Finite Group Databasebeta

← #37545522

Subgroups — #37545522

Order 23168.

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

normal subgroups only

This group has 444 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#26042095C212yesmaximal, complemented
2#26042094C212yesmaximal, complemented
3#26041658C211yesmaximal, characteristic, complemented
4#26041553C212yesmaximal, complemented
5#26041552C212yesmaximal, complemented
6#26041463C212yesmaximal, complemented
7#26041461C211yesmaximal, characteristic, complemented
8#26041393C212yesmaximal, complemented
9#456×C181C211yesmaximal, characteristic, complemented
10#19153864C2211yescharacteristic
11#19153863C2212yes—
12#19153849C2212yes—
13#19153848—22no—
14#19153847—22no—
15#19153847C2212yes—
16#19153846—22no—
17#19153846C2212yes—
18#19153845—22no—
19#19153837—22no—
20#19153836—22no—
21#19153836C2212yes—
22C2×#15668439—22no—
23C2×#15668439C2212yes—
24C2×#15668439C2212yes—
25Q16×D181—22no—
26#19153824C2212yes—
27#19153823C2212yes—
28#19153816C2211yescharacteristic
29#19153751C2212yes—
30#19153744C2212yes—
31#19153735C2211yescharacteristic
32#19153734C2212yes—
33C4×#13089678C2211yescharacteristic
342−1+4×C181C2211yescharacteristic
35#137×C181C2212yes—
36C4⋊Q8×C181C2211yescharacteristic
37#104×C181C2212yes—
38#101×C181C2211yescharacteristic
39#15668450—22no—
40#15668449—22no—
41#15668449—41no—
42#15668449C2312yes—
43#15668448—22no—
44#15668448—42no—
45C2×Dic362—22no—
46C2×Dic362C2312yes—
47Q8×D181—22no—
48Q8×D181—22no—
49Dic724—22no—
50#15668439—22no—
51#15668439—22no—
52#15668439—22no—
53#15668439—22no—
54#15668439—22no—
55#15668439—22no—
56#15668439D412yes—
57#15668438—22no—
58#15668437—22no—
59#15668437D412yes—
60#15668436—22no—
61#15668436D412yes—
62#15668434D412yes—
63#15668433—22no—
64#15668433C2312yes—
65#15668432—22no—
66C2×#13089678—22no—
67C2×#13089678C2312yes—
68C8×D181—22no—
69C4⋊Dic181—41no—
70#41×C181—22no—
71#41×C181—41no—
72#41×C181C2312yes—
73C2×Q8×C181—22no—
74C2×Q8×C181C2311yescharacteristic
75C2×Q8×C181C2311yescharacteristic
76Q16×C181—22no—
77SD16×C181—22no—
78M16×C181C2312yes—
79C4⋊C4×C181—41no—
80C42×C181C2311yescharacteristic
81Dic362—22no—
82Dic362—22no—
83Dic362—41no—
84Dic362—42no—
85Dic362C2×D412yes—
86D724—41no—
87D724C2×D412yes—
88C22⋊D181—22no—
89C22⋊D181—42no—
90C2×Dic181—22no—
91C2×Dic181—42no—
92C4×D181—22no—
93C4×D181—42no—
94C4×D181—42no—
95#13089678—22no—
96#13089678C2×D412yes—
97#13089678C2×D412yes—
98Q8×C181—22no—
99Q8×C181—22no—
100Q8×C181—41no—
101Q8×C181C2×D412yes—
102D4×C181—22no—
103D4×C181—41no—
104D4×C181C2×D412yes—
105C2×C4×C181—21no—
106C2×C4×C181—22no—
107C2×C4×C181—21no—
108C2×C4×C181—21no—
109C2×C4×C181—42no—
110C2×C4×C181C2411yescharacteristic
111C8×C181—22no—
112C22×C181—22no—
113C22×C181H(5,𝔽2)11yescharacteristic
114C2×D181—22no—
115C4×C181—22no—
116C4×C181—22no—
117C4×C181—41no—
118C4×C181—41no—
119C4×C181C22×D412yes—
120Dic181—22no—
121Dic181—42no—
122C2×C181—21no—
