Finite Group Databasebeta

← #36352501

Subgroups — #36352501

Order 22144.

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

normal subgroups only

This group has 266 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#26012120C211yesmaximal, characteristic, complemented
2#26011642C211yesmaximal, characteristic
3#26011405C211yesmaximal, characteristic
4#26011229C211yesmaximal, characteristic, complemented
5#26011228C211yesmaximal, characteristic, complemented
6#26011216C211yesmaximal, characteristic, complemented
7#450×C173C211yesmaximal, characteristic
8C2×#14583814—21no—
9#19149678—21no—
10#19149677—21no—
11#19149669—21no—
12#19149669—21no—
13#19149658C2211yescharacteristic
14#19149656—21no—
15#19149635—21no—
16C2×#14583805C2211yescharacteristic
17C2×C22⋊Dic173—21no—
18C2×C22⋊Dic173—21no—
19#19149600—21no—
20C22⋊#13063436C2211yescharacteristic
21#19149586C2211yescharacteristic
22C22×D4×C173C2211yescharacteristic
23C2×SD16×C173—21no—
24C22⋊Q8×C173C2211yescharacteristic
25#102×C173—21no—
26C22⋊C8×C173C2211yescharacteristic
27C2×Dic346—21no—
28C2×Dic346D411yescharacteristic
29C22×Dic173—21no—
30C22×Dic173D411yescharacteristic
31C22×Dic173D411yescharacteristic
32#14583814—42no—
33C2×#13063436—21no—
34C4⋊Dic173—21no—
35C4⋊Dic173D411yescharacteristic
36#14583805—21no—
37#14583805—21no—
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41#14583805—21no—
42#14583805—21no—
43C22⋊Dic173—21no—
44C22⋊Dic173—42no—
45C22⋊Dic173—42no—
46C4×Dic173—21no—
47C4×Dic173—21no—
48C24×C173—21no—
49C2×Q8×C173D411yescharacteristic
50C2×D4×C173—21no—
51C2×D4×C173—21no—
52C2×D4×C173—42no—
53C22×C4×C173C2311yescharacteristic
54SD16×C173—42no—
55C2×C8×C173—21no—
56C4⋊C4×C173—21no—
57C4⋊C4×C173D411yescharacteristic
58C22⋊C4×C173—21no—
59Dic346—41no—
60Dic346—81no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C2×Dic173—41no—
67C2×Dic173—41no—
68C2×Dic173—41no—
69C2×Dic173—41no—
70C2×Dic173SD1612yescomplemented
71#13063436—41no—
72C23×C173—22no—
73C23×C173—41no—
74C23×C173—41no—
75C23×C173—41no—
76C23×C173—41no—
77C23×C173—41no—
78C23×C173—41no—
79C23×C173C2×D411yescharacteristic
80Q8×C173—41no—
81D4×C173—22no—
82D4×C173—22no—
83D4×C173—41no—
84D4×C173—41no—
85C2×C4×C173—21no—
86C2×C4×C173—21no—
87C2×C4×C173—41no—
88C2×C4×C173C2×D411yescharacteristic
89C2×C4×C173C2×D411yescharacteristic
90C8×C173—41no—
91C22×C173—21no—
92C22×C173—21no—
93C22×C173—42no—
94C22×C173—42no—
95C22×C173—42no—
96C22×C173—81no—
97C22×C173—81no—
98C22×C173—81no—
99C22×C173—81no—
100C22×C173#12011yescharacteristic
101C22×C173C2×SD1611yescharacteristic
102C22×C173#13711yescharacteristic
103C4×C173—21no—
104C4×C173—21no—
105C4×C173—41no—
106C4×C173—41no—
107Dic173—22no—
108Dic173—41no—
109Dic173—41no—
110Dic173—41no—
111Dic173—81no—
112C2×C173—21no—
113C2×C173—21no—
114C2×C173—82no—
115C2×C173#44811yescharacteristic
116C2×C173#45411yescharacteristic
117C2×C173B(4,𝔽2)11yescharacteristic
118C173#159911yescharacteristic, complemented, Sylow
119#1599—1731nomaximal, Sylow
120#544—1731no—
121#450—1731no—
122#450D17311yescharacteristic
123#394—1731no—
124#328—1731no—
125#328—1731no—
126#324—1731no—
127C22×D4C2×D17311yescharacteristic
128C2×SD16—21no—
129C2×SD16—3461no—
130C4⋊Q8—3461no—
131C4⋊Q8—3461no—
132#125—3461no—
133#125—3461no—
134C22⋊Q8—3461no—
135C22⋊Q8—1731no—
136C22⋊Q8C2×D17311yescharacteristic
137#117—3461no—
138C2×C4⋊C4—1731no—
139C2×C22⋊C4—3461no—
140C2×C22⋊C4—3461no—
141#102—21no—
142#102—3461no—
143C22⋊C8—1731no—
144C22⋊C8C2×D17311yescharacteristic
145#95—1731no—
146C24—21no—
147C2×Q8—3461no—
148C2×Q8—1731no—
149C2×Q8C22⋊D17311yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C22×C4—3461no—
154C22×C4—1731no—
155C22×C4—1731no—
156C22×C4C22×D17311yescharacteristic
157SD16—42no—
158SD16—6922no—
159C2×C8—21no—
160C2×C8—3461no—
161C4⋊C4—21no—
162C4⋊C4—3461no—
163C4⋊C4—3461no—
164C4⋊C4—3461no—
165C4⋊C4—3461no—
166C4⋊C4—3461no—
167C4⋊C4—3461no—
168C4⋊C4—3461no—
169C4⋊C4—3461no—
170C4⋊C4—1731no—
171C4⋊C4C22⋊D17311yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—3461no—
174C22⋊C4—6922no—
175C22⋊C4—6922no—
176C42—3461no—
177C42—3461no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23D4×D17311yescharacteristic
186Q8—41no—
187Q8—13841no—
188Q8—6921no—
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—3461no—
197C2×C4—3461no—
198C2×C4—3461no—
199C2×C4—3461no—
200C2×C4—3461no—
201C2×C4—1732no—
202C2×C4—6921no—
203C2×C4—6921no—
204C2×C4—6921no—
205C2×C4—6921no—
206C2×C4D4×D17311yescharacteristic
207C2×C4C2×C22⋊D17311yescharacteristic
208C8—41no—
209C8—6921no—
210C22—21no—
211C22—21no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22#1914964711yescharacteristic
220C22SD16×D17311yescharacteristic
221C22#1914970811yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—3462no—
227C4—13841no—
228C4—6921no—
229C4—6921no—
230C4—6921no—
231C2—21no—
232C2—21no—
233C2—82no—
234C2#2601162811yescharacteristic
235C2#2601166211yescharacteristic
236C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#450×C173C2
socleC22×C173#120
prime radicalC22×D4×C173C22
hypercenter#450D173
hypocenterC173#1599
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26012120C2semidirect
4. #26011229C2semidirect
5. #26011228C2semidirect
6. #26011216C2semidirect
70. C2×Dic173SD16semidirect
118. C173#1599semidirect

Maximal subgroups

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

subgroupindex
#260121202
#260116422
#260114052
#260112292
#260112282
#260112162
#450×C1732
#1599173