Finite Group Databasebeta

← #36352568

Subgroups — #36352568

Order 22144.

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

normal subgroups only

This group has 254 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#26011805C212yesmaximal, complemented
3#26011421C211yesmaximal, 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
12C2×#14583805—21no—
13#19149613—22no—
14#19149601—22no—
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#14583805—42no—
33#14583803—22no—
34#14583803—22no—
35C22×C4×C173D411yescharacteristic
36C22×C4×C173D411yescharacteristic
37C22×C4×C173C2311yescharacteristic
38C2×C8×C173—22no—
39C4⋊C4×C173—22no—
40C22×D173—22no—
41Dic346—42no—
42D692—42no—
43D692—42no—
44D692—81no—
45C22⋊D173—42no—
46C22⋊D173—42no—
47C2×Dic173—22no—
48C2×Dic173—22no—
49C2×Dic173—41no—
50C2×Dic173—41no—
51C2×Dic173—41no—
52C2×Dic173—41no—
53C2×Dic173—41no—
54C4×D173—44no—
55C23×C173C2×D411yescharacteristic
56C2×C4×C173—21no—
57C2×C4×C173—21no—
58C2×C4×C173—21no—
59C2×C4×C173—41no—
60C2×C4×C173—41no—
61C2×C4×C173D812yes—
62C2×C4×C173C2×D412yes—
63C8×C173—42no—
64C22×C173—21no—
65C22×C173—21no—
66C22×C173#12011yescharacteristic
67C22×C173C2×D811yescharacteristic
68C22×C173AGL(1,ℤ/8)11yescharacteristic
69C2×D173—42no—
70C2×D173—42no—
71C2×D173—42no—
72C4×C173—22no—
73C4×C173—22no—
74C4×C173—41no—
75Dic173—42no—
76Dic173—42no—
77Dic173—81no—
78C2×C173—21no—
79C2×C173—21no—
80C2×C173#44711yescharacteristic
81C2×C173#45611yescharacteristic
82C2×C173#45811yescharacteristic
83D173—82no—
84C173#161011yescharacteristic, complemented, Sylow
85#1610—1731nomaximal, Sylow
86#537—1731no—
87#480—1732no—
88#398—1731no—
89#327—1732no—
90#324D17311yescharacteristic, complemented
91C2×#41—3461no—
92C22⋊Q8—3462no—
93#121—3462no—
94#121—3461no—
95#121—1732no—
96C2×C4⋊C4—3461no—
97C2×C4⋊C4—1731no—
98C2×C4⋊C4C2×D17311yescharacteristic
99#107—3462no—
100#102—3462no—
101C22⋊C8C2×D17312yes—
102#95—3461no—
103#95—1731no—
104#41—6922no—
105#41—6922no—
106C2×Q8—3461no—
107C2×D4—3461no—
108C2×D4—3461no—
109C2×D4—3462no—
110C2×D4—1732no—
111C22×C4—3462no—
112C22×C4—3461no—
113C22×C4—3461no—
114C22×C4D69211yescharacteristic
115C22×C4D69211yescharacteristic
116C22×C4C22×D17311yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—3461no—
120C4⋊C4—3462no—
121C4⋊C4—1732no—
122C4⋊C4—6922no—
123C22⋊C4—3462no—
124C22⋊C4—3462no—
125C23—3462no—
126C23C2×D69211yescharacteristic
127Q8—6922no—
128D4—13841no—
129D4—6922no—
130D4—6922no—
131D4—6922no—
132D4—6922no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—3462no—
139C2×C4—3462no—
140C2×C4—6924no—
141C2×C4—6921no—
142C2×C4—6921no—
143C2×C4—6921no—
144C2×C4—6921no—
145C2×C4—6921no—
146C2×C4D138412yes—
147C2×C4D4×D17312yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—6922no—
152C22—6922no—
153C22—6922no—
154C22#1914964911yescharacteristic
155C22C2×D138411yescharacteristic
156C22#1914970711yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—13841no—
161C4—6922no—
162C4—6922no—
163C2—21no—
164C2—21no—
165C2—13842no—
166C2#2601162711yescharacteristic
167C2#2601167611yescharacteristic
168C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#324×C173C2
socleC22×C173#120
prime radical#26012072C2
hypercenter#324D173
hypocenterC173#1610
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. #26011805C2semidirect
3. #26011421C2semidirect
4. #26011226C2semidirect
5. #324×C173C2semidirect
84. C173#1610semidirect
90. #324D173semidirect

Maximal subgroups

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

subgroupindex
#260120722
#260118052
#260114212
#260112262
#324×C1732
#1610173