Finite Group Databasebeta

← #36369726

Subgroups — #36369726

Order 22144.

336 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 74 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×#19149744C214yesmaximal, complemented
2C4×#14583836C211yesmaximal, characteristic, complemented
3C4×C4⋊#10621924C211yesmaximal, characteristic, complemented
4#26012042C211yesmaximal, characteristic
5C2×#14583836C2212yes—
6C2×C4⋊#10621924—21no—
7C2×C4×#10621924C2212yes—
8#19149744—28no—
9#19149734—21no—
10C4×#13063443—21no—
11(C2×#10621924)⋊C4—21no—
12C2×#14583827C2212yes—
13#19149676C411yescharacteristic, complemented
14#19149668C412yescomplemented
15#19149654—21no—
16#19149654—21no—
17#19149641—21no—
18#19149641—21no—
19C42×D173C2211yescharacteristic
20C4⋊Q8×C173C411yescharacteristic, complemented
21C22×#10621924—22no—
22#14583836—22no—
23#14583836D414yescomplemented
24C2×#13063443—22no—
25C4⋊#10621924—24no—
26C4×#10621924—24no—
27#14583827—24no—
28#14583827—24no—
29C2×D692—22no—
30C2×D692—21no—
31C2×D692—21no—
32C2×D692C2×C412yes—
33C2×C4×D173—22no—
34C2×C4×D173D411yescharacteristic
35C2×C4×D173D411yescharacteristic
36C2×C4×D173C2311yescharacteristic
37#14583803—22no—
38#14583803—22no—
39C4×Dic173C2×C411yescharacteristic
40C4×Dic173D411yescharacteristic
41C4×Dic173D411yescharacteristic
42C2×Q8×C173C2×C412yes—
43C4⋊C4×C173—21no—
44C4⋊C4×C173—21no—
45C42×C173C2×C411yescharacteristic
46C2×#10621924—24no—
47C2×#10621924—42no—
48C2×#10621924—42no—
49#13063443—24no—
50C22×D173—22no—
51C22×D173C2×D411yescharacteristic
52D692—24no—
53D692—42no—
54D692—42no—
55D692—42no—
56D692—42no—
57C2×Dic173C22⋊C411yescharacteristic
58C2×Dic173C22⋊C411yescharacteristic
59C2×Dic173C2×D411yescharacteristic
60C4×D173—22no—
61C4×D173—41no—
62C4×D173—44no—
63C4×D173C2×D412yes—
64C4×D173#4112yes—
65Q8×C173—24no—
66C2×C4×C173—22no—
67C2×C4×C173C22⋊C411yescharacteristic
68C2×C4×C173C22⋊C411yescharacteristic
69C2×C4×C173C22×C411yescharacteristic
70C22×C173C2×C22⋊C411yescharacteristic
71C2×D173—21no—
72C2×D173—21no—
73C2×D173—42no—
74C2×D173—42no—
75C2×D173—42no—
76C2×D173#12411yescharacteristic
77C2×D173#12711yescharacteristic
78#10621924—44no—
79C4×C173—21no—
80C4×C173—42no—
81C4×C173#10412yescomplemented
82C4×C173C4×D412yes—
83Dic173—21no—
84Dic173—21no—
85Dic173#12112yes—
86C2×C173#39011yescharacteristic
87C2×C173C2×#10411yescharacteristic
88C2×C173#42111yescharacteristic
89D173—21no—
90D173—21no—
91D173—82no—
92C173#198011yescharacteristic, complemented, Sylow
93#1980—1731nomaximal, Sylow
94#532—1731no—
95C2×#104—1734no—
96C4×M16—1731no—
97C4×C4⋊C4—1731no—
98C2×#41—1732no—
99C2×M16—1732no—
100C4⋊Q8#1062192411yescharacteristic, complemented
101#127—1731no—
102#124—1732no—
103#121—3461no—
104#121—3461no—
105C4×D4—3461no—
106C4×D4—3461no—
107C2×C4⋊C4—3461no—
108C2×C42—1732no—
109C2×C42—1731no—
110#104—3468no—
111#97—3461no—
112C4×C8—3461no—
113#95—3461no—
114#41—3464no—
115#41—3464no—
116C2×Q8C2×#1062192412yes—
117C2×D4—3462no—
118C2×D4—3461no—
119C2×D4—3461no—
120C2×D4—1732no—
121C22×C4—3462no—
122C22×C4—3462no—
123C22×C4—1731no—
124C22×C4—1731no—
125C22×C4—1731no—
126M16—3462no—
127M16—1734no—
128C2×C8—3462no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—3464no—
132C22⋊C4—3462no—
133C22⋊C4—3462no—
134C42—3464no—
135C42—1731no—
136C42—1731no—
137C42—1731no—
138C42C2×#1062192411yescharacteristic
139C23—3462no—
140C23—1731no—
141Q8—24no—
142D4—3464no—
143D4—6922no—
144D4—6922no—
145D4—6922no—
146D4—6922no—
147C2×C4—22no—
148C2×C4—3462no—
149C2×C4—3464no—
150C2×C4—1731no—
151C2×C4—1731no—
152C2×C4—1731no—
153C2×C4—1732no—
154C2×C4—1732no—
155C2×C4—6921no—
156C2×C4—6924no—
157C2×C4—6922no—
158C2×C4—6922no—
159C2×C4C22⋊#1062192411yescharacteristic
160C2×C4C22⋊#1062192411yescharacteristic
161C2×C4C22×#1062192411yescharacteristic
162C8—3464no—
163C22—3461no—
164C22—3461no—
165C22—1731no—
166C22—1731no—
167C22—6922no—
168C22—6922no—
169C22—6922no—
170C22C2×C22⋊#1062192411yescharacteristic
171C4—21no—
172C4—42no—
173C4—3461no—
174C4—3461no—
175C4—1732no—
176C4—6924no—
177C4#1914974412yescomplemented
178C4D4×#1062192412yes—
179C2—3461no—
180C2—3461no—
181C2—13842no—
182C2#2601244311yescharacteristic
183C2C2×#1914974411yescharacteristic
184C2#2601249011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4⋊Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012042C2
cosocleC2×C4×D173C23
hypercenterC4⋊Q8#10621924
hypocenterC173#1980
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#19149744C2semidirect
2. C4×#14583836C2semidirect
3. C4×C4⋊#10621924C2semidirect
13. #19149676C4semidirect
14. #19149668C4semidirect
20. C4⋊Q8×C173C4semidirect
23. #14583836D4semidirect
81. C4×C173#104semidirect
92. C173#1980semidirect
100. C4⋊Q8#10621924semidirect
177. C4#19149744semidirect

Maximal subgroups

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

subgroupindex
C2×#191497442
C4×#145838362
C4×C4⋊#106219242
#260120422
#1980173