Finite Group Databasebeta

← #37583264

Subgroups — #37583264

Order 24448.

358 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 121 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#27133218C211yesmaximal, characteristic
2#27133218C211yesmaximal, characteristic
3#27133059C211yesmaximal, characteristic, complemented
4#27133042C211yesmaximal, characteristic, complemented
5#27133014C211yesmaximal, characteristic, complemented
6#27132920C214yesmaximal
7#27132767C214yesmaximal, complemented
8C4×#15690270C211yesmaximal, characteristic
9#487×C191C211yesmaximal, characteristic, complemented
10C2×#15690277C2214yes—
11C2×#15690270C2212yes—
12#20259322C2211yescharacteristic
13#20259322C2211yescharacteristic
14#20259313—21no—
15#20259313—21no—
16#20259302—21no—
17#20259302C2212yes—
18#20259298—21no—
19#20259298C2212yes—
20Q8×Dic191—21no—
21Q8×Dic191C2212yes—
22#20259287—21no—
23#20259287C2212yes—
24#20259280—21no—
25#20259280—21no—
26C42×D191C2211yescharacteristic
27#20259253C2214yes—
28#20259247C2214yes—
29#20259232C2212yes—
30C4×C191⋊C8C2211yescharacteristic
31C4⋊Q8×C191C2211yescharacteristic
32C4⋊Q8×C191C2211yescharacteristic
33#103×C191C2214yes—
34C4×C8×C191C2211yescharacteristic
35C2×Dic382—21no—
36C2×Dic382C2312yes—
37C2×C4×D191D411yescharacteristic
38C2×C4×D191D411yescharacteristic
39C2×C4×D191C2311yescharacteristic
40#15690277—28no—
41#15690270—24no—
42C2×C191⋊C8C2312yes—
43C4⋊Dic191—22no—
44C4⋊Dic191—22no—
45C4⋊Dic191—21no—
46C4⋊Dic191—21no—
47C4⋊Dic191—21no—
48C4⋊Dic191C2312yes—
49Dic191⋊C4—22no—
50Dic191⋊C4—22no—
51(C2×D191)⋊C4—22no—
52(C2×D191)⋊C4—22no—
53C4×Dic191—22no—
54C4×Dic191—22no—
55C4×Dic191D411yescharacteristic
56C4×Dic191D411yescharacteristic
57C4×Dic191C2311yescharacteristic
58C2×Q8×C191—21no—
59C2×Q8×C191C2312yes—
60C2×C8×C191C2312yes—
61C4⋊C4×C191—21no—
62C4⋊C4×C191—21no—
63C4⋊C4×C191—21no—
64C4⋊C4×C191C2312yes—
65C42×C191C2311yescharacteristic
66C22×D191C2×D411yescharacteristic
67Dic382—24no—
68Dic382—42no—
69C2×Dic191—22no—
70C2×Dic191—22no—
71C2×Dic191C2×D411yescharacteristic
72C2×Dic191C2×D411yescharacteristic
73C2×Dic191C2×D411yescharacteristic
74C4×D191—22no—
75C4×D191—22no—
76C4×D191—22no—
77C191⋊C8#4114yes—
78Q8×C191—24no—
79Q8×C191—42no—
80C2×C4×C191—22no—
81C2×C4×C191—22no—
82C2×C4×C191C2×D411yescharacteristic
83C2×C4×C191C2×D411yescharacteristic
84C2×C4×C191C2411yescharacteristic
85C8×C191—22no—
86C22×C191C22×D411yescharacteristic
87C2×D191—21no—
88C2×D191—21no—
89C2×D191—21no—
90C4×C191—42no—
91C4×C191—42no—
92C4×C191#13712yes—
93C4×C191#13712yes—
94C4×C191C2×#4112yes—
95Dic191—21no—
96Dic191—21no—
97Dic191—21no—
98Dic191—42no—
99Dic191—42no—
100C2×C191#53211yescharacteristic
101C2×C191C2×#13711yescharacteristic
102C2×C191C2×#13711yescharacteristic
103D191—41no—
104C191#318511yescharacteristic, complemented, Sylow
105#3185—1911nomaximal, Sylow
106#533—1911no—
107#533—1911no—
108#498—1911no—
109#494—1911no—
110#487—1911no—
111#487D19111yescharacteristic, complemented
112#470—1914no—
113#441—1914no—
114C4×M16—1911no—
115C2×Q16—1914no—
116C2×M16—1912no—
117C4⋊Q8—1911no—
118C4⋊Q8—1911no—
119C4⋊Q8C2×D19111yescharacteristic
120C4⋊Q8C2×D19111yescharacteristic
121#125—3821no—
122#125—3821no—
123C22⋊Q8—3821no—
124C22⋊Q8—3821no—
125C22⋊Q8—1912no—
126C22⋊Q8—1912no—
127C4×Q8—3821no—
128C4×Q8—3821no—
129C4×Q8—1912no—
130C4×Q8—1912no—
131#117—3821no—
132#117—3821no—
133C2×C42—1911no—
134#106—1914no—
135#103—1914no—
136#103C2×D19114yes—
137#97—1912no—
138C4×C8—1911no—
139C4×C8C2×D19111yescharacteristic
140C2×Q8—21no—
141C2×Q8—3821no—
142C2×Q8—1912no—
143C2×Q8C22×D19112yes—
144C22×C4—1911no—
145C22×C4—1911no—
146C22×C4—1911no—
147Q16—3828no—
148M16—3824no—
149C2×C8—1912no—
150C2×C8C22×D19112yes—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—3822no—
155C4⋊C4—3822no—
156C4⋊C4—3822no—
157C4⋊C4—3822no—
158C4⋊C4—3821no—
159C4⋊C4—3821no—
160C4⋊C4—3821no—
161C4⋊C4—1912no—
162C4⋊C4C22×D19112yes—
163C22⋊C4—3822no—
164C22⋊C4—3822no—
165C42—3822no—
166C42—3822no—
167C42—1911no—
168C42—1911no—
169C42—1911no—
170C42C22×D19111yescharacteristic
171C23—1911no—
172Q8—24no—
173Q8—42no—
174Q8—3824no—
175Q8—7642no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—3822no—
179C2×C4—3822no—
180C2×C4—3822no—
181C2×C4—3822no—
182C2×C4—3822no—
183C2×C4—1911no—
184C2×C4—1911no—
185C2×C4—1911no—
186C2×C4D4×D19111yescharacteristic
187C2×C4D4×D19111yescharacteristic
188C2×C4C23×D19111yescharacteristic
189C8—22no—
190C8—1914no—
191C22—3821no—
192C22—3821no—
193C22—3821no—
194C22C2×D4×D19111yescharacteristic
195C4—42no—
196C4—42no—
197C4—3821no—
198C4—3821no—
199C4—3821no—
200C4—7642no—
201C4—7642no—
202C4#2025935112yes—
203C4#2025935412yes—
204C4#41×D19112yes—
205C2—7641no—
206C2#2713321211yescharacteristic
207C2C2×#2025935111yescharacteristic
208C2C2×#2025935411yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#487×C191C2
socleC22×C191C22×D4
prime radicalC22×D191C2×D4
hypercenter#487D191
hypocenterC191#3185
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #27133059C2semidirect
4. #27133042C2semidirect
5. #27133014C2semidirect
7. #27132767C2semidirect
9. #487×C191C2semidirect
104. C191#3185semidirect
111. #487D191semidirect

Maximal subgroups

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

subgroupindex
#271332182
#271330592
#271330422
#271330142
#271329202
#271327672
C4×#156902702
#487×C1912
#3185191