Finite Group Databasebeta

← #37574549

Subgroups — #37574549

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#27132846C212yesmaximal
3#27132462C211yesmaximal, characteristic, complemented
4#27132461C212yesmaximal, complemented
5#456×C191C211yesmaximal, characteristic
6#20259352—22no—
7#20259321—22no—
8#20259320—21no—
9#20259320C2212yes—
10C4×Dic382—21no—
11#20259287—22no—
12#20259249—22no—
13#20259241C2212yes—
142−1+4×C191C2211yescharacteristic
15#137×C191—21no—
16C4⋊Q8×C191C2211yescharacteristic
17#104×C191—21no—
18#101×C191C2211yescharacteristic
19C2×Dic382—21no—
20C2×Dic382—21no—
21C2×Dic382D412yes—
22#15690277—42no—
23#15690275—42no—
24#15690271—22no—
25C4⋊Dic191—21no—
26Dic191⋊C4—22no—
27Dic191⋊C4—22no—
28Dic191⋊C4—22no—
29C4×Dic191—21no—
30C4×Dic191—22no—
31C4×Dic191D412yes—
32#41×C191—22no—
33#41×C191—21no—
34#41×C191—41no—
35C2×Q8×C191—41no—
36C2×Q8×C191D411yescharacteristic
37C2×Q8×C191C2311yescharacteristic
38Q16×C191—41no—
39SD16×C191—41no—
40M16×C191—21no—
41C4⋊C4×C191—22no—
42C42×C191D411yescharacteristic
43Dic382—24no—
44Dic382—42no—
45Dic382—81no—
46C2×Dic191—22no—
47C2×Dic191—22no—
48C191⋊C8—42no—
49Q8×C191—21no—
50Q8×C191—22no—
51Q8×C191—41no—
52Q8×C191—41no—
53D4×C191—21no—
54D4×C191—22no—
55D4×C191—41no—
56C2×C4×C191—21no—
57C2×C4×C191—21no—
58C2×C4×C191—41no—
59C2×C4×C191—42no—
60C2×C4×C191C2×D412yes—
61C2×C4×C191C2×D411yescharacteristic
62C8×C191—41no—
63C22×C191—41no—
64C22×C191#12011yescharacteristic
65C4×C191—22no—
66C4×C191—21no—
67C4×C191—22no—
68C4×C191—41no—
69C4×C191—41no—
70Dic191—24no—
71Dic191—42no—
72C2×C191—21no—
73C2×C191—81no—
74C2×C191B(4,𝔽2)11yescharacteristic
75C191#220511yescharacteristic, complemented, Sylow
76#2205—1911nomaximal, Sylow
77Q82—1911no—
78#456—1912no—
79#456D19111yescharacteristic
80#356—1911no—
81#356—1912no—
822−1+4C2×D19111yescharacteristic
83#137—21no—
84#137—3822no—
85C4⋊Q8—3822no—
86C4⋊Q8—3821no—
87C4⋊Q8—1912no—
88C4⋊Q8C2×D19111yescharacteristic
89C4×Q8—3822no—
90C4×Q8—3821no—
91#104—21no—
92#104—3822no—
93#101—1912no—
94#101C2×D19111yescharacteristic
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—3821no—
100C2×Q8—3821no—
101C2×Q8—1912no—
102C2×Q8C22⋊D19111yescharacteristic
103C2×Q8C22×D19111yescharacteristic
104Q16—41no—
105Q16—7642no—
106SD16—41no—
107SD16—7642no—
108M16—21no—
109M16—3822no—
110C4⋊C4—22no—
111C4⋊C4—3822no—
112C4⋊C4—3821no—
113C4⋊C4—3822no—
114C4⋊C4—3822no—
115C42—3822no—
116C42—3821no—
117C42—1912no—
118C42C22⋊D19111yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—3824no—
124Q8—15281no—
125Q8—7642no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—3822no—
134C2×C4—3822no—
135C2×C4D4×D19112yes—
136C2×C4C2×C22⋊D19111yescharacteristic
137C8—41no—
138C8—7642no—
139C22—41no—
140C22#2025929011yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—3824no—
147C4—7642no—
148C2—21no—
149C2—81no—
150C2#2713284911yescharacteristic
subgroupG/H
centerC2#27132849
commutator, cosocleC2×Q8×C191C23
FrattiniC2×Q8C22×D191
Fitting#456×C191C2
socleC2×C191B(4,𝔽2)
prime radical2−1+4×C191C22
hypercenter#456D191
hypocenterC191#2205
LCSG ▷ C2×Q8×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #27133393C2semidirect
3. #27132462C2semidirect
4. #27132461C2semidirect
75. C191#2205semidirect

Maximal subgroups

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

subgroupindex
#271333932
#271328462
#271324622
#271324612
#456×C1912
#2205191