Finite Group Databasebeta

← #36352564

Subgroups — #36352564

Order 22144.

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

normal subgroups only

This group has 254 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#26012068C211yesmaximal, characteristic
2#26011804C211yesmaximal, characteristic, complemented
3#26011422C211yesmaximal, characteristic, complemented
4#26011226C211yesmaximal, characteristic, complemented
5#26011225C211yesmaximal, characteristic, complemented
6#26011216C211yesmaximal, characteristic, complemented
7#480×C173C211yesmaximal, characteristic, complemented
8C2×#14583825—21no—
9#19149657—21no—
10#19149656—21no—
11#19149653C2211yescharacteristic
12#19149652—21no—
13#19149651—21no—
14#19149651—21no—
15C2×#14583805—21no—
16C2×#14583805C2211yescharacteristic
17#19149612—21no—
18#19149599—21no—
19C22⋊#13063436C2211yescharacteristic
20#19149586—21no—
21#19149586C2211yescharacteristic
22#121×C173C2211yescharacteristic
23C2×C4⋊C4×C173C2211yescharacteristic
24#107×C173—21no—
25#102×C173—21no—
26C22⋊C8×C173C2211yescharacteristic
27#14583825—42no—
28#14583825—42no—
29C2×Dic346—21no—
30C2×D692D411yescharacteristic
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C22×Dic173—21no—
35C22×Dic173—21no—
36C22×Dic173D411yescharacteristic
37C22×Dic173D411yescharacteristic
38C2×C4×D173—21no—
39C2×#13063436—21no—
40C4⋊Dic173D411yescharacteristic
41#14583805—21no—
42#14583805—21no—
43#14583805—21no—
44#14583805—42no—
45C22⋊Dic173—21no—
46#14583803—21no—
47#14583803—21no—
48C2×D4×C173—21no—
49C2×D4×C173D411yescharacteristic
50C22×C4×C173—21no—
51C22×C4×C173C2311yescharacteristic
52C2×C8×C173—21no—
53C4⋊C4×C173—21no—
54C4⋊C4×C173—21no—
55C4⋊C4×C173D411yescharacteristic
56C22⋊C4×C173—21no—
57C22×D173—21no—
58Dic346—42no—
59D692—41no—
60C22⋊D173—42no—
61C22⋊D173—41no—
62C22⋊D173—41no—
63C22⋊D173—81no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C2×Dic173—22no—
67C2×Dic173—21no—
68C2×Dic173—21no—
69C2×Dic173—42no—
70C2×Dic173—41no—
71C2×Dic173—41no—
72C2×Dic173—41no—
73C2×Dic173D812yes—
74C4×D173—42no—
75#13063436—41no—
76C23×C173—21no—
77C23×C173C2×D411yescharacteristic
78D4×C173—41no—
79D4×C173—41no—
80D4×C173—41no—
81C2×C4×C173—21no—
82C2×C4×C173—41no—
83C2×C4×C173—41no—
84C2×C4×C173—41no—
85C2×C4×C173—41no—
86C2×C4×C173C2×D411yescharacteristic
87C2×C4×C173C2×D411yescharacteristic
88C8×C173—41no—
89C22×C173—21no—
90C22×C173—21no—
91C22×C173—41no—
92C22×C173—41no—
93C22×C173—41no—
94C22×C173#12011yescharacteristic
95C22×C173C2×D811yescharacteristic
96C22×C173AGL(1,ℤ/8)11yescharacteristic
97C2×D173—41no—
98C2×D173—41no—
99C2×D173—41no—
100C4×C173—21no—
101C4×C173—21no—
102C4×C173—41no—
103C4×C173—81no—
104Dic173—22no—
105Dic173—42no—
106Dic173—41no—
107Dic173—41no—
108C2×C173—21no—
109C2×C173—21no—
110C2×C173—81no—
111C2×C173#44711yescharacteristic
112C2×C173#45611yescharacteristic
113C2×C173#45811yescharacteristic
114D173—81no—
115C173#161011yescharacteristic, complemented, Sylow
116#1610—1731nomaximal, Sylow
117#537—1731no—
118#480—1731no—
119#480D17311yescharacteristic, complemented
120#398—1731no—
121#327—1731no—
122#327—1731no—
123#324—1731no—
124C2×#41—3461no—
125C22⋊Q8—3461no—
126C22⋊Q8—3461no—
127#121—3461no—
128#121—3461no—
129#121—3461no—
130#121—1731no—
131#121C2×D17311yescharacteristic
132C2×C4⋊C4—3461no—
133C2×C4⋊C4—1731no—
134C2×C4⋊C4C2×D17311yescharacteristic
135#107—21no—
136#107—3461no—
137#102—21no—
138#102—3461no—
139C22⋊C8—1731no—
140C22⋊C8C2×D17311yescharacteristic
141#95—3461no—
142#95—1731no—
143#41—6922no—
144#41—6922no—
145C2×Q8—3461no—
146C2×D4—21no—
147C2×D4—3461no—
148C2×D4—3461no—
149C2×D4—3461no—
150C2×D4—1731no—
151C2×D4C22⋊D17311yescharacteristic
152C22×C4—21no—
153C22×C4—3461no—
154C22×C4—3461no—
155C22×C4—3461no—
156C22×C4—1731no—
157C22×C4—1731no—
158C22×C4C22×D17311yescharacteristic
159C2×C8—21no—
160C2×C8—3461no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—3461no—
164C4⋊C4—3461no—
165C4⋊C4—3461no—
166C4⋊C4—1731no—
167C4⋊C4—6922no—
168C4⋊C4C22⋊D17311yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—3461no—
171C22⋊C4—3461no—
172C22⋊C4—3461no—
173C23—21no—
174C23—3461no—
175C23D4×D17311yescharacteristic
176Q8—6922no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—13841no—
181D4—6921no—
182D4—6922no—
183D4—6921no—
184D4—6921no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—3461no—
191C2×C4—3461no—
192C2×C4—3462no—
193C2×C4—3461no—
194C2×C4—3461no—
195C2×C4—1732no—
196C2×C4—6922no—
197C2×C4—6922no—
198C2×C4—6921no—
199C2×C4—6921no—
200C2×C4—6921no—
201C2×C4D4×D17311yescharacteristic
202C2×C4C2×C22⋊D17311yescharacteristic
203C8—41no—
204C8—6921no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—6921no—
211C22—6921no—
212C22—6921no—
213C22#1914964711yescharacteristic
214C22D8×D17311yescharacteristic
215C22#1914970311yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—81no—
220C4—3462no—
221C4—6922no—
222C4—6921no—
223C4—6921no—
224C2—21no—
225C2—21no—
226C2—81no—
227C2—13841no—
228C2#2601162311yescharacteristic
229C2#2601167211yescharacteristic
230C2#2601168011yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#480×C173C2
socleC22×C173#120
prime radical#26012068C2
hypercenter#480D173
hypocenterC173#1610
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #26011804C2semidirect
3. #26011422C2semidirect
4. #26011226C2semidirect
5. #26011225C2semidirect
6. #26011216C2semidirect
7. #480×C173C2semidirect
115. C173#1610semidirect
119. #480D173semidirect

Maximal subgroups

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

subgroupindex
#260120682
#260118042
#260114222
#260112262
#260112252
#260112162
#480×C1732
#1610173