123C2×C181—42no—
124C2×C181D4211yescharacteristic
125D181—42no—
126C181#329311yescharacteristic, complemented, Sylow
127#3293—1811nomaximal, Sylow
128#578—1812no—
129#578—1812no—
130#494—1811no—
131#473—1812no—
132#473—1812no—
133#456—1811no—
134#456—1812no—
135#456D18111yescharacteristic, complemented
136#443—1812no—
1372−1+4—1812no—
1382−1+4—1811no—
1392−1+4C2×D18111yescharacteristic
140#137—3622no—
141#137—3622no—
142#137—3622no—
143#137—3622no—
144#137—1812no—
145#137—1812no—
146#137—1812no—
147#137C2×D18112yes—
148#135—3622no—
149#135—3622no—
150#135—1812no—
151C2×Q16—3622no—
152C2×Q16—3622no—
153C2×Q16—1812no—
154C2×Q16—1812no—
155#131—1812no—
156#131—1812no—
157C4⋊Q8—1811no—
158C4⋊Q8C2×D18111yescharacteristic
159#108—1812no—
160#104—1812no—
161#104C2×D18112yes—
162#101—1811no—
163#101—1812no—
164#101C2×D18111yescharacteristic
165C4×C8—1811no—
166#41—22no—
167#41—41no—
168#41—3622no—
169#41—3622no—
170#41—3622no—
171#41—1812no—
172#41—7241no—
173#41—7242no—
174#41C22×D18112yes—
175C2×Q8—22no—
176C2×Q8—3622no—
177C2×Q8—3622no—
178C2×Q8—3622no—
179C2×Q8—1812no—
180C2×Q8C22×D18111yescharacteristic
181C2×Q8C22×D18111yescharacteristic
182Q16—22no—
183Q16—3622no—
184Q16—3622no—
185Q16—3622no—
186Q16—3622no—
187Q16—3622no—
188Q16—3622no—
189Q16—3622no—
190Q16—1812no—
191SD16—22no—
192SD16—3622no—
193SD16—3622no—
194SD16—3622no—
195SD16—1812no—
196SD16—1812no—
197D8—1812no—
198M16—3622no—
199M16—3622no—
200M16—1812no—
201M16C22×D18112yes—
202C2×C8—3622no—
203C2×C8—3622no—
204C2×C8—1812no—
205C4⋊C4—41no—
206C4⋊C4—7241no—
207C42C22×D18111yescharacteristic
208Q8—22no—
209Q8—22no—
210Q8—41no—
211Q8—3622no—
212Q8—3622no—
213Q8—1812no—
214Q8—7242no—
215Q8—7241no—
216Q8D4×D18112yes—
217D4—22no—
218D4—41no—
219D4—3622no—
220D4—1812no—
221D4—7241no—
222D4—7242no—
223D4D4×D18112yes—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—3622no—
230C2×C4—3622no—
231C2×C4—7242no—
232C2×C4—7242no—
233C2×C4—7242no—
234C2×C4C23×D18111yescharacteristic
235C8—22no—
236C8—3622no—
237C8—1812no—
238C8—1812no—
239C22—22no—
240C22—3622no—
241C22#1915386111yescharacteristic
242C4—22no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—3622no—
247C4—7242no—
248C4C2×D4×D18112yes—
249C2—21no—
250C2—42no—
251C2—7242no—
252C2#2604191111yescharacteristic
subgroupG/H
centerC2#26041911
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#456×C181C2
socleC2×C181D42
hypercenter#456D181
hypocenterC181#3293
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #26042095C2semidirect
2. #26042094C2semidirect
3. #26041658C2semidirect
4. #26041553C2semidirect
5. #26041552C2semidirect
6. #26041463C2semidirect
7. #26041461C2semidirect
8. #26041393C2semidirect
9. #456×C181C2semidirect
126. C181#3293semidirect
135. #456D181semidirect

Maximal subgroups

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

subgroupindex
#260420952
#260420942
#260416582
#260415532
#260415522
#260414632
#260414612
#260413932
#456×C1812
#3293181