Finite Group Databasebeta

← #36352561

Subgroups — #36352561

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#26012086C211yesmaximal, characteristic
2#26011822C211yesmaximal, characteristic, complemented
3#26011802C211yesmaximal, characteristic
4#26011423C211yesmaximal, 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⋊Dic173—21no—
16C2×C4⋊Dic173C2211yescharacteristic
17C2×#14583805C2211yescharacteristic
18#19149612—21no—
19#19149611—21no—
20#19149604—21no—
21#19149600—21no—
22C22⋊#13063436C2211yescharacteristic
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⋊Dic173—42no—
37C4⋊Dic173D411yescharacteristic
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41#14583805—21no—
42#14583805—21no—
43C22⋊Dic173—21no—
44C22⋊Dic173—21no—
45C22⋊Dic173—21no—
46C22⋊Dic173—21no—
47C4×Dic173—21no—
48C2×D4×C173—21no—
49C2×D4×C173D411yescharacteristic
50C22×C4×C173—21no—
51C22×C4×C173C2311yescharacteristic
52C2×C8×C173—21no—
53C4⋊C4×C173D411yescharacteristic
54C22⋊C4×C173—21no—
55Dic346—41no—
56C2×Dic173—21no—
57C2×Dic173—21no—
58C2×Dic173—21no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—41no—
63C2×Dic173—41no—
64C2×Dic173—41no—
65C2×Dic173—41no—
66C2×Dic173—81no—
67#13063436—41no—
68C23×C173—21no—
69C23×C173C2×D411yescharacteristic
70D4×C173—41no—
71D4×C173—41no—
72D4×C173—41no—
73C2×C4×C173—22no—
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—41no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173#12011yescharacteristic
87C22×C173#13511yescharacteristic
88C22×C173AGL(1,ℤ/8)11yescharacteristic
89C4×C173—21no—
90C4×C173—21no—
91C4×C173—42no—
92C4×C173—41no—
93Dic173—41no—
94Dic173—41no—
95Dic173—41no—
96Dic173—41no—
97Dic173—81no—
98C2×C173—21no—
99C2×C173—21no—
100C2×C173—81no—
101C2×C173#44911yescharacteristic
102C2×C173#45611yescharacteristic
103C2×C173#45811yescharacteristic
104C173#160911yescharacteristic, complemented, Sylow
105#1609—1731nomaximal, Sylow
106#539—1731no—
107#483—1731no—
108#480—1731no—
109#398—1731no—
110#328—1731no—
111#327D17311yescharacteristic
112#324—1731no—
113#126—3461no—
114#125—3461no—
115#123—3461no—
116#123—3461no—
117C22⋊Q8—3461no—
118C22⋊Q8—1731no—
119#121C2×D17311yescharacteristic
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×Q8—1731no—
133C2×D4—21no—
134C2×D4D69211yescharacteristic
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—3461no—
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—21no—
161C23D4×D17311yescharacteristic
162Q8—6921no—
163D4—41no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—3461no—
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—41no—
189C22—41no—
190C22—41no—
191C22#1914964911yescharacteristic
192C22#1914970011yescharacteristic
193C22#1914970311yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—13841no—
199C4—6921no—
200C4—6921no—
201C4—6921no—
202C4—6921no—
203C2—21no—
204C2—21no—
205C2—81no—
206C2#2601163711yescharacteristic
207C2#2601167511yescharacteristic
208C2#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#1609
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. #26011822C2semidirect
4. #26011423C2semidirect
5. #26011227C2semidirect
6. #26011216C2semidirect
104. C173#1609semidirect

Maximal subgroups

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

subgroupindex
#260120862
#260118222
#260118022
#260114232
#260112272
#260112162
#327×C1732
#1609173