Finite Group Databasebeta

← #36352599

Subgroups — #36352599

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#26011809C212yesmaximal, complemented
3#26011224C212yesmaximal, complemented
4#26011217C211yesmaximal, characteristic, complemented
5#400×C173C211yesmaximal, characteristic, complemented
6C2×#14583826—21no—
7#19149659—22no—
8#19149654—22no—
9#19149653—21no—
10#19149653C2212yes—
11#19149610—22no—
12#19149599—22no—
13C22⋊#13063436C2212yes—
14C2×C4⋊C4×C173C2211yescharacteristic
15C2×C4⋊C4×C173C2211yescharacteristic
16#95×C173—21no—
17#95×C173—21no—
18#95×C173C2211yescharacteristic
19#14583826—42no—
20#14583826—42no—
21C2×Dic346—21no—
22C2×D692—21no—
23C2×D692—21no—
24C2×D692D412yes—
25C2×C22⋊D173—22no—
26C2×C4×D173—22no—
27C2×#13063436—22no—
28C4⋊Dic173—21no—
29C4⋊Dic173D412yes—
30#14583803—22no—
31#14583803—22no—
32C22×C4×C173—21no—
33C22×C4×C173—21no—
34C22×C4×C173D411yescharacteristic
35C22×C4×C173D411yescharacteristic
36C22×C4×C173C2311yescharacteristic
37C4⋊C4×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—
47C4×D173—44no—
48#13063436—42no—
49C23×C173C2×D411yescharacteristic
50C2×C4×C173—21no—
51C2×C4×C173—21no—
52C2×C4×C173—21no—
53C2×C4×C173—41no—
54C2×C4×C173—41no—
55C2×C4×C173—41no—
56C2×C4×C173—41no—
57C2×C4×C173—42no—
58C2×C4×C173—41no—
59C2×C4×C173—41no—
60C2×C4×C173SD1612yes—
61C2×C4×C173C2×D412yes—
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—
73C4×C173—81no—
74C4×C173—81no—
75Dic173—42no—
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#400D17311yescharacteristic, complemented
87#327—1732no—
88#324—1731no—
89C2×#41—3461no—
90C22⋊Q8—3462no—
91#121—3462no—
92#121—3461no—
93#121—1732no—
94C2×C4⋊C4C2×D17311yescharacteristic
95C2×C4⋊C4C2×D17311yescharacteristic
96#106—3462no—
97#102—3462no—
98C22⋊C8—1732no—
99#95—21no—
100#95—21no—
101#95C2×D17311yescharacteristic
102#41—6922no—
103#41—6922no—
104C2×Q8—3461no—
105C2×D4—3461no—
106C2×D4—3461no—
107C2×D4—3462no—
108C2×D4—1732no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—3462no—
112C22×C4C22⋊D17311yescharacteristic
113C22×C4C22⋊D17311yescharacteristic
114C22×C4C22×D17311yescharacteristic
115C2×C8—3462no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—3461no—
119C4⋊C4—1732no—
120C22⋊C4—3462no—
121C22⋊C4—3462no—
122C23—3462no—
123C23C2×C22⋊D17311yescharacteristic
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—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—3462no—
141C2×C4—6924no—
142C2×C4#1458381312yes—
143C2×C4D4×D17312yes—
144C8—6922no—
145C22—21no—
146C22—21no—
147C22—6922no—
148C22—6922no—
149C22—6922no—
150C22#1914964711yescharacteristic
151C22C2×#1458381311yescharacteristic
152C22#1914970411yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—81no—
157C4—81no—
158C4—6922no—
159C2—21no—
160C2—21no—
161C2—13842no—
162C2#2601164111yescharacteristic
163C2#2601166311yescharacteristic
164C2#2601168011yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#400×C173C2
socleC22×C173#120
prime radical#26012072C2
hypercenter#400D173
hypocenterC173#1616
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
2. #26011809C2semidirect
3. #26011224C2semidirect
4. #26011217C2semidirect
5. #400×C173C2semidirect
82. C173#1616semidirect
86. #400D173semidirect

Maximal subgroups

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

subgroupindex
#260120722
#260118092
#260112242
#260112172
#400×C1732
#1616173