Finite Group Databasebeta

← #37583168

Subgroups — #37583168

Order 24448.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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#27133264C212yesmaximal, complemented
2#27133063C211yesmaximal, characteristic, complemented
3#27133038C211yesmaximal, characteristic, complemented
4#27133026C212yesmaximal, complemented
5#27132917C211yesmaximal, characteristic, complemented
6#27132913C211yesmaximal, characteristic, complemented
7#27132898C212yesmaximal, complemented
8#27132776C211yesmaximal, characteristic, complemented
9C4×#15690272C212yesmaximal, complemented
10#27132620C211yesmaximal, characteristic, complemented
11#442×C191C211yesmaximal, characteristic, complemented
12C2×D1528C2211yescharacteristic
13C2×#15690272C2212yes—
14C2×#15690270C2211yescharacteristic
15C2×C8×D191C2211yescharacteristic
16#20259318C2212yes—
17#20259313C2212yes—
18#20259305—22no—
19#20259304—22no—
20#20259297C2212yes—
21#20259296—22no—
22#20259295—22no—
23#20259293C2212yes—
24C4×D764C2212yes—
25#20259279C2211yescharacteristic
26#20259255C2212yes—
27#20259254C2211yescharacteristic
28#20259244C2212yes—
29#20259243C2212yes—
30#20259242C2212yes—
31#20259232C2211yescharacteristic
32C4×C191⋊C8C2211yescharacteristic
33C8×Dic191C2211yescharacteristic
34C2×D8×C191C2211yescharacteristic
35C4×D4×C191C2212yes—
36#106×C191C2211yescharacteristic
37#102×C191C2212yes—
38#97×C191C2211yescharacteristic
39C2×D764—22no—
40C2×D764C2312yes—
41C2×C22⋊D191—22no—
42C2×C22⋊D191—22no—
43C2×C22⋊D191—22no—
44C2×C4×D191—22no—
45C2×C4×D191C2311yescharacteristic
46D1528—22no—
47#15690272—24no—
48#15690270—22no—
49C2×C191⋊C8C2311yescharacteristic
50C2×C191⋊C8C2311yescharacteristic
51C8×D191—22no—
52C4⋊Dic191—22no—
53C4⋊Dic191C2312yes—
54Dic191⋊C4—22no—
55Dic191⋊C4D412yes—
56C22⋊Dic191—22no—
57(C2×D191)⋊C4—22no—
58(C2×D191)⋊C4—22no—
59(C2×D191)⋊C4D412yes—
60C4×Dic191C2311yescharacteristic
61C2×D4×C191C2312yes—
62C22×C4×C191—22no—
63D8×C191—22no—
64C2×C8×C191C2311yescharacteristic
65C2×C8×C191C2311yescharacteristic
66C4⋊C4×C191C2312yes—
67C22⋊C4×C191—22no—
68C42×C191C2311yescharacteristic
69C22×D191—22no—
70C22×D191C2×D411yescharacteristic
71D764—22no—
72D764—22no—
73D764—82no—
74C22⋊D191—82no—
75C22⋊D191—82no—
76C22⋊D191—82no—
77C2×Dic191—22no—
78C2×Dic191C2×D412yes—
79C2×Dic191C2×D411yescharacteristic
80C4×D191—21no—
81C4×D191—21no—
82C4×D191—82no—
83C191⋊C8—21no—
84C191⋊C8#4112yes—
85C23×C191—22no—
86D4×C191—22no—
87D4×C191—22no—
88C2×C4×C191—22no—
89C2×C4×C191—82no—
90C2×C4×C191C2×D412yes—
91C2×C4×C191C2411yescharacteristic
92C8×C191—21no—
93C8×C191#4112yes—
94C22×C191—42no—
95C22×C191—42no—
96C22×C191—42no—
97C22×C191C22×D411yescharacteristic
98C2×D191—21no—
99C2×D191—22no—
100C2×D191—42no—
101C2×D191—42no—
102C2×D191—42no—
103C4×C191—22no—
104C4×C191—42no—
105C4×C191C2×#4111yescharacteristic
106C4×C191C2×#4111yescharacteristic
107Dic191—21no—
108Dic191—22no—
109Dic191—42no—
110C2×C191—82no—
