Finite Group Databasebeta

← #36351560

Subgroups — #36351560

Order 22144.

222 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 37 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#26011258C211yesmaximal, characteristic, complemented
2#26011215C211yesmaximal, characteristic, complemented
3#392×C173C211yesmaximal, characteristic
4C2×C22⋊Dic173—21no—
5C22⋊#13063436—21no—
6C22×D4×C173—21no—
7C22×D4×C173C411yescharacteristic, complemented
8C2×C22⋊C4×C173—21no—
9C2×C22⋊C4×C173C2211yescharacteristic
10#95×C173C411yescharacteristic, complemented
11C22×Dic173—41no—
12C2×#13063436—41no—
13C22⋊Dic173—22no—
14C22⋊Dic173—41no—
15C24×C173—21no—
16C24×C173D411yescharacteristic
17C2×D4×C173—42no—
18C2×D4×C173—41no—
19C2×D4×C173—42no—
20C2×D4×C173—41no—
21C2×D4×C173—41no—
22C2×D4×C173—42no—
23C22×C4×C173—21no—
24C22×C4×C173C2×C411yescharacteristic
25C22×C4×C173D411yescharacteristic
26C22⋊C4×C173—41no—
27C22⋊C4×C173—42no—
28C2×Dic173—42no—
29C2×Dic173—81no—
30C2×Dic173—81no—
31#13063436—81no—
32C23×C173—21no—
33C23×C173—21no—
34C23×C173—22no—
35C23×C173—41no—
36C23×C173—41no—
37C23×C173—41no—
38C23×C173—41no—
39C23×C173—41no—
40C23×C173—41no—
41C23×C173—41no—
42C23×C173—41no—
43C23×C173C22⋊C411yescharacteristic
44C23×C173D811yescharacteristic
45C23×C173SD1611yescharacteristic
46D4×C173—42no—
47D4×C173—44no—
48D4×C173—44no—
49D4×C173—42no—
50C2×C4×C173—22no—
51C2×C4×C173—21no—
52C2×C4×C173—21no—
53C2×C4×C173—41no—
54C2×C4×C173—41no—
55C2×C4×C173—41no—
56C2×C4×C173—41no—
57C22×C173—21no—
58C22×C173—21no—
59C22×C173—21no—
60C22×C173—21no—
61C22×C173—41no—
62C22×C173—41no—
63C22×C173—42no—
64C22×C173—42no—
65C22×C173—41no—
66C22×C173—41no—
67C22×C173—42no—
68C22×C173—81no—
69C22×C173—81no—
70C22×C173—81no—
71C22×C173—81no—
72C22×C173—81no—
73C22×C173#9911yescharacteristic
74C22×C173#10211yescharacteristic
75C22×C173#10411yescharacteristic
76C4×C173—42no—
77C4×C173—41no—
78C4×C173—41no—
79Dic173—82no—
80C2×C173—21no—
81C2×C173—21no—
82C2×C173—41no—
83C2×C173—41no—
84C2×C173—82no—
85C2×C173#32711yescharacteristic
86C2×C173AΣL(1,𝔽16)11yescharacteristic
87C2×C173#35311yescharacteristic
88C173#134311yescharacteristic, complemented, Sylow
89#1343—1731nomaximal, Sylow
90#392D17311yescharacteristic
91#342—1731no—
92#323—1731no—
93C22×D4—21no—
94C22×D4Dic17311yescharacteristic, complemented
95C2×C22⋊C4—21no—
96C2×C22⋊C4—3461no—
97C2×C22⋊C4C2×D17311yescharacteristic
98C22⋊C8—3461no—
99#95Dic17311yescharacteristic, complemented
100C24—21no—
101C24C22⋊D17311yescharacteristic
102C2×D4—41no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—41no—
106C2×D4—41no—
107C2×D4—42no—
108C22×C4—21no—
109C22×C4—6921no—
110C22×C4C2×Dic17311yescharacteristic
111C22×C4C22⋊D17311yescharacteristic
112C2×C8—6921no—
113C22⋊C4—41no—
114C22⋊C4—42no—
115C22⋊C4—3462no—
116C22⋊C4—6921no—
117C23—21no—
118C23—21no—
119C23—22no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23C22⋊Dic17311yescharacteristic
129C23#1458381111yescharacteristic
130C23#1458381411yescharacteristic
131D4—42no—
132D4—44no—
133D4—44no—
134D4—42no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—13841no—
143C2×C4—13841no—
144C2×C4—6922no—
145C8—13841no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—41no—
151C22—41no—
152C22—42no—
153C22—42no—
154C22—41no—
155C22—41no—
156C22—42no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22#1914959411yescharacteristic
163C22#1914960011yescharacteristic
164C22#1914960611yescharacteristic
165C4—42no—
166C4—41no—
167C4—41no—
168C4—13842no—
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—82no—
174C2#2601122511yescharacteristic
175C2#2601127811yescharacteristic
176C2#2601128411yescharacteristic
subgroupG/H
centerC22#19149594
commutatorC22×C4×C173C2×C4
FrattiniC2×C22⋊C4C2×D173
Fitting, prime radical#392×C173C2
socleC22×C173#99
cosocleC2×C22⋊C4×C173C22
hypercenter#392D173
hypocenterC173#1343
LCSG ▷ C22×C4×C173 ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #26011258C2semidirect
2. #26011215C2semidirect
7. C22×D4×C173C4semidirect
10. #95×C173C4semidirect
88. C173#1343semidirect
94. C22×D4Dic173semidirect
99. #95Dic173semidirect

Maximal subgroups

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

subgroupindex
#260112582
#260112152
#392×C1732
#1343173