Finite Group Databasebeta

← #37574509

Subgroups — #37574509

Order 24448.

258 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 41 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#27133343C211yesmaximal, characteristic, complemented
2#27132853C211yesmaximal, characteristic, complemented
3#27132850C211yesmaximal, characteristic
4#27132458C211yesmaximal, characteristic, complemented
5#27132454C211yesmaximal, characteristic, complemented
6#27132451C211yesmaximal, characteristic, complemented
7#455×C191C211yesmaximal, characteristic, complemented
8#20259315—21no—
9#20259304—21no—
10#20259303—21no—
11#20259303C2211yescharacteristic
12#20259298—21no—
13#20259290—21no—
14#20259290C2211yescharacteristic
15(C22⋊D191)⋊C4—21no—
16#20259279—21no—
17C22⋊C4×D191—21no—
18#20259237—21no—
19#20259237—21no—
20#20259237C2211yescharacteristic
21#20259237C2211yescharacteristic
22H(5,𝔽2)×C191C2211yescharacteristic
23#137×C191—21no—
24#124×C191C2211yescharacteristic
25#104×C191—21no—
26#100×C191C2211yescharacteristic
27C23×D191D411yescharacteristic
28C2×C22⋊D191—21no—
29C2×C22⋊D191—41no—
30C22×Dic191D411yescharacteristic, complemented
31C2×C4×D191—21no—
32C2×C4×D191—21no—
33C4⋊Dic191—21no—
34Dic191⋊C4—21no—
35Dic191⋊C4—21no—
36Dic191⋊C4—21no—
37Dic191⋊C4—41no—
38C22⋊Dic191—21no—
39C22⋊Dic191—21no—
40C22⋊Dic191—21no—
41C22⋊Dic191—41no—
42C22⋊Dic191D411yescharacteristic
43C22⋊Dic191D411yescharacteristic, complemented
44(C2×D191)⋊C4—21no—
45(C2×D191)⋊C4—21no—
46(C2×D191)⋊C4—21no—
47(C2×D191)⋊C4—41no—
48C4×Dic191—21no—
49#41×C191—21no—
50#41×C191—41no—
51C2×Q8×C191D411yescharacteristic, complemented
52C2×D4×C191—21no—
53C2×D4×C191—21no—
54C2×D4×C191—41no—
55C2×D4×C191C2311yescharacteristic
56Q16×C191—41no—
57SD16×C191—41no—
58M16×C191—21no—
59C22⋊C4×C191—21no—
60C22⋊C4×C191—21no—
61C42×C191D411yescharacteristic, complemented
62C22×D191—21no—
63C22×D191—42no—
64C22×D191—41no—
65C22⋊D191—42no—
66C22⋊D191—82no—
67C2×Dic191—21no—
68C2×Dic191—21no—
69C2×Dic191—21no—
70C2×Dic191—41no—
71C2×Dic191—41no—
72C2×Dic191—41no—
73C4×D191—42no—
74C4×D191—81no—
75C23×C191—21no—
76C23×C191—21no—
77C23×C191C2×D411yescharacteristic
78C23×C191C2×D411yescharacteristic
79Q8×C191—21no—
80Q8×C191—41no—
81D4×C191—21no—
82D4×C191—41no—
83D4×C191—41no—
84D4×C191—41no—
85D4×C191—41no—
86C2×C4×C191—21no—
87C2×C4×C191—21no—
88C2×C4×C191—21no—
89C2×C4×C191—21no—
90C2×C4×C191—41no—
91C2×C4×C191C2×D411yescharacteristic
92C8×C191—41no—
93C22×C191—21no—
94C22×C191—21no—
95C22×C191—41no—
96C22×C191—41no—
97C22×C191—41no—
98C22×C191—81no—
99C22×C191—81no—
100C22×C191#12011yescharacteristic
101C2×D191—22no—
102C2×D191—41no—
103C2×D191—41no—
104C2×D191—42no—
105C2×D191—81no—
106C4×C191—21no—
107C4×C191—41no—
108C4×C191—41no—
109C4×C191—41no—
110Dic191—41no—
111Dic191—41no—
112Dic191—41no—
113Dic191—81no—
114Dic191—81no—
115C2×C191—21no—
116C2×C191—41no—
117C2×C191—41no—
118C2×C191—81no—
