Finite Group Databasebeta

← #36351876

Subgroups — #36351876

Order 22144.

264 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#19149637C211yesmaximal, characteristic
2#26011541C212yesmaximal, complemented
3#26011217C211yesmaximal, characteristic, complemented
4#26011214C212yesmaximal
5#324×C173C211yesmaximal, characteristic, complemented
6C22×C4×D173C411yescharacteristic
7#19149637—22no—
8#19149637—22no—
9C2×C4⋊Dic173C411yescharacteristic
10C2×#14583803C411yescharacteristic
11C2×#14583803C2211yescharacteristic
12C2×C4×Dic173C411yescharacteristic
13C2×C4×Dic173C2211yescharacteristic
14#19149608—22no—
15C22⋊#13063436C2212yes—
16C8×Dic173—22no—
17C2×C4⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yes—
19C23×D173C2×C411yescharacteristic
20C22×Dic173C2×C411yescharacteristic
21C22×Dic173C2×C411yescharacteristic
22C22×Dic173C2×C411yescharacteristic
23C2×C4×D173—21no—
24C2×C4×D173—21no—
25C2×C4×D173—22no—
26C2×C4×D173C814yescomplemented
27C2×#13063436—22no—
28C4⋊Dic173—22no—
29#14583803—22no—
30#14583803—22no—
31C4×Dic173—21no—
32C4×Dic173—21no—
33C4×Dic173D412yes—
34C4×Dic173D412yes—
35C22×C4×C173C2×C411yescharacteristic
36C22×C4×C173C2×C411yescharacteristic
37C22×C4×C173C2311yescharacteristic
38C2×C8×C173—22no—
39C4⋊C4×C173—22no—
40C22×D173—21no—
41C22×D173—21no—
42C22×D173—41no—
43C22×D173—41no—
44C22×D173C2×C812yes—
45C2×Dic173—21no—
46C2×Dic173—41no—
47C2×Dic173—41no—
48C2×Dic173C22⋊C412yes—
49C2×Dic173C22⋊C411yescharacteristic
50C2×Dic173C22⋊C411yescharacteristic
51C2×Dic173C2×C812yes—
52C2×Dic173M1612yes—
53C4×D173—24no—
54C4×D173—24no—
55#13063436—42no—
56C23×C173C22×C411yescharacteristic
57C2×C4×C173—21no—
58C2×C4×C173—21no—
59C2×C4×C173—21no—
60C2×C4×C173—41no—
61C2×C4×C173—41no—
62C2×C4×C173C2×C812yes—
63C2×C4×C173C2×D412yes—
64C8×C173—42no—
65C22×C173—21no—
66C22×C173—21no—
67C22×C173C2×C22⋊C411yescharacteristic
68C22×C173C22×C811yescharacteristic
69C22×C173C2×M1611yescharacteristic
70C2×D173—22no—
71C2×D173—42no—
72C2×D173—41no—
73C2×D173—41no—
74C2×D173—42no—
75C4×C173—22no—
76C4×C173—22no—
77C4×C173—41no—
78Dic173—22no—
79Dic173—41no—
80Dic173C22⋊C812yes—
81Dic173C22⋊C812yes—
82C2×C173—21no—
83C2×C173—21no—
84C2×C173C2×C22⋊C811yescharacteristic
85C2×C173#41011yescharacteristic
86C2×C173#41311yescharacteristic
87D173—42no—
88C173#145811yescharacteristic, complemented, Sylow
89#1458—1731nomaximal, Sylow
90C2×#117—1731no—
91#431—1732no—
92#324—1731no—
93#324D17311yescharacteristic, complemented
94#323—1732no—
95C23×C4—1731no—
96#117—3462no—
97#117—3462no—
98C2×C4⋊C4—1731no—
99C2×C4⋊C4C2×D17311yescharacteristic
100C2×C22⋊C4—1731no—
101C2×C22⋊C4—1731no—
102C2×C42—1731no—
103C2×C42—1731no—
104C4⋊C8—3462no—
105C22⋊C8—1732no—
106C22⋊C8C2×D17312yes—
107C4×C8—3462no—
108C24—1731no—
109C22×C4—3461no—
110C22×C4—3461no—
111C22×C4—3462no—
112C22×C4—1734no—
113C22×C4—1731no—
114C22×C4—1731no—
115C22×C4—1731no—
116C22×C4C4×D17311yescharacteristic
117C22×C4C4×D17311yescharacteristic
118C22×C4C22×D17311yescharacteristic
119C2×C8—22no—
120C2×C8—3462no—
121C4⋊C4—22no—
122C4⋊C4—3462no—
123C22⋊C4—3462no—
124C22⋊C4—3462no—
125C42—3461no—
126C42—3461no—
127C42—1732no—
128C42—1732no—
129C23—3461no—
130C23—3461no—
131C23—1732no—
132C23—6921no—
133C23—6921no—
134C23C2×C4×D17311yescharacteristic
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—3464no—
141C2×C4—3464no—
142C2×C4—3461no—
143C2×C4—1732no—
144C2×C4—1731no—
145C2×C4—1731no—
146C2×C4—1732no—
147C2×C4—1732no—
148C2×C4—6921no—
149C2×C4—6921no—
150C2×C4C8×D17312yes—
151C2×C4D4×D17312yes—
152C8—42no—
153C8—6922no—
154C22—21no—
155C22—21no—
156C22—3462no—
157C22—6922no—
158C22—6922no—
159C22—6921no—
160C22—6921no—
161C22C22⋊C4×D17311yescharacteristic
162C22C2×C8×D17311yescharacteristic
163C22M16×D17311yescharacteristic
164C4—22no—
165C4—22no—
166C4—41no—
167C4—3462no—
168C4—1732no—
169C4—1732no—
170C4—6921no—
171C2—21no—
172C2—21no—
173C2—6922no—
174C2C22⋊C8×D17311yescharacteristic
175C2#2601146311yescharacteristic
176C2#2601147411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC22×C173C22×C8
FrattiniC22×C4C22×D173
Fitting#324×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC23×D173C2×C4
cosocleC22×C4×C173C23
hypercenter#324D173
hypocenterC173#1458
LCSG ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #26011541C2semidirect
3. #26011217C2semidirect
5. #324×C173C2semidirect
26. C2×C4×D173C8semidirect
88. C173#1458semidirect
93. #324D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496372
#260115412
#260112172
#260112142
#324×C1732
#1458173