Finite Group Databasebeta

← #36352594

Subgroups — #36352594

Order 22144.

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

normal subgroups only

This group has 208 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#26011824C211yesmaximal, characteristic, complemented
2#26011807C211yesmaximal, characteristic
3#26011430C211yesmaximal, characteristic, complemented
4#26011228C211yesmaximal, characteristic, complemented
5#26011225C211yesmaximal, characteristic
6#26011217C211yesmaximal, characteristic, complemented
7#539×C173C211yesmaximal, characteristic
8C2×C4⋊Dic173C2211yescharacteristic
9#19149612—21no—
10#19149610—21no—
11#19149603—21no—
12#19149600—21no—
13C22⋊#13063436C2211yescharacteristic
14C22⋊#13063436C2211yescharacteristic
15#19149586—21no—
16#19149586—21no—
17#19149586C2211yescharacteristic
18#126×C173—21no—
19#125×C173—21no—
20#123×C173—21no—
21#123×C173—21no—
22C22⋊Q8×C173—21no—
23C22⋊Q8×C173C2211yescharacteristic
24#121×C173C2211yescharacteristic
25C4×D4×C173—21no—
26C2×C4⋊C4×C173C2211yescharacteristic
27C22×Dic173—21no—
28C22×Dic173—21no—
29C2×#13063436—21no—
30C2×#13063436—21no—
31C4⋊Dic173—21no—
32C4⋊Dic173—21no—
33C2×Q8×C173D411yescharacteristic
34C2×D4×C173—21no—
35C2×D4×C173D411yescharacteristic
36C22×C4×C173—21no—
37C22×C4×C173D411yescharacteristic
38C22×C4×C173D411yescharacteristic
39C22×C4×C173C2311yescharacteristic
40C4⋊C4×C173—21no—
41C4⋊C4×C173—21no—
42C4⋊C4×C173—21no—
43C4⋊C4×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173D411yescharacteristic
47C4⋊C4×C173D411yescharacteristic
48C22⋊C4×C173—21no—
49C22⋊C4×C173—21no—
50C22⋊C4×C173—21no—
51C22⋊C4×C173—21no—
52C22⋊C4×C173—21no—
53C42×C173—21no—
54C2×Dic173—41no—
55C2×Dic173—41no—
56C2×Dic173—42no—
57C2×Dic173—41no—
58C2×Dic173—41no—
59#13063436—41no—
60#13063436—41no—
61C23×C173—21no—
62C23×C173C2×D411yescharacteristic
63Q8×C173—41no—
64D4×C173—41no—
65D4×C173—41no—
66D4×C173—41no—
67C2×C4×C173—21no—
68C2×C4×C173—21no—
69C2×C4×C173—21no—
70C2×C4×C173—21no—
71C2×C4×C173—21no—
72C2×C4×C173—21no—
73C2×C4×C173—21no—
74C2×C4×C173—41no—
75C2×C4×C173—41no—
76C2×C4×C173—81no—
77C2×C4×C173C2×D411yescharacteristic
78C2×C4×C173C2×D411yescharacteristic
79C22×C173—21no—
80C22×C173—21no—
81C22×C173—41no—
82C22×C173—41no—
83C22×C173—41no—
84C22×C173#12011yescharacteristic
85C22×C173#13511yescharacteristic
86C22×C173#13711yescharacteristic
87C4×C173—21no—
88C4×C173—21no—
89C4×C173—41no—
90C4×C173—41no—
91C4×C173—41no—
92C4×C173—41no—
93C4×C173—41no—
94Dic173—81no—
95Dic173—81no—
96C2×C173—21no—
97C2×C173—21no—
98C2×C173—81no—
99C2×C173#45211yescharacteristic
100C2×C173#45411yescharacteristic
101C2×C173#45811yescharacteristic
102C173#161511yescharacteristic, complemented, Sylow
103#1615—1731nomaximal, Sylow
104#539D17311yescharacteristic
105#484—1731no—
106#481—1731no—
107#400—1731no—
108#328—1731no—
109#327—1731no—
110#324—1731no—
111#126—21no—
112#125—21no—
113#123—21no—
114#123—21no—
115C22⋊Q8—21no—
116C22⋊Q8C2×D17311yescharacteristic
117#121C2×D17311yescharacteristic
118C4×D4—21no—
119C2×C4⋊C4—1731no—
120C2×C4⋊C4C2×D17311yescharacteristic
121#107—3461no—
122#106—3461no—
123#103—3461no—
124#102—3461no—
125C22⋊C8—1731no—
126C22⋊C8—1731no—
127#95—3461no—
128#95—3461no—
129#95—1731no—
130C2×Q8C22⋊D17311yescharacteristic
131C2×D4—21no—
132C2×D4C22⋊D17311yescharacteristic
133C22×C4—21no—
134C22×C4—3461no—
135C22×C4—3461no—
136C22×C4C22⋊D17311yescharacteristic
137C22×C4C22⋊D17311yescharacteristic
138C22×C4C22×D17311yescharacteristic
139C2×C8—3461no—
140C2×C8—3461no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—3461no—
148C4⋊C4—3461no—
149C4⋊C4C22⋊D17311yescharacteristic
150C4⋊C4C22⋊D17311yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—21no—
156C42—21no—
157C23—21no—
158C23C2×C22⋊D17311yescharacteristic
159Q8—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—81no—
173C2×C4—6921no—
174C2×C4—6921no—
175C2×C4—6922no—
176C2×C4—6921no—
177C2×C4—6921no—
178C2×C4C2×C22⋊D17311yescharacteristic
179C2×C4C2×C22⋊D17311yescharacteristic
180C8—6921no—
181C8—6921no—
182C22—21no—
183C22—21no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22#1914964811yescharacteristic
188C22#1914969911yescharacteristic
189C22#1914970911yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—41no—
194C4—41no—
195C4—41no—
196C4—41no—
197C4—13841no—
198C4—13841no—
199C2—21no—
200C2—21no—
201C2—81no—
202C2#2601165311yescharacteristic
203C2#2601166411yescharacteristic
204C2#2601168111yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#539×C173C2
socleC22×C173#120
prime radical#121×C173C22
hypercenter#539D173
hypocenterC173#1615
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #26011824C2semidirect
3. #26011430C2semidirect
4. #26011228C2semidirect
6. #26011217C2semidirect
102. C173#1615semidirect

Maximal subgroups

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

subgroupindex
#260118242
#260118072
#260114302
#260112282
#260112252
#260112172
#539×C1732
#1615173