Finite Group Databasebeta

← #37571486

Subgroups — #37571486

Order 24448.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#27133262C211yesmaximal, characteristic
2#27132818C211yesmaximal, characteristic, complemented
3#27132804C211yesmaximal, characteristic
4#27132584C211yesmaximal, characteristic, complemented
5#27132396C211yesmaximal, characteristic
6#27132388C211yesmaximal, characteristic, complemented
7#328×C191C211yesmaximal, characteristic, complemented
8C22×Dic382C2211yescharacteristic
9C2×#15690277—21no—
10C2×#15690276—21no—
11#20259317—21no—
12#20259313—21no—
13#20259307—21no—
14#20259306—21no—
15#20259298—21no—
16#20259296C2211yescharacteristic
17(C22⋊D191)⋊C4—21no—
18C2×C4⋊Dic191—21no—
19C2×Dic191⋊C4C2211yescharacteristic
20#20259247—21no—
21#20259245—21no—
22C22⋊(C191⋊C8)C2211yescharacteristic
23C191⋊#95—21no—
24C22⋊Q8×C191C2211yescharacteristic
25C22⋊C8×C191C2211yescharacteristic
26#95×C191C2211yescharacteristic
27C2×Dic382—21no—
28C2×Dic382—21no—
29C2×Dic382—42no—
30C2×D764D411yescharacteristic
31C2×C22⋊D191—21no—
32C22×Dic191—21no—
33C22×Dic191D411yescharacteristic
34C22×Dic191D411yescharacteristic
35C2×C4×D191—21no—
36#15690277—42no—
37#15690276—42no—
38C2×C191⋊C8—21no—
39C4⋊Dic191—21no—
40C4⋊Dic191—21no—
41C4⋊Dic191—42no—
42C4⋊Dic191D411yescharacteristic
43Dic191⋊C4—21no—
44Dic191⋊C4—21no—
45Dic191⋊C4—21no—
46(C2×D191)⋊C4—21no—
47(C2×D191)⋊C4—21no—
48(C2×D191)⋊C4—21no—
49(C2×D191)⋊C4—21no—
50C4×Dic191—21no—
51C2×Q8×C191D411yescharacteristic
52C22×C4×C191—21no—
53C22×C4×C191C2311yescharacteristic
54C2×C8×C191—21no—
55C4⋊C4×C191—21no—
56C4⋊C4×C191D411yescharacteristic
57C22⋊C4×C191—21no—
58C22×D191—21no—
59Dic382—22no—
60Dic382—22no—
61Dic382—41no—
62Dic382—41no—
63D764—41no—
64C22⋊D191—41no—
65C22⋊D191—41no—
66C2×Dic191—22no—
67C2×Dic191—21no—
68C2×Dic191—21no—
69C2×Dic191—21no—
70C2×Dic191—21no—
71C2×Dic191—21no—
72C2×Dic191—41no—
73C2×Dic191—41no—
74C2×Dic191—41no—
75C4×D191—81no—
76C191⋊C8—41no—
77C23×C191C2×D411yescharacteristic
78Q8×C191—41no—
79C2×C4×C191—22no—
80C2×C4×C191—21no—
81C2×C4×C191—21no—
82C2×C4×C191—41no—
83C2×C4×C191—41no—
84C2×C4×C191—41no—
85C2×C4×C191C2×D411yescharacteristic
86C2×C4×C191C2×D411yescharacteristic
87C8×C191—41no—
88C22×C191—21no—
89C22×C191—21no—
90C22×C191#12011yescharacteristic
91C22×C191#13511yescharacteristic
92C22×C191#13711yescharacteristic
93C2×D191—41no—
94C2×D191—41no—
95C2×D191—41no—
96C4×C191—21no—
97C4×C191—21no—
98C4×C191—42no—
99C4×C191—41no—
100C4×C191—41no—
101Dic191—41no—
102Dic191—41no—
103Dic191—41no—
104Dic191—42no—
105C2×C191—21no—
106C2×C191—21no—
107C2×C191#45211yescharacteristic
108C2×C191#45611yescharacteristic
109C2×C191B(4,𝔽2)11yescharacteristic
110D191—81no—
111C191#160311yescharacteristic, complemented, Sylow
112#1603—1911nomaximal, Sylow
113#539—1911no—
114#451—1911no—
115#448—1911no—
116#395—1911no—
117#328D19111yescharacteristic, complemented
118#327—1911no—
119#324—1911no—
120C22×Q8—1911no—
121C2×Q16—3821no—
122C2×SD16—3821no—
123#126—3821no—
124#125—3821no—
125#123—3821no—
126#123—3821no—
127C22⋊Q8—3821no—
128C22⋊Q8C2×D19111yescharacteristic
129#121—1911no—
130C4×D4—3821no—
131C2×C4⋊C4—3821no—
132C2×C4⋊C4—1911no—
133#103—3821no—
134#103—3821no—
135C22⋊C8—1911no—
136C22⋊C8C2×D19111yescharacteristic
137#95—3821no—
138#95C2×D19111yescharacteristic
139C2×Q8—3821no—
140C2×Q8—3821no—
141C2×Q8—7642no—
142C2×Q8D76411yescharacteristic
143C2×D4—3821no—
144C2×D4—1911no—
145C22×C4—21no—
146C22×C4—3821no—
147C22×C4—3821no—
148C22×C4—1911no—
149C22×C4—1911no—
150C22×C4C22×D19111yescharacteristic
151Q16—7642no—
152SD16—7642no—
153C2×C8—21no—
154C2×C8—3821no—
155C4⋊C4—21no—
156C4⋊C4—3821no—
157C4⋊C4—3821no—
158C4⋊C4—3821no—
159C4⋊C4—3821no—
160C4⋊C4—3821no—
161C4⋊C4—1911no—
162C4⋊C4—7642no—
163C4⋊C4D76411yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—3821no—
166C22⋊C4—3821no—
167C22⋊C4—3821no—
168C22⋊C4—3821no—
169C42—3821no—
170C23—3821no—
171C23D4×D19111yescharacteristic
172Q8—41no—
173Q8—3822no—
174Q8—3822no—
175Q8—7641no—
176Q8—7641no—
177D4—7641no—
178D4—7641no—
179D4—7641no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3822no—
187C2×C4—3821no—
188C2×C4—3821no—
189C2×C4—3821no—
190C2×C4—3821no—
191C2×C4—3821no—
192C2×C4—15281no—
193C2×C4—7641no—
194C2×C4—7641no—
195C2×C4—7641no—
196C2×C4D4×D19111yescharacteristic
197C2×C4C2×D76411yescharacteristic
198C8—41no—
199C8—7641no—
200C22—21no—
201C22—21no—
202C22—7641no—
203C22—7641no—
204C22—7641no—
205C22#2025929211yescharacteristic
206C22#2025934311yescharacteristic
207C22#2025935411yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—41no—
213C4—7641no—
214C4—7641no—
215C4—7641no—
216C4—7642no—
217C2—21no—
218C2—21no—
219C2—15281no—
220C2#2713282811yescharacteristic
221C2#2713284711yescharacteristic
222C2#2713285111yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#328×C191C2
socleC22×C191#120
prime radical#20259296C22
hypercenter#328D191
hypocenterC191#1603
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27132818C2semidirect
4. #27132584C2semidirect
6. #27132388C2semidirect
7. #328×C191C2semidirect
111. C191#1603semidirect
117. #328D191semidirect

Maximal subgroups

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

subgroupindex
#271332622
#271328182
#271328042
#271325842
#271323962
#271323882
#328×C1912
#1603191