Finite Group Databasebeta

← #36352563

Subgroups — #36352563

Order 22144.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#26012090C211yesmaximal, characteristic
2#26011819C211yesmaximal, characteristic, complemented
3#26011805C211yesmaximal, characteristic
4#26011423C211yesmaximal, characteristic, complemented
5#26011224C211yesmaximal, characteristic
6#26011216C211yesmaximal, characteristic, complemented
7#328×C173C211yesmaximal, characteristic, complemented
8#19149674—21no—
9#19149670—21no—
10#19149664—21no—
11#19149663—21no—
12#19149655—21no—
13#19149653C2211yescharacteristic
14#19149639—21no—
15C2×C4⋊Dic173—21no—
16C2×C4⋊Dic173C2211yescharacteristic
17C2×#14583805C2211yescharacteristic
18#19149613—21no—
19#19149610—21no—
20#19149603—21no—
21#19149601—21no—
22C22⋊#13063436C2211yescharacteristic
23#19149586—21no—
24C22⋊Q8×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C2×D692D411yescharacteristic
28C2×C22⋊D173—21no—
29C22×Dic173—21no—
30C22×Dic173D411yescharacteristic
31C22×Dic173D411yescharacteristic
32C2×C4×D173—21no—
33C2×#13063436—21no—
34C4⋊Dic173—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173—21no—
37C4⋊Dic173—21no—
38C4⋊Dic173—42no—
39C4⋊Dic173D411yescharacteristic
40#14583805—21no—
41#14583805—21no—
42#14583805—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—21no—
47C4×Dic173—21no—
48C2×Q8×C173D411yescharacteristic
49C22×C4×C173—21no—
50C22×C4×C173C2311yescharacteristic
51C2×C8×C173—21no—
52C4⋊C4×C173—21no—
53C4⋊C4×C173D411yescharacteristic
54C22⋊C4×C173—21no—
55C22×D173—21no—
56D692—41no—
57C22⋊D173—41no—
58C22⋊D173—41no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C2×Dic173—21no—
64C2×Dic173—41no—
65C2×Dic173—41no—
66C2×Dic173—41no—
67C2×Dic173—41no—
68C4×D173—81no—
69#13063436—41no—
70C23×C173C2×D411yescharacteristic
71Q8×C173—41no—
72C2×C4×C173—22no—
73C2×C4×C173—21no—
74C2×C4×C173—21no—
75C2×C4×C173—41no—
76C2×C4×C173—41no—
77C2×C4×C173—41no—
78C2×C4×C173C2×D411yescharacteristic
79C2×C4×C173C2×D411yescharacteristic
80C8×C173—41no—
81C22×C173—21no—
82C22×C173—21no—
83C22×C173#12011yescharacteristic
84C22×C173#13511yescharacteristic
85C22×C173AGL(1,ℤ/8)11yescharacteristic
86C2×D173—41no—
87C2×D173—41no—
88C2×D173—41no—
89C4×C173—21no—
90C4×C173—21no—
91C4×C173—42no—
92C4×C173—41no—
93C4×C173—41no—
94Dic173—41no—
95Dic173—41no—
96Dic173—41no—
97Dic173—81no—
98C2×C173—21no—
99C2×C173—21no—
100C2×C173#44911yescharacteristic
101C2×C173#45611yescharacteristic
102C2×C173#45811yescharacteristic
103D173—81no—
104C173#160911yescharacteristic, complemented, Sylow
105#1609—1731nomaximal, Sylow
106#539—1731no—
107#483—1731no—
108#480—1731no—
109#398—1731no—
110#328D17311yescharacteristic, complemented
111#327—1731no—
112#324—1731no—
113#126—3461no—
114#125—3461no—
115#123—3461no—
116#123—3461no—
117C22⋊Q8—3461no—
118C22⋊Q8C2×D17311yescharacteristic
119#121—1731no—
120C4×D4—3461no—
121C2×C4⋊C4—3461no—
122C2×C4⋊C4—1731no—
123C2×C4⋊C4—1731no—
124#107—3461no—
125#106—3461no—
126#103—3461no—
127#102—3461no—
128C22⋊C8—1731no—
129C22⋊C8C2×D17311yescharacteristic
130#95—3461no—
131#95C2×D17311yescharacteristic
132C2×Q8D69211yescharacteristic
133C2×D4—3461no—
134C2×D4—1731no—
135C22×C4—21no—
136C22×C4—3461no—
137C22×C4—3461no—
138C22×C4—1731no—
139C22×C4—1731no—
140C22×C4C22×D17311yescharacteristic
141C2×C8—21no—
142C2×C8—3461no—
143C4⋊C4—21no—
144C4⋊C4—3461no—
145C4⋊C4—3461no—
146C4⋊C4—3461no—
147C4⋊C4—3461no—
148C4⋊C4—3461no—
149C4⋊C4—3461no—
150C4⋊C4—3461no—
151C4⋊C4—1731no—
152C4⋊C4—6922no—
153C4⋊C4D69211yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—3461no—
156C22⋊C4—3461no—
157C22⋊C4—3461no—
158C22⋊C4—3461no—
159C42—3461no—
160C23—3461no—
161C23D4×D17311yescharacteristic
162Q8—41no—
163D4—6921no—
164D4—6921no—
165D4—6921no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—3461no—
173C2×C4—3461no—
174C2×C4—3461no—
175C2×C4—3461no—
176C2×C4—3461no—
177C2×C4—13841no—
178C2×C4—6921no—
179C2×C4—6921no—
180C2×C4—6921no—
181C2×C4—6921no—
182C2×C4D4×D17311yescharacteristic
183C2×C4C2×D69211yescharacteristic
184C8—41no—
185C8—6921no—
186C22—21no—
187C22—21no—
188C22—6921no—
189C22—6921no—
190C22—6921no—
191C22#1914964911yescharacteristic
192C22#1914970211yescharacteristic
193C22#1914970611yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—13841no—
200C4—6921no—
201C4—6921no—
202C4—6921no—
203C2—21no—
204C2—21no—
205C2—13841no—
206C2#2601163911yescharacteristic
207C2#2601167511yescharacteristic
208C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173#120
prime radical#19149653C22
hypercenter#328D173
hypocenterC173#1609
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #26011819C2semidirect
4. #26011423C2semidirect
6. #26011216C2semidirect
7. #328×C173C2semidirect
104. C173#1609semidirect
110. #328D173semidirect

Maximal subgroups

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

subgroupindex
#260120902
#260118192
#260118052
#260114232
#260112242
#260112162
#328×C1732
#1609173