Finite Group Databasebeta

← #36352082

Subgroups — #36352082

Order 22144.

294 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 77 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×#19149658C211yesmaximal, characteristic
2#26011455C211yesmaximal, characteristic
3#26011227C214yesmaximal, complemented
4C4×C22⋊C4×C173C211yesmaximal, characteristic
5C22×Dic346C411yescharacteristic, complemented
6C2×#14583810—21no—
7#19149658—21no—
8#19149658—21no—
9#19149658C412yescomplemented
10#19149658C2212yes—
11C2×C4⋊Dic173C411yescharacteristic, complemented
12C2×#14583805—21no—
13C2×C22⋊Dic173—21no—
14C22⋊#13063436—21no—
15C22⋊#13063436C2212yes—
16C23×C4×C173C2211yescharacteristic
17C2×C22⋊C4×C173—21no—
18C2×C42×C173—21no—
19#95×C173C2212yes—
20C2×Dic346—21no—
21C2×Dic346—42no—
22C2×Dic346C2×C412yes—
23C22×Dic173—21no—
24C22×Dic173—21no—
25#14583810—42no—
26C2×#13063436—22no—
27C4⋊Dic173—21no—
28C4⋊Dic173C2×C412yes—
29#14583805—22no—
30#14583805—22no—
31C22⋊Dic173—22no—
32C22⋊Dic173—22no—
33C24×C173D411yescharacteristic
34C22×C4×C173—22no—
35C22×C4×C173—42no—
36C22×C4×C173C2×C411yescharacteristic
37C22×C4×C173C2×C411yescharacteristic
38C22×C4×C173D411yescharacteristic
39C22×C4×C173D411yescharacteristic
40C22×C4×C173D411yescharacteristic
41C22×C4×C173C2311yescharacteristic
42C22⋊C4×C173—24no—
43C42×C173—42no—
44Dic346—42no—
45Dic346—42no—
46C2×Dic173—22no—
47C2×Dic173—22no—
48C2×Dic173—42no—
49C2×Dic173—42no—
50#13063436—42no—
51C23×C173—22no—
52C23×C173—22no—
53C23×C173—41no—
54C23×C173C22⋊C411yescharacteristic
55C23×C173C22⋊C411yescharacteristic
56C23×C173C2×D411yescharacteristic
57C2×C4×C173—22no—
58C2×C4×C173—22no—
59C2×C4×C173—22no—
60C2×C4×C173—24no—
61C2×C4×C173—41no—
62C2×C4×C173—41no—
63C2×C4×C173—41no—
64C2×C4×C173—42no—
65C2×C4×C173—42no—
66C2×C4×C173C22⋊C411yescharacteristic
67C2×C4×C173C22⋊C411yescharacteristic
68C2×C4×C173C22×C411yescharacteristic
69C2×C4×C173C2×D411yescharacteristic
70C22×C173—21no—
71C22×C173—21no—
72C22×C173—21no—
73C22×C173—21no—
74C22×C173—21no—
75C22×C173—21no—
76C22×C173—21no—
77C22×C173—44no—
78C22×C173#10412yes—
79C22×C173C2×C22⋊C411yescharacteristic
80C22×C173#13712yes—
81C4×C173—21no—
82C4×C173—21no—
83C4×C173—21no—
84C4×C173—44no—
85C4×C173#9912yes—
86Dic173—42no—
87Dic173—42no—
88C2×C173—22no—
89C2×C173—22no—
90C2×C173—41no—
91C2×C173C2×#9911yescharacteristic
92C2×C173#41911yescharacteristic
93C2×C173C2×#10411yescharacteristic
94C173#150811yescharacteristic, complemented, Sylow
95#1508—1731nomaximal, Sylow
96C2×C22⋊Q8—1731no—
97#407—1731no—
98C4×C22⋊C4D17311yescharacteristic
99#328—1734no—
100C22×Q8—1731no—
101C23×C4C2×D17311yescharacteristic
102C2×M16—3461no—
103C22⋊Q8—3461no—
104C22⋊Q8—3461no—
105C22⋊Q8—1732no—
106C22⋊Q8—1732no—
107C2×C4⋊C4—3461no—
108C2×C4⋊C4—1731no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—3461no—
111C2×C42—21no—
112C22⋊C8—3461no—
113C22⋊C8—1732no—
114#95C2×D17312yes—
115C24C22⋊D17311yescharacteristic
116C2×Q8—3461no—
117C2×Q8—1732no—
118C2×Q8—6922no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4—3461no—
122C22×C4—3461no—
123C22×C4C4×D17311yescharacteristic
124C22×C4C4×D17311yescharacteristic
125C22×C4C22⋊D17311yescharacteristic
126C22×C4D69211yescharacteristic
127C22×C4D69211yescharacteristic
128C22×C4C22×D17311yescharacteristic
129M16—6922no—
130C2×C8—3462no—
131C4⋊C4—3462no—
132C4⋊C4—3462no—
133C4⋊C4—3461no—
134C4⋊C4—1732no—
135C22⋊C4—24no—
136C22⋊C4—3462no—
137C22⋊C4—3462no—
138C42—42no—
139C23—22no—
140C23—22no—
141C23—41no—
142C23#1458380311yescharacteristic
143C23#1458380311yescharacteristic
144C23C2×C22⋊D17311yescharacteristic
145Q8—6922no—
146Q8—6922no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—3462no—
157C2×C4—3462no—
158C2×C4—6922no—
159C2×C4—6922no—
160C2×C4#1458380311yescharacteristic
161C2×C4#1458380311yescharacteristic
162C2×C4C2×C4×D17311yescharacteristic
163C2×C4C2×D69211yescharacteristic
164C8—6922no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—44no—
173C22#1914960712yes—
174C22C2×#1458380311yescharacteristic
175C22#1914970912yes—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—44no—
180C4—6922no—
181C4—6922no—
182C4#1914959312yes—
183C2—22no—
184C2—22no—
185C2—41no—
186C2C2×#1914959311yescharacteristic
187C2#2601149911yescharacteristic
188C2C2×#1914960711yescharacteristic
subgroupG/H
centerC22C2×#14583803
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
FittingC4×C22⋊C4×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC24×C173D4
cosocleC22×C4×C173C23
hypercenterC4×C22⋊C4D173
hypocenterC173#1508
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #26011227C2semidirect
5. C22×Dic346C4semidirect
9. #19149658C4semidirect
11. C2×C4⋊Dic173C4semidirect
94. C173#1508semidirect

Maximal subgroups

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

subgroupindex
C2×#191496582
#260114552
#260112272
C4×C22⋊C4×C1732
#1508173