Finite Group Databasebeta

← #36353642

Subgroups — #36353642

Order 22144.

316 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 81 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×C4⋊C4×D173C211yesmaximal, characteristic
2#26011454C212yesmaximal
3#26011260C211yesmaximal, characteristic, complemented
4#26011259C212yesmaximal, complemented
5#343×C173C211yesmaximal, characteristic, complemented
6C22×C4×D173C412yes—
7C22×C4×D173C2211yescharacteristic
8C2×#14583810C2212yes—
9C2×C4⋊Dic173C2211yescharacteristic
10C2×#14583805C412yes—
11C4⋊C4×D173—22no—
12C4⋊C4×D173—22no—
13#19149591—22no—
14#19149591—22no—
15C2×M16×C173C2212yes—
16C2×C4⋊C4×C173C2211yescharacteristic
17C23×D173C2×C411yescharacteristic
18C22×Dic173C2×C411yescharacteristic
19C22×Dic173C2×C412yes—
20C2×C4×D173—22no—
21C2×C4×D173—22no—
22C2×C4×D173—21no—
23C2×C4×D173—22no—
24C2×C4×D173—21no—
25C2×C4×D173—44no—
26C2×C4×D173D412yes—
27C2×C4×D173D412yes—
28#14583810—22no—
29#14583810—22no—
30C2×#13063436—22no—
31C4⋊Dic173—22no—
32#14583805—24no—
33C22×C4×C173C2×C412yes—
34C22×C4×C173C2311yescharacteristic
35M16×C173—22no—
36M16×C173—22no—
37C2×C8×C173—22no—
38C4⋊C4×C173—22no—
39C22×D173—21no—
40C22×D173—21no—
41C22×D173—22no—
42C22×D173—22no—
43C22×D173C22⋊C411yescharacteristic
44C22×D173C22⋊C411yescharacteristic
45C2×Dic173—21no—
46C2×Dic173—21no—
47C2×Dic173—22no—
48C2×Dic173—42no—
49C2×Dic173C22⋊C412yes—
50C4×D173—22no—
51C4×D173—22no—
52C4×D173—22no—
53C4×D173—22no—
54C4×D173—44no—
55C4×D173—44no—
56#13063436—42no—
57C23×C173C22×C411yescharacteristic
58C2×C4×C173—22no—
59C2×C4×C173—42no—
60C2×C4×C173C2×D412yes—
61C2×C4×C173#4111yescharacteristic
62C2×C4×C173#4112yes—
63C2×C4×C173#4111yescharacteristic
64C8×C173—42no—
65C22×C173—21no—
66C22×C173—21no—
67C22×C173C2×C22⋊C411yescharacteristic
68C22×C173#11711yescharacteristic
69C22×C173#11711yescharacteristic
70C2×D173—24no—
71C2×D173—24no—
72C2×D173—21no—
73C2×D173—21no—
74C2×D173—21no—
75C2×D173—21no—
76C2×D173#10012yes—
77C2×D173#10112yes—
78C4×C173—42no—
79C4×C173#12312yes—
80C4×C173#12312yes—
81Dic173—22no—
82Dic173—42no—
83C2×C173—21no—
84C2×C173—21no—
85C2×C173#38111yescharacteristic
86C2×C173C2×#10011yescharacteristic
87C2×C173C2×#10111yescharacteristic
88D173—22no—
89D173—22no—
90C173#182111yescharacteristic, complemented, Sylow
91#1821—1731nomaximal, Sylow
92C22×C4⋊C4—1731no—
93#407—1732no—
94#343—1732no—
95#343—1731no—
96#343D17311yescharacteristic, complemented
97C23×C4—1731no—
98C23×C4—1732no—
99C2×M16—1732no—
100C2×M16C2×D17312yes—
101C2×C4⋊C4—3462no—
102C2×C4⋊C4—3462no—
103C2×C4⋊C4—1732no—
104C2×C4⋊C4—1731no—
105C2×C4⋊C4C2×D17311yescharacteristic
106C22⋊C8—3462no—
107C22⋊C8—3462no—
108C24—1731no—
109C22×C4—3462no—
110C22×C4—3462no—
111C22×C4—3461no—
112C22×C4—3462no—
113C22×C4—3461no—
114C22×C4—1731no—
115C22×C4—1732no—
116C22×C4—1732no—
117C22×C4—1732no—
118C22×C4—6924no—
119C22×C4C4×D17312yes—
120C22×C4C22×D17311yescharacteristic
121M16—22no—
122M16—22no—
123M16—3462no—
124M16—3462no—
125C2×C8—22no—
126C2×C8—3462no—
127C4⋊C4—22no—
128C4⋊C4—3464no—
129C4⋊C4—3462no—
130C23—3462no—
131C23—3462no—
132C23—3461no—
133C23—3461no—
134C23—1731no—
135C23—1731no—
136C23C2×C4×D17311yescharacteristic
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—3461no—
140C2×C4—3461no—
141C2×C4—3462no—
142C2×C4—3462no—
143C2×C4—3462no—
144C2×C4—3462no—
145C2×C4—3462no—
146C2×C4—1732no—
147C2×C4—6924no—
148C2×C4—6924no—
149C2×C4—6922no—
150C2×C4D4×D17312yes—
151C2×C4#1458382511yescharacteristic
152C2×C4#1458382612yes—
153C2×C4#1458382711yescharacteristic
154C8—42no—
155C8—6922no—
156C22—21no—
157C22—21no—
158C22—3464no—
159C22—3464no—
160C22—3461no—
161C22—3461no—
162C22—3461no—
163C22—3461no—
164C22—1732no—
165C22—1732no—
166C22C22⋊C4×D17311yescharacteristic
167C22#1914963611yescharacteristic
168C22#1914963711yescharacteristic
169C4—42no—
170C4—3462no—
171C4—6922no—
172C4#1914966112yes—
173C4#1914966412yes—
174C2—21no—
175C2—21no—
176C2—3462no—
177C2—3462no—
178C2#2601135311yescharacteristic
179C2#100×D17311yescharacteristic
180C2#101×D17311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC23×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#343×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC23×D173C2×C4
cosocleC22×C4×C173C23
hypercenter#343D173
hypocenterC173#1821
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
3. #26011260C2semidirect
4. #26011259C2semidirect
5. #343×C173C2semidirect
90. C173#1821semidirect
96. #343D173semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1732
#260114542
#260112602
#260112592
#343×C1732
#1821173