Finite Group Databasebeta

← #37574474

Subgroups — #37574474

Order 24448.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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#27133465C211yesmaximal, characteristic, complemented
2C2×#20259263C211yesmaximal, characteristic, complemented
3#27132930C211yesmaximal, characteristic, complemented
4#27132477C211yesmaximal, characteristic, complemented
5#27132470C211yesmaximal, characteristic, complemented
6#27132448C211yesmaximal, characteristic, complemented
7#509×C191C211yesmaximal, characteristic, complemented
8#20259368—21no—
9#20259349—22no—
10#20259348—21no—
11#20259347—21no—
12#20259345—21no—
13C2×D1528C2211yescharacteristic
14D8×D191—21no—
15#20259328C2211yescharacteristic
16#20259263—22no—
17C2×C191⋊C16C2211yescharacteristic, complemented
18#20259258C2211yescharacteristic, complemented
19#20259239—21no—
20#135×C191C2211yescharacteristic
21C2×D8×C191C2211yescharacteristic, complemented
22SD32×C191—21no—
23D16×C191—21no—
24M32×C191C2211yescharacteristic, complemented
25#15690288—22no—
26#15690287—41no—
27#15690287D411yescharacteristic
28C2×D764—21no—
29C2×D764—41no—
30D4×D191—22no—
31D4×D191—42no—
32#15690275—41no—
33#15690274—22no—
34D1528—21no—
35D1528D412yes—
36#15690272—22no—
37#15690272—41no—
38#15690271—21no—
39#15690271D411yescharacteristic
40#15690270—21no—
41C8×D191—21no—
42C191⋊C16—21no—
43#41×C191—21no—
44C2×D4×C191—21no—
45Q16×C191D411yescharacteristic
46SD16×C191—21no—
47D8×C191—21no—
48D8×C191—21no—
49D8×C191D411yescharacteristic
50C2×C8×C191C2311yescharacteristic
51C16×C191—21no—
52C22×D191—42no—
53C22×D191—41no—
54Dic382D811yescharacteristic
55D764—22no—
56D764—42no—
57D764—41no—
58D764D811yescharacteristic
59C22⋊D191—21no—
60C22⋊D191—42no—
61C4×D191—21no—
62C4×D191—42no—
63C191⋊C8—21no—
64C191⋊C8—41no—
65C23×C191—41no—
66Q8×C191—21no—
67D4×C191—21no—
68D4×C191—41no—
69D4×C191—41no—
70D4×C191—41no—
71C2×C4×C191—41no—
72C2×C4×C191C2×D411yescharacteristic
73C8×C191C2×D411yescharacteristic
74C8×C191C2×D411yescharacteristic
75C22×C191—41no—
76C22×C191—81no—
77C22×C191—161no—
78C22×C191AGL(1,ℤ/8)11yescharacteristic
79C2×D191—21no—
80C2×D191—42no—
81C2×D191—162no—
82C2×D191—161no—
83C4×C191—41no—
84C4×C191#12011yescharacteristic
85C4×C191C2×D811yescharacteristic
86Dic191—21no—
87C2×C191—21no—
88C2×C191—81no—
89C2×C191—161no—
90C2×C191#44711yescharacteristic
91D191—41no—
92D191—82no—
93C191#219211yescharacteristic, complemented, Sylow
94#2192—1911nomaximal, Sylow
95#576—1911no—
96#509D19111yescharacteristic, complemented
97C2×D16—1911no—
98#472—1911no—
99#361—1911no—
100#359—1911no—
101#350—1911no—
102H(5,𝔽2)—3821no—
103AGL(1,ℤ/8)—3822no—
104AGL(1,ℤ/8)—3821no—
105AGL(1,ℤ/8)—3821no—
106#135—3821no—
107#135C2×D19111yescharacteristic
108C2×D8—3821no—
109C2×D8—1911no—
110C2×D8C2×D19111yescharacteristic, complemented
111#131—1911no—
112SD32—21no—
113D16—21no—
114D16—3822no—
115M32C2×D19111yescharacteristic, complemented
116C2×C16—1911no—
117#108—1911no—
118#100—3821no—
119#41—21no—
120#41—3822no—
121#41—1911no—
122#41—7641no—
123C2×D4—21no—
124C2×D4—3822no—
125C2×D4—3821no—
126C2×D4—7641no—
127C2×D4—7642no—
128Q16C22⋊D19111yescharacteristic
129SD16—21no—
130SD16—3822no—
131SD16—7641no—
132D8—21no—
133D8—21no—
134D8—3822no—
135D8—3821no—
136D8—1912no—
137D8—7641no—
138D8C22⋊D19111yescharacteristic
139M16—3821no—
140M16—3821no—
141M16—1911no—
142C2×C8—3821no—
143C2×C8C22×D19111yescharacteristic
144C16—21no—
145C16—3821no—
146C23—41no—
147C23—7642no—
148C23—7641no—
149Q8—21no—
150Q8—1911no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—3821no—
156D4—3822no—
157D4—1911no—
158D4—7642no—
159D4—7642no—
160D4—7641no—
161C2×C4—41no—
162C2×C4—3821no—
163C2×C4—7642no—
164C2×C4D4×D19111yescharacteristic
165C8—3821no—
166C8—7641no—
167C8D4×D19111yescharacteristic
168C8C2×C22⋊D19111yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—3821no—
173C22—30561no—
174C22—30562no—
175C22—7642no—
176C22#2025934611yescharacteristic
177C4—41no—
178C4—3821no—
179C4#2025929011yescharacteristic
180C4D8×D19111yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—15282no—
185C2—7641no—
186C2#2713279511yescharacteristic
subgroupG/H
centerC2#27132795
commutator, cosocleC2×C8×C191C23
FrattiniC2×C8C22×D191
Fitting#509×C191C2
socleC2×C191#447
hypercenter#509D191
hypocenterC191#2192
LCSG ▷ C2×C8×C191 ▷ C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #27133465C2semidirect
2. C2×#20259263C2semidirect
3. #27132930C2semidirect
4. #27132477C2semidirect
5. #27132470C2semidirect
6. #27132448C2semidirect
7. #509×C191C2semidirect
17. C2×C191⋊C16C22semidirect
18. #20259258C22semidirect
21. C2×D8×C191C22semidirect
24. M32×C191C22semidirect
93. C191#2192semidirect
96. #509D191semidirect
110. C2×D8C2×D191semidirect
115. M32C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271334652
C2×#202592632
#271329302
#271324772
#271324702
#271324482
#509×C1912
#2192191