Finite Group Databasebeta

← #37571726

Subgroups — #37571726

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#27133373C211yesmaximal, characteristic
2#27133015C211yesmaximal, characteristic, complemented
3#27132865C211yesmaximal, characteristic
4#27132411C211yesmaximal, characteristic, complemented
5Q8⋊(C191⋊C8)C211yesmaximal, characteristic, complemented
6#27132390C211yesmaximal, characteristic
7#475×C191C211yesmaximal, characteristic, complemented
8C2×#15690274—21no—
9#20259322C2211yescharacteristic
10#20259314—21no—
11#20259300—22no—
12#20259298—21no—
13C4×D764—21no—
14C4×D764C2211yescharacteristic
15C4⋊C4×D191—21no—
16C4⋊(C191⋊C8)C2211yescharacteristic
17#20259246—21no—
18#20259245—22no—
19C4×C191⋊C8C2211yescharacteristic
20C4⋊Q8×C191C2211yescharacteristic
21C4×Q8×C191C2211yescharacteristic
22#106×C191—21no—
23C4⋊C8×C191C2211yescharacteristic
24#103×C191—21no—
25C2×Dic382D412yes—
26C2×D764D411yescharacteristic
27C2×C4×D191—22no—
28C2×C4×D191—21no—
29#15690274—42no—
30C2×C191⋊C8—21no—
31C2×C191⋊C8—21no—
32C4⋊Dic191—21no—
33C4⋊Dic191—21no—
34C4⋊Dic191D411yescharacteristic
35Dic191⋊C4—22no—
36Dic191⋊C4—22no—
37(C2×D191)⋊C4—22no—
38(C2×D191)⋊C4—21no—
39C2×Q8×C191—21no—
40C2×Q8×C191D411yescharacteristic
41C2×C8×C191—21no—
42C4⋊C4×C191—21no—
43C4⋊C4×C191—21no—
44C4⋊C4×C191—21no—
45C4⋊C4×C191D411yescharacteristic
46C42×C191—21no—
47C42×C191C2311yescharacteristic
48C22×D191—21no—
49Dic382—42no—
50D764—21no—
51D764SD1612yes—
52C2×Dic191—21no—
53C2×Dic191—22no—
54C4×D191—41no—
55C4×D191—41no—
56C4×D191—82no—
57C191⋊C8—41no—
58C191⋊C8—41no—
59Q8×C191—21no—
60Q8×C191—21no—
61Q8×C191—41no—
62Q8×C191—41no—
63C2×C4×C191—21no—
64C2×C4×C191—21no—
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—41no—
76C4×C191—41no—
77C4×C191—81no—
78C4×C191C2×SD1611yescharacteristic
79C4×C191#13511yescharacteristic
80C4×C191#13711yescharacteristic
81C4×C191#13711yescharacteristic
82Dic191—41no—
83Dic191—42no—
84C2×C191#44811yescharacteristic
85C2×C191#45211yescharacteristic
86C2×C191#45511yescharacteristic
87D191—42no—
88C191#164311yescharacteristic, complemented, Sylow
89#1643—1911nomaximal, Sylow
90#554—1911no—
91#487—1911no—
92#475D19111yescharacteristic, complemented
93#461—1911no—
94#333—1911no—
95Q8⋊C8—1911no—
96D4⋊C8—1911no—
97C2×SD16—3821no—
98C4⋊Q8—1911no—
99C4⋊Q8C2×D19111yescharacteristic
100#125—3821no—
101C22⋊Q8—3822no—
102C22⋊Q8—3821no—
103C4×Q8C2×D19111yescharacteristic
104C4×D4—3821no—
105C4×D4—1911no—
106C2×C4⋊C4—3821no—
107#106—21no—
108C4⋊C8—1911no—
109C4⋊C8C2×D19111yescharacteristic
110#103—21no—
111#103—3822no—
112#103—3821no—
113C4×C8—1911no—
114C2×Q8—21no—
115C2×Q8—1912no—
116C2×Q8C22⋊D19111yescharacteristic
117C2×D4—1911no—
118C22×C4—3822no—
119C22×C4—3821no—
120SD16—7642no—
121C2×C8—21no—
122C2×C8—3821no—
123C2×C8—3821no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—3822no—
128C4⋊C4—3822no—
129C4⋊C4—3821no—
130C4⋊C4—3821no—
131C4⋊C4—1911no—
132C4⋊C4C22⋊D19111yescharacteristic
133C22⋊C4—3822no—
134C22⋊C4—3821no—
135C42—21no—
136C42C22×D19111yescharacteristic
137C23—3821no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—7642no—
143D4—3821no—
144D4—1912no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—3821no—
149C2×C4—3822no—
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—41no—
165C4—41no—
166C4—81no—
167C4—7641no—
168C4—7642no—
169C4SD16×D19111yescharacteristic
170C4#2025934211yescharacteristic
171C4#2025935111yescharacteristic
172C4#2025935311yescharacteristic
173C2—7642no—
174C2#2713280011yescharacteristic
175C2#2713282311yescharacteristic
176C2#2713283911yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#475×C191C2
socleC22×C191#120
prime radicalC2×D764D4
hypercenter#475D191
hypocenterC191#1643
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #27133015C2semidirect
4. #27132411C2semidirect
5. Q8⋊(C191⋊C8)C2semidirect
7. #475×C191C2semidirect
88. C191#1643semidirect
92. #475D191semidirect

Maximal subgroups

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

subgroupindex
#271333732
#271330152
#271328652
#271324112
Q8⋊(C191⋊C8)2
#271323902
#475×C1912
#1643191