Finite Group Databasebeta

← #37583240

Subgroups — #37583240

Order 24448.

402 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 125 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#27133218C211yesmaximal, characteristic, complemented
2#27133214C211yesmaximal, characteristic
3#27133036C211yesmaximal, characteristic, complemented
4#27133015C211yesmaximal, characteristic, complemented
5#27133011C211yesmaximal, characteristic, complemented
6#27132910C214yesmaximal, complemented
7#27132762C214yesmaximal, complemented
8C4×#15690270C211yesmaximal, characteristic, complemented
9#500×C191C211yesmaximal, characteristic, complemented
10C2×#15690288—21no—
11C2×#15690276C2214yes—
12C2×#15690270C2212yes—
13#20259322C2211yescharacteristic
14#20259319C2211yescharacteristic
15#20259313—21no—
16#20259311—21no—
17#20259302C2212yes—
18#20259298—21no—
19#20259297—21no—
20#20259297C2212yes—
21Q8×Dic191—21no—
22#20259287C2212yes—
23#20259284—21no—
24#20259284C2212yes—
25#20259280—21no—
26C42×D191C2211yescharacteristic
27#20259245C2214yes—
28#20259242C2214yes—
29#20259232C2212yes—
30C4×C191⋊C8C2211yescharacteristic
31C4⋊Q8×C191C2211yescharacteristic
32C4⋊Q8×C191C2211yescharacteristic
33#107×C191C2214yes—
34C4×C8×C191C2211yescharacteristic
35#15690288—44no—
36C2×Dic382C2312yes—
37C2×D764—22no—
38C2×D764—21no—
39C2×D764—21no—
40C2×D764C2312yes—
41C2×C4×D191—22no—
42C2×C4×D191D411yescharacteristic
43C2×C4×D191D411yescharacteristic
44C2×C4×D191C2311yescharacteristic
45#15690276—28no—
46#15690270—24no—
47C2×C191⋊C8C2312yes—
48C4⋊Dic191—22no—
49C4⋊Dic191—21no—
50C4⋊Dic191—21no—
51Dic191⋊C4—22no—
52(C2×D191)⋊C4—22no—
53(C2×D191)⋊C4—22no—
54(C2×D191)⋊C4—22no—
55C4×Dic191—22no—
56C4×Dic191D411yescharacteristic
57C4×Dic191D411yescharacteristic
58C4×Dic191C2311yescharacteristic
59C2×Q8×C191—21no—
60C2×Q8×C191—21no—
61C2×C8×C191C2312yes—
62C4⋊C4×C191—21no—
63C4⋊C4×C191—21no—
64C4⋊C4×C191C2312yes—
65C4⋊C4×C191C2312yes—
66C42×C191C2311yescharacteristic
67C22×D191—22no—
68C22×D191C2×D411yescharacteristic
69Dic382—24no—
70D764—24no—
71D764—44no—
72D764—42no—
73D764—42no—
74C2×Dic191—22no—
75C2×Dic191C2×D411yescharacteristic
76C2×Dic191C2×D411yescharacteristic
77C2×Dic191C2×D411yescharacteristic
78C4×D191—22no—
79C4×D191—22no—
80C4×D191—22no—
81C4×D191—44no—
82C191⋊C8—22no—
83Q8×C191—42no—
84Q8×C191—42no—
85C2×C4×C191—22no—
86C2×C4×C191—22no—
87C2×C4×C191C2×D411yescharacteristic
88C2×C4×C191C2×D411yescharacteristic
89C2×C4×C191C2411yescharacteristic
90C8×C191#4114yescomplemented
91C22×C191C22×D411yescharacteristic
92C2×D191—21no—
93C2×D191—21no—
94C2×D191—21no—
95C2×D191—42no—
96C2×D191—42no—
97C2×D191—42no—
98C4×C191—42no—
99C4×C191—42no—
100C4×C191AGL(1,ℤ/8)12yes—
101C4×C191#13712yes—
102C4×C191C2×#4112yes—
103Dic191—21no—
104Dic191—21no—
105Dic191—21no—
106Dic191—42no—
107C2×C191#53211yescharacteristic
108C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
109C2×C191C2×#13711yescharacteristic
