Finite Group Databasebeta

← #36368999

Subgroups — #36368999

Order 22144.

340 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 70 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
1D173⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
2#26012433C211yesmaximal, characteristic, complemented
3#26012342C214yesmaximal, complemented
4C2×#19149650C211yesmaximal, characteristic
5C2×D173⋊C8—21no—
6C2×C4⋊#10621924—21no—
7C2×C22⋊#10621924—21no—
8#19149736C2212yescomplemented
9C22⋊#13063443—21no—
10(C2×#10621924)⋊C4C2212yescomplemented
11C22×C22⋊D173—21no—
12C22×C4×D173C2211yescharacteristic
13#19149650—21no—
14#19149650—21no—
15#19149650C412yescomplemented
16#19149650C2212yes—
17C2×C4⋊Dic173C411yescharacteristic, complemented
18C2×C22⋊Dic173—21no—
19C22×D4×C173C411yescharacteristic, complemented
20C22×#10621924—22no—
21D173⋊C8—42no—
22C2×#13063443—22no—
23C4⋊#10621924—42no—
24C22⋊#10621924—42no—
25C23×D173D411yescharacteristic
26C2×C22⋊D173—22no—
27C2×C22⋊D173—22no—
28C2×C22⋊D173—42no—
29C2×C22⋊D173—42no—
30C22×Dic173—21no—
31C22×Dic173D411yescharacteristic
32C2×C4×D173—42no—
33C2×C4×D173C2×C411yescharacteristic
34C2×C4×D173D411yescharacteristic
35C2×C4×D173D411yescharacteristic
36C2×C4×D173C2311yescharacteristic
37C4⋊Dic173—21no—
38C4⋊Dic173C2×C412yes—
39C22⋊Dic173—22no—
40C22⋊Dic173—22no—
41C24×C173—21no—
42C2×D4×C173—21no—
43C2×D4×C173—42no—
44C2×D4×C173C2×C412yes—
45C22×C4×C173C2×C411yescharacteristic
46C2×#10621924—42no—
47C2×#10621924—42no—
48C2×#10621924—42no—
49#13063443—42no—
50C22×D173—41no—
51C22×D173—41no—
52C22×D173C22⋊C411yescharacteristic
53C22×D173D812yes—
54C22×D173SD1612yes—
55C22×D173C2×D411yescharacteristic
56C22⋊D173—44no—
57C22⋊D173—42no—
58C22⋊D173—42no—
59C2×Dic173—22no—
60C2×Dic173—41no—
61C2×Dic173—42no—
62C2×Dic173C22⋊C411yescharacteristic
63C2×Dic173C2×D411yescharacteristic
64C4×D173—22no—
65C4×D173—22no—
66C4×D173—41no—
67C4×D173—41no—
68C23×C173—22no—
69C23×C173—42no—
70C23×C173—42no—
71C23×C173—42no—
72C23×C173C22⋊C411yescharacteristic
73D4×C173—42no—
74D4×C173—42no—
75C2×C4×C173—22no—
76C2×C4×C173C22⋊C411yescharacteristic
77C2×C4×C173C22×C411yescharacteristic
78C22×C173—21no—
79C22×C173—21no—
80C22×C173—42no—
81C22×C173—42no—
82C22×C173—42no—
83C22×C173—84no—
84C22×C173#10212yes—
85C22×C173C2×C22⋊C411yescharacteristic
86C2×D173—21no—
87C2×D173—21no—
88C2×D173—21no—
89C2×D173—21no—
90C2×D173—42no—
91C2×D173—42no—
92C2×D173#10212yes—
93C2×D173C2×D811yescharacteristic
94C2×D173C2×SD1611yescharacteristic
95#10621924—82no—
96C4×C173—21no—
97C4×C173#9912yes—
98Dic173—21no—
99Dic173—21no—
100Dic173—42no—
101C2×C173—22no—
102C2×C173—82no—
103C2×C173C2×#9911yescharacteristic
104C2×C173C2×#10211yescharacteristic
