Finite Group Databasebeta

← #37573051

Subgroups — #37573051

Order 24448.

362 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 79 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#27133232C211yesmaximal, characteristic
2#27133137C211yesmaximal, characteristic, complemented
3#27132681C211yesmaximal, characteristic, complemented
4#27132679C211yesmaximal, characteristic, complemented
5C2×#20259238C211yesmaximal, characteristic, complemented
6#27132419C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C191C211yesmaximal, characteristic, complemented
8C2×#15690287C2211yescharacteristic
9C22×D764C2211yescharacteristic
10C2×#15690271C2211yescharacteristic
11#20259319C412yescomplemented
12#20259310C412yescomplemented
13#20259297—22no—
14#20259292—22no—
15C4×D764—22no—
16#20259284—22no—
17#20259281C2211yescharacteristic
18C22⋊C4×D191—22no—
19#20259250—22no—
20#20259248—22no—
21#20259238—22no—
22C2×M16×C191C2211yescharacteristic
23#117×C191C2211yescharacteristic
24#117×C191C2211yescharacteristic
25C23×D191D412yes—
26#15690287—21no—
27#15690287—21no—
28#15690287—21no—
29#15690287—21no—
30C2×Dic382C2×C411yescharacteristic, complemented
31C2×D764—24no—
32C2×D764—21no—
33C2×D764—21no—
34C2×D764—21no—
35C2×D764—21no—
36C2×D764C2×C412yes—
37C2×D764C2×C411yescharacteristic, complemented
38C2×D764D412yes—
39C2×C22⋊D191—21no—
40C2×C4×D191—24no—
41C2×C4×D191—21no—
42#15690271—21no—
43#15690271—21no—
44C2×C191⋊C8—21no—
45C4⋊Dic191—21no—
46C22⋊Dic191—21no—
47(C2×D191)⋊C4—24no—
48(C2×D191)⋊C4—24no—
49C4×Dic191D412yescomplemented
50C22×C4×C191C2311yescharacteristic
51M16×C191—21no—
52M16×C191—21no—
53C2×C8×C191—21no—
54C4⋊C4×C191—21no—
55C4⋊C4×C191—21no—
56C22⋊C4×C191—21no—
57C22⋊C4×C191—21no—
58C42×C191C2×C412yescomplemented
59C42×C191D412yescomplemented
60C22×D191—22no—
61C22×D191—22no—
62C22×D191—21no—
63C22×D191—82no—
64C22×D191C22⋊C414yes—
65Dic382—21no—
66Dic382—21no—
67D764—22no—
68D764—22no—
69D764—21no—
70D764—21no—
71D764—22no—
72D764—22no—
73D764—84no—
74C22⋊D191—41no—
75C22⋊D191—41no—
76C2×Dic191—21no—
77C2×Dic191—22no—
78C4×D191—41no—
79C4×D191—41no—
80C4×D191—84no—
81C191⋊C8—41no—
82C23×C191C2×D411yescharacteristic
83C2×C4×C191—22no—
84C2×C4×C191—22no—
85C2×C4×C191C22×C411yescharacteristic
86C2×C4×C191C2×D411yescharacteristic
87C2×C4×C191C2×D411yescharacteristic
88C2×C4×C191C2×D411yescharacteristic
89C2×C4×C191#4111yescharacteristic
90C2×C4×C191#4111yescharacteristic
91C8×C191—41no—
92C22×C191—41no—
93C22×C191C2×C22⋊C411yescharacteristic
94C22×C191#12011yescharacteristic
95C22×C191#12311yescharacteristic
96C2×D191—24no—
97C2×D191—44no—
98C2×D191—44no—
99C2×D191—42no—
100C2×D191—41no—
101C2×D191—82no—
102C4×C191—42no—
103C4×C191—42no—
104C4×C191C4×D411yescharacteristic
105C4×C191C4×D411yescharacteristic
106C4×C191#12111yescharacteristic
107C4×C191#12111yescharacteristic
108Dic191—42no—
109Dic191—41no—
110C2×C191—21no—
111C2×C191—21no—
112C2×C191—21no—
113C2×C191#38611yescharacteristic
114D191—44no—
115D191—81no—
116C191#191111yescharacteristic, complemented, Sylow
117#1911—1911nomaximal, Sylow
118#535—1911no—
119#518—1911no—
120#421—1911no—
