Finite Group Databasebeta

← #36352517

Subgroups — #36352517

Order 22144.

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

normal subgroups only

This group has 308 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#26011632C212yesmaximal, complemented
3#26011403C211yesmaximal, characteristic, complemented
4#26011226C212yesmaximal, complemented
5#323×C173C211yesmaximal, characteristic, complemented
6C22×Dic346C2211yescharacteristic, complemented
7C22×C22⋊D173—21no—
8C2×#14583815—22no—
9#19149663—22no—
10#19149653—21no—
11#19149653C2212yes—
12#19149649—22no—
13C2×C22⋊Dic173—21no—
14#19149604—22no—
15#19149586—21no—
16#19149586C2211yescharacteristic, complemented
17C2×C22⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yescomplemented
19C23×D173—21no—
20C2×Dic346—22no—
21C2×Dic346—42no—
22C2×D692—21no—
23C2×D692D412yes—
24C2×C22⋊D173—22no—
25C2×C22⋊D173—22no—
26C2×C22⋊D173—42no—
27C2×C22⋊D173—42no—
28C22×Dic173—21no—
29C22×Dic173—21no—
30C22×Dic173—21no—
31#14583815—44no—
32C4⋊Dic173—21no—
33C4⋊Dic173D412yes—
34C22⋊Dic173—42no—
35#14583803—22no—
36#14583803—22no—
37C24×C173D411yescharacteristic
38C22×C4×C173D411yescharacteristic
39C22×C4×C173C2311yescharacteristic
40C2×C8×C173—22no—
41C22⋊C4×C173—22no—
42C22×D173—22no—
43C22×D173—42no—
44C22×D173—81no—
45C22×D173—81no—
46Dic346—24no—
47Dic346—42no—
48D692—42no—
49D692—81no—
50C22⋊D173—44no—
51C22⋊D173—42no—
52C22⋊D173—42no—
53C2×Dic173—22no—
54C2×Dic173—22no—
55C2×Dic173—42no—
56C2×Dic173—41no—
57C2×Dic173—41no—
58C2×Dic173—41no—
59C2×Dic173—41no—
60C2×Dic173—41no—
61C23×C173—21no—
62C23×C173—21no—
63C23×C173—41no—
64C23×C173—41no—
65C23×C173SD1612yescomplemented
66C23×C173C2×D411yescharacteristic
67C2×C4×C173—21no—
68C2×C4×C173—41no—
69C2×C4×C173—41no—
70C2×C4×C173C2×D412yes—
71C8×C173—42no—
72C22×C173—21no—
73C22×C173—21no—
74C22×C173—22no—
75C22×C173—42no—
76C22×C173—41no—
77C22×C173—41no—
78C22×C173—81no—
79C22×C173#12011yescharacteristic
80C22×C173C2×SD1611yescharacteristic
81C22×C173AGL(1,ℤ/8)11yescharacteristic
82C2×D173—42no—
83C2×D173—42no—
84C2×D173—42no—
85C2×D173—84no—
86C4×C173—22no—
87C4×C173—41no—
88Dic173—42no—
89Dic173—42no—
90Dic173—81no—
91C2×C173—21no—
92C2×C173—21no—
93C2×C173—42no—
94C2×C173#45011yescharacteristic
95C2×C173#45511yescharacteristic
96C2×C173B(4,𝔽2)11yescharacteristic
97D173—82no—
98C173#160111yescharacteristic, complemented, Sylow
99#1601—1731nomaximal, Sylow
100#534—1731no—
101#448—1732no—
102#393—1731no—
103#327—1732no—
104#323D17311yescharacteristic, complemented
105C22×Q8—1731no—
106C22×D4—3461no—
107C2×SD16—3462no—
108#123—3462no—
109#121—3461no—
110#121—1732no—
111#120—3462no—
112C2×C22⋊C4—3461no—
113C2×C22⋊C4C2×D17311yescharacteristic
114#103—3462no—
115C22⋊C8C2×D17312yescomplemented
116#95—3461no—
117#95—1731no—
118C24—3461no—
119C24D69211yescharacteristic
120C2×Q8—3462no—
121C2×Q8—6922no—
122C2×D4—3461no—
123C2×D4—3462no—
124C2×D4—3462no—
125C2×D4—1732no—
126C2×D4—6922no—
127C2×D4—6922no—
128C22×C4—3461no—
129C22×C4—3461no—
130C22×C4—3461no—
131C22×C4D69211yescharacteristic
132C22×C4C22×D17311yescharacteristic
133SD16—6924no—
134C2×C8—22no—
135C4⋊C4—3461no—
136C4⋊C4—1732no—
137C22⋊C4—22no—
138C22⋊C4—3462no—
139C22⋊C4—3462no—
140C22⋊C4—6922no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—3462no—
146C23—13841no—
147C23—13841no—
148C23—6922no—
149C23#1458381512yescomplemented
150C23C2×D69211yescharacteristic
151Q8—3464no—
152Q8—6922no—
153D4—13841no—
154D4—6922no—
155D4—6924no—
156D4—6922no—
157D4—6922no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—3462no—
162C2×C4—3462no—
163C2×C4—6921no—
164C2×C4—6921no—
165C2×C4—6921no—
166C2×C4—6922no—
167C2×C4—6921no—
168C2×C4—6921no—
169C2×C4D4×D17312yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—13844no—
179C22—6922no—
180C22—6922no—
181C22—6922no—
182C22#1914964911yescharacteristic
183C22C2×#1458381511yescharacteristic
184C22#1914970711yescharacteristic
185C4—22no—
186C4—41no—
187C4—13841no—
188C4—6922no—
189C4—6922no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—13842no—
194C2#2601164411yescharacteristic
195C2#2601167111yescharacteristic
196C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#323×C173C2
socleC22×C173#120
prime radical#26012053C2
hypercenter#323D173
hypocenterC173#1601
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. #26011632C2semidirect
3. #26011403C2semidirect
4. #26011226C2semidirect
5. #323×C173C2semidirect
6. C22×Dic346C22semidirect
16. #19149586C22semidirect
18. C22⋊C8×C173C22semidirect
65. C23×C173SD16semidirect
98. C173#1601semidirect
104. #323D173semidirect
115. C22⋊C8C2×D173semidirect
149. C23#14583815semidirect

Maximal subgroups

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

subgroupindex
#260120532
#260116322
#260114032
#260112262
#323×C1732
#1601173