Finite Group Databasebeta

← #37574543

Subgroups — #37574543

Order 24448.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#27133413C211yesmaximal, characteristic, complemented
2#27132848C212yesmaximal, complemented
3#27132835C211yesmaximal, characteristic
4#27132461C212yesmaximal, complemented
5#355×C191C211yesmaximal, characteristic, complemented
6#20259370C2211yescharacteristic
7#20259355—22no—
8#20259348—21no—
9#20259320C2212yes—
10#20259315—22no—
11#20259314—21no—
12#20259303—21no—
13#20259299—22no—
14#20259250—21no—
15#20259248—22no—
16#20259241C2211yescharacteristic
17#124×C191C2211yescharacteristic
18#101×C191C2212yes—
19#15690288—41no—
20#15690287—21no—
21#15690287—22no—
22C2×Dic382D412yes—
23Q8×D191—41no—
24C22×Dic191—21no—
25Dic764—42no—
26#15690276—42no—
27#15690275—41no—
28#15690272—41no—
29#15690271—21no—
30C4⋊Dic191—21no—
31C4⋊Dic191—21no—
32Dic191⋊C4—22no—
33Dic191⋊C4—22no—
34Dic191⋊C4—22no—
35C22⋊Dic191—22no—
36C22⋊Dic191—21no—
37C4×Dic191D412yes—
38C2×Q8×C191C2311yescharacteristic
39C2×D4×C191D411yescharacteristic
40M16×C191—22no—
41C22⋊C4×C191—22no—
42C42×C191D411yescharacteristic
43Dic382—21no—
44Dic382—22no—
45Dic382—42no—
46D764—21no—
47D764—22no—
48C22⋊D191—41no—
49C2×Dic191—22no—
50C2×Dic191—22no—
51C2×Dic191—81no—
52C4×D191—42no—
53C4×D191—41no—
54C191⋊C8—41no—
55C23×C191—21no—
56Q8×C191—41no—
57D4×C191—41no—
58C2×C4×C191—21no—
59C2×C4×C191C2×D412yes—
60C2×C4×C191C2×D411yescharacteristic
61C8×C191—42no—
62C22×C191—42no—
63C22×C191—41no—
64C22×C191#12011yescharacteristic
65C2×D191—41no—
66C4×C191—22no—
67C4×C191—21no—
68C4×C191—41no—
69Dic191—42no—
70Dic191—41no—
71Dic191—42no—
72C2×C191—21no—
73C2×C191—81no—
74C2×C191B(4,𝔽2)11yescharacteristic
75D191—81no—
76C191#220311yescharacteristic, complemented, Sylow
77#2203—1911nomaximal, Sylow
78#563—1911no—
79#456—1912no—
80#454—1911no—
81#356—1912no—
82#355D19111yescharacteristic, complemented
832−1+4—1911no—
84#137—3822no—
85AGL(1,ℤ/8)—3821no—
86C4⋊Q8—1912no—
87#126—3822no—
88#125—3821no—
89#124C2×D19111yescharacteristic
90#123—3821no—
91C22⋊Q8—3822no—
92#104—3822no—
93#104—3821no—
94#101—1911no—
95#101C2×D19112yes—
96#41—3822no—
97#41—3821no—
98#41—7641no—
99C2×Q8—1912no—
100C2×Q8—7641no—
101C2×Q8C22×D19111yescharacteristic
102C2×D4D76411yescharacteristic
103C22×C4—3821no—
104Q16—7642no—
105SD16—7642no—
106SD16—7641no—
107D8—7641no—
108M16—22no—
109M16—3821no—
110C4⋊C4—3822no—
111C4⋊C4—3821no—
112C4⋊C4—3821no—
113C4⋊C4—3822no—
114C4⋊C4—3822no—
115C22⋊C4—22no—
116C22⋊C4—3822no—
117C22⋊C4—3821no—
118C42—1912no—
119C42D76411yescharacteristic
120C23—21no—
121Q8—41no—
122Q8—3821no—
123Q8—3822no—
124Q8—7642no—
125D4—41no—
126D4—3821no—
127D4—3822no—
128D4—7641no—
129C2×C4—21no—
130C2×C4—3822no—
131C2×C4—3822no—
132C2×C4—15281no—
133C2×C4—7642no—
134C2×C4—7641no—
135C2×C4D4×D19112yes—
136C2×C4C2×D76411yescharacteristic
137C8—42no—
138C8—7641no—
139C22—42no—
140C22—41no—
141C22—7641no—
142C22#2025929211yescharacteristic
143C4—22no—
144C4—21no—
145C4—41no—
146C4—7642no—
147C4—7642no—
148C4—7641no—
149C2—21no—
150C2—81no—
151C2—15281no—
152C2#2713285111yescharacteristic
subgroupG/H
centerC2#27132851
commutator, cosocleC2×Q8×C191C23
FrattiniC2×Q8C22×D191
Fitting#355×C191C2
socleC2×C191B(4,𝔽2)
prime radical#27132835C2
hypercenter#355D191
hypocenterC191#2203
LCSG ▷ C2×Q8×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #27133413C2semidirect
2. #27132848C2semidirect
4. #27132461C2semidirect
5. #355×C191C2semidirect
76. C191#2203semidirect
82. #355D191semidirect

Maximal subgroups

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

subgroupindex
#271334132
#271328482
#271328352
#271324612
#355×C1912
#2203191