Finite Group Databasebeta

← #37571475

Subgroups — #37571475

Order 24448.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#27133288C211yesmaximal, characteristic
2#27132819C211yesmaximal, characteristic, complemented
3#27132574C211yesmaximal, characteristic
4#27132401C211yesmaximal, characteristic, complemented
5#27132400C211yesmaximal, characteristic, complemented
6#27132386C211yesmaximal, characteristic
7#451×C191C211yesmaximal, characteristic, complemented
8C2×#15690277—21no—
9#20259302—21no—
10#20259301C2211yescharacteristic
11#20259300—21no—
12#20259300—21no—
13#20259299—21no—
14#20259298—21no—
15C2×(C2×D191)⋊C4—21no—
16C2×(C2×D191)⋊C4C2211yescharacteristic
17C22⋊C4×D191—21no—
18#20259246—21no—
19C22⋊(C191⋊C8)C2211yescharacteristic
20C191⋊#95—21no—
21C191⋊#95C2211yescharacteristic
22C22×Q8×C191C2211yescharacteristic
23C2×Q16×C191—21no—
24C22⋊Q8×C191C2211yescharacteristic
25#103×C191—21no—
26C22⋊C8×C191C2211yescharacteristic
27C23×D191D411yescharacteristic
28C2×Dic382—21no—
29C2×Dic382D411yescharacteristic
30C22×Dic191—21no—
31C22×Dic191D411yescharacteristic
32C2×C4×D191—21no—
33C2×C4×D191—21no—
34#15690277—42no—
35C2×C191⋊C8—21no—
36C4⋊Dic191—21no—
37C4⋊Dic191D411yescharacteristic
38Dic191⋊C4—21no—
39Dic191⋊C4—21no—
40Dic191⋊C4—21no—
41C22⋊Dic191—21no—
42(C2×D191)⋊C4—21no—
43(C2×D191)⋊C4—21no—
44(C2×D191)⋊C4—21no—
45(C2×D191)⋊C4—21no—
46(C2×D191)⋊C4—42no—
47C2×Q8×C191—21no—
48C2×Q8×C191—21no—
49C2×Q8×C191—42no—
50C2×Q8×C191D411yescharacteristic
51C22×C4×C191—21no—
52C22×C4×C191C2311yescharacteristic
53Q16×C191—42no—
54C2×C8×C191—21no—
55C4⋊C4×C191—21no—
56C4⋊C4×C191D411yescharacteristic
57C22⋊C4×C191—21no—
58C22×D191—21no—
59C22×D191—21no—
60C22×D191—41no—
61C22×D191—41no—
62C22×D191Q1612yescomplemented
63Dic382—41no—
64Dic382—81no—
65C2×Dic191—21no—
66C2×Dic191—21no—
67C2×Dic191—21no—
68C2×Dic191—41no—
69C2×Dic191—41no—
70C2×Dic191—41no—
71C2×Dic191—41no—
72C4×D191—81no—
73C4×D191—81no—
74C191⋊C8—41no—
75C23×C191C2×D411yescharacteristic
76Q8×C191—22no—
77Q8×C191—22no—
78Q8×C191—41no—
79Q8×C191—41no—
80Q8×C191—41no—
81C2×C4×C191—22no—
82C2×C4×C191—21no—
83C2×C4×C191—21no—
84C2×C4×C191—41no—
85C2×C4×C191—41no—
86C2×C4×C191—41no—
87C2×C4×C191C2×D411yescharacteristic
88C2×C4×C191C2×D411yescharacteristic
89C8×C191—41no—
90C22×C191—21no—
91C22×C191—21no—
92C22×C191#12011yescharacteristic
93C22×C191C2×Q1611yescharacteristic
94C22×C191#13711yescharacteristic
95C2×D191—22no—
96C2×D191—41no—
97C2×D191—41no—
98C2×D191—42no—
99C2×D191—81no—
100C4×C191—21no—
101C4×C191—21no—
102C4×C191—41no—
103C4×C191—42no—
104C4×C191—41no—
105Dic191—41no—
106Dic191—41no—
107Dic191—41no—
108Dic191—81no—
109C2×C191—21no—
110C2×C191—21no—
111C2×C191#45111yescharacteristic
112C2×C191#45511yescharacteristic
