Finite Group Databasebeta

← #36368900

Subgroups — #36368900

Order 22144.

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

normal subgroups only

This group has 266 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#26012494C211yesmaximal, characteristic, complemented
2C4×#14583835C211yesmaximal, characteristic, complemented
3#26012341C212yesmaximal, complemented
4#26012339C212yesmaximal, complemented
5#26011959C211yesmaximal, characteristic
6C2×#14583835—21no—
7C2×#14583835—21no—
8#19149746—21no—
9C4⋊#13063443C2212yes—
10#19149734—21no—
11C4×#13063443C2212yes—
12C2×#14583826C411yescharacteristic
13#19149644C411yescharacteristic
14#19149644C2211yescharacteristic
15#19149641C411yescharacteristic
16#19149641C2211yescharacteristic
17#19149639—21no—
18#19149639—21no—
19#19149636—21no—
20C2×C4×Dic173C2211yescharacteristic
21C42×D173—21no—
22#117×C173C411yescharacteristic
23#14583835—24no—
24#14583835—42no—
25C2×#13063443—22no—
26C2×#13063443—22no—
27#14583826—22no—
28#14583826—22no—
29C2×Dic346C2×C411yescharacteristic
30C2×D692C2×C411yescharacteristic
31C2×C22⋊D173—21no—
32C22×Dic173D411yescharacteristic
33C22×Dic173D411yescharacteristic
34C2×C4×D173—21no—
35C2×C4×D173—21no—
36C2×C4×D173—21no—
37#14583805—21no—
38#14583805—21no—
39#14583803—21no—
40#14583803—21no—
41C4×Dic173—21no—
42C4×Dic173—21no—
43C4×Dic173C2×C411yescharacteristic
44C4×Dic173D411yescharacteristic
45C4×Dic173D411yescharacteristic
46C4×Dic173C2311yescharacteristic
47C22×C4×C173C2×C411yescharacteristic
48C4⋊C4×C173C2×C411yescharacteristic
49C4⋊C4×C173C2×C411yescharacteristic
50C22⋊C4×C173—21no—
51C42×C173—21no—
52#13063443—24no—
53#13063443—42no—
54C22×D173—21no—
55Dic346—21no—
56Dic346M1612yescomplemented
57D692—21no—
58D692M1612yescomplemented
59C22⋊D173—24no—
60C2×Dic173—21no—
61C2×Dic173—22no—
62C2×Dic173—21no—
63C2×Dic173—21no—
64C2×Dic173C22⋊C411yescharacteristic
65C2×Dic173C22⋊C411yescharacteristic
66C2×Dic173C2×D411yescharacteristic
67C2×Dic173C2×D411yescharacteristic
68C4×D173—24no—
69C4×D173—42no—
70C4×D173—41no—
71C4×D173—41no—
72C23×C173C22⋊C411yescharacteristic
73C2×C4×C173—21no—
74C2×C4×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173—21no—
77C2×C4×C173C22⋊C411yescharacteristic
78C2×C4×C173C22×C411yescharacteristic
79C22×C173—21no—
80C22×C173—21no—
81C22×C173—21no—
82C22×C173C2×C22⋊C411yescharacteristic
83C2×D173—22no—
84C2×D173—41no—
85C2×D173—41no—
86C4×C173—21no—
87C4×C173—41no—
88C4×C173—41no—
89C4×C173C2×M1611yescharacteristic
90C4×C173C2×M1611yescharacteristic
91Dic173—22no—
92Dic173—21no—
93Dic173—21no—
94Dic173#10412yes—
95Dic173AGL(1,ℤ/8)11yescharacteristic
96Dic173#13711yescharacteristic
97C2×C173—41no—
98C2×C173#40711yescharacteristic
99C2×C173#41711yescharacteristic
100C2×C173C2×#10411yescharacteristic
101D173—42no—
102C173#148911yescharacteristic, complemented, Sylow
103#1489—1731nomaximal, Sylow
104C4×#41—1731no—
105#423—1731no—
106C4×M16—1731no—
107Q8⋊C8—1732no—
108D4⋊C8—1732no—
109C2×#41—1731no—
110C2×M16—3461no—
111C2×M16—3461no—
112C4×Q8—1731no—
113C4×Q8—1731no—
114C4×D4—3461no—
115C4×D4—3461no—
116C4×D4—1731no—
117C4×D4—1731no—
118#117—3461no—
119#117#1062192411yescharacteristic
120C2×C42—3461no—
121C2×C42—1731no—
122C4⋊C8—3461no—
123C4⋊C8—1732no—
124#97—3461no—
125C4×C8—1732no—
126#41—3462no—
127#41—3462no—
128C2×Q8—1731no—
129C2×D4—3461no—
130C2×D4—1731no—
131C22×C4—3461no—
132C22×C4—3461no—
133C22×C4—3461no—
134C22×C4—1731no—
135C22×C4—1731no—
136C22×C4C2×#1062192411yescharacteristic
137M16—3464no—
138M16—6922no—
139C2×C8—3462no—
140C2×C8—3462no—
141C4⋊C4—3461no—
142C4⋊C4—3461no—
143C4⋊C4C2×#1062192411yescharacteristic
144C4⋊C4C2×#1062192411yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—3461no—
147C22⋊C4—3461no—
148C42—21no—
149C42—3461no—
150C42—3461no—
151C42—1731no—
152C42—1731no—
153C42—1731no—
154C42—1731no—
155C23—3461no—
156C23C22⋊#1062192411yescharacteristic
157Q8—3461no—
158Q8—1732no—
159D4—3464no—
160D4—3461no—
161D4—1732no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—3464no—
167C2×C4—3461no—
168C2×C4—3462no—
169C2×C4—3461no—
170C2×C4—3461no—
171C2×C4—1731no—
172C2×C4—1731no—
173C2×C4—1731no—
174C2×C4—1731no—
175C2×C4—6921no—
176C2×C4—6921no—
177C2×C4—6922no—
178C2×C4C22⋊#1062192411yescharacteristic
179C2×C4C22×#1062192411yescharacteristic
180C8—3464no—
181C8—6922no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—3462no—
186C22—6921no—
187C22—6921no—
188C22C2×C22⋊#1062192411yescharacteristic
189C4—21no—
190C4—41no—
191C4—41no—
192C4—3462no—
193C4—3461no—
194C4—3461no—
195C4—1731no—
196C4—1731no—
197C4—1732no—
198C4D173⋊M1611yescharacteristic
199C4D173⋊M1611yescharacteristic
200C2—41no—
201C2—6922no—
202C2#2601246311yescharacteristic
203C2#2601248211yescharacteristic
204C2D173⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#117×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×#14583826C4
cosocleC4×Dic173C23
hypercenter#117#10621924
hypocenterC173#1489
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #26012494C2semidirect
2. C4×#14583835C2semidirect
3. #26012341C2semidirect
4. #26012339C2semidirect
56. Dic346M16semidirect
58. D692M16semidirect
102. C173#1489semidirect

Maximal subgroups

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

subgroupindex
#260124942
C4×#145838352
#260123412
#260123392
#260119592
#1489173