Finite Group Databasebeta

← #36355584

Subgroups — #36355584

Order 22144.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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#26011673C212yesmaximal, complemented
2#26011663C211yesmaximal, characteristic
3#26011290C212yesmaximal, complemented
4#26011288C211yesmaximal, characteristic, complemented
5#563×C173C211yesmaximal, characteristic, complemented
6#19149727C2211yescharacteristic
7#19149710—22no—
8#19149705—21no—
9#19149607—21no—
10#19149607—22no—
11#19149598C2211yescharacteristic
12#19149598C2212yes—
13C4⋊Q8×C173C2212yes—
14#126×C173—22no—
15#125×C173—21no—
16#124×C173C2211yescharacteristic
17#123×C173—21no—
18C22⋊Q8×C173—22no—
19#14583827—41no—
20#14583826—21no—
21#14583826—22no—
22Q8×D173—41no—
23#14583816—42no—
24#14583814—41no—
25#14583813—42no—
26#14583811—41no—
27#14583810—21no—
28#14583810—22no—
29C2×Q8×C173D412yes—
30C2×Q8×C173C2311yescharacteristic
31C2×D4×C173D411yescharacteristic
32C22×C4×C173—21no—
33C4⋊C4×C173—21no—
34C4⋊C4×C173—22no—
35C4⋊C4×C173—21no—
36C4⋊C4×C173—22no—
37C4⋊C4×C173—22no—
38C22⋊C4×C173—22no—
39C22⋊C4×C173—21no—
40C22⋊C4×C173—22no—
41C42×C173D412yes—
42C42×C173D411yescharacteristic
43Dic346—21no—
44Dic346—22no—
45D692—21no—
46D692—22no—
47C22⋊D173—41no—
48C4×D173—41no—
49C4×D173—42no—
50#13063436—41no—
51#13063436—42no—
52C23×C173—21no—
53Q8×C173—41no—
54Q8×C173—42no—
55D4×C173—41no—
56C2×C4×C173—22no—
57C2×C4×C173—21no—
58C2×C4×C173—22no—
59C2×C4×C173—81no—
60C2×C4×C173C2×D412yes—
61C2×C4×C173C2×D411yescharacteristic
62C22×C173—42no—
63C22×C173—41no—
64C22×C173#12011yescharacteristic
65C2×D173—41no—
66C4×C173—21no—
67C4×C173—22no—
68C4×C173—42no—
69C4×C173—41no—
70C4×C173—42no—
71Dic173—41no—
72C2×C173—21no—
73C2×C173—81no—
74C2×C173B(4,𝔽2)11yescharacteristic
75D173—81no—
76C173#220311yescharacteristic, complemented, Sylow
77#2203—1731nomaximal, Sylow
78#563D17311yescharacteristic, complemented
79#456—1732no—
80#454—1731no—
81#356—1732no—
82#355—1731no—
832−1+4—1731no—
84#137—3462no—
85AGL(1,ℤ/8)—3461no—
86C4⋊Q8C2×D17312yes—
87#126—22no—
88#125—21no—
89#124C2×D17311yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—3461no—
93#104—3462no—
94#101—1732no—
95#101—1731no—
96#41—3461no—
97#41—3462no—
98#41—6921no—
99C2×Q8—6921no—
100C2×Q8C22⋊D17312yes—
101C2×Q8C22×D17311yescharacteristic
102C2×D4C22⋊D17311yescharacteristic
103C22×C4—21no—
104Q16—6922no—
105SD16—6921no—
106SD16—6922no—
107D8—6921no—
108M16—3461no—
109M16—3462no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D17311yescharacteristic
119C42C22⋊D17312yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—3461no—
124Q8—3462no—
125D4—41no—
126D4—3461no—
127D4—3462no—
128D4—6921no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—6921no—
134C2×C4—6922no—
135C2×C4C2×C22⋊D17312yes—
136C2×C4C2×C22⋊D17311yescharacteristic
137C8—6921no—
138C8—6922no—
139C22—42no—
140C22—41no—
141C22—6921no—
142C22#1914964811yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—6921no—
149C2—21no—
150C2—81no—
151C2—13841no—
152C2#2601167811yescharacteristic
subgroupG/H
centerC2#26011678
commutator, cosocleC2×Q8×C173C23
FrattiniC2×Q8C22×D173
Fitting#563×C173C2
socleC2×C173B(4,𝔽2)
prime radical#26011663C2
hypercenter#563D173
hypocenterC173#2203
LCSG ▷ C2×Q8×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #26011673C2semidirect
3. #26011290C2semidirect
4. #26011288C2semidirect
5. #563×C173C2semidirect
76. C173#2203semidirect
78. #563D173semidirect

Maximal subgroups

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

subgroupindex
#260116732
#260116632
#260112902
#260112882
#563×C1732
#2203173