Finite Group Databasebeta

← #37535752

Subgroups — #37535752

Order 23168.

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#26042009C211yesmaximal, characteristic, complemented
2#26041464C212yesmaximal
3#26041461C211yesmaximal, characteristic
4#26041077C212yesmaximal, complemented
5#356×C181C211yesmaximal, characteristic, complemented
6#19153864C2211yescharacteristic
7#19153849—22no—
8#19153847—21no—
9#19153815—22no—
10#19153814—21no—
11#19153814C2212yes—
12C4×Dic362—21no—
13#19153781—22no—
14#19153744—21no—
15#19153742—22no—
16#19153735C2211yescharacteristic
17C4⋊Q8×C181C2211yescharacteristic
18#101×C181C2212yes—
19#15668450—41no—
20#15668449—22no—
21#15668449—21no—
22C2×Dic362—21no—
23C2×Dic362—21no—
24C2×Dic362D412yes—
25Q8×D181—41no—
26Dic724—42no—
27#15668439—41no—
28#15668438—42no—
29#15668436—41no—
30#15668433—21no—
31C4⋊Dic181—21no—
32#15668428—22no—
33#15668428—22no—
34#15668428—22no—
35C4×Dic181—21no—
36C4×Dic181—22no—
37C4×Dic181D412yes—
38C2×Q8×C181D411yescharacteristic
39C2×Q8×C181C2311yescharacteristic
40M16×C181—22no—
41C4⋊C4×C181—22no—
42C42×C181D411yescharacteristic
43Dic362—24no—
44Dic362—21no—
45Dic362—22no—
46Dic362—42no—
47Dic362—81no—
48D724—21no—
49D724—22no—
50C22⋊D181—41no—
51C2×Dic181—22no—
52C2×Dic181—22no—
53C4×D181—42no—
54C4×D181—41no—
55#13089678—41no—
56Q8×C181—41no—
57Q8×C181—41no—
58C2×C4×C181—21no—
59C2×C4×C181—21no—
60C2×C4×C181C2×D412yes—
61C2×C4×C181C2×D411yescharacteristic
62C8×C181—42no—
63C22×C181#12011yescharacteristic
64C2×D181—41no—
65C4×C181—22no—
66C4×C181—22no—
67C4×C181—21no—
68C4×C181—41no—
69Dic181—24no—
70Dic181—41no—
71Dic181—42no—
72C2×C181—21no—
73C2×C181B(4,𝔽2)11yescharacteristic
74D181—81no—
75C181#220511yescharacteristic, complemented, Sylow
76#2205—1811nomaximal, Sylow
77Q82—1811no—
78#456—1811no—
79#456—1812no—
80#356—1812no—
81#356D18111yescharacteristic, complemented
822−1+4—1811no—
83#137—3622no—
84#137—3621no—
85C4⋊Q8—3622no—
86C4⋊Q8—3621no—
87C4⋊Q8—1812no—
88C4⋊Q8C2×D18111yescharacteristic
89C4×Q8—3622no—
90C4×Q8—3621no—
91#104—3622no—
92#104—3621no—
93#101—1811no—
94#101C2×D18112yes—
95#41—3622no—
96#41—3621no—
97#41—7241no—
98C2×Q8—3621no—
99C2×Q8—3621no—
100C2×Q8—1812no—
101C2×Q8—7241no—
102C2×Q8D72411yescharacteristic
103C2×Q8C22×D18111yescharacteristic
104Q16—7242no—
105Q16—7241no—
106SD16—7242no—
107SD16—7241no—
108M16—22no—
109M16—3621no—
110C4⋊C4—22no—
111C4⋊C4—3622no—
112C4⋊C4—3621no—
113C4⋊C4—3622no—
114C4⋊C4—3622no—
115C42—3622no—
116C42—3621no—
117C42—1812no—
118C42D72411yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—3624no—
122Q8—3622no—
123Q8—3621no—
124Q8—14481no—
125Q8—7242no—
126D4—3622no—
127D4—3621no—
128D4—7241no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—3622no—
132C2×C4—3622no—
133C2×C4—7242no—
134C2×C4—7241no—
135C2×C4D4×D18112yes—
136C2×C4C2×D72411yescharacteristic
137C8—42no—
138C8—7241no—
139C22—7241no—
140C22#1915378611yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—3624no—
146C4—7242no—
147C4—7241no—
148C2—21no—
149C2—14481no—
150C2#2604146711yescharacteristic
subgroupG/H
centerC2#26041467
commutator, cosocleC2×Q8×C181C23
FrattiniC2×Q8C22×D181
Fitting#356×C181C2
socleC2×C181B(4,𝔽2)
prime radical#19153864C22
hypercenter#356D181
hypocenterC181#2205
LCSG ▷ C2×Q8×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #26042009C2semidirect
4. #26041077C2semidirect
5. #356×C181C2semidirect
75. C181#2205semidirect
81. #356D181semidirect

Maximal subgroups

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

subgroupindex
#260420092
#260414642
#260414612
#260410772
#356×C1812
#2205181