Finite Group Databasebeta

← #36355576

Subgroups — #36355576

Order 22144.

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

normal subgroups only

This group has 226 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#26011680C212yesmaximal
2#26011668C211yesmaximal, characteristic
3#26011286C211yesmaximal, characteristic, complemented
4#26011281C212yesmaximal, complemented
5#563×C173C211yesmaximal, characteristic, complemented
6#19149724C2211yescharacteristic
7#19149710—21no—
8#19149662—22no—
9#19149607—21no—
10#19149596C2211yescharacteristic
11#19149594C2212yes—
12#19149593—22no—
13C4⋊Q8×C173—21no—
14#126×C173—22no—
15#125×C173—21no—
16#124×C173C2211yescharacteristic
17#123×C173C2212yes—
18C22⋊Q8×C173—22no—
19#14583826—21no—
20#14583825—41no—
21C2×C22⋊D173—22no—
22D4×D173—41no—
23#14583816—41no—
24#14583813—41no—
25#14583810—21no—
26#14583805—42no—
27C22⋊Dic173—22no—
28#14583803—42no—
29C2×Q8×C173—21no—
30C2×Q8×C173D411yescharacteristic
31C2×D4×C173C2311yescharacteristic
32C22×C4×C173D412yes—
33C4⋊C4×C173—22no—
34C4⋊C4×C173—21no—
35C4⋊C4×C173—22no—
36C4⋊C4×C173—21no—
37C4⋊C4×C173—22no—
38C22⋊C4×C173—22no—
39C22⋊C4×C173—22no—
40C22⋊C4×C173D412yes—
41C42×C173—21no—
42C42×C173D411yescharacteristic
43C22×D173—22no—
44Dic346—21no—
45D692—21no—
46C22⋊D173—42no—
47C22⋊D173—41no—
48C2×Dic173—22no—
49C2×Dic173—42no—
50C4×D173—41no—
51#13063436—41no—
52C23×C173C2×D412yes—
53Q8×C173—41no—
54Q8×C173—81no—
55D4×C173—41no—
56C2×C4×C173—21no—
57C2×C4×C173—22no—
58C2×C4×C173—21no—
59C2×C4×C173—22no—
60C2×C4×C173—42no—
61C2×C4×C173C2×D411yescharacteristic
62C22×C173—22no—
63C22×C173—42no—
64C22×C173#12011yescharacteristic
65C2×D173—41no—
66C2×D173—82no—
67C4×C173—21no—
68C4×C173—42no—
69C4×C173—41no—
70C4×C173—42no—
71C4×C173—41no—
72Dic173—41no—
73Dic173—82no—
74C2×C173—21no—
75C2×C173—42no—
76C2×C173B(4,𝔽2)11yescharacteristic
77D173—81no—
78C173#220111yescharacteristic, complemented, Sylow
79#2201—1731nomaximal, Sylow
80#563D17311yescharacteristic, complemented
81#458—1732no—
82#455—1731no—
83#354—1731no—
84#352—1732no—
85H(5,𝔽2)—1731no—
86#137—3461no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D17311yescharacteristic
91#123—3462no—
92#123C2×D17312yes—
93C22⋊Q8—22no—
94#104—3461no—
95#100—1731no—
96#99—3462no—
97#99—1732no—
98#41—3461no—
99#41—6921no—
100C2×Q8—21no—
101C2×Q8C22⋊D17311yescharacteristic
102C2×D4—3462no—
103C2×D4—6921no—
104C2×D4C22×D17311yescharacteristic
105C22×C4C22⋊D17312yes—
106Q16—6921no—
107SD16—6921no—
108M16—3461no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—6922no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—3462no—
118C22⋊C4—6922no—
119C22⋊C4C22⋊D17312yes—
120C42—21no—
121C42C22⋊D17311yescharacteristic
122C23—3462no—
123C23C2×C22⋊D17312yes—
124Q8—41no—
125Q8—81no—
126Q8—3461no—
127D4—41no—
128D4—3461no—
129D4—6922no—
130D4—6921no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—3462no—
137C2×C4—6921no—
138C2×C4—6922no—
139C2×C4C2×C22⋊D17311yescharacteristic
140C8—6921no—
141C22—22no—
142C22—42no—
143C22—13842no—
144C22—6921no—
145C22#1914964811yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—13842no—
152C4—6921no—
153C2—21no—
154C2—42no—
155C2—13841no—
156C2#2601167811yescharacteristic
subgroupG/H
centerC2#26011678
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#563×C173C2
socleC2×C173B(4,𝔽2)
prime radical#19149724C22
hypercenter#563D173
hypocenterC173#2201
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #26011286C2semidirect
4. #26011281C2semidirect
5. #563×C173C2semidirect
78. C173#2201semidirect
80. #563D173semidirect

Maximal subgroups

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

subgroupindex
#260116802
#260116682
#260112862
#260112812
#563×C1732
#2201173