Finite Group Databasebeta

← #36352585

Subgroups — #36352585

Order 22144.

288 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 53 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#26012053C211yesmaximal, characteristic
2#26011810C212yesmaximal, complemented
3#26011427C211yesmaximal, characteristic, complemented
4#26011226C212yesmaximal, complemented
5#323×C173C211yesmaximal, characteristic, complemented
6C22×C22⋊D173—21no—
7#19149663—22no—
8#19149653—21no—
9#19149653C2212yes—
10#19149649—22no—
11C2×C4⋊Dic173C2211yescharacteristic, complemented
12C2×#14583805—21no—
13C2×C22⋊Dic173—21no—
14#19149611—22no—
15#19149601—22no—
16#19149586C2211yescharacteristic, complemented
17C2×C22⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yescomplemented
19C23×D173—21no—
20C2×D692—21no—
21C2×D692D412yes—
22C2×C22⋊D173—22no—
23C2×C22⋊D173—22no—
24C2×C22⋊D173—42no—
25C2×C22⋊D173—42no—
26C22×Dic173—21no—
27C22×Dic173—21no—
28C22×Dic173—21no—
29C4⋊Dic173—21no—
30C4⋊Dic173—22no—
31C4⋊Dic173D412yes—
32#14583805—42no—
33C22⋊Dic173—42no—
34#14583803—22no—
35#14583803—22no—
36C24×C173D411yescharacteristic
37C22×C4×C173D411yescharacteristic
38C22×C4×C173C2311yescharacteristic
39C2×C8×C173—22no—
40C22⋊C4×C173—22no—
41C22×D173—22no—
42C22×D173—42no—
43C22×D173—81no—
44C22×D173—81no—
45D692—42no—
46D692—81no—
47C22⋊D173—44no—
48C22⋊D173—42no—
49C22⋊D173—42no—
50C2×Dic173—22no—
51C2×Dic173—22no—
52C2×Dic173—41no—
53C2×Dic173—42no—
54C2×Dic173—41no—
55C2×Dic173—41no—
56C2×Dic173—41no—
57C2×Dic173—41no—
58C23×C173—21no—
59C23×C173—21no—
60C23×C173—41no—
61C23×C173—41no—
62C23×C173SD1612yes—
63C23×C173C2×D411yescharacteristic
64C2×C4×C173—21no—
65C2×C4×C173—41no—
66C2×C4×C173—41no—
67C2×C4×C173C2×D412yes—
68C8×C173—42no—
69C22×C173—21no—
70C22×C173—21no—
71C22×C173—22no—
72C22×C173—42no—
73C22×C173—41no—
74C22×C173—41no—
75C22×C173—81no—
76C22×C173#12011yescharacteristic
77C22×C173C2×SD1611yescharacteristic
78C22×C173AGL(1,ℤ/8)11yescharacteristic
79C2×D173—42no—
80C2×D173—42no—
81C2×D173—42no—
82C2×D173—84no—
83C4×C173—22no—
84C4×C173—41no—
85Dic173—42no—
86Dic173—42no—
87Dic173—81no—
88C2×C173—21no—
89C2×C173—21no—
90C2×C173—42no—
91C2×C173#45011yescharacteristic
92C2×C173#45311yescharacteristic
93C2×C173#45811yescharacteristic
94D173—82no—
95C173#161311yescharacteristic, complemented, Sylow
96#1613—1731nomaximal, Sylow
97#534—1731no—
98#481—1732no—
99#399—1731no—
100#327—1732no—
101#323D17311yescharacteristic, complemented
102C22×D4—3461no—
103#123—3462no—
104#121—3461no—
105#121—1732no—
106#120—3462no—
107C2×C4⋊C4—3461no—
108C2×C4⋊C4—1731no—
109C2×C22⋊C4—3461no—
110C2×C22⋊C4C2×D17311yescharacteristic
111#106—3462no—
112#102—3462no—
113C22⋊C8C2×D17312yescomplemented
114#95—1731no—
115C24—3461no—
116C24D69211yescharacteristic
117C2×D4—3461no—
118C2×D4—3462no—
119C2×D4—3462no—
120C2×D4—1732no—
121C2×D4—6922no—
122C2×D4—6922no—
123C22×C4—3461no—
124C22×C4—3461no—
125C22×C4—3461no—
126C22×C4D69211yescharacteristic
127C22×C4C22×D17311yescharacteristic
128C2×C8—22no—
129C4⋊C4—3461no—
130C4⋊C4—3462no—
131C4⋊C4—1732no—
132C4⋊C4—6922no—
133C22⋊C4—22no—
134C22⋊C4—3462no—
135C22⋊C4—3462no—
136C22⋊C4—6922no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—3462no—
142C23—13841no—
143C23—13841no—
144C23—6922no—
145C23#1458381512yes—
146C23C2×D69211yescharacteristic
147D4—13841no—
148D4—6922no—
149D4—6924no—
150D4—6922no—
151D4—6922no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—3462no—
156C2×C4—3462no—
157C2×C4—6921no—
158C2×C4—6921no—
159C2×C4—6921no—
160C2×C4—6921no—
161C2×C4—6922no—
162C2×C4—6921no—
163C2×C4D4×D17312yes—
164C8—42no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—13844no—
173C22—6922no—
174C22—6922no—
175C22—6922no—
176C22#1914964911yescharacteristic
177C22C2×#1458381511yescharacteristic
178C22#1914970711yescharacteristic
179C4—22no—
180C4—41no—
181C4—13841no—
182C4—6922no—
183C4—6922no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—13842no—
188C2#2601164411yescharacteristic
189C2#2601166111yescharacteristic
190C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#323×C173C2
socleC22×C173#120
prime radical#26012053C2
hypercenter#323D173
hypocenterC173#1613
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #26011810C2semidirect
3. #26011427C2semidirect
4. #26011226C2semidirect
5. #323×C173C2semidirect
11. C2×C4⋊Dic173C22semidirect
16. #19149586C22semidirect
18. C22⋊C8×C173C22semidirect
95. C173#1613semidirect
101. #323D173semidirect
113. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120532
#260118102
#260114272
#260112262
#323×C1732
#1613173