—3621no—
180C2×D4—3621no—
181C2×D4—1811no—
182C22×C4—21no—
183C22×C4—3621no—
184C22×C4—3621no—
185C22×C4C22×D18111yescharacteristic
186Q16—3624no—
187SD16—3624no—
188M16—21no—
189M16—3621no—
190M16—3621no—
191M16D72412yescomplemented
192C2×C8—21no—
193C2×C8—3621no—
194C4⋊C4D72412yes—
195C22⋊C4—21no—
196C22⋊C4—3628no—
197C22⋊C4—3624no—
198C42—21no—
199C42—21no—
200C42D72412yescomplemented
201C23—3622no—
202C23—3622no—
203C23—3622no—
204C23—3621no—
205C23—7244no—
206C23D4×D18111yescharacteristic
207Q8—3622no—
208Q8—3621no—
209Q8—3622no—
210Q8—3621no—
211Q8—7241no—
212Q8—7241no—
213D4—3621no—
214D4—3621no—
215D4—14484no—
216D4—7241no—
217D4—7241no—
218D4—7241no—
219D4—7241no—
220D4—7241no—
221D4—7241no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—3622no—
226C2×C4—3621no—
227C2×C4—7241no—
228C2×C4—7241no—
229C2×C4—7241no—
230C2×C4—7241no—
231C2×C4D4×D18111yescharacteristic
232C2×C4D4×D18111yescharacteristic
233C2×C4C2×D72411yescharacteristic
234C2×C4C2×D72411yescharacteristic
235C2×C4C2×D72411yescharacteristic
236C2×C4#1566845011yescharacteristic
237C8—22no—
238C8—7241no—
239C22—22no—
240C22—14482no—
241C22—7244no—
242C22—7241no—
243C22—7242no—
244C22—7242no—
245C22—7244no—
246C22#1915378611yescharacteristic
247C22#1915378611yescharacteristic
248C22#1915379111yescharacteristic
249C4—42no—
250C4—42no—
251C4—7241no—
252C4—7242no—
253C4#1915378611yescharacteristic
254C4#1915379111yescharacteristic
255C4#1915379111yescharacteristic
256C4#1915381311yescharacteristic
257C2—21no—
258C2—21no—
259C2—14482no—
260C2—14481no—
261C2#2604118911yescharacteristic
262C2#2604145812yes—
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#339×C181C2
socleC22×C181#120
prime radical#26041848C2
hypercenter#339D181
hypocenterC181#2004
LCSG ▷ C22×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×#19153849C2semidirect
3. C2×#19153804C2semidirect
4. C2×#19153744C2semidirect
5. #26041287C2semidirect
6. C2×#19153732C2semidirect
7. #339×C181C2semidirect
40. C2×Dic362D4semidirect
62. M16×C181D4semidirect
68. C42×C181D4semidirect
131. C181#2004semidirect
139. #339D181semidirect
191. M16D724semidirect
200. C42D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538492
#260418482
C2×#191538042
C2×#191537442
#260412872
C2×#191537322
#339×C1812
#2004181