110D191—41no—
111D191—82no—
112C191#318211yescharacteristic, complemented, Sylow
113#3182—1911nomaximal, Sylow
114#533—1911no—
115#532—1911no—
116#500D19111yescharacteristic, complemented
117#492—1911no—
118#487—1911no—
119#486—1911no—
120#468—1914no—
121#440—1914no—
122C4×M16—1911no—
123C2×#41—3821no—
124C2×SD16—1914no—
125C2×M16—1912no—
126C4⋊Q8—1911no—
127C4⋊Q8C2×D19111yescharacteristic
128C4⋊Q8C2×D19111yescharacteristic
129#127—1911no—
130#125—3821no—
131#124—3821no—
132C22⋊Q8—3821no—
133C22⋊Q8—1912no—
134#121—3821no—
135#121—1912no—
136C4×Q8—3821no—
137C4×Q8—1912no—
138C4×D4—3821no—
139C4×D4—1912no—
140#117—3821no—
141C2×C42—1911no—
142#107C2×D19114yes—
143#103—1914no—
144#102—1914no—
145#97—1912no—
146C4×C8—1911no—
147C4×C8C2×D19111yescharacteristic
148#41—7644no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—1912no—
152C2×D4—3822no—
153C2×D4—3821no—
154C2×D4—3821no—
155C2×D4—1912no—
156C22×C4—3822no—
157C22×C4—1911no—
158C22×C4—1911no—
159C22×C4—1911no—
160SD16—3828no—
161M16—3824no—
162C2×C8—1912no—
163C2×C8C22×D19112yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—3822no—
167C4⋊C4—3822no—
168C4⋊C4—3821no—
169C4⋊C4—3821no—
170C4⋊C4C22×D19112yes—
171C4⋊C4C22×D19112yes—
172C22⋊C4—3822no—
173C22⋊C4—3822no—
174C22⋊C4—3822no—
175C42—3822no—
176C42—1911no—
177C42—1911no—
178C42—1911no—
179C42C22×D19111yescharacteristic
180C23—3822no—
181C23—1911no—
182Q8—42no—
183Q8—42no—
184Q8—3824no—
185D4—3824no—
186D4—7644no—
187D4—7642no—
188D4—7642no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—3822no—
192C2×C4—3822no—
193C2×C4—3822no—
194C2×C4—3822no—
195C2×C4—1911no—
196C2×C4—1911no—
197C2×C4—1911no—
198C2×C4—7644no—
199C2×C4D4×D19111yescharacteristic
200C2×C4D4×D19111yescharacteristic
201C2×C4C23×D19111yescharacteristic
202C8—3822no—
203C8#1569028814yescomplemented
204C22—3821no—
205C22—3821no—
206C22—3821no—
207C22—7642no—
208C22—7642no—
209C22—7642no—
210C22C2×D4×D19111yescharacteristic
211C4—42no—
212C4—42no—
213C4—3821no—
214C4—3821no—
215C4—3821no—
216C4—7642no—
217C4#2025934612yes—
218C4#2025935412yes—
219C4C2×#1569028812yes—
220C2—15282no—
221C2—7641no—
222C2#2713321411yescharacteristic
223C2C2×#2025934611yescharacteristic
224C2C2×#2025935411yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#500×C191C2
socleC22×C191C22×D4
prime radical#27133214C2
hypercenter#500D191
hypocenterC191#3182
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #27133218C2semidirect
3. #27133036C2semidirect
4. #27133015C2semidirect
5. #27133011C2semidirect
6. #27132910C2semidirect
7. #27132762C2semidirect
8. C4×#15690270C2semidirect
9. #500×C191C2semidirect
90. C8×C191#41semidirect
112. C191#3182semidirect
116. #500D191semidirect
203. C8#15690288semidirect

Maximal subgroups

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

subgroupindex
#271332182
#271332142
#271330362
#271330152
#271330112
#271329102
#271327622
C4×#156902702
#500×C1912
#3182191