Finite Group Databasebeta

← #37582040

Subgroups — #37582040

Order 24448.

490 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 133 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×#20259355C211yesmaximal, characteristic, complemented
2C22×#15690275C211yesmaximal, characteristic, complemented
3#27133230C211yesmaximal, characteristic
4#27133140C211yesmaximal, characteristic, complemented
5#27132884C214yesmaximal, complemented
6#27132862C214yesmaximal, complemented
7#27132691C211yesmaximal, characteristic, complemented
8#27132657C211yesmaximal, characteristic, complemented
9#418×C191C211yesmaximal, characteristic, complemented
10C2×#15690287C2211yescharacteristic
11C22×Dic382C2211yescharacteristic
12#20259355—24no—
13C2×Dic764C2212yescomplemented
14C2×#15690276C2212yescomplemented
15C2×#15690275—24no—
16C2×#15690275C2214yes—
17C22×C191⋊C8C2211yescharacteristic
18#20259318C2212yes—
19#20259309C2212yes—
20#20259293—22no—
21#20259291—22no—
22C4×Dic382—22no—
23#20259287C2214yescomplemented
24#20259281C2211yescharacteristic
25#20259278—22no—
26Dic191⋊C8C2214yescomplemented
27#20259245C2212yescomplemented
28#20259242C2212yescomplemented
29C22×D4×C191C2211yescharacteristic
30C2×M16×C191C2211yescharacteristic
31#117×C191C2211yescharacteristic
32#102×C191C2214yes—
33#15690287—22no—
34#15690287—22no—
35C2×Dic382—21no—
36C2×Dic382—22no—
37C2×Dic382D414yescomplemented
38C2×Dic382C2311yescharacteristic
39C2×Dic382C2312yes—
40C2×D764C2311yescharacteristic
41C2×C22⋊D191—24no—
42C2×C22⋊D191—21no—
43C22×Dic191—21no—
44C2×C4×D191—21no—
45Dic764—24no—
46#15690276—24no—
47#15690275—28no—
48C2×C191⋊C8—22no—
49C2×C191⋊C8C2312yes—
50C4⋊Dic191D412yes—
51Dic191⋊C4—24no—
52C22⋊Dic191—24no—
53C22⋊Dic191D412yes—
54C4×Dic191—24no—
55C4×Dic191C2312yes—
56C24×C191—21no—
57C2×D4×C191—22no—
58C2×D4×C191—21no—
59C2×D4×C191—22no—
60C2×D4×C191C2312yes—
61C22×C4×C191C2311yescharacteristic
62M16×C191—22no—
63C2×C8×C191C2312yes—
64C4⋊C4×C191C2312yes—
65C22⋊C4×C191—21no—
66C42×C191—21no—
67C22×D191—21no—
68Dic382—24no—
69Dic382—22no—
70Dic382—21no—
71Dic382—21no—
72Dic382C2×D414yes—
73D764—21no—
74D764—21no—
75C22⋊D191—42no—
76C22⋊D191—84no—
77C2×Dic191—24no—
78C2×Dic191—22no—
79C2×Dic191—21no—
80C2×Dic191C2×D414yes—
81C4×D191—42no—
82C191⋊C8—22no—
83C23×C191—22no—
84C23×C191—42no—
85C23×C191—82no—
86C23×C191C2×D411yescharacteristic
87D4×C191—22no—
88D4×C191—22no—
89D4×C191—42no—
90C2×C4×C191—22no—
91C2×C4×C191C2×D411yescharacteristic
92C2×C4×C191C2×D412yes—
93C2×C4×C191#4112yes—
94C2×C4×C191C2411yescharacteristic
95C8×C191—22no—
96C22×C191—22no—
97C22×C191—42no—
98C22×C191—44no—
99C22×C191—84no—
100C22×C191#12112yes—
101C22×C191C22×D411yescharacteristic
102C2×D191—41no—
103C2×D191—42no—
104C4×C191—42no—
105C4×C191#12112yes—
106C4×C191C22×D411yescharacteristic
107C4×C191C2×#4111yescharacteristic
108Dic191—24no—
109Dic191—24no—
110Dic191—41no—
111C2×C191—22no—
112C2×C191—82no—
113C2×C191C2×#12111yescharacteristic
114C2×C191#57712yes—
115D191—81no—
