Finite Group Databasebeta

← #37571731

Subgroups — #37571731

Order 24448.

208 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 55 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#27133374C211yesmaximal, characteristic
2#27132940C211yesmaximal, characteristic
3#27132870C211yesmaximal, characteristic, complemented
4#27132412C211yesmaximal, characteristic, complemented
5#27132395C211yesmaximal, characteristic, complemented
6#27132392C211yesmaximal, characteristic
7#487×C191C211yesmaximal, characteristic, complemented
8C2×#15690277—21no—
9#20259313—21no—
10#20259302—21no—
11#20259298—22no—
12C4×Dic382C2211yescharacteristic
13C4×D764C2211yescharacteristic
14#20259284—21no—
15C4⋊C4×D191—21no—
16#20259255—21no—
17C4⋊(C191⋊C8)C2211yescharacteristic
18C4⋊(C191⋊C8)C2211yescharacteristic
19#20259246—21no—
20#20259242—21no—
21C4⋊Q8×C191C2211yescharacteristic
22C4⋊Q8×C191C2211yescharacteristic
23#103×C191—22no—
24C4×C8×C191C2211yescharacteristic
25C2×Dic382D411yescharacteristic
26C2×D764D411yescharacteristic
27C2×C4×D191—22no—
28C2×C4×D191—21no—
29#15690277—42no—
30C2×C191⋊C8—21no—
31C2×C191⋊C8—21no—
32C4⋊Dic191—22no—
33C4⋊Dic191D411yescharacteristic
34C4⋊Dic191D411yescharacteristic
35Dic191⋊C4—22no—
36Dic191⋊C4—21no—
37(C2×D191)⋊C4—22no—
38(C2×D191)⋊C4—21no—
39C4×Dic191—21no—
40C2×Q8×C191—21no—
41C2×Q8×C191D412yes—
42C2×C8×C191—21no—
43C4⋊C4×C191—21no—
44C4⋊C4×C191—21no—
45C4⋊C4×C191—21no—
46C4⋊C4×C191—21no—
47C42×C191C2311yescharacteristic
48C22×D191—21no—
49Dic382—21no—
50Dic382—21no—
51D764—21no—
52D764Q1612yescomplemented
53C2×Dic191—21no—
54C2×Dic191—21no—
55C2×Dic191—21no—
56C4×D191—41no—
57C4×D191—41no—
58C4×D191—82no—
59C191⋊C8—41no—
60C191⋊C8—41no—
61Q8×C191—42no—
62Q8×C191—41no—
63Q8×C191—41no—
64C2×C4×C191—22no—
65C2×C4×C191—41no—
66C2×C4×C191C2×D411yescharacteristic
67C2×C4×C191C2×D411yescharacteristic
68C2×C4×C191C2×D411yescharacteristic
69C8×C191—41no—
70C22×C191#12011yescharacteristic
71C2×D191—22no—
72C2×D191—41no—
73C2×D191—41no—
74C4×C191—21no—
75C4×C191—42no—
76C4×C191—81no—
77C4×C191C2×Q1611yescharacteristic
78C4×C191#13511yescharacteristic
79C4×C191AGL(1,ℤ/8)11yescharacteristic
80C4×C191#13711yescharacteristic
81Dic191—41no—
82Dic191—41no—
83Dic191—41no—
84C2×C191#44911yescharacteristic
85C2×C191#45111yescharacteristic
86C2×C191#45511yescharacteristic
87D191—42no—
88C191#164411yescharacteristic, complemented, Sylow
89#1644—1911nomaximal, Sylow
90#554—1911no—
91#487D19111yescharacteristic, complemented
92#474—1911no—
93#462—1911no—
94#333—1911no—
95Q8⋊C8—1911no—
96D4⋊C8—1911no—
97C2×Q16—3821no—
98C4⋊Q8C2×D19111yescharacteristic
99C4⋊Q8C2×D19111yescharacteristic
100#125—3821no—
101C22⋊Q8—3821no—
102C22⋊Q8—3822no—
103C4×Q8—1911no—
104C4×D4—3821no—
105C4×D4—1911no—
106C2×C4⋊C4—3821no—
107#107—3821no—
108C4⋊C8—1911no—
109C4⋊C8—1911no—
110#103—22no—
111#103—3821no—
112#102—3821no—
113C4×C8C2×D19111yescharacteristic
114C2×Q8—21no—
115C2×Q8—1911no—
116C2×Q8C22⋊D19112yes—
117C2×D4—1911no—
118C22×C4—3822no—
119C22×C4—3821no—
120Q16—7642no—
121C2×C8—21no—
122C2×C8—3821no—
123C2×C8—3821no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—3822no—
129C4⋊C4—3822no—
130C4⋊C4—3821no—
131C4⋊C4—1911no—
132C4⋊C4—1911no—
133C22⋊C4—3822no—
134C22⋊C4—3821no—
135C42—3821no—
136C42C22×D19111yescharacteristic
137C23—3821no—
138Q8—42no—
139Q8—41no—
140Q8—41no—
141Q8—3821no—
142Q8—3821no—
143D4—3821no—
144D4—1912no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—3821no—
148C2×C4—3821no—
149C2×C4—3821no—
150C2×C4—15282no—
151C2×C4—7641no—
152C2×C4—7641no—
153C2×C4D4×D19111yescharacteristic
154C2×C4D4×D19111yescharacteristic
155C2×C4C2×C22⋊D19111yescharacteristic
156C8—41no—
157C8—7641no—
158C8—7641no—
159C22—3822no—
160C22—7641no—
161C22—7641no—
162C22#2025929011yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—7641no—
167C4—7641no—
168C4—7641no—
169C4Q16×D19111yescharacteristic
170C4#2025934311yescharacteristic
171C4#2025934911yescharacteristic
172C4#2025935411yescharacteristic
173C2—7642no—
174C2#2713280711yescharacteristic
175C2#2713281711yescharacteristic
176C2#2713284011yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#487×C191C2
socleC22×C191#120
prime radicalC2×D764D4
hypercenter#487D191
hypocenterC191#1644
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #27132870C2semidirect
4. #27132412C2semidirect
5. #27132395C2semidirect
7. #487×C191C2semidirect
52. D764Q16semidirect
88. C191#1644semidirect
91. #487D191semidirect

Maximal subgroups

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

subgroupindex
#271333742
#271329402
#271328702
#271324122
#271323952
#271323922
#487×C1912
#1644191