Finite Group Databasebeta

← #36368942

Subgroups — #36368942

Order 22144.

284 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 68 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#26012509C211yesmaximal, characteristic, complemented
2C4×C4⋊#10621924C211yesmaximal, characteristic, complemented
3#26012343C212yesmaximal, complemented
4#26012342C212yesmaximal, complemented
5#26012038C211yesmaximal, characteristic
6C2×C4⋊#10621924—21no—
7C2×C4×#10621924—21no—
8#19149746—21no—
9#19149736C2212yes—
10C4×#13063443—21no—
11(C2×#10621924)⋊C4C2212yes—
12C2×#14583826—21no—
13#19149678C411yescharacteristic, complemented
14#19149675C411yescharacteristic, complemented
15#19149665—21no—
16#19149654C2211yescharacteristic
17#19149652—21no—
18#19149650C2211yescharacteristic
19C4×D692C411yescharacteristic, complemented
20C4×C22⋊D173—21no—
21C42×D173C2211yescharacteristic
22C4×D4×C173C411yescharacteristic, complemented
23C22×#10621924—22no—
24C2×#13063443—22no—
25C4⋊#10621924—24no—
26C4×#10621924—42no—
27#14583826—42no—
28#14583826—42no—
29C2×Dic346—21no—
30C2×D692—21no—
31C2×D692C2×C411yescharacteristic
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C2×C22⋊D173—21no—
35C2×C4×D173—21no—
36C2×C4×D173D411yescharacteristic
37C2×C4×D173D411yescharacteristic
38C2×C4×D173C2311yescharacteristic
39C4⋊Dic173C2×C411yescharacteristic
40#14583805—21no—
41C22⋊Dic173—21no—
42#14583803—21no—
43#14583803—21no—
44C4×Dic173C2×C411yescharacteristic
45C4×Dic173D411yescharacteristic
46C4×Dic173D411yescharacteristic
47C2×D4×C173C2×C411yescharacteristic
48C22×C4×C173—21no—
49C4⋊C4×C173C2×C411yescharacteristic
50C22⋊C4×C173—21no—
51C42×C173C2×C411yescharacteristic
52C2×#10621924—24no—
53C2×#10621924—42no—
54C2×#10621924—42no—
55#13063443—42no—
56C22×D173—21no—
57C22×D173C2×D411yescharacteristic
58Dic346—42no—
59D692—41no—
60D692—41no—
61D692—41no—
62C22⋊D173—41no—
63C22⋊D173—41no—
64C22⋊D173—42no—
65C22⋊D173—42no—
66C2×Dic173—21no—
67C2×Dic173C22⋊C411yescharacteristic
68C2×Dic173C22⋊C411yescharacteristic
69C2×Dic173C2×D411yescharacteristic
70C4×D173—22no—
71C4×D173—41no—
72C4×D173—42no—
73C4×D173D812yes—
74C4×D173SD1612yes—
75C23×C173—21no—
76D4×C173—41no—
77C2×C4×C173—21no—
78C2×C4×C173—42no—
79C2×C4×C173C22⋊C411yescharacteristic
80C2×C4×C173C22⋊C411yescharacteristic
81C2×C4×C173C22×C411yescharacteristic
82C22×C173—41no—
83C22×C173—41no—
84C22×C173—41no—
85C22×C173C2×C22⋊C411yescharacteristic
86C2×D173—21no—
87C2×D173—21no—
88C2×D173—41no—
89C2×D173—41no—
90C2×D173—41no—
91C2×D173C2×D811yescharacteristic
92C2×D173C2×SD1611yescharacteristic
93#10621924—44no—
94C4×C173—21no—
95C4×C173—21no—
96C4×C173—41no—
97C4×C173#10212yes—
98Dic173—21no—
99Dic173—41no—
100Dic173#10212yes—
101Dic173#10412yes—
102C2×C173—81no—
103C2×C173#41011yescharacteristic
