Finite Group Databasebeta

← #36368993

Subgroups — #36368993

Order 22144.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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#26012510C211yesmaximal, characteristic, complemented
2#26012435C211yesmaximal, characteristic, complemented
3#26012343C212yesmaximal, complemented
4#26012342C212yesmaximal, complemented
5#26012038C211yesmaximal, characteristic
6C2×C4⋊#10621924—21no—
7C4⋊#13063443—21no—
8#19149736C2212yes—
9C4×#13063443—21no—
10(C2×#10621924)⋊C4—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—42no—
26#14583826—42no—
27#14583826—42no—
28C2×Dic346—21no—
29C2×D692—21no—
30C2×D692C2×C411yescharacteristic
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C2×C4×D173—21no—
35C2×C4×D173D411yescharacteristic
36C2×C4×D173D411yescharacteristic
37C2×C4×D173C2311yescharacteristic
38C4⋊Dic173C2×C411yescharacteristic
39#14583805—21no—
40C22⋊Dic173—21no—
41#14583803—21no—
42#14583803—21no—
43C4×Dic173C2×C411yescharacteristic
44C4×Dic173D411yescharacteristic
45C4×Dic173D411yescharacteristic
46C2×D4×C173C2×C411yescharacteristic
47C22×C4×C173—21no—
48C4⋊C4×C173C2×C411yescharacteristic
49C22⋊C4×C173—21no—
50C42×C173C2×C411yescharacteristic
51C2×#10621924—42no—
52C2×#10621924—42no—
53C2×#10621924—42no—
54#13063443—42no—
55C22×D173—21no—
56C22×D173C2×D411yescharacteristic
57Dic346—42no—
58D692—41no—
59D692—41no—
60D692—41no—
61C22⋊D173—42no—
62C22⋊D173—41no—
63C22⋊D173—41no—
64C22⋊D173—42no—
65C2×Dic173—21no—
66C2×Dic173C22⋊C411yescharacteristic
67C2×Dic173C22⋊C411yescharacteristic
68C2×Dic173C2×D411yescharacteristic
69C4×D173—41no—
70C4×D173—41no—
71C4×D173—42no—
72C4×D173D812yes—
73C4×D173SD1612yes—
74C23×C173—21no—
75D4×C173—41no—
76C2×C4×C173—21no—
77C2×C4×C173—42no—
78C2×C4×C173C22⋊C411yescharacteristic
79C2×C4×C173C22⋊C411yescharacteristic
80C2×C4×C173C22×C411yescharacteristic
81C22×C173—41no—
82C22×C173—41no—
83C22×C173—41no—
84C22×C173C2×C22⋊C411yescharacteristic
85C2×D173—21no—
86C2×D173—21no—
87C2×D173—41no—
88C2×D173—41no—
89C2×D173—41no—
90C2×D173C2×D811yescharacteristic
91C2×D173C2×SD1611yescharacteristic
92#10621924—82no—
93C4×C173—21no—
94C4×C173—21no—
95C4×C173—41no—
96C4×C173#10212yes—
97Dic173—21no—
98Dic173—21no—
99Dic173—41no—
100Dic173#10212yes—
101C2×C173—81no—
102C2×C173#41011yescharacteristic
103C2×C173C2×#10211yescharacteristic
104C2×C173#42111yescharacteristic
105D173—21no—
106D173—21no—
107D173—81no—
108C173#151711yescharacteristic, complemented, Sylow
109#1517—1731nomaximal, Sylow
110#532—1731no—
111#431—1731no—
112#382—1731no—
113#327—1732no—
114#327—1732no—
115C2×#41—3461no—
116C4⋊Q8—1731no—
117#127—1731no—
118#124—3461no—
119#121—3461no—
120#121—1731no—
121#121—1731no—
122C4×D4—3461no—
123C4×D4—1731no—
124C4×D4#1062192411yescharacteristic, complemented
125C2×C4⋊C4—3461no—
126C2×C42—1731no—
127C4⋊C8—3461no—
128C22⋊C8—1732no—
129C4×C8—3461no—
130#95—3461no—
131#95—1732no—
132#41—6922no—
133#41—6922no—
134C2×Q8—3461no—
135C2×D4—3461no—
136C2×D4—3461no—
137C2×D4—3461no—
138C2×D4—3461no—
139C2×D4—1731no—
140C2×D4C2×#1062192411yescharacteristic
141C22×C4—21no—
142C22×C4—3461no—
143C22×C4—3462no—
144C22×C4—1731no—
145C22×C4—1731no—
146C22×C4—1731no—
147C2×C8—3462no—
148C4⋊C4—3461no—
149C4⋊C4—1731no—
150C4⋊C4—6922no—
151C4⋊C4C2×#1062192411yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—3461no—
154C22⋊C4—3461no—
155C22⋊C4—3461no—
156C42—1731no—
157C42—1731no—
158C42—1731no—
159C42C2×#1062192411yescharacteristic
160C23—21no—
161C23—3461no—
162C23—1731no—
163Q8—6922no—
164D4—41no—
165D4—6921no—
166D4—6921no—
167D4—6921no—
168D4—6922no—
169D4—6922no—
170D4—6921no—
171D4—6921no—
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—3461no—
175C2×C4—1731no—
176C2×C4—1731no—
177C2×C4—1731no—
178C2×C4—1732no—
179C2×C4—1732no—
180C2×C4—6921no—
181C2×C4—6921no—
182C2×C4—6922no—
183C2×C4—6922no—
184C2×C4—6922no—
185C2×C4—6922no—
186C2×C4C22⋊#1062192411yescharacteristic
187C2×C4C22⋊#1062192411yescharacteristic
188C2×C4C22×#1062192411yescharacteristic
189C8—6922no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—3461no—
194C22—3461no—
195C22—1731no—
196C22—1731no—
197C22—6921no—
198C22—6921no—
199C22—6921no—
200C22C2×C22⋊#1062192411yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—3461no—
205C4—3461no—
206C4—13842no—
207C4—1732no—
208C4—6921no—
209C4#1914974112yes—
210C2—81no—
211C2—3461no—
212C2—3461no—
213C2—13841no—
214C2#2601247011yescharacteristic
215C2C2×#1914974111yescharacteristic
216C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#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#1517
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #26012510C2semidirect
2. #26012435C2semidirect
3. #26012343C2semidirect
4. #26012342C2semidirect
13. #19149678C4semidirect
14. #19149675C4semidirect
19. C4×D692C4semidirect
22. C4×D4×C173C4semidirect
108. C173#1517semidirect
124. C4×D4#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125102
#260124352
#260123432
#260123422
#260120382
#1517173