Finite Group Databasebeta

← #37574541

Subgroups — #37574541

Order 24448.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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#27132845C212yesmaximal, complemented
2#27132835C211yesmaximal, characteristic
3#27132462C212yesmaximal, complemented
4#27132460C211yesmaximal, characteristic, complemented
5#563×C191C211yesmaximal, characteristic, complemented
6#20259370C2211yescharacteristic
7#20259353—22no—
8#20259348—21no—
9#20259250—21no—
10#20259250—22no—
11#20259241C2212yes—
12#20259241C2211yescharacteristic
13C4⋊Q8×C191C2212yes—
14#126×C191—22no—
15#125×C191—21no—
16#124×C191C2211yescharacteristic
17#123×C191—21no—
18C22⋊Q8×C191—22no—
19#15690288—41no—
20#15690287—21no—
21#15690287—22no—
22Q8×D191—41no—
23#15690277—42no—
24#15690275—41no—
25#15690274—42no—
26#15690272—41no—
27#15690271—21no—
28#15690271—22no—
29C2×Q8×C191D412yes—
30C2×Q8×C191C2311yescharacteristic
31C2×D4×C191D411yescharacteristic
32C22×C4×C191—21no—
33C4⋊C4×C191—21no—
34C4⋊C4×C191—22no—
35C4⋊C4×C191—21no—
36C4⋊C4×C191—22no—
37C4⋊C4×C191—22no—
38C22⋊C4×C191—22no—
39C22⋊C4×C191—21no—
40C22⋊C4×C191—22no—
41C42×C191D411yescharacteristic
42C42×C191D412yes—
43Dic382—21no—
44Dic382—22no—
45D764—21no—
46D764—22no—
47C22⋊D191—41no—
48C4×D191—41no—
49C4×D191—42no—
50C191⋊C8—41no—
51C191⋊C8—42no—
52C23×C191—21no—
53Q8×C191—41no—
54Q8×C191—42no—
55D4×C191—41no—
56C2×C4×C191—21no—
57C2×C4×C191—22no—
58C2×C4×C191—22no—
59C2×C4×C191—81no—
60C2×C4×C191C2×D412yes—
61C2×C4×C191C2×D411yescharacteristic
62C22×C191—42no—
63C22×C191—41no—
64C22×C191#12011yescharacteristic
65C2×D191—41no—
66C4×C191—21no—
67C4×C191—22no—
68C4×C191—41no—
69C4×C191—42no—
70C4×C191—42no—
71Dic191—41no—
72C2×C191—21no—
73C2×C191—81no—
74C2×C191B(4,𝔽2)11yescharacteristic
75D191—81no—
76C191#220311yescharacteristic, complemented, Sylow
77#2203—1911nomaximal, Sylow
78#563D19111yescharacteristic, complemented
79#456—1912no—
80#454—1911no—
81#356—1912no—
82#355—1911no—
832−1+4—1911no—
84#137—3822no—
85AGL(1,ℤ/8)—3821no—
86C4⋊Q8C2×D19112yes—
87#126—22no—
88#125—21no—
89#124C2×D19111yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—3821no—
93#104—3822no—
94#101—1912no—
95#101—1911no—
96#41—3821no—
97#41—3822no—
98#41—7641no—
99C2×Q8—7641no—
100C2×Q8C22⋊D19112yes—
101C2×Q8C22×D19111yescharacteristic
102C2×D4C22⋊D19111yescharacteristic
103C22×C4—21no—
104Q16—7642no—
105SD16—7641no—
106SD16—7642no—
107D8—7641no—
108M16—3821no—
109M16—3822no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D19111yescharacteristic
119C42C22⋊D19112yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—3821no—
124Q8—3822no—
125D4—41no—
126D4—3821no—
127D4—3822no—
128D4—7641no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—7641no—
134C2×C4—7642no—
135C2×C4C2×C22⋊D19112yes—
136C2×C4C2×C22⋊D19111yescharacteristic
137C8—7641no—
138C8—7642no—
139C22—42no—
140C22—41no—
141C22—7641no—
142C22#2025929111yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—7641no—
149C2—21no—
150C2—81no—
151C2—15281no—
152C2#2713285011yescharacteristic
subgroupG/H
centerC2#27132850
commutator, cosocleC2×Q8×C191C23
FrattiniC2×Q8C22×D191
Fitting#563×C191C2
socleC2×C191B(4,𝔽2)
prime radical#27132835C2
hypercenter#563D191
hypocenterC191#2203
LCSG ▷ C2×Q8×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #27132845C2semidirect
3. #27132462C2semidirect
4. #27132460C2semidirect
5. #563×C191C2semidirect
76. C191#2203semidirect
78. #563D191semidirect

Maximal subgroups

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

subgroupindex
#271328452
#271328352
#271324622
#271324602
#563×C1912
#2203191