Finite Group Databasebeta

← #36352595

Subgroups — #36352595

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#26012086C211yesmaximal, characteristic
2#26011827C211yesmaximal, characteristic, complemented
3#26011807C211yesmaximal, characteristic
4#26011431C211yesmaximal, characteristic, complemented
5#26011227C211yesmaximal, characteristic, complemented
6#26011216C211yesmaximal, characteristic, complemented
7#327×C173C211yesmaximal, characteristic
8#19149672—21no—
9#19149669—21no—
10#19149660—21no—
11#19149660—21no—
12#19149658C2211yescharacteristic
13#19149656—21no—
14D4×Dic173—21no—
15C2×C4⋊Dic173C2211yescharacteristic
16C2×#14583805C2211yescharacteristic
17#19149613—21no—
18#19149610—21no—
19#19149604—21no—
20#19149600—21no—
21C22⋊#13063436C2211yescharacteristic
22#19149586—21no—
23#19149586—21no—
24#121×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C2×Dic346D411yescharacteristic
28C22×Dic173—21no—
29C22×Dic173—21no—
30C22×Dic173D411yescharacteristic
31C22×Dic173D411yescharacteristic
32C2×#13063436—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173D411yescharacteristic
37#14583805—21no—
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41#14583805—21no—
42C22⋊Dic173—21no—
43C22⋊Dic173—21no—
44C22⋊Dic173—21no—
45C22⋊Dic173—21no—
46C4×Dic173—21no—
47C2×D4×C173—21no—
48C2×D4×C173D411yescharacteristic
49C22×C4×C173—21no—
50C22×C4×C173C2311yescharacteristic
51C2×C8×C173—21no—
52C4⋊C4×C173D411yescharacteristic
53C22⋊C4×C173—21no—
54Dic346—41no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C2×Dic173—21no—
58C2×Dic173—21no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—41no—
62C2×Dic173—42no—
63C2×Dic173—41no—
64C2×Dic173—81no—
65#13063436—41no—
66C23×C173—21no—
67C23×C173C2×D411yescharacteristic
68D4×C173—41no—
69D4×C173—41no—
70D4×C173—41no—
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—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—81no—
91Dic173—41no—
92Dic173—41no—
93Dic173—41no—
94Dic173—41no—
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#539—1731no—
105#484—1731no—
106#481—1731no—
107#400—1731no—
108#328—1731no—
109#327D17311yescharacteristic
110#324—1731no—
111#126—3461no—
112#125—3461no—
113#123—3461no—
114#123—3461no—
115C22⋊Q8—3461no—
116C22⋊Q8—1731no—
117#121C2×D17311yescharacteristic
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×Q8—1731no—
131C2×D4—21no—
132C2×D4D69211yescharacteristic
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—3461no—
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—21no—
158C23D4×D17311yescharacteristic
159Q8—6921no—
160D4—41no—
161D4—41no—
162D4—41no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—3461no—
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—41no—
185C22—41no—
186C22—41no—
187C22#1914964911yescharacteristic
188C22#1914969811yescharacteristic
189C22#1914970811yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—81no—
194C4—13841no—
195C4—6921no—
196C4—6921no—
197C4—6921no—
198C4—6921no—
199C2—21no—
200C2—21no—
201C2—81no—
202C2#2601165411yescharacteristic
203C2#2601166511yescharacteristic
204C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#327×C173C2
socleC22×C173#120
prime radical#121×C173C22
hypercenter#327D173
hypocenterC173#1615
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #26011827C2semidirect
4. #26011431C2semidirect
5. #26011227C2semidirect
6. #26011216C2semidirect
102. C173#1615semidirect

Maximal subgroups

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

subgroupindex
#260120862
#260118272
#260118072
#260114312
#260112272
#260112162
#327×C1732
#1615173