Finite Group Databasebeta

← #37534739

Subgroups — #37534739

Order 23168.

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

normal subgroups only

This group has 478 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×#19153844C211yesmaximal, characteristic, complemented
2#26041848C211yesmaximal, characteristic, complemented
3C2×#19153813C211yesmaximal, characteristic, complemented
4C2×#19153744C211yesmaximal, characteristic, complemented
5#26041283C211yesmaximal, characteristic, complemented
6C2×#19153732C211yesmaximal, characteristic, complemented
7#339×C181C211yesmaximal, characteristic, complemented
8C2×#15668449C2211yescharacteristic
9C22×D724—22no—
10C22×D724C2211yescharacteristic
11C22×D724C2211yescharacteristic
12#19153844—22no—
13#19153844—22no—
14C2×#15668438—21no—
15C2×D1448—21no—
16C2×#15668433C2211yescharacteristic
17#19153813—22no—
18#19153813—21no—
19#19153813—22no—
20#19153804—21no—
21#19153791—22no—
22#19153786—22no—
23#19153744—22no—
24#19153736—22no—
25#19153732—22no—
26C2×M16×C181C2211yescharacteristic
27#117×C181C2211yescharacteristic
28C2×C42×C181C2211yescharacteristic
29C23×D181—21no—
30C23×D181D412yes—
31#15668449—21no—
32#15668449—21no—
33#15668449—21no—
34#15668449—21no—
35C2×Dic362D411yescharacteristic, complemented
36C2×D724—28no—
37C2×D724—24no—
38C2×D724—21no—
39C2×D724—21no—
40C2×D724—21no—
41C2×D724—21no—
42C2×D724—21no—
43C2×D724—21no—
44C2×D724—21no—
45C2×D724—21no—
46C2×D724—21no—
47C2×D724—21no—
48C2×D724—21no—
49C2×D724—48no—
50C2×D724D411yescharacteristic, complemented
51C2×D724D412yescomplemented
52C2×C22⋊D181—21no—
53C2×C4×D181—21no—
54#15668438—24no—
55D1448—24no—
56#15668433—21no—
57#15668433—21no—
58C2×#13089678—21no—
59#15668426—24no—
60C22×C4×C181—21no—
61C22×C4×C181C2311yescharacteristic
62M16×C181—21no—
63M16×C181D412yescomplemented
64C2×C8×C181—21no—
65C4⋊C4×C181D412yescomplemented
66C22⋊C4×C181—21no—
67C42×C181—21no—
68C42×C181—21no—
69C42×C181D412yescomplemented
70C22×D181—22no—
71C22×D181—21no—
72C22×D181—22no—
73C22×D181—22no—
74C22×D181—22no—
75C22×D181—44no—
76C22×D181—44no—
77C22×D181—41no—
78C22×D181—41no—
79Dic362—21no—
80Dic362—21no—
81D724—21no—
82D724—22no—
83D724—22no—
84D724—21no—
85D724—416no—
86D724—41no—
87D724—41no—
88D724—41no—
89D724—41no—
90D724—41no—
91D724—41no—
92D724—84no—
93C22⋊D181—41no—
94C22⋊D181—41no—
95C2×Dic181—21no—
96C4×D181—41no—
97C4×D181—41no—
98#13089678—41no—
99C23×C181C2×D411yescharacteristic
100C2×C4×C181—24no—
101C2×C4×C181—22no—
102C2×C4×C181—22no—
103C2×C4×C181C2×D411yescharacteristic
104C2×C4×C181C2×D411yescharacteristic
105C2×C4×C181C2×D411yescharacteristic
106C2×C4×C181C2×D411yescharacteristic
107C2×C4×C181C2×D411yescharacteristic
108C2×C4×C181#4111yescharacteristic
109C8×C181—22no—
110C22×C181—22no—
111C22×C181#12011yescharacteristic
112C22×C181#12011yescharacteristic
113C22×C181#12111yescharacteristic
114C2×D181—44no—
115C2×D181—44no—
116C2×D181—41no—
117C2×D181—42no—
118C2×D181—42no—
119C2×D181—42no—
120C2×D181—44no—
121C2×D181—82no—
122C2×D181—82no—
123C2×D181—82no—
124C4×C181—24no—
125C4×C181—42no—
126C4×C181#12011yescharacteristic
127C4×C181#12111yescharacteristic
128C4×C181#12111yescharacteristic
129C4×C181#12711yescharacteristic
130Dic181—41no—
131C2×C181—21no—
132C2×C181—21no—
133C2×C181#39211yescharacteristic
134C2×C181#45312yes—
135D181—81no—
136D181—82no—
137D181—82no—
138C181#200211yescharacteristic, complemented, Sylow
