Finite Group Databasebeta

← #37535749

Subgroups — #37535749

Order 23168.

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#26041995C211yesmaximal, characteristic, complemented
2#26041461C211yesmaximal, characteristic, complemented
3#26041454C212yesmaximal, complemented
4#26041075C212yesmaximal, complemented
5#356×C181C211yesmaximal, characteristic, complemented
6#19153864C2211yescharacteristic
7#19153847—21no—
8#19153844—22no—
9#19153805—21no—
10#19153805C2212yes—
11#19153791—22no—
12C4×D724—21no—
13#19153774—22no—
14#19153744—21no—
15#19153742—22no—
16#19153735C2211yescharacteristic
17C4⋊Q8×C181C2211yescharacteristic
18#101×C181C2212yes—
19#15668450—41no—
20#15668449—22no—
21#15668449—21no—
22Q8×D181—41no—
23C2×D724—21no—
24C2×D724—21no—
25C2×D724D412yes—
26C2×C4×D181—22no—
27#15668439—41no—
28#15668438—42no—
29#15668436—41no—
30D1448—42no—
31#15668433—21no—
32C4⋊Dic181—21no—
33#15668426—22no—
34#15668426—22no—
35#15668426—22no—
36C4×Dic181—21no—
37C4×Dic181D412yes—
38C2×Q8×C181D411yescharacteristic
39C2×Q8×C181C2311yescharacteristic
40M16×C181—22no—
41C4⋊C4×C181—22no—
42C42×C181D411yescharacteristic
43C22×D181—22no—
44Dic362—22no—
45Dic362—21no—
46D724—22no—
47D724—21no—
48D724—42no—
49D724—42no—
50D724—81no—
51C22⋊D181—41no—
52C2×Dic181—22no—
53C4×D181—44no—
54C4×D181—42no—
55C4×D181—41no—
56#13089678—41no—
57Q8×C181—41no—
58Q8×C181—41no—
59C2×C4×C181—21no—
60C2×C4×C181—21no—
61C2×C4×C181C2×D412yes—
62C2×C4×C181C2×D411yescharacteristic
63C8×C181—42no—
64C22×C181#12011yescharacteristic
65C2×D181—42no—
66C2×D181—42no—
67C2×D181—41no—
68C2×D181—42no—
69C4×C181—22no—
70C4×C181—22no—
71C4×C181—21no—
72C4×C181—41no—
73Dic181—42no—
74Dic181—41no—
75C2×C181—21no—
76C2×C181B(4,𝔽2)11yescharacteristic
77D181—81no—
78D181—82no—
79C181#220411yescharacteristic, complemented, Sylow
80#2204—1811nomaximal, Sylow
81#555—1811no—
82#456—1811no—
83#454—1812no—
84#356D18111yescharacteristic, complemented
85#355—1812no—
862−1+4—1811no—
87#137—3621no—
88AGL(1,ℤ/8)—3622no—
89C4⋊Q8C2×D18111yescharacteristic
90#124—3621no—
91#124—1812no—
92#121—3622no—
93C4×D4—3621no—
94#117—3622no—
95#104—3622no—
96#104—3621no—
97#101—1811no—
98#101C2×D18112yes—
99#41—3622no—
100#41—3621no—
101#41—7241no—
102C2×Q8—7241no—
103C2×Q8D72411yescharacteristic
104C2×Q8C22×D18111yescharacteristic
105C2×D4—3621no—
106C2×D4—3621no—
107C2×D4—1812no—
108C22×C4—3622no—
109Q16—7241no—
110SD16—7242no—
111SD16—7241no—
112D8—7242no—
113M16—22no—
114M16—3621no—
115C4⋊C4—22no—
116C4⋊C4—3621no—
117C22⋊C4—3622no—
118C22⋊C4—3622no—
119C22⋊C4—3622no—
120C42—3621no—
121C42—1812no—
122C42D72411yescharacteristic
123C23—3622no—
124Q8—41no—
125Q8—41no—
126Q8—3622no—
127Q8—3621no—
128D4—3622no—
129D4—3621no—
130D4—14481no—
131D4—7242no—
132D4—7242no—
133D4—7241no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—3622no—
137C2×C4—7244no—
138C2×C4—7242no—
139C2×C4—7241no—
140C2×C4D4×D18112yes—
141C2×C4C2×D72411yescharacteristic
142C8—42no—
143C8—7241no—
144C22—7242no—
145C22—7242no—
146C22—7241no—
147C22—7242no—
148C22#1915378611yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—41no—
153C4—7242no—
154C4—7241no—
155C2—21no—
156C2—14482no—
157C2—14481no—
158C2#2604146711yescharacteristic
subgroupG/H
centerC2#26041467
commutator, cosocleC2×Q8×C181C23
FrattiniC2×Q8C22×D181
Fitting#356×C181C2
socleC2×C181B(4,𝔽2)
hypercenter#356D181
hypocenterC181#2204
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. #26041995C2semidirect
2. #26041461C2semidirect
3. #26041454C2semidirect
4. #26041075C2semidirect
5. #356×C181C2semidirect
79. C181#2204semidirect
84. #356D181semidirect

Maximal subgroups

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

subgroupindex
#260419952
#260414612
#260414542
#260410752
#356×C1812
#2204181