113C2×C191B(4,𝔽2)11yescharacteristic
114D191—42no—
115C191#160211yescharacteristic, complemented, Sylow
116#1602—1911nomaximal, Sylow
117#543—1911no—
118#451—1911no—
119#451D19111yescharacteristic, complemented
120#393—1911no—
121#328—1911no—
122#328—1911no—
123#323—1911no—
124C22×Q8C2×D19111yescharacteristic
125C2×Q16—21no—
126C2×Q16—3821no—
127C22⋊Q8—3821no—
128C22⋊Q8—3821no—
129C22⋊Q8—3821no—
130C22⋊Q8—3821no—
131C22⋊Q8—3821no—
132C22⋊Q8—1911no—
133C22⋊Q8C2×D19111yescharacteristic
134C2×C22⋊C4—3821no—
135C2×C22⋊C4—3821no—
136C2×C22⋊C4—1911no—
137#103—21no—
138#103—3821no—
139C22⋊C8—1911no—
140C22⋊C8C2×D19111yescharacteristic
141#95—3821no—
142#95—1911no—
143C24—1911no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8—3821no—
148C2×Q8—1911no—
149C2×Q8C22⋊D19111yescharacteristic
150C22×C4—21no—
151C22×C4—3821no—
152C22×C4—3821no—
153C22×C4—3821no—
154C22×C4—1911no—
155C22×C4C22×D19111yescharacteristic
156Q16—42no—
157Q16—7642no—
158C2×C8—21no—
159C2×C8—3821no—
160C4⋊C4—21no—
161C4⋊C4—3821no—
162C4⋊C4—3821no—
163C4⋊C4—3821no—
164C4⋊C4—3821no—
165C4⋊C4—1911no—
166C4⋊C4C22⋊D19111yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—3821no—
169C22⋊C4—3821no—
170C22⋊C4—3821no—
171C22⋊C4—3821no—
172C22⋊C4—3821no—
173C22⋊C4—7642no—
174C23—3821no—
175C23—3821no—
176C23—1912no—
177C23—7641no—
178C23—7641no—
179C23D4×D19111yescharacteristic
180Q8—22no—
181Q8—22no—
182Q8—41no—
183Q8—41no—
184Q8—41no—
185Q8—15281no—
186Q8—7641no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—3821no—
194C2×C4—3821no—
195C2×C4—3821no—
196C2×C4—15281no—
197C2×C4—15281no—
198C2×C4—7641no—
199C2×C4—7641no—
200C2×C4—7641no—
201C2×C4—7641no—
202C2×C4D4×D19111yescharacteristic
203C2×C4C2×C22⋊D19111yescharacteristic
204C8—41no—
205C8—7641no—
206C22—21no—
207C22—21no—
208C22—3822no—
209C22—15281no—
210C22—7642no—
211C22—7641no—
212C22—7641no—
213C22#2025929011yescharacteristic
214C22Q16×D19111yescharacteristic
215C22#2025935411yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—15281no—
222C4—7641no—
223C4—7641no—
224C4—7641no—
225C2—21no—
226C2—21no—
227C2—7642no—
228C2#2713281711yescharacteristic
229C2#2713283911yescharacteristic
230C2#2713284911yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#451×C191C2
socleC22×C191#120
prime radicalC23×D191D4
hypercenter#451D191
hypocenterC191#1602
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #27132819C2semidirect
4. #27132401C2semidirect
5. #27132400C2semidirect
7. #451×C191C2semidirect
62. C22×D191Q16semidirect
115. C191#1602semidirect
119. #451D191semidirect

Maximal subgroups

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

subgroupindex
#271332882
#271328192
#271325742
#271324012
#271324002
#271323862
#451×C1912
#1602191