139#2002—1811nomaximal, Sylow
140C2×AGL(1,ℤ/8)—1811no—
141#535—1811no—
142C2×#127—1811no—
143C2×#104—1811no—
144#418—1811no—
145C2×#100—1811no—
146#339D18111yescharacteristic, complemented
147C2×#41—1811no—
148C22×D4—3622no—
149C22×D4—1811no—
150C22×D4—1811no—
151AGL(1,ℤ/8)—3622no—
152AGL(1,ℤ/8)—3622no—
153C2×SD16—3621no—
154C2×D8—3621no—
155C2×M16—1811no—
156C2×M16C2×D18111yescharacteristic
157#127—3622no—
158#127—3621no—
159#127—3622no—
160#124—3621no—
161#121—3622no—
162#120—3622no—
163#117C2×D18111yescharacteristic
164C2×C42C2×D18111yescharacteristic
165#104—3622no—
166#102—3622no—
167#100—3622no—
168C24—3621no—
169C24—1812no—
170#41—3621no—
171#41—3621no—
172#41—3621no—
173#41—3621no—
174C2×Q8—1811no—
175C2×D4—3624no—
176C2×D4—3628no—
177C2×D4—3621no—
178C2×D4—3621no—
179C2×D4—3621no—
180C2×D4—3621no—
181C2×D4—3621no—
182C2×D4—3621no—
183C2×D4—3621no—
184C2×D4—3621no—
185C2×D4—3621no—
186C2×D4—3621no—
187C2×D4—3621no—
188C2×D4—3621no—
189C2×D4—1811no—
190C2×D4—1812no—
191C2×D4—7248no—
192C22×C4—21no—
193C22×C4—3621no—
194C22×C4C22×D18111yescharacteristic
195SD16—3624no—
196D8—3624no—
197M16—21no—
198M16—3621no—
199M16—3621no—
200M16D72412yescomplemented
201C2×C8—21no—
202C2×C8—3621no—
203C4⋊C4D72412yescomplemented
204C22⋊C4—21no—
205C22⋊C4—3624no—
206C42—21no—
207C42—21no—
208C42D72412yescomplemented
209C23—3622no—
210C23—3622no—
211C23—3622no—
212C23—3622no—
213C23—3621no—
214C23—7244no—
215C23—7244no—
216C23—7241no—
217C23—7241no—
218C23D4×D18111yescharacteristic
219Q8—3621no—
220Q8—3621no—
221D4—3621no—
222D4—3622no—
223D4—3622no—
224D4—3621no—
225D4—14484no—
226D4—72416no—
227D4—7241no—
228D4—7241no—
229D4—7241no—
230D4—7241no—
231D4—7241no—
232D4—7241no—
233D4—7241no—
234D4—7241no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—3621no—
239C2×C4—7241no—
240C2×C4—7241no—
241C2×C4D4×D18111yescharacteristic
242C2×C4D4×D18111yescharacteristic
243C2×C4C2×D72411yescharacteristic
244C2×C4C2×D72411yescharacteristic
245C2×C4C2×D72411yescharacteristic
246C2×C4#1566845011yescharacteristic
247C8—22no—
248C8—7241no—
249C22—22no—
250C22—14482no—
251C22—14482no—
252C22—14482no—
253C22—7244no—
254C22—7244no—
255C22—7241no—
256C22—7242no—
257C22—7242no—
258C22—7242no—
259C22—7244no—
260C22#1915378611yescharacteristic
261C22#1915378611yescharacteristic
262C22#1915379111yescharacteristic
263C4—24no—
264C4—42no—
265C4—7241no—
266C4#1915378611yescharacteristic
267C4#1915379111yescharacteristic
268C4#1915379111yescharacteristic
269C4#1915381311yescharacteristic
270C2—21no—
271C2—21no—
272C2—14482no—
273C2—14482no—
274C2—14481no—
275C2#2604118911yescharacteristic
276C2#2604144812yes—
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#339×C181C2
socleC22×C181#120
hypercenter#339D181
hypocenterC181#2002
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×#19153844C2semidirect
2. #26041848C2semidirect
3. C2×#19153813C2semidirect
4. C2×#19153744C2semidirect
5. #26041283C2semidirect
6. C2×#19153732C2semidirect
7. #339×C181C2semidirect
35. C2×Dic362D4semidirect
50. C2×D724D4semidirect
51. C2×D724D4semidirect
63. M16×C181D4semidirect
65. C4⋊C4×C181D4semidirect
69. C42×C181D4semidirect
138. C181#2002semidirect
146. #339D181semidirect
200. M16D724semidirect
203. C4⋊C4D724semidirect
208. C42D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538442
#260418482
C2×#191538132
C2×#191537442
#260412832
C2×#191537322
#339×C1812
#2002181