119C2×C191B(4,𝔽2)11yescharacteristic
120D191—42no—
121C191#219711yescharacteristic, complemented, Sylow
122#2197—1911nomaximal, Sylow
123#551—1911no—
124#458—1911no—
125B(4,𝔽2)—1911no—
126#455D19111yescharacteristic, complemented
127#354—1911no—
128#352—1911no—
129AΣL(1,𝔽16)—1911no—
130H(5,𝔽2)C2×D19111yescharacteristic
131#137—21no—
132#126—3821no—
133#124C2×D19111yescharacteristic
134#123—3821no—
135#123—3821no—
136#123—1911no—
137C22⋊Q8—3821no—
138#120—3821no—
139#120—1911no—
140C4×D4—3821no—
141#117—3821no—
142C2×C22⋊C4—3821no—
143#104—21no—
144#100C2×D19111yescharacteristic
145#99—3821no—
146#99—3821no—
147#99—1911no—
148#99—1911no—
149C24—1911no—
150#41—21no—
151#41—41no—
152C2×Q8C22⋊D19111yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—3821no—
157C2×D4—7641no—
158C2×D4C22×D19111yescharacteristic
159C22×C4—3821no—
160C22×C4—3821no—
161C22×C4—1911no—
162Q16—41no—
163SD16—41no—
164M16—21no—
165C4⋊C4—3821no—
166C4⋊C4—3821no—
167C4⋊C4—3821no—
168C4⋊C4—3821no—
169C4⋊C4—7641no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—3821no—
173C22⋊C4—3821no—
174C22⋊C4—3821no—
175C22⋊C4—3821no—
176C22⋊C4—3821no—
177C22⋊C4—3821no—
178C22⋊C4—1911no—
179C22⋊C4—1911no—
180C22⋊C4—7641no—
181C22⋊C4—7641no—
182C42—3821no—
183C42C22⋊D19111yescharacteristic, complemented
184C23—21no—
185C23—21no—
186C23—3821no—
187C23—7641no—
188C23—7642no—
189C23D4×D19111yescharacteristic
190C23D4×D19111yescharacteristic
191Q8—21no—
192Q8—41no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—41no—
197D4—41no—
198D4—15282no—
199D4—7642no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—3821no—
206C2×C4—3821no—
207C2×C4—3821no—
208C2×C4—15281no—
209C2×C4—7642no—
210C2×C4—7641no—
211C2×C4—7641no—
212C2×C4—7641no—
213C2×C4C2×C22⋊D19111yescharacteristic
214C8—41no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22—3822no—
223C22—15281no—
224C22—7641no—
225C22—7642no—
226C22—7641no—
227C22#2025929011yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—15281no—
233C4—15281no—
234C4—7641no—
235C4—7641no—
236C4—7641no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—81no—
241C2—7642no—
242C2#2713284911yescharacteristic
subgroupG/H
centerC2#27132849
commutator, cosocleC2×D4×C191C23
FrattiniC2×D4C22×D191
Fitting#455×C191C2
socleC2×C191B(4,𝔽2)
prime radical#27132850C2
hypercenter#455D191
hypocenterC191#2197
LCSG ▷ C2×D4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #27133343C2semidirect
2. #27132853C2semidirect
4. #27132458C2semidirect
5. #27132454C2semidirect
6. #27132451C2semidirect
7. #455×C191C2semidirect
30. C22×Dic191D4semidirect
43. C22⋊Dic191D4semidirect
51. C2×Q8×C191D4semidirect
61. C42×C191D4semidirect
121. C191#2197semidirect
126. #455D191semidirect
152. C2×Q8C22⋊D191semidirect
183. C42C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
#271333432
#271328532
#271328502
#271324582
#271324542
#271324512
#455×C1912
#2197191