Finite Group Databasebeta

← #36352602

Subgroups — #36352602

Order 22144.

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

normal subgroups only

This group has 246 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#26012072C211yesmaximal, characteristic
2#26011810C212yesmaximal, complemented
3#26011430C211yesmaximal, characteristic, complemented
4#26011226C212yesmaximal, complemented
5#324×C173C211yesmaximal, characteristic, complemented
6C2×#14583826—21no—
7#19149659—22no—
8#19149654—22no—
9#19149653—21no—
10#19149653C2212yes—
11C2×C4⋊Dic173C2211yescharacteristic
12#19149611—22no—
13#19149601—22no—
14#19149586—21no—
15#19149586—21no—
16#19149586C2211yescharacteristic
17C2×C4⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yes—
19#14583826—42no—
20#14583826—42no—
21C2×Dic346—21no—
22C2×D692—21no—
23C2×D692—21no—
24C2×D692D412yes—
25C2×C22⋊D173—22no—
26C22×Dic173—21no—
27C22×Dic173—21no—
28C2×C4×D173—22no—
29C4⋊Dic173—21no—
30C4⋊Dic173—22no—
31C4⋊Dic173D412yes—
32#14583803—22no—
33#14583803—22no—
34C22×C4×C173D411yescharacteristic
35C22×C4×C173D411yescharacteristic
36C22×C4×C173C2311yescharacteristic
37C2×C8×C173—22no—
38C4⋊C4×C173—22no—
39C22×D173—22no—
40Dic346—42no—
41D692—42no—
42D692—42no—
43D692—81no—
44C22⋊D173—42no—
45C22⋊D173—42no—
46C2×Dic173—22no—
47C2×Dic173—41no—
48C2×Dic173—41no—
49C2×Dic173—42no—
50C2×Dic173—41no—
51C2×Dic173—41no—
52C4×D173—44no—
53C23×C173C2×D411yescharacteristic
54C2×C4×C173—21no—
55C2×C4×C173—21no—
56C2×C4×C173—21no—
57C2×C4×C173—41no—
58C2×C4×C173—41no—
59C2×C4×C173SD1612yes—
60C2×C4×C173C2×D412yes—
61C8×C173—42no—
62C22×C173—21no—
63C22×C173—21no—
64C22×C173#12011yescharacteristic
65C22×C173C2×SD1611yescharacteristic
66C22×C173AGL(1,ℤ/8)11yescharacteristic
67C2×D173—42no—
68C2×D173—42no—
69C2×D173—42no—
70C4×C173—22no—
71C4×C173—22no—
72C4×C173—41no—
73Dic173—42no—
74Dic173—81no—
75Dic173—81no—
76C2×C173—21no—
77C2×C173—21no—
78C2×C173#45011yescharacteristic
79C2×C173#45411yescharacteristic
80C2×C173#45811yescharacteristic
81D173—82no—
82C173#161611yescharacteristic, complemented, Sylow
83#1616—1731nomaximal, Sylow
84#537—1731no—
85#481—1732no—
86#400—1731no—
87#327—1732no—
88#324D17311yescharacteristic, complemented
89C2×#41—3461no—
90C22⋊Q8—3462no—
91#121—3462no—
92#121—3461no—
93#121—1732no—
94C2×C4⋊C4—1731no—
95C2×C4⋊C4C2×D17311yescharacteristic
96#106—3462no—
97#102—3462no—
98C22⋊C8C2×D17312yes—
99#95—3461no—
100#95—3461no—
101#95—1731no—
102#41—6922no—
103#41—6922no—
104C2×Q8—3461no—
105C2×D4—3461no—
106C2×D4—3461no—
107C2×D4—3462no—
108C2×D4—1732no—
109C22×C4—3462no—
110C22×C4—3461no—
111C22×C4—3461no—
112C22×C4D69211yescharacteristic
113C22×C4D69211yescharacteristic
114C22×C4C22×D17311yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—3461no—
118C4⋊C4—3462no—
119C4⋊C4—1732no—
120C22⋊C4—3462no—
121C22⋊C4—3462no—
122C23—3462no—
123C23C2×D69211yescharacteristic
124Q8—6922no—
125D4—13841no—
126D4—6922no—
127D4—6922no—
128D4—6922no—
129D4—6922no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—3462no—
136C2×C4—6924no—
137C2×C4—6921no—
138C2×C4—6921no—
139C2×C4—6922no—
140C2×C4—6921no—
141C2×C4—6921no—
142C2×C4#1458381512yes—
143C2×C4D4×D17312yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—6922no—
148C22—6922no—
149C22—6922no—
150C22#1914964911yescharacteristic
151C22C2×#1458381511yescharacteristic
152C22#1914970711yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—13841no—
157C4—13841no—
158C4—6922no—
159C2—21no—
160C2—21no—
161C2—13842no—
162C2#2601164411yescharacteristic
163C2#2601166611yescharacteristic
164C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#324×C173C2
socleC22×C173#120
prime radical#26012072C2
hypercenter#324D173
hypocenterC173#1616
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
2. #26011810C2semidirect
3. #26011430C2semidirect
4. #26011226C2semidirect
5. #324×C173C2semidirect
82. C173#1616semidirect
88. #324D173semidirect

Maximal subgroups

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

subgroupindex
#260120722
#260118102
#260114302
#260112262
#324×C1732
#1616173