Finite Group Databasebeta

← #37574548

Subgroups — #37574548

Order 24448.

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

normal subgroups only

This group has 236 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#27133379C211yesmaximal, characteristic, complemented
2#27132845C211yesmaximal, characteristic, complemented
3#27132838C212yesmaximal, complemented
4#27132459C212yesmaximal, complemented
5#356×C191C211yesmaximal, characteristic, complemented
6#20259370C2211yescharacteristic
7#20259353—21no—
8#20259350—22no—
9#20259311—21no—
10#20259311C2212yes—
11#20259297—22no—
12C4×D764—21no—
13#20259280—22no—
14#20259250—21no—
15#20259248—22no—
16#20259241C2211yescharacteristic
17C4⋊Q8×C191C2211yescharacteristic
18#101×C191C2212yes—
19#15690288—41no—
20#15690287—22no—
21#15690287—21no—
22Q8×D191—41no—
23C2×D764—21no—
24C2×D764—21no—
25C2×D764D412yes—
26C2×C4×D191—22no—
27#15690277—41no—
28#15690276—42no—
29#15690274—41no—
30D1528—42no—
31#15690271—21no—
32C4⋊Dic191—21no—
33(C2×D191)⋊C4—22no—
34(C2×D191)⋊C4—22no—
35(C2×D191)⋊C4—22no—
36C4×Dic191—21no—
37C4×Dic191D412yes—
38C2×Q8×C191D411yescharacteristic
39C2×Q8×C191C2311yescharacteristic
40M16×C191—22no—
41C4⋊C4×C191—22no—
42C42×C191D411yescharacteristic
43C22×D191—22no—
44Dic382—22no—
45Dic382—21no—
46D764—22no—
47D764—21no—
48D764—42no—
49D764—42no—
50D764—81no—
51C22⋊D191—41no—
52C2×Dic191—22no—
53C4×D191—44no—
54C4×D191—42no—
55C4×D191—41no—
56C191⋊C8—41no—
57Q8×C191—41no—
58Q8×C191—41no—
59C2×C4×C191—21no—
60C2×C4×C191—21no—
61C2×C4×C191C2×D412yes—
62C2×C4×C191C2×D411yescharacteristic
63C8×C191—42no—
64C22×C191#12011yescharacteristic
65C2×D191—42no—
66C2×D191—42no—
67C2×D191—41no—
68C2×D191—42no—
69C4×C191—22no—
70C4×C191—22no—
71C4×C191—21no—
72C4×C191—41no—
73Dic191—42no—
74Dic191—41no—
75C2×C191—21no—
76C2×C191B(4,𝔽2)11yescharacteristic
77D191—81no—
78D191—82no—
79C191#220411yescharacteristic, complemented, Sylow
80#2204—1911nomaximal, Sylow
81#555—1911no—
82#456—1911no—
83#454—1912no—
84#356D19111yescharacteristic, complemented
85#355—1912no—
862−1+4—1911no—
87#137—3821no—
88AGL(1,ℤ/8)—3822no—
89C4⋊Q8C2×D19111yescharacteristic
90#124—3821no—
91#124—1912no—
92#121—3822no—
93C4×D4—3821no—
94#117—3822no—
95#104—3822no—
96#104—3821no—
97#101—1911no—
98#101C2×D19112yes—
99#41—3822no—
100#41—3821no—
101#41—7641no—
102C2×Q8—7641no—
103C2×Q8D76411yescharacteristic
104C2×Q8C22×D19111yescharacteristic
105C2×D4—3821no—
106C2×D4—3821no—
107C2×D4—1912no—
108C22×C4—3822no—
109Q16—7641no—
110SD16—7642no—
111SD16—7641no—
112D8—7642no—
113M16—22no—
114M16—3821no—
115C4⋊C4—22no—
116C4⋊C4—3821no—
117C22⋊C4—3822no—
118C22⋊C4—3822no—
119C22⋊C4—3822no—
120C42—3821no—
121C42—1912no—
122C42D76411yescharacteristic
123C23—3822no—
124Q8—41no—
125Q8—41no—
126Q8—3822no—
127Q8—3821no—
128D4—3822no—
129D4—3821no—
130D4—15281no—
131D4—7642no—
132D4—7642no—
133D4—7641no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—3822no—
137C2×C4—7644no—
138C2×C4—7642no—
139C2×C4—7641no—
140C2×C4D4×D19112yes—
141C2×C4C2×D76411yescharacteristic
142C8—42no—
143C8—7641no—
144C22—7642no—
145C22—7642no—
146C22—7641no—
147C22—7642no—
148C22#2025929211yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—41no—
153C4—7642no—
154C4—7641no—
155C2—21no—
156C2—15282no—
157C2—15281no—
158C2#2713285111yescharacteristic
subgroupG/H
centerC2#27132851
commutator, cosocleC2×Q8×C191C23
FrattiniC2×Q8C22×D191
Fitting#356×C191C2
socleC2×C191B(4,𝔽2)
hypercenter#356D191
hypocenterC191#2204
LCSG ▷ C2×Q8×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #27133379C2semidirect
2. #27132845C2semidirect
3. #27132838C2semidirect
4. #27132459C2semidirect
5. #356×C191C2semidirect
79. C191#2204semidirect
84. #356D191semidirect

Maximal subgroups

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

subgroupindex
#271333792
#271328452
#271328382
#271324592
#356×C1912
#2204191