104C2×C173C2×#10211yescharacteristic
105C2×C173C2×#10411yescharacteristic
106D173—21no—
107D173—21no—
108D173—81no—
109C173#150211yescharacteristic, complemented, Sylow
110#1502—1731nomaximal, Sylow
111#532—1731no—
112#431—1731no—
113C4×C4⋊C4—1731no—
114#327—1732no—
115#327—1732no—
116C2×#41—3461no—
117C4⋊Q8—1731no—
118#127—1731no—
119#124—3461no—
120#121—3461no—
121#121—1731no—
122#121—1731no—
123C4×D4—3461no—
124C4×D4—1731no—
125C4×D4#1062192411yescharacteristic, complemented
126C2×C4⋊C4—3461no—
127C2×C42—3461no—
128C2×C42—1731no—
129C4⋊C8—3461no—
130C22⋊C8—1732no—
131C4×C8—3461no—
132#95—1732no—
133#41—6922no—
134#41—6922no—
135C2×Q8—3461no—
136C2×D4—3461no—
137C2×D4—3461no—
138C2×D4—3461no—
139C2×D4—3461no—
140C2×D4—1731no—
141C2×D4C2×#1062192411yescharacteristic
142C22×C4—21no—
143C22×C4—3462no—
144C22×C4—3461no—
145C22×C4—1731no—
146C22×C4—1731no—
147C22×C4—1731no—
148C2×C8—3462no—
149C4⋊C4—3461no—
150C4⋊C4—3464no—
151C4⋊C4—1731no—
152C4⋊C4C2×#1062192411yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—3461no—
155C22⋊C4—3461no—
156C22⋊C4—3461no—
157C42—1731no—
158C42—1731no—
159C42—1731no—
160C42—6922no—
161C42C2×#1062192411yescharacteristic
162C23—21no—
163C23—3461no—
164C23—1731no—
165Q8—6922no—
166D4—41no—
167D4—6921no—
168D4—6921no—
169D4—6921no—
170D4—6922no—
171D4—6922no—
172D4—6921no—
173D4—6921no—
174C2×C4—21no—
175C2×C4—42no—
176C2×C4—3462no—
177C2×C4—3464no—
178C2×C4—3461no—
179C2×C4—1731no—
180C2×C4—1731no—
181C2×C4—1731no—
182C2×C4—1732no—
183C2×C4—1732no—
184C2×C4—6921no—
185C2×C4—6922no—
186C2×C4—6922no—
187C2×C4—6922no—
188C2×C4C22⋊#1062192411yescharacteristic
189C2×C4C22⋊#1062192411yescharacteristic
190C2×C4C22×#1062192411yescharacteristic
191C8—6922no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—3461no—
196C22—3461no—
197C22—1731no—
198C22—1731no—
199C22—6921no—
200C22—6921no—
201C22—6921no—
202C22C2×C22⋊#1062192411yescharacteristic
203C4—21no—
204C4—21no—
205C4—41no—
206C4—3461no—
207C4—1732no—
208C4—1732no—
209C4—6924no—
210C4—6921no—
211C4#1914974112yes—
212C2—81no—
213C2—3461no—
214C2—3461no—
215C2—13841no—
216C2#2601247011yescharacteristic
217C2C2×#1914974111yescharacteristic
218C2D173⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4×D4×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012038C2
cosocleC2×C4×D173C23
hypercenterC4×D4#10621924
hypocenterC173#1502
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #26012509C2semidirect
2. C4×C4⋊#10621924C2semidirect
3. #26012343C2semidirect
4. #26012342C2semidirect
13. #19149678C4semidirect
14. #19149675C4semidirect
19. C4×D692C4semidirect
22. C4×D4×C173C4semidirect
109. C173#1502semidirect
125. C4×D4#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125092
C4×C4⋊#106219242
#260123432
#260123422
#260120382
#1502173