Finite Group Databasebeta

← #37572552

Subgroups — #37572552

Order 24448.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#15690270C211yesmaximal, characteristic
2C2×#20259255C211yesmaximal, characteristic, complemented
3C2×#20259253C211yesmaximal, characteristic, complemented
4#27132520C211yesmaximal, characteristic
5#27132520C211yesmaximal, characteristic
6#27132426C211yesmaximal, characteristic, complemented
7#340×C191C211yesmaximal, characteristic, complemented
8C22×C4×D191C2211yescharacteristic
9C2×#15690270—22no—
10C2×#15690270—22no—
11C2×#15690270C414yescomplemented
12C22×C191⋊C8C2211yescharacteristic
13C2×C4⋊Dic191C2211yescharacteristic
14C2×C4⋊Dic191C2211yescharacteristic
15C2×(C2×D191)⋊C4—21no—
16C2×(C2×D191)⋊C4—21no—
17#20259255—22no—
18#20259253—22no—
19C191⋊#95—21no—
20C191⋊#95—21no—
21C22×C8×C191C2211yescharacteristic
22C2×C4⋊C4×C191C2211yescharacteristic
23C2×C4⋊C4×C191C2211yescharacteristic
24C23×D191D411yescharacteristic
25C22×Dic191—21no—
26C22×Dic191—21no—
27C22×Dic191D411yescharacteristic
28C2×C4×D191—21no—
29C2×C4×D191—21no—
30C2×C4×D191—21no—
31C2×C4×D191—21no—
32C2×C4×D191C2×C412yes—
33C2×C4×D191Q812yes—
34#15690270—28no—
35C2×C191⋊C8C2×C412yes—
36C2×C191⋊C8D412yes—
37C2×C191⋊C8D412yes—
38C4⋊Dic191—21no—
39C4⋊Dic191—21no—
40C4⋊Dic191—21no—
41C4⋊Dic191—21no—
42(C2×D191)⋊C4—42no—
43(C2×D191)⋊C4—42no—
44C22×C4×C191—21no—
45C22×C4×C191—21no—
46C22×C4×C191C2311yescharacteristic
47C2×C8×C191—21no—
48C2×C8×C191—21no—
49C2×C8×C191C2×C412yes—
50C4⋊C4×C191—21no—
51C4⋊C4×C191—21no—
52C4⋊C4×C191—21no—
53C4⋊C4×C191—21no—
54C22×D191—21no—
55C22×D191—21no—
56C22×D191—21no—
57C22×D191—21no—
58C22×D191—21no—
59C22×D191—21no—
60C22×D191C4⋊C412yes—
61C2×Dic191—21no—
62C2×Dic191—21no—
63C2×Dic191—41no—
64C2×Dic191—41no—
65C2×Dic191—41no—
66C2×Dic191—41no—
67C2×Dic191—41no—
68C2×Dic191—41no—
69C2×Dic191C4⋊C412yes—
70C4×D191—22no—
71C4×D191—22no—
72C4×D191—22no—
73C4×D191—22no—
74C191⋊C8C22⋊C414yescomplemented
75C23×C191C2×D411yescharacteristic
76C2×C4×C191—41no—
77C2×C4×C191—41no—
78C2×C4×C191—41no—
79C2×C4×C191—41no—
80C2×C4×C191—41no—
81C2×C4×C191—41no—
82C2×C4×C191C22×C411yescharacteristic
83C2×C4×C191C2×D411yescharacteristic
84C2×C4×C191C2×D411yescharacteristic
85C2×C4×C191C2×Q811yescharacteristic
86C2×C4×C191#4111yescharacteristic
87C2×C4×C191#4111yescharacteristic
88C8×C191—22no—
89C22×C191C2×C4⋊C411yescharacteristic
90C22×C191#12111yescharacteristic
91C22×C191#12111yescharacteristic
92C22×C191AGL(1,ℤ/8)11yescharacteristic
93C22×C191AGL(1,ℤ/8)11yescharacteristic
94C22×C191#13711yescharacteristic
95C22×C191#13711yescharacteristic
96C2×D191—22no—
97C2×D191—22no—
98C2×D191—22no—
99C2×D191—41no—
100C2×D191—41no—
101C2×D191—41no—
102C2×D191—41no—
103C4×C191—81no—
104C4×C191—81no—
105C4×C191C2×C22⋊C411yescharacteristic
106C4×C191#11711yescharacteristic
107C4×C191C22⋊Q811yescharacteristic
108C4×C191C22⋊Q811yescharacteristic
109Dic191—22no—
110Dic191—81no—
111Dic191—81no—
112C2×C191#38011yescharacteristic
113C2×C191#42811yescharacteristic
114C2×C191#42811yescharacteristic
115C2×C191#46811yescharacteristic
116C2×C191#46811yescharacteristic
117C2×C191#46911yescharacteristic
118C2×C191#47011yescharacteristic
119D191—42no—