116C191#304311yescharacteristic, complemented, Sylow
117#3043—1911nomaximal, Sylow
118C2×#137—1911no—
119C22×SD16—1911no—
120#535—1911no—
121#519—1911no—
122#464—1914no—
123#460—1914no—
124#424—1911no—
125#418D19111yescharacteristic, complemented
126#416—1911no—
127C2×#41—1911no—
128C22×Q8—1911no—
129C22×D4C2×D19111yescharacteristic
130#137—3824no—
131C2×Q16—1912no—
132C2×SD16—3824no—
133C2×SD16—1914no—
134C2×SD16—1912no—
135C2×M16C2×D19111yescharacteristic
136C22×C8—1911no—
137#127—1912no—
138#124—1912no—
139#121—3822no—
140#120—3822no—
141C4×Q8—3822no—
142C4×Q8—1914no—
143#117—3822no—
144#117—1911no—
145#117C2×D19111yescharacteristic
146C4⋊C8—1914no—
147#103—1912no—
148#102—1912no—
149#102C2×D19114yes—
150C24—21no—
151#41—3822no—
152#41—3822no—
153C2×Q8—3821no—
154C2×Q8—3822no—
155C2×Q8—1914no—
156C2×Q8—1911no—
157C2×Q8—1912no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—3821no—
162C2×D4—3824no—
163C2×D4—1911no—
164C2×D4C22×D19112yes—
165C22×C4—3821no—
166C22×C4—3821no—
167C22×C4C22×D19111yescharacteristic
168Q16—3824no—
169SD16—3824no—
170SD16—3828no—
171M16—22no—
172C2×C8—3822no—
173C2×C8—1912no—
174C2×C8C22×D19112yes—
175C4⋊C4—3824no—
176C4⋊C4—1912no—
177C4⋊C4C22×D19112yes—
178C22⋊C4—21no—
179C22⋊C4—3824no—
180C22⋊C4—1912no—
181C42—21no—
182C42—3824no—
183C42—1912no—
184C23—22no—
185C23—42no—
186C23—82no—
187C23—3821no—
188C23D4×D19111yescharacteristic
189Q8—3822no—
190Q8—3824no—
191Q8—3821no—
192Q8—3821no—
193Q8—1914no—
194D4—22no—
195D4—22no—
196D4—42no—
197D4—3821no—
198D4—3821no—
199D4—15284no—
200D4—7642no—
201C2×C4—22no—
202C2×C4—3822no—
203C2×C4—3824no—
204C2×C4—3821no—
205C2×C4—1914no—
206C2×C4—7642no—
207C2×C4D4×D19112yes—
208C2×C4D4×D19111yescharacteristic
209C2×C4#1569028712yes—
210C2×C4C23×D19111yescharacteristic
211C8—22no—
212C8—3822no—
213C22—22no—
214C22—44no—
215C22—42no—
216C22—84no—
217C22—7642no—
218C22—7641no—
219C22#2025929512yes—
220C22C2×D4×D19111yescharacteristic
221C4—42no—
222C4—3824no—
223C4—3824no—
224C4—7641no—
225C4#2025929512yes—
226C4C2×D4×D19111yescharacteristic
227C4C2×#1569028711yescharacteristic
228C2—22no—
229C2—82no—
230C2—15281no—
231C2C2×#2025929511yescharacteristic
232C2#2713347312yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#418×C191C2
socleC22×C191C22×D4
prime radical#27133230C2
hypercenter#418D191
hypocenterC191#3043
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#20259355C2semidirect
2. C22×#15690275C2semidirect
4. #27133140C2semidirect
5. #27132884C2semidirect
6. #27132862C2semidirect
7. #27132691C2semidirect
8. #27132657C2semidirect
9. #418×C191C2semidirect
13. C2×Dic764C22semidirect
14. C2×#15690276C22semidirect
23. #20259287C22semidirect
26. Dic191⋊C8C22semidirect
27. #20259245C22semidirect
28. #20259242C22semidirect
37. C2×Dic382D4semidirect
116. C191#3043semidirect
125. #418D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593552
C22×#156902752
#271332302
#271331402
#271328842
#271328622
#271326912
#271326572
#418×C1912
#3043191