121#421—1911no—
122C2×#100—1911no—
123H(3,ℤ/4)—1911no—
124H(3,ℤ/4)D19111yescharacteristic, complemented
125C2×#41—1911no—
126C22×D4—1911no—
127C2×M16—1911no—
128C2×M16C2×D19111yescharacteristic
129#127—1912no—
130#124—1912no—
131#121—3822no—
132#120—3822no—
133C4×D4—3822no—
134C4×D4—3822no—
135#117—1911no—
136#117C2×D19111yescharacteristic
137#117C2×D19111yescharacteristic
138C2×C22⋊C4—3822no—
139#104—3822no—
140#104—3822no—
141#100—3822no—
142C24—1912no—
143#41—3821no—
144#41—3821no—
145#41—3821no—
146#41—3821no—
147C2×Q8—1911no—
148C2×D4—3824no—
149C2×D4—3821no—
150C2×D4—3821no—
151C2×D4—3821no—
152C2×D4—3821no—
153C2×D4—3821no—
154C2×D4—1911no—
155C2×D4—1912no—
156C2×D4—1912no—
157C22×C4—3824no—
158C22×C4—3821no—
159C22×C4C22×D19111yescharacteristic
160M16—21no—
161M16—21no—
162M16—3821no—
163M16—3821no—
164C2×C8—21no—
165C2×C8—3821no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—3821no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—3824no—
172C22⋊C4—3824no—
173C22⋊C4—3821no—
174C42—1912no—
175C42C4×D19112yescomplemented
176C42D76412yescomplemented
177C23—3822no—
178C23—3822no—
179C23—3821no—
180C23—1914no—
181C23—15282no—
182C23D4×D19111yescharacteristic
183Q8—3821no—
184Q8—3821no—
185D4—3822no—
186D4—3821no—
187D4—3822no—
188D4—3821no—
189D4—3822no—
190D4—3822no—
191D4—15284no—
192D4—7641no—
193D4—7641no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—3821no—
197C2×C4—3822no—
198C2×C4—15284no—
199C2×C4—7641no—
200C2×C4—7641no—
201C2×C4C2×C4×D19111yescharacteristic
202C2×C4D4×D19111yescharacteristic
203C2×C4D4×D19111yescharacteristic
204C2×C4C2×D76411yescharacteristic
205C2×C4#1569028711yescharacteristic
206C2×C4#1569028811yescharacteristic
207C8—41no—
208C8—7641no—
209C22—41no—
210C22—3824no—
211C22—15282no—
212C22—7642no—
213C22—7644no—
214C22—7644no—
215C22—7641no—
216C22C22⋊C4×D19111yescharacteristic
217C22#2025929211yescharacteristic
218C22#2025930711yescharacteristic
219C4—42no—
220C4—42no—
221C4—7642no—
222C4—7641no—
223C4#2025928411yescharacteristic
224C4C4×D76411yescharacteristic
225C4#2025929511yescharacteristic
226C4#2025929711yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—15281no—
231C2—7644no—
232C2#2713254111yescharacteristic
subgroupG/H
centerC2#27132541
commutatorC2×C4×C191C22×C4
FrattiniC22×C4C22×D191
FittingH(3,ℤ/4)×C191C2
socleC2×C191#386
prime radical#27133232C2
cosocleC22×C4×C191C23
hypercenterH(3,ℤ/4)D191
hypocenterC191#1911
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #27133137C2semidirect
3. #27132681C2semidirect
4. #27132679C2semidirect
5. C2×#20259238C2semidirect
6. #27132419C2semidirect
7. H(3,ℤ/4)×C191C2semidirect
11. #20259319C4semidirect
12. #20259310C4semidirect
30. C2×Dic382C2×C4semidirect
37. C2×D764C2×C4semidirect
49. C4×Dic191D4semidirect
58. C42×C191C2×C4semidirect
59. C42×C191D4semidirect
116. C191#1911semidirect
124. H(3,ℤ/4)D191semidirect
175. C42C4×D191semidirect
176. C42D764semidirect

Maximal subgroups

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

subgroupindex
#271332322
#271331372
#271326812
#271326792
C2×#202592382
#271324192
H(3,ℤ/4)×C1912
#1911191