Finite Group Databasebeta

← #37574551

Subgroups — #37574551

Order 24448.

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

normal subgroups only

This group has 224 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#27133393C211yesmaximal, characteristic, complemented
2#27132848C212yesmaximal
3#27132845C211yesmaximal, characteristic
4#27132461C212yesmaximal, complemented
5#356×C191C211yesmaximal, characteristic, complemented
6#20259370C2211yescharacteristic
7#20259355—22no—
8#20259353—21no—
9#20259321—22no—
10#20259320—21no—
11#20259320C2212yes—
12C4×Dic382—21no—
13#20259287—22no—
14#20259250—21no—
15#20259248—22no—
16#20259241C2211yescharacteristic
17C4⋊Q8×C191C2211yescharacteristic
18#101×C191C2212yes—
19#15690288—41no—
20#15690287—21no—
21#15690287—22no—
22C2×Dic382—21no—
23C2×Dic382—21no—
24C2×Dic382D412yes—
25Q8×D191—41no—
26Dic764—42no—
27#15690277—41no—
28#15690276—42no—
29#15690274—41no—
30#15690271—21no—
31C4⋊Dic191—21no—
32Dic191⋊C4—22no—
33Dic191⋊C4—22no—
34Dic191⋊C4—22no—
35C4×Dic191—21no—
36C4×Dic191—22no—
37C4×Dic191D412yes—
38C2×Q8×C191D411yescharacteristic
39C2×Q8×C191C2311yescharacteristic
40M16×C191—22no—
41C4⋊C4×C191—22no—
42C42×C191D411yescharacteristic
43Dic382—24no—
44Dic382—21no—
45Dic382—22no—
46Dic382—42no—
47Dic382—81no—
48D764—21no—
49D764—22no—
50C22⋊D191—41no—
51C2×Dic191—22no—
52C2×Dic191—22no—
53C4×D191—42no—
54C4×D191—41no—
55C191⋊C8—41no—
56Q8×C191—41no—
57Q8×C191—41no—
58C2×C4×C191—21no—
59C2×C4×C191—21no—
60C2×C4×C191C2×D412yes—
61C2×C4×C191C2×D411yescharacteristic
62C8×C191—42no—
63C22×C191#12011yescharacteristic
64C2×D191—41no—
65C4×C191—22no—
66C4×C191—22no—
67C4×C191—21no—
68C4×C191—41no—
69Dic191—24no—
70Dic191—41no—
71Dic191—42no—
72C2×C191—21no—
73C2×C191B(4,𝔽2)11yescharacteristic
74D191—81no—
75C191#220511yescharacteristic, complemented, Sylow
76#2205—1911nomaximal, Sylow
77Q82—1911no—
78#456—1911no—
79#456—1912no—
80#356—1912no—
81#356D19111yescharacteristic, complemented
822−1+4—1911no—
83#137—3822no—
84#137—3821no—
85C4⋊Q8—3822no—
86C4⋊Q8—3821no—
87C4⋊Q8—1912no—
88C4⋊Q8C2×D19111yescharacteristic
89C4×Q8—3822no—
90C4×Q8—3821no—
91#104—3822no—
92#104—3821no—
93#101—1911no—
94#101C2×D19112yes—
95#41—3822no—
96#41—3821no—
97#41—7641no—
98C2×Q8—3821no—
99C2×Q8—3821no—
100C2×Q8—1912no—
101C2×Q8—7641no—
102C2×Q8D76411yescharacteristic
103C2×Q8C22×D19111yescharacteristic
104Q16—7642no—
105Q16—7641no—
106SD16—7642no—
107SD16—7641no—
108M16—22no—
109M16—3821no—
110C4⋊C4—22no—
111C4⋊C4—3822no—
112C4⋊C4—3821no—
113C4⋊C4—3822no—
114C4⋊C4—3822no—
115C42—3822no—
116C42—3821no—
117C42—1912no—
118C42D76411yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—3824no—
122Q8—3822no—
123Q8—3821no—
124Q8—15281no—
125Q8—7642no—
126D4—3822no—
127D4—3821no—
128D4—7641no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—3822no—
132C2×C4—3822no—
133C2×C4—7642no—
134C2×C4—7641no—
135C2×C4D4×D19112yes—
136C2×C4C2×D76411yescharacteristic
137C8—42no—
138C8—7641no—
139C22—7641no—
140C22#2025929211yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—3824no—
146C4—7642no—
147C4—7641no—
148C2—21no—
149C2—15281no—
150C2#2713285111yescharacteristic
subgroupG/H
centerC2#27132851
commutator, cosocleC2×Q8×C191C23
FrattiniC2×Q8C22×D191
Fitting#356×C191C2
socleC2×C191B(4,𝔽2)
prime radical#20259370C22
hypercenter#356D191
hypocenterC191#2205
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. #27133393C2semidirect
4. #27132461C2semidirect
5. #356×C191C2semidirect
75. C191#2205semidirect
81. #356D191semidirect

Maximal subgroups

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

subgroupindex
#271333932
#271328482
#271328452
#271324612
#356×C1912
#2205191