Finite Group Databasebeta

← #36353753

Subgroups — #36353753

Order 22144.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C22×C4⋊Dic173C211yesmaximal, characteristic
2C2×#19149613C212yesmaximal, complemented
3#26011351C211yesmaximal, characteristic
4#26011254C212yesmaximal, complemented
5C2×C22⋊C8×C173C211yesmaximal, characteristic
6C23×Dic173—21no—
7C2×C4⋊Dic173—24no—
8C2×C4⋊Dic173—21no—
9C2×C4⋊Dic173C2211yescharacteristic
10C2×C4⋊Dic173C2211yescharacteristic
11C2×C4⋊Dic173C2212yes—
12C2×C22⋊Dic173—21no—
13#19149613—24no—
14#19149586—21no—
15C23×C4×C173C2211yescharacteristic
16C22×C8×C173C2212yes—
17C22⋊C8×C173C414yescomplemented
18C22×Dic173—24no—
19C22×Dic173—22no—
20C22×Dic173—21no—
21C22×Dic173—21no—
22C22×Dic173—42no—
23C22×Dic173—42no—
24C4⋊Dic173—24no—
25C4⋊Dic173—22no—
26C4⋊Dic173—22no—
27C4⋊Dic173—22no—
28C4⋊Dic173D414yes—
29C22⋊Dic173—42no—
30C24×C173D411yescharacteristic
31C22×C4×C173—22no—
32C22×C4×C173—22no—
33C22×C4×C173C2×C412yes—
34C22×C4×C173D411yescharacteristic
35C22×C4×C173Q812yes—
36C22×C4×C173C2311yescharacteristic
37C2×C8×C173—22no—
38C2×C8×C173—22no—
39C2×C8×C173C2×C414yescomplemented
40C2×Dic173—24no—
41C2×Dic173—48no—
42C2×Dic173—42no—
43C2×Dic173—42no—
44C2×Dic173—41no—
45C2×Dic173—41no—
46C2×Dic173—41no—
47C2×Dic173—41no—
48C2×Dic173—41no—
49C2×Dic173—41no—
50C23×C173—21no—
51C23×C173—21no—
52C23×C173—21no—
53C23×C173—21no—
54C23×C173C4⋊C412yes—
55C23×C173D812yes—
56C23×C173Q1612yes—
57C23×C173C2×D411yescharacteristic
58C2×C4×C173—21no—
59C2×C4×C173—21no—
60C2×C4×C173—24no—
61C2×C4×C173—24no—
62C2×C4×C173C4⋊C412yes—
63C2×C4×C173C22×C411yescharacteristic
64C2×C4×C173C2×D412yes—
65C2×C4×C173C2×Q811yescharacteristic
66C2×C4×C173#4112yes—
67C8×C173—24no—
68C22×C173—22no—
69C22×C173—22no—
70C22×C173—22no—
71C22×C173—22no—
72C22×C173—22no—
73C22×C173—22no—
74C22×C173#10714yescomplemented
75C22×C173C2×C4⋊C411yescharacteristic
76C22×C173#12011yescharacteristic
77C22×C173#12311yescharacteristic
78C22×C173C2×D811yescharacteristic
79C22×C173C2×Q1611yescharacteristic
80C22×C173AGL(1,ℤ/8)11yescharacteristic
81C22×C173#13711yescharacteristic
82C4×C173—22no—
83C4×C173C4×D412yes—
84C4×C173C22⋊Q812yes—
85Dic173—44no—
86Dic173—81no—
87Dic173—81no—
88C2×C173—24no—
89C2×C173#38511yescharacteristic
90C2×C173C2×#10711yescharacteristic
91C2×C173#42811yescharacteristic
92C2×C173#44711yescharacteristic
93C2×C173#45111yescharacteristic
94C2×C173#48011yescharacteristic
95C2×C173#48411yescharacteristic
96C173#185211yescharacteristic, complemented, Sylow
97#1852—1731nomaximal, Sylow
98C22×C4⋊C4—1731no—
99C2×#107—1732no—
100C2×C22⋊C8D17311yescharacteristic
101#380—1731no—
102#340—1732no—
103C23×C4—3461no—
104C23×C4C2×D17311yescharacteristic
105C22×C8C2×D17312yes—
106C2×C4⋊C4—3461no—
107C2×C4⋊C4—3464no—
108C2×C4⋊C4—1731no—
109C2×C4⋊C4—1731no—
110C2×C4⋊C4—1732no—
111C2×C22⋊C4—3461no—
112#107—3464no—
113C22⋊C8Dic17314yescomplemented
114#95—3461no—
115C24D69211yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—3464no—
119C22×C4—3462no—
120C22×C4—3461no—
121C22×C4—3461no—
122C22×C4—6922no—
123C22×C4—6922no—
124C22×C4C2×Dic17312yes—
125C22×C4D69211yescharacteristic
126C22×C4Dic34612yes—
127C22×C4C22×D17311yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8C2×Dic17314yescomplemented
131C4⋊C4—3464no—
132C4⋊C4—3462no—
133C4⋊C4—3462no—
134C4⋊C4—3462no—
135C4⋊C4—1734no—
136C22⋊C4—6922no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23C4⋊Dic17312yes—
142C23D138412yes—
143C23Dic69212yes—
144C23C2×D69211yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—3464no—
150C2×C4—6921no—
151C2×C4—6921no—
152C2×C4—6921no—
153C2×C4—6921no—
154C2×C4—6921no—
155C2×C4—6921no—
156C2×C4—6928no—
157C2×C4—6922no—
158C2×C4—6922no—
159C2×C4C4⋊Dic17312yes—
160C2×C4C22×Dic17311yescharacteristic
161C2×C4D4×D17312yes—
162C2×C4C2×Dic34611yescharacteristic
163C2×C4#1458382512yes—
164C8—24no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#1914961314yescomplemented
172C22C2×C4⋊Dic17311yescharacteristic
173C22#1914964911yescharacteristic
174C22#1914966311yescharacteristic
175C22C2×D138411yescharacteristic
176C22C2×Dic69211yescharacteristic
177C22#1914970711yescharacteristic
178C22#1914971211yescharacteristic
179C4—22no—
180C4—13841no—
181C4—13841no—
182C4—6924no—
183C4D4×Dic17312yes—
184C4#1914965612yes—
185C2—24no—
186C2#2601136611yescharacteristic
187C2C2×#1914961311yescharacteristic
188C2#2601153311yescharacteristic
189C2#2601162711yescharacteristic
190C2#2601164911yescharacteristic
191C2#2601180511yescharacteristic
192C2#2601182711yescharacteristic
subgroupG/H
centerC23C2×D692
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
FittingC2×C22⋊C8×C173C2
socleC23×C173C2×D4
prime radicalC24×C173D4
cosocleC22×C4×C173C23
hypercenterC2×C22⋊C8D173
hypocenterC173#1852
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#19149613C2semidirect
4. #26011254C2semidirect
17. C22⋊C8×C173C4semidirect
39. C2×C8×C173C2×C4semidirect
74. C22×C173#107semidirect
96. C173#1852semidirect
113. C22⋊C8Dic173semidirect
130. C2×C8C2×Dic173semidirect
171. C22#19149613semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic1732
C2×#191496132
#260113512
#260112542
C2×C22⋊C8×C1732
#1852173