Finite Group Databasebeta

← #36352539

Subgroups — #36352539

Order 22144.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 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#26012122C211yesmaximal, characteristic
2#26011649C212yesmaximal
3#26011411C211yesmaximal, characteristic
4#26011229C212yesmaximal
5#324×C173C211yesmaximal, characteristic
6C22×Dic346C2211yescharacteristic
7C2×Dic692—22no—
8#19149678—22no—
9#19149670—22no—
10#19149658—21no—
11#19149658C2212yes—
12#19149638—21no—
13C2×#14583805—21no—
14#19149604—22no—
15#19149586—21no—
16#19149586C2211yescharacteristic
17C2×C4⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yes—
19C2×Dic346—21no—
20C2×Dic346—22no—
21C2×Dic346—42no—
22C2×Dic346D412yes—
23C22×Dic173—21no—
24C22×Dic173—21no—
25Dic692—44no—
26C4⋊Dic173—21no—
27C4⋊Dic173—21no—
28C4⋊Dic173—21no—
29C4⋊Dic173D412yes—
30#14583805—22no—
31#14583805—22no—
32#14583805—42no—
33C22⋊Dic173—22no—
34C4×Dic173—22no—
35C22×C4×C173D411yescharacteristic
36C22×C4×C173D411yescharacteristic
37C22×C4×C173C2311yescharacteristic
38C2×C8×C173—22no—
39C4⋊C4×C173—22no—
40Dic346—24no—
41Dic346—42no—
42Dic346—42no—
43Dic346—81no—
44C2×Dic173—22no—
45C2×Dic173—22no—
46C2×Dic173—22no—
47C2×Dic173—22no—
48C2×Dic173—41no—
49C2×Dic173—41no—
50C2×Dic173—41no—
51C2×Dic173—41no—
52C23×C173C2×D411yescharacteristic
53C2×C4×C173—21no—
54C2×C4×C173—21no—
55C2×C4×C173—21no—
56C2×C4×C173—41no—
57C2×C4×C173—41no—
58C2×C4×C173Q1612yescomplemented
59C2×C4×C173C2×D412yes—
60C8×C173—42no—
61C22×C173—21no—
62C22×C173—21no—
63C22×C173#12011yescharacteristic
64C22×C173C2×Q1611yescharacteristic
65C22×C173#13711yescharacteristic
66C4×C173—22no—
67C4×C173—22no—
68C4×C173—41no—
69Dic173—42no—
70Dic173—42no—
71Dic173—42no—
72Dic173—42no—
73C2×C173—21no—
74C2×C173—21no—
75C2×C173#45111yescharacteristic
76C2×C173#45611yescharacteristic
77C2×C173B(4,𝔽2)11yescharacteristic
78C173#160511yescharacteristic, complemented, Sylow
79#1605—1731nomaximal, Sylow
80#544—1731no—
81#451—1732no—
82#395—1731no—
83#328—1732no—
84#324D17311yescharacteristic
85C22×Q8—1731no—
86C2×Q16—3462no—
87C4⋊Q8—3462no—
88#125—3462no—
89C22⋊Q8—3461no—
90C22⋊Q8—1732no—
91#117—3461no—
92C2×C4⋊C4—3461no—
93C2×C4⋊C4C2×D17311yescharacteristic
94#103—3462no—
95C22⋊C8C2×D17312yes—
96#95—3461no—
97#95—1731no—
98C2×Q8—3461no—
99C2×Q8—3462no—
100C2×Q8—1732no—
101C2×Q8—6922no—
102C22×C4—3461no—
103C22×C4—3461no—
104C22×C4D69211yescharacteristic
105C22×C4D69211yescharacteristic
106C22×C4C22×D17311yescharacteristic
107Q16—6924no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—3461no—
111C4⋊C4—3461no—
112C4⋊C4—3461no—
113C4⋊C4—3462no—
114C4⋊C4—3462no—
115C4⋊C4—1732no—
116C4⋊C4—6922no—
117C22⋊C4—3462no—
118C42—3462no—
119C23C2×D69211yescharacteristic
120Q8—3464no—
121Q8—13841no—
122Q8—6922no—
123Q8—6922no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—3462no—
130C2×C4—3462no—
131C2×C4—3462no—
132C2×C4—3462no—
133C2×C4—6921no—
134C2×C4—6921no—
135C2×C4—6921no—
136C2×C4—6921no—
137C2×C4Dic69212yescomplemented
138C2×C4D4×D17312yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#1914964911yescharacteristic
143C22C2×Dic69211yescharacteristic
144C22#1914971211yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—6922no—
149C4—6922no—
150C4—6922no—
151C4—6922no—
152C2—21no—
153C2—21no—
154C2#2601164911yescharacteristic
155C2#2601167611yescharacteristic
156C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#324×C173C2
socleC22×C173#120
prime radicalC23×C173C2×D4
hypercenter#324D173
hypocenterC173#1605
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
58. C2×C4×C173Q16semidirect
78. C173#1605semidirect
137. C2×C4Dic692semidirect

Maximal subgroups

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

subgroupindex
#260121222
#260116492
#260114112
#260112292
#324×C1732
#1605173