Finite Group Databasebeta

← #36351561

Subgroups — #36351561

Order 22144.

222 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 35 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#26011400C211yesmaximal, characteristic
2#26011258C211yesmaximal, characteristic, complemented
3#323×C173C211yesmaximal, characteristic, complemented
4C22×D692C411yescharacteristic, complemented
5C22×C22⋊D173—21no—
6C2×C22⋊Dic173—21no—
7C2×#14583803—21no—
8#19149586C411yescharacteristic, complemented
9C2×C22⋊C4×C173C2211yescharacteristic
10C22⋊C8×C173—21no—
11C23×D173—21no—
12C2×D692—41no—
13C2×D692—42no—
14C2×C22⋊D173—42no—
15C2×C22⋊D173—42no—
16C2×C22⋊D173—41no—
17C2×C22⋊D173—41no—
18C22×Dic173—21no—
19C22×Dic173—41no—
20C22⋊Dic173—22no—
21C22⋊Dic173—41no—
22#14583803—42no—
23C24×C173D411yescharacteristic
24C22×C4×C173C2×C411yescharacteristic
25C22×C4×C173D411yescharacteristic
26C2×C8×C173—41no—
27C22⋊C4×C173—41no—
28C22×D173—22no—
29C22×D173—41no—
30C22×D173—41no—
31C22×D173—41no—
32C22×D173—41no—
33C22×D173—41no—
34C22×D173—41no—
35D692—42no—
36D692—42no—
37C22⋊D173—44no—
38C22⋊D173—44no—
39C2×Dic173—22no—
40C2×Dic173—41no—
41C2×Dic173—41no—
42C2×Dic173—42no—
43C2×Dic173—81no—
44C2×Dic173—81no—
45C23×C173—21no—
46C23×C173—21no—
47C23×C173—41no—
48C23×C173—41no—
49C23×C173C22⋊C411yescharacteristic
50C23×C173D811yescharacteristic
51C23×C173SD1611yescharacteristic
52C2×C4×C173—21no—
53C2×C4×C173—21no—
54C2×C4×C173—41no—
55C2×C4×C173—41no—
56C8×C173—81no—
57C22×C173—21no—
58C22×C173—21no—
59C22×C173—21no—
60C22×C173—21no—
61C22×C173—41no—
62C22×C173—41no—
63C22×C173—41no—
64C22×C173—41no—
65C22×C173—81no—
66C22×C173#9911yescharacteristic
67C22×C173#10211yescharacteristic
68C22×C173#10411yescharacteristic
69C2×D173—42no—
70C2×D173—42no—
71C2×D173—42no—
72C2×D173—81no—
73C2×D173—81no—
74C2×D173—81no—
75C2×D173—81no—
76C4×C173—41no—
77C4×C173—41no—
78Dic173—42no—
79Dic173—82no—
80C2×C173—21no—
81C2×C173—21no—
82C2×C173—41no—
83C2×C173—41no—
84C2×C173#32711yescharacteristic
85C2×C173AΣL(1,𝔽16)11yescharacteristic
86C2×C173#35311yescharacteristic
87D173—82no—
88C173#134311yescharacteristic, complemented, Sylow
89#1343—1731nomaximal, Sylow
90#392—1731no—
91#342—1731no—
92#323D17311yescharacteristic, complemented
93C22×D4—3461no—
94C22×D4—1731no—
95C2×C22⋊C4—3461no—
96C2×C22⋊C4—3461no—
97C2×C22⋊C4C2×D17311yescharacteristic
98C22⋊C8—21no—
99#95—1731no—
100C24—3461no—
101C24D69211yescharacteristic
102C2×D4—6921no—
103C2×D4—6922no—
104C2×D4—6922no—
105C2×D4—6922no—
106C2×D4—6921no—
107C2×D4—6921no—
108C22×C4—3461no—
109C22×C4—6921no—
110C22×C4C4×D17311yescharacteristic
111C22×C4C22⋊D17311yescharacteristic
112C2×C8—41no—
113C22⋊C4—41no—
114C22⋊C4—3462no—
115C22⋊C4—6922no—
116C22⋊C4—6921no—
117C23—21no—
118C23—21no—
119C23—41no—
120C23—41no—
121C23—3462no—
122C23—6921no—
123C23—6921no—
124C23—6921no—
125C23—6921no—
126C23—6921no—
127C23—6921no—
128C23#1458380311yescharacteristic
129C23D138411yescharacteristic
130C23#1458381511yescharacteristic
131D4—6922no—
132D4—6922no—
133D4—6924no—
134D4—6924no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—3462no—
140C2×C4—13841no—
141C2×C4—13841no—
142C2×C4—6922no—
143C2×C4—6921no—
144C2×C4—6921no—
145C8—81no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22—41no—
154C22—81no—
155C22—13841no—
156C22—13841no—
157C22—13841no—
158C22—13841no—
159C22—6922no—
160C22—6922no—
161C22—6922no—
162C22#1914959311yescharacteristic
163C22#1914960111yescharacteristic
164C22#1914960511yescharacteristic
165C4—41no—
166C4—41no—
167C4—13842no—
168C4—6922no—
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—13842no—
174C2#2601122611yescharacteristic
175C2#2601127911yescharacteristic
176C2#2601128311yescharacteristic
subgroupG/H
centerC22#19149593
commutatorC22×C4×C173C2×C4
FrattiniC2×C22⋊C4C2×D173
Fitting#323×C173C2
socleC22×C173#99
prime radical#26011400C2
cosocleC2×C22⋊C4×C173C22
hypercenter#323D173
hypocenterC173#1343
LCSG ▷ C22×C4×C173 ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #26011258C2semidirect
3. #323×C173C2semidirect
4. C22×D692C4semidirect
8. #19149586C4semidirect
88. C173#1343semidirect
92. #323D173semidirect

Maximal subgroups

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

subgroupindex
#260114002
#260112582
#323×C1732
#1343173