Finite Group Databasebeta

← #36352518

Subgroups — #36352518

Order 22144.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#26012116C211yesmaximal, characteristic
2#26011647C211yesmaximal, characteristic, complemented
3#26011402C211yesmaximal, characteristic
4#26011229C211yesmaximal, characteristic, complemented
5#26011228C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic
7#451×C173C211yesmaximal, characteristic, complemented
8C2×#14583816—21no—
9#19149659—21no—
10#19149658C2211yescharacteristic
11#19149657—21no—
12#19149657—21no—
13#19149656—21no—
14#19149655—21no—
15C2×#14583803—21no—
16C2×#14583803C2211yescharacteristic
17C22⋊C4×D173—21no—
18#19149603—21no—
19C22⋊#13063436C2211yescharacteristic
20#19149586—21no—
21#19149586C2211yescharacteristic
22C22×Q8×C173C2211yescharacteristic
23C2×Q16×C173—21no—
24C22⋊Q8×C173C2211yescharacteristic
25#103×C173—21no—
26C22⋊C8×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28C2×Dic346—21no—
29C2×Dic346D411yescharacteristic
30C22×Dic173—21no—
31C22×Dic173D411yescharacteristic
32C2×C4×D173—21no—
33C2×C4×D173—21no—
34#14583816—42no—
35C2×#13063436—21no—
36C4⋊Dic173—21no—
37C4⋊Dic173D411yescharacteristic
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41C22⋊Dic173—21no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—42no—
47C2×Q8×C173—21no—
48C2×Q8×C173—21no—
49C2×Q8×C173—42no—
50C2×Q8×C173D411yescharacteristic
51C22×C4×C173—21no—
52C22×C4×C173C2311yescharacteristic
53Q16×C173—42no—
54C2×C8×C173—21no—
55C4⋊C4×C173—21no—
56C4⋊C4×C173D411yescharacteristic
57C22⋊C4×C173—21no—
58C22×D173—21no—
59C22×D173—21no—
60C22×D173—41no—
61C22×D173—41no—
62C22×D173Q1612yescomplemented
63Dic346—41no—
64Dic346—81no—
65C2×Dic173—21no—
66C2×Dic173—21no—
67C2×Dic173—21no—
68C2×Dic173—41no—
69C2×Dic173—41no—
70C2×Dic173—41no—
71C2×Dic173—41no—
72C4×D173—81no—
73C4×D173—81no—
74#13063436—41no—
75C23×C173C2×D411yescharacteristic
76Q8×C173—22no—
77Q8×C173—22no—
78Q8×C173—41no—
79Q8×C173—41no—
80Q8×C173—41no—
81C2×C4×C173—22no—
82C2×C4×C173—21no—
83C2×C4×C173—21no—
84C2×C4×C173—41no—
85C2×C4×C173—41no—
86C2×C4×C173—41no—
87C2×C4×C173C2×D411yescharacteristic
88C2×C4×C173C2×D411yescharacteristic
89C8×C173—41no—
90C22×C173—21no—
91C22×C173—21no—
92C22×C173#12011yescharacteristic
93C22×C173C2×Q1611yescharacteristic
94C22×C173#13711yescharacteristic
95C2×D173—22no—
96C2×D173—41no—
97C2×D173—41no—
98C2×D173—42no—
99C2×D173—81no—
100C4×C173—21no—
101C4×C173—21no—
102C4×C173—41no—
103C4×C173—42no—
104C4×C173—41no—
105Dic173—41no—
106Dic173—41no—
107Dic173—41no—
108Dic173—81no—
109C2×C173—21no—
110C2×C173—21no—
111C2×C173#45111yescharacteristic
112C2×C173#45511yescharacteristic
113C2×C173B(4,𝔽2)11yescharacteristic
114D173—42no—
115C173#160211yescharacteristic, complemented, Sylow
116#1602—1731nomaximal, Sylow
117#543—1731no—
118#451—1731no—
119#451D17311yescharacteristic, complemented
120#393—1731no—
121#328—1731no—
122#328—1731no—
123#323—1731no—
124C22×Q8C2×D17311yescharacteristic
125C2×Q16—21no—
126C2×Q16—3461no—
127C22⋊Q8—3461no—
128C22⋊Q8—3461no—
129C22⋊Q8—3461no—
130C22⋊Q8—3461no—
131C22⋊Q8—3461no—
132C22⋊Q8—1731no—
133C22⋊Q8C2×D17311yescharacteristic
134C2×C22⋊C4—3461no—
135C2×C22⋊C4—3461no—
136C2×C22⋊C4—1731no—
137#103—21no—
138#103—3461no—
139C22⋊C8—1731no—
140C22⋊C8C2×D17311yescharacteristic
141#95—3461no—
142#95—1731no—
143C24—1731no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8—3461no—
148C2×Q8—1731no—
149C2×Q8C22⋊D17311yescharacteristic
150C22×C4—21no—
151C22×C4—3461no—
152C22×C4—3461no—
153C22×C4—3461no—
154C22×C4—1731no—
155C22×C4C22×D17311yescharacteristic
156Q16—42no—
157Q16—6922no—
158C2×C8—21no—
159C2×C8—3461no—
160C4⋊C4—21no—
161C4⋊C4—3461no—
162C4⋊C4—3461no—
163C4⋊C4—3461no—
164C4⋊C4—3461no—
165C4⋊C4—1731no—
166C4⋊C4C22⋊D17311yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—3461no—
169C22⋊C4—3461no—
170C22⋊C4—3461no—
171C22⋊C4—3461no—
172C22⋊C4—3461no—
173C22⋊C4—6922no—
174C23—3461no—
175C23—3461no—
176C23—1732no—
177C23—6921no—
178C23—6921no—
179C23D4×D17311yescharacteristic
180Q8—22no—
181Q8—22no—
182Q8—41no—
183Q8—41no—
184Q8—41no—
185Q8—13841no—
186Q8—6921no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—3461no—
194C2×C4—3461no—
195C2×C4—3461no—
196C2×C4—13841no—
197C2×C4—13841no—
198C2×C4—6921no—
199C2×C4—6921no—
200C2×C4—6921no—
201C2×C4—6921no—
202C2×C4D4×D17311yescharacteristic
203C2×C4C2×C22⋊D17311yescharacteristic
204C8—41no—
205C8—6921no—
206C22—21no—
207C22—21no—
208C22—3462no—
209C22—13841no—
210C22—6922no—
211C22—6921no—
212C22—6921no—
213C22#1914964711yescharacteristic
214C22Q16×D17311yescharacteristic
215C22#1914971111yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—13841no—
222C4—6921no—
223C4—6921no—
224C4—6921no—
225C2—21no—
226C2—21no—
227C2—6922no—
228C2#2601164511yescharacteristic
229C2#2601166711yescharacteristic
230C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#451×C173C2
socleC22×C173#120
prime radicalC23×D173D4
hypercenter#451D173
hypocenterC173#1602
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #26011647C2semidirect
4. #26011229C2semidirect
5. #26011228C2semidirect
7. #451×C173C2semidirect
62. C22×D173Q16semidirect
115. C173#1602semidirect
119. #451D173semidirect

Maximal subgroups

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

subgroupindex
#260121162
#260116472
#260114022
#260112292
#260112282
#260112142
#451×C1732
#1602173