Finite Group Databasebeta

← #36352531

Subgroups — #36352531

Order 22144.

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

normal subgroups only

This group has 282 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#26011630C212yesmaximal, complemented
3#26011224C212yesmaximal, complemented
4#26011217C211yesmaximal, characteristic, complemented
5#395×C173C211yesmaximal, characteristic, complemented
6C2×#14583826—21no—
7C2×#14583813—22no—
8#19149659—22no—
9#19149654—22no—
10#19149653—21no—
11#19149653C2212yes—
12#19149603—22no—
13C22⋊#13063436C2212yes—
14C22×Q8×C173C2211yescharacteristic
15C2×C4⋊C4×C173—21no—
16C2×C4⋊C4×C173C2211yescharacteristic
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—
27#14583813—44no—
28C2×#13063436—22no—
29C4⋊Dic173—21no—
30C4⋊Dic173D412yes—
31#14583803—22no—
32#14583803—22no—
33C2×Q8×C173—22no—
34C2×Q8×C173—42no—
35C22×C4×C173—21no—
36C22×C4×C173—21no—
37C22×C4×C173D411yescharacteristic
38C22×C4×C173D411yescharacteristic
39C22×C4×C173C2311yescharacteristic
40C4⋊C4×C173—22no—
41C4⋊C4×C173—42no—
42C22×D173—22no—
43Dic346—42no—
44D692—42no—
45D692—42no—
46D692—81no—
47C22⋊D173—42no—
48C22⋊D173—42no—
49C2×Dic173—22no—
50C4×D173—44no—
51#13063436—42no—
52C23×C173C2×D411yescharacteristic
53Q8×C173—24no—
54Q8×C173—42no—
55C2×C4×C173—21no—
56C2×C4×C173—21no—
57C2×C4×C173—22no—
58C2×C4×C173—22no—
59C2×C4×C173—21no—
60C2×C4×C173—41no—
61C2×C4×C173—41no—
62C2×C4×C173—41no—
63C2×C4×C173—41no—
64C2×C4×C173—41no—
65C2×C4×C173—41no—
66C2×C4×C173SD1612yescomplemented
67C2×C4×C173C2×D412yes—
68C22×C173—21no—
69C22×C173—21no—
70C22×C173#12011yescharacteristic
71C22×C173C2×SD1611yescharacteristic
72C22×C173AGL(1,ℤ/8)11yescharacteristic
73C2×D173—42no—
74C2×D173—42no—
75C2×D173—42no—
76C4×C173—22no—
77C4×C173—22no—
78C4×C173—42no—
79C4×C173—42no—
80C4×C173—41no—
81Dic173—42no—
82C2×C173—21no—
83C2×C173—21no—
84C2×C173#45011yescharacteristic
85C2×C173#45611yescharacteristic
86C2×C173B(4,𝔽2)11yescharacteristic
87D173—82no—
88C173#160411yescharacteristic, complemented, Sylow
89#1604—1731nomaximal, Sylow
90#537—1731no—
91#448—1732no—
92#395D17311yescharacteristic, complemented
93#327—1732no—
94#324—1731no—
95C2×#41—3461no—
96C22×Q8C2×D17311yescharacteristic
97C2×SD16—3462no—
98C22⋊Q8—3462no—
99#121—3462no—
100#121—3461no—
101#121—1732no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C2×D17311yescharacteristic
104#103—3462no—
105C22⋊C8—1732no—
106#95—21no—
107#95C2×D17311yescharacteristic
108#41—6922no—
109#41—6922no—
110C2×Q8—22no—
111C2×Q8—42no—
112C2×Q8—3461no—
113C2×D4—3461no—
114C2×D4—3461no—
115C2×D4—3462no—
116C2×D4—1732no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—3462no—
120C22×C4C22⋊D17311yescharacteristic
121C22×C4C22⋊D17311yescharacteristic
122C22×C4C22×D17311yescharacteristic
123SD16—6924no—
124C2×C8—3462no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—3461no—
128C4⋊C4—1732no—
129C22⋊C4—3462no—
130C22⋊C4—3462no—
131C23—3462no—
132C23C2×C22⋊D17311yescharacteristic
133Q8—24no—
134Q8—42no—
135Q8—6922no—
136D4—13841no—
137D4—6922no—
138D4—6922no—
139D4—6922no—
140D4—6922no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—3462no—
153C2×C4—6924no—
154C2×C4#1458381312yescomplemented
155C2×C4D4×D17312yes—
156C8—6922no—
157C22—21no—
158C22—21no—
159C22—6922no—
160C22—6922no—
161C22—6922no—
162C22#1914964711yescharacteristic
163C22C2×#1458381311yescharacteristic
164C22#1914970411yescharacteristic
165C4—22no—
166C4—22no—
167C4—42no—
168C4—42no—
169C4—41no—
170C4—6922no—
171C2—21no—
172C2—21no—
173C2—13842no—
174C2#2601164111yescharacteristic
175C2#2601167311yescharacteristic
176C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#395×C173C2
socleC22×C173#120
prime radical#26012072C2
hypercenter#395D173
hypocenterC173#1604
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
2. #26011630C2semidirect
3. #26011224C2semidirect
4. #26011217C2semidirect
5. #395×C173C2semidirect
66. C2×C4×C173SD16semidirect
88. C173#1604semidirect
92. #395D173semidirect
154. C2×C4#14583813semidirect

Maximal subgroups

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

subgroupindex
#260120722
#260116302
#260112242
#260112172
#395×C1732
#1604173