111C2×C191#53211yescharacteristic
112C2×C191#57612yes—
113D191—41no—
114D191—82no—
115C191#317611yescharacteristic, complemented, Sylow
116#3176—1911nomaximal, Sylow
117#540—1912no—
118#499—1911no—
119#493—1911no—
120#490—1912no—
121#469—1911no—
122#469—1911no—
123#466—1912no—
124#442—1911no—
125#442D19111yescharacteristic, complemented
126C4×D8—1912no—
127#405—1911no—
128C2×D8—1912no—
129C2×D8—1911no—
130C2×D8C2×D19111yescharacteristic
131C2×M16—1911no—
132C22×C8—1911no—
133#127—1912no—
134#125—1912no—
135#123—3822no—
136#123—3822no—
137#121—3822no—
138#121—3822no—
139#121—1912no—
140#121—1912no—
141C4×D4—1912no—
142C4×D4C2×D19112yes—
143#117—1911no—
144#107—1912no—
145#106—1911no—
146#106C2×D19111yescharacteristic
147#102—1912no—
148#102—1912no—
149#102—1912no—
150#102C2×D19112yes—
151#97—1911no—
152#97C2×D19111yescharacteristic
153C4×C8—1911no—
154C4×C8—1911no—
155C2×D4—3822no—
156C2×D4—3822no—
157C2×D4—3822no—
158C2×D4—3822no—
159C2×D4—1912no—
160C2×D4C22×D19112yes—
161C22×C4—22no—
162C22×C4—3822no—
163C22×C4—1911no—
164D8—22no—
165D8—3824no—
166D8—3822no—
167M16—3822no—
168C2×C8—3822no—
169C2×C8—1911no—
170C2×C8—1911no—
171C2×C8C22×D19111yescharacteristic
172C2×C8C22×D19111yescharacteristic
173C4⋊C4—3822no—
174C4⋊C4—3822no—
175C4⋊C4—1912no—
176C4⋊C4—1912no—
177C4⋊C4C22×D19112yes—
178C22⋊C4—22no—
179C22⋊C4—3822no—
180C22⋊C4—3822no—
181C22⋊C4—3822no—
182C22⋊C4—1912no—
183C42—1911no—
184C42C22×D19111yescharacteristic
185C23—22no—
186C23—3822no—
187C23—1911no—
188D4—22no—
189D4—22no—
190D4—3822no—
191D4—3822no—
192D4—15282no—
193D4—15282no—
194D4—15282no—
195D4—15282no—
196C2×C4—22no—
197C2×C4—82no—
198C2×C4—3821no—
199C2×C4—3821no—
200C2×C4—3822no—
201C2×C4—1912no—
202C2×C4—1911no—
203C2×C4—15282no—
204C2×C4D4×D19112yes—
205C2×C4C23×D19111yescharacteristic
206C8—21no—
207C8—3821no—
208C8—1912no—
209C8#1569028712yes—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—3821no—
214C22—3822no—
215C22—7642no—
216C22—7642no—
217C22—7642no—
218C22C2×D4×D19111yescharacteristic
219C4—22no—
220C4—42no—
221C4—3821no—
222C4—3822no—
223C4—7642no—
224C4#41×D19111yescharacteristic
225C4C2×#1569028711yescharacteristic
226C2—82no—
227C2—15282no—
228C2—7641no—
229C2#2713321011yescharacteristic
230C2#2713346712yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#442×C191C2
socleC22×C191C22×D4
hypercenter#442D191
hypocenterC191#3176
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #27133264C2semidirect
2. #27133063C2semidirect
3. #27133038C2semidirect
4. #27133026C2semidirect
5. #27132917C2semidirect
6. #27132913C2semidirect
7. #27132898C2semidirect
8. #27132776C2semidirect
9. C4×#15690272C2semidirect
10. #27132620C2semidirect
11. #442×C191C2semidirect
115. C191#3176semidirect
125. #442D191semidirect

Maximal subgroups

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

subgroupindex
#271332642
#271330632
#271330382
#271330262
#271329172
#271329132
#271328982
#271327762
C4×#156902722
#271326202
#442×C1912
#3176191