105C2×C173#42111yescharacteristic
106D173—21no—
107D173—22no—
108D173—21no—
109C173#151911yescharacteristic, complemented, Sylow
110#1519—1731nomaximal, Sylow
111C2×#121—1731no—
112C2×C22⋊C8—1731no—
113#380—1731no—
114#327—1734no—
115C22×D4—3461no—
116C22×D4#1062192411yescharacteristic, complemented
117C23×C4—1731no—
118C22×C8—3461no—
119#121—3461no—
120#121—3461no—
121#121—1732no—
122#121—1732no—
123C2×C4⋊C4—3461no—
124C2×C4⋊C4—1731no—
125C2×C22⋊C4—3461no—
126C2×C22⋊C4—3461no—
127C22⋊C8—3461no—
128C22⋊C8—1732no—
129#95—1732no—
130C24—21no—
131C24—1731no—
132C2×D4—21no—
133C2×D4—42no—
134C2×D4—3462no—
135C2×D4—3462no—
136C2×D4—6922no—
137C2×D4—6922no—
138C2×D4C2×#1062192412yes—
139C22×C4—3461no—
140C22×C4—3462no—
141C22×C4—1731no—
142C22×C4—1731no—
143C22×C4—1731no—
144C22×C4—1731no—
145C22×C4—1731no—
146C22×C4—6922no—
147C22×C4C2×#1062192411yescharacteristic
148C2×C8—3462no—
149C2×C8—6922no—
150C4⋊C4—3461no—
151C4⋊C4—1732no—
152C4⋊C4—6922no—
153C22⋊C4—3462no—
154C22⋊C4—3462no—
155C22⋊C4—6922no—
156C23—22no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—1731no—
161C23—1731no—
162C23—1732no—
163C23—1732no—
164C23—6921no—
165C23—6921no—
166C23C22⋊#1062192411yescharacteristic
167D4—42no—
168D4—42no—
169D4—6922no—
170D4—6924no—
171D4—6922no—
172C2×C4—22no—
173C2×C4—3462no—
174C2×C4—3462no—
175C2×C4—3462no—
176C2×C4—1731no—
177C2×C4—1731no—
178C2×C4—6921no—
179C2×C4—6921no—
180C2×C4—6921no—
181C2×C4—6922no—
182C2×C4—6922no—
183C2×C4—6922no—
184C2×C4—6922no—
185C2×C4C22⋊#1062192411yescharacteristic
186C2×C4C22×#1062192411yescharacteristic
187C8—6922no—
188C22—21no—
189C22—21no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—84no—
194C22—3461no—
195C22—3461no—
196C22—3461no—
197C22—3461no—
198C22—1731no—
199C22—1732no—
200C22—1731no—
201C22—6922no—
202C22—6922no—
203C22#1914974112yes—
204C22C2×C22⋊#1062192411yescharacteristic
205C4—21no—
206C4—3461no—
207C4—3461no—
208C4—13842no—
209C4—6922no—
210C4#1914973712yes—
211C2—22no—
212C2—82no—
213C2—3461no—
214C2—3462no—
215C2—3461no—
216C2C2×#1914973711yescharacteristic
217C2C2×#1914974111yescharacteristic
218C2#2601248911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22×D4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×#19149650C2
cosocleC2×C4×D173C23
hypercenterC22×D4#10621924
hypocenterC173#1519
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. D173⋊(C22⋊C8)C2semidirect
2. #26012433C2semidirect
3. #26012342C2semidirect
8. #19149736C22semidirect
10. (C2×#10621924)⋊C4C22semidirect
15. #19149650C4semidirect
17. C2×C4⋊Dic173C4semidirect
19. C22×D4×C173C4semidirect
109. C173#1519semidirect
116. C22×D4#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊(C22⋊C8)2
#260124332
#260123422
C2×#191496502
#1519173