120C191#180911yescharacteristic, complemented, Sylow
121#1809—1911nomaximal, Sylow
122C22×M16—1911no—
123C2×#107—1911no—
124C2×#106—1911no—
125#380—1911no—
126#380—1911no—
127#340—1911no—
128#340D19111yescharacteristic, complemented
129C23×C4—1911no—
130C2×M16—3822no—
131C2×M16—3822no—
132C2×M16—1914no—
133C22×C8—1911no—
134C22×C8C2×D19111yescharacteristic
135C2×C4⋊C4—1911no—
136C2×C4⋊C4—1911no—
137C2×C4⋊C4C2×D19111yescharacteristic
138C2×C4⋊C4C2×D19111yescharacteristic
139C2×C22⋊C4—3821no—
140C2×C22⋊C4—3821no—
141#107—3822no—
142#106—3822no—
143#95—3821no—
144#95—3821no—
145C24—1911no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—3821no—
149C22×C4—3821no—
150C22×C4—3821no—
151C22×C4—3821no—
152C22×C4—3821no—
153C22×C4—3821no—
154C22×C4—1911no—
155C22×C4—1912no—
156C22×C4—1912no—
157C22×C4C22×D19111yescharacteristic
158M16—3828no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—1912no—
162C2×C8—1912no—
163C2×C8—1912no—
164C2×C8C4×D19112yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—3821no—
170C4⋊C4—3821no—
171C4⋊C4—3821no—
172C4⋊C4—3821no—
173C22⋊C4—7642no—
174C22⋊C4—7642no—
175C23—3821no—
176C23—3821no—
177C23—3821no—
178C23—3821no—
179C23—3821no—
180C23—3821no—
181C23—1912no—
182C23D4×D19111yescharacteristic
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—3821no—
190C2×C4—3821no—
191C2×C4—3822no—
192C2×C4—3822no—
193C2×C4—3822no—
194C2×C4—3822no—
195C2×C4—1912no—
196C2×C4—7641no—
197C2×C4—7641no—
198C2×C4—7641no—
199C2×C4—7641no—
200C2×C4—7641no—
201C2×C4—7641no—
202C2×C4C2×C4×D19111yescharacteristic
203C2×C4D4×D19111yescharacteristic
204C2×C4D4×D19111yescharacteristic
205C2×C4Q8×D19111yescharacteristic
206C2×C4#1569028711yescharacteristic
207C2×C4#1569028711yescharacteristic
208C8—22no—
209C8—1914no—
210C22—3822no—
211C22—3822no—
212C22—3822no—
213C22—7641no—
214C22—7641no—
215C22—7641no—
216C22—7641no—
217C22C4⋊C4×D19111yescharacteristic
218C22#2025929511yescharacteristic
219C22#2025929511yescharacteristic
220C22#2025934611yescharacteristic
221C22#2025934911yescharacteristic
222C22#2025935111yescharacteristic
223C22#2025935411yescharacteristic
224C4—81no—
225C4—81no—
226C4—3822no—
227C4—15281no—
228C4—15281no—
229C4C22⋊C4×D19111yescharacteristic
230C4#2025927911yescharacteristic
231C4#2025930011yescharacteristic
232C4#2025930011yescharacteristic
233C2—7642no—
234C2#2713252211yescharacteristic
235C2#2713270211yescharacteristic
236C2#2713270411yescharacteristic
237C2#2713290811yescharacteristic
238C2#2713291211yescharacteristic
239C2#2713291511yescharacteristic
240C2#2713292011yescharacteristic
subgroupG/H
centerC23D4×D191
commutatorC2×C4×C191C22×C4
FrattiniC22×C4C22×D191
Fitting#340×C191C2
socleC23×C191C2×D4
prime radicalC23×D191D4
cosocleC22×C4×C191C23
hypercenter#340D191
hypocenterC191#1809
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#20259255C2semidirect
3. C2×#20259253C2semidirect
6. #27132426C2semidirect
7. #340×C191C2semidirect
11. C2×#15690270C4semidirect
74. C191⋊C8C22⋊C4semidirect
120. C191#1809semidirect
128. #340D191semidirect

Maximal subgroups

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

subgroupindex
C22×#156902702
C2×#202592552
C2×#202592532
#271325202
#271324262
#340×C1912
#1809191