Finite Group Databasebeta

← #36352597

Subgroups — #36352597

Order 22144.

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

normal subgroups only

This group has 208 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#26011824C211yesmaximal, characteristic, complemented
3#26011810C211yesmaximal, characteristic
4#26011431C211yesmaximal, 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⋊Dic173C2211yescharacteristic
16C2×#14583805C2211yescharacteristic
17#19149612—21no—
18#19149611—21no—
19#19149603—21no—
20#19149601—21no—
21C22⋊#13063436C2211yescharacteristic
22#19149586—21no—
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⋊Dic173D411yescharacteristic
39#14583805—21no—
40#14583805—21no—
41#14583805—21no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46C4×Dic173—21no—
47C2×Q8×C173D411yescharacteristic
48C22×C4×C173—21no—
49C22×C4×C173C2311yescharacteristic
50C2×C8×C173—21no—
51C4⋊C4×C173—21no—
52C4⋊C4×C173D411yescharacteristic
53C22⋊C4×C173—21no—
54C22×D173—21no—
55D692—41no—
56C22⋊D173—41no—
57C22⋊D173—41no—
58C2×Dic173—21no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C2×Dic173—41no—
64C2×Dic173—42no—
65C2×Dic173—41no—
66C4×D173—81no—
67#13063436—41no—
68C23×C173C2×D411yescharacteristic
69Q8×C173—41no—
70C2×C4×C173—21no—
71C2×C4×C173—21no—
72C2×C4×C173—41no—
73C2×C4×C173—41no—
74C2×C4×C173—41no—
75C2×C4×C173—41no—
76C2×C4×C173C2×D411yescharacteristic
77C2×C4×C173C2×D411yescharacteristic
78C8×C173—41no—
79C22×C173—21no—
80C22×C173—21no—
81C22×C173#12011yescharacteristic
82C22×C173#13511yescharacteristic
83C22×C173#13711yescharacteristic
84C2×D173—41no—
85C2×D173—41no—
86C2×D173—41no—
87C4×C173—21no—
88C4×C173—21no—
89C4×C173—41no—
90C4×C173—41no—
91C4×C173—81no—
92Dic173—41no—
93Dic173—41no—
94Dic173—41no—
95Dic173—81no—
96C2×C173—21no—
97C2×C173—21no—
98C2×C173#45211yescharacteristic
99C2×C173#45411yescharacteristic
100C2×C173#45811yescharacteristic
101D173—81no—
102C173#161511yescharacteristic, complemented, Sylow
103#1615—1731nomaximal, Sylow
104#539—1731no—
105#484—1731no—
106#481—1731no—
107#400—1731no—
108#328D17311yescharacteristic, complemented
109#327—1731no—
110#324—1731no—
111#126—3461no—
112#125—3461no—
113#123—3461no—
114#123—3461no—
115C22⋊Q8—3461no—
116C22⋊Q8C2×D17311yescharacteristic
117#121—1731no—
118C4×D4—3461no—
119C2×C4⋊C4—1731no—
120C2×C4⋊C4—1731no—
121#107—3461no—
122#106—3461no—
123#103—3461no—
124#102—3461no—
125C22⋊C8—1731no—
126C22⋊C8C2×D17311yescharacteristic
127#95—3461no—
128#95—3461no—
129#95C2×D17311yescharacteristic
130C2×Q8D69211yescharacteristic
131C2×D4—3461no—
132C2×D4—1731no—
133C22×C4—21no—
134C22×C4—3461no—
135C22×C4—3461no—
136C22×C4—1731no—
137C22×C4—1731no—
138C22×C4C22×D17311yescharacteristic
139C2×C8—21no—
140C2×C8—3461no—
141C4⋊C4—21no—
142C4⋊C4—3461no—
143C4⋊C4—3461no—
144C4⋊C4—3461no—
145C4⋊C4—3461no—
146C4⋊C4—3461no—
147C4⋊C4—3461no—
148C4⋊C4—3461no—
149C4⋊C4—1731no—
150C4⋊C4D69211yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—3461no—
153C22⋊C4—3461no—
154C22⋊C4—3461no—
155C22⋊C4—3461no—
156C42—3461no—
157C23—3461no—
158C23D4×D17311yescharacteristic
159Q8—41no—
160D4—6921no—
161D4—6921no—
162D4—6921no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3461no—
170C2×C4—3461no—
171C2×C4—3461no—
172C2×C4—3461no—
173C2×C4—3461no—
174C2×C4—13841no—
175C2×C4—6922no—
176C2×C4—6921no—
177C2×C4—6921no—
178C2×C4D4×D17311yescharacteristic
179C2×C4C2×D69211yescharacteristic
180C8—41no—
181C8—6921no—
182C22—21no—
183C22—21no—
184C22—6921no—
185C22—6921no—
186C22—6921no—
187C22#1914964911yescharacteristic
188C22#1914970011yescharacteristic
189C22#1914971111yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—41no—
194C4—81no—
195C4—13841no—
196C4—6921no—
197C4—6921no—
198C4—6921no—
199C2—21no—
200C2—21no—
201C2—13841no—
202C2#2601165611yescharacteristic
203C2#2601166511yescharacteristic
204C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173#120
prime radical#19149653C22
hypercenter#328D173
hypocenterC173#1615
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. #26011824C2semidirect
4. #26011431C2semidirect
6. #26011216C2semidirect
7. #328×C173C2semidirect
102. C173#1615semidirect
108. #328D173semidirect

Maximal subgroups

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

subgroupindex
#260120902
#260118242
#260118102
#260114312
#260112242
#260112162
#328×C1732
#1615173