Finite Group Databasebeta

← #36355583

Subgroups — #36355583

Order 22144.

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

normal subgroups only

This group has 216 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#26012241C211yesmaximal, characteristic, complemented
2#26011674C212yesmaximal, complemented
3#26011289C212yesmaximal, complemented
4#26011288C211yesmaximal, characteristic, complemented
5#454×C173C211yesmaximal, characteristic
6#19149709—22no—
7#19149677C2212yes—
8#19149672—22no—
9#19149671—21no—
10#19149660—21no—
11#19149656—22no—
12#19149606—22no—
13#19149598C2212yes—
142−1+4×C173C2211yescharacteristic
15AGL(1,ℤ/8)×C173—21no—
16#124×C173C2211yescharacteristic
17#104×C173—21no—
18#101×C173C2211yescharacteristic
19C2×Dic346D412yes—
20C22×Dic173—21no—
21#14583816—42no—
22#14583814—42no—
23#14583810—22no—
24C4⋊Dic173—21no—
25C4⋊Dic173—21no—
26#14583805—22no—
27#14583805—22no—
28#14583805—22no—
29C22⋊Dic173—22no—
30C22⋊Dic173—21no—
31C4×Dic173D412yes—
32#41×C173—21no—
33#41×C173—22no—
34#41×C173—41no—
35C2×Q8×C173—41no—
36C2×Q8×C173C2311yescharacteristic
37C2×D4×C173D411yescharacteristic
38SD16×C173—41no—
39D8×C173—41no—
40M16×C173—21no—
41C22⋊C4×C173—22no—
42C42×C173D411yescharacteristic
43Dic346—42no—
44C2×Dic173—22no—
45C2×Dic173—22no—
46C2×Dic173—81no—
47#13063436—42no—
48C23×C173—21no—
49Q8×C173—22no—
50Q8×C173—21no—
51Q8×C173—41no—
52D4×C173—22no—
53D4×C173—21no—
54D4×C173—41no—
55D4×C173—41no—
56C2×C4×C173—21no—
57C2×C4×C173—41no—
58C2×C4×C173—42no—
59C2×C4×C173C2×D412yes—
60C2×C4×C173C2×D411yescharacteristic
61C8×C173—41no—
62C22×C173—42no—
63C22×C173—41no—
64C22×C173—41no—
65C22×C173#12011yescharacteristic
66C4×C173—21no—
67C4×C173—22no—
68C4×C173—41no—
69C4×C173—41no—
70Dic173—42no—
71Dic173—42no—
72C2×C173—21no—
73C2×C173—81no—
74C2×C173—81no—
75C2×C173B(4,𝔽2)11yescharacteristic
76C173#220311yescharacteristic, complemented, Sylow
77#2203—1731nomaximal, Sylow
78#563—1731no—
79#456—1732no—
80#454D17311yescharacteristic
81#356—1732no—
82#355—1731no—
832−1+4C2×D17311yescharacteristic
84#137—3462no—
85AGL(1,ℤ/8)—21no—
86C4⋊Q8—1732no—
87#126—3462no—
88#125—3461no—
89#124C2×D17311yescharacteristic
90#123—3461no—
91C22⋊Q8—3462no—
92#104—21no—
93#104—3462no—
94#101—1732no—
95#101C2×D17311yescharacteristic
96#41—22no—
97#41—21no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8—1732no—
101C2×Q8C22×D17311yescharacteristic
102C2×D4C22⋊D17311yescharacteristic
103C22×C4—3461no—
104Q16—6922no—
105SD16—41no—
106SD16—6922no—
107D8—41no—
108M16—21no—
109M16—3462no—
110C4⋊C4—3461no—
111C4⋊C4—3462no—
112C4⋊C4—3461no—
113C4⋊C4—3462no—
114C4⋊C4—3462no—
115C22⋊C4—22no—
116C22⋊C4—3462no—
117C22⋊C4—3461no—
118C42—1732no—
119C42C22⋊D17311yescharacteristic
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—6922no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—3462no—
133C2×C4—3462no—
134C2×C4—13841no—
135C2×C4D4×D17312yes—
136C2×C4C2×C22⋊D17311yescharacteristic
137C8—41no—
138C8—6922no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#1914964711yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—41no—
147C4—6922no—
148C4—6922no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×Q8×C173C23
FrattiniC2×Q8C22×D173
Fitting, prime radical#454×C173C2
socleC2×C173B(4,𝔽2)
hypercenter#454D173
hypocenterC173#2203
LCSG ▷ C2×Q8×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #26012241C2semidirect
2. #26011674C2semidirect
3. #26011289C2semidirect
4. #26011288C2semidirect
76. C173#2203semidirect

Maximal subgroups

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

subgroupindex
#260122412
#260116742
#260112892
#260112882
#454×C1732
#2203173