Finite Group Databasebeta

← #392×D173 #36356757

Subgroups — #392×D173 #36356757

Order 22144.

1812 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 223 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
1C22×D4×D173C213yesmaximal, complemented
2C2×C22⋊C4×D173C213yesmaximal, complemented
3#26011401C211yesmaximal, characteristic, complemented
4#26011400C213yesmaximal, complemented
5#26011399C213yesmaximal, complemented
6#95×D173C211yesmaximal, characteristic, complemented
7#392×C173C211yesmaximal, characteristic, complemented
8C24×D173C2213yes—
9C22×D692C2213yescomplemented
10C22×C22⋊D173C2216yescomplemented
11C2×D4×D173—224no—
12C2×D4×D173—212no—
13C22×C4×D173C2211yescharacteristic
14C22×C4×D173C2213yes—
15C2×C22⋊Dic173C2213yescomplemented
16C2×#14583803C2213yescomplemented
17C2×#14583803C2213yescomplemented
18C22⋊C4×D173—212no—
19#19149586C2213yescomplemented
20C22×D4×C173C2213yescomplemented
21C2×C22⋊C4×C173C2213yescomplemented
22#95×C173C2211yescharacteristic, complemented
23C23×D173—26no—
24C23×D173—212no—
25C23×D173—212no—
26C23×D173—26no—
27C23×D173D4112yescomplemented
28C23×D173C2313yes—
29C23×D173C2311yescharacteristic
30C2×D692—212no—
31C2×D692—26no—
32C2×C22⋊D173—212no—
33C2×C22⋊D173—212no—
34C2×C22⋊D173—212no—
35D4×D173—296no—
36C22×Dic173C2311yescharacteristic
37C22×Dic173C2313yes—
38C2×C4×D173—212no—
39C2×C4×D173—26no—
40C2×C4×D173D4112yescomplemented
41C22⋊Dic173—26no—
42#14583803—26no—
43#14583803—26no—
44C24×C173C2313yes—
45C2×D4×C173—212no—
46C2×D4×C173—26no—
47C22×C4×C173C2311yescharacteristic
48C22×C4×C173C2313yes—
49C22⋊C4×C173—26no—
50C22×D173—26no—
51C22×D173—26no—
52C22×D173—23no—
53C22×D173—23no—
54C22×D173—248no—
55C22×D173—224no—
56C22×D173—412no—
57C22×D173—424no—
58C22×D173—412no—
59C22×D173C2×D416yes—
60C22×D173C2×D416yes—
61C22×D173C2×D416yescomplemented
62C22×D173#4112yes—
63D692—224no—
64C22⋊D173—248no—
65C2×Dic173—26no—
66C2×Dic173—23no—
67C2×Dic173C2×D416yescomplemented
68C4×D173—224no—
69C4×D173—44no—
70C23×C173—26no—
71C23×C173—26no—
72C23×C173—23no—
73C23×C173—23no—
74C23×C173C2×D416yescomplemented
75C23×C173C2411yescharacteristic
76D4×C173—224no—
77C2×C4×C173—26no—
78C2×C4×C173—23no—
79C2×C4×C173C2×D416yescomplemented
80C22×C173—212no—
81C22×C173—26no—
82C22×C173—43no—
83C22×C173—46no—
84C22×C173—43no—
85C22×C173C22×D413yes—
86C22×C173C22×D413yes—
87C22×C173C2×#4111yescharacteristic
88C2×D173—212no—
89C2×D173—26no—
90C2×D173—43no—
91C2×D173—46no—
92C2×D173—43no—
93C2×D173—448no—
94C2×D173#120112yes—
95C2×D173#121112yes—
96C2×D173#12714yes—
97C4×C173—26no—
98C4×C173—41no—
99Dic173—26no—
100Dic173—41no—
101C2×C173—46no—
102C2×C173C2×#12013yes—
103C2×C173C2×#12113yes—
104C2×C173C2×#12711yescharacteristic
105D173—46no—
106D173#39218yesdirect product factor
107C173C2×#39211yescharacteristic, complemented, Sylow
108C2×#392—1731nomaximal, Sylow
109C23×D4—1733no—
110C22×C22⋊C4—1733no—
111#392—1733no—
112#392—1733no—
113#392—1731no—
114#392D17311yescharacteristic, direct product factor
115C2×#95—1731no—
116C25—1733no—
117C22×D4—34624no—
118C22×D4—34612no—
119C22×D4—1736no—
120C22×D4—1733no—
121C22×D4C2×D17313yescomplemented
122C23×C4—1731no—
123C23×C4—1733no—
124C2×C22⋊C4—34612no—
125C2×C22⋊C4—1733no—
126C2×C22⋊C4—1733no—
127C2×C22⋊C4—1733no—
128C2×C22⋊C4C2×D17313yescomplemented
129#95—1733no—
130#95C2×D17311yescharacteristic, complemented
131C24—3466no—
132C24—34612no—
133C24—34612no—
134C24—3466no—
135C24—17312no—
136C24—1733no—
137C24—1731no—
138C24C22×D17313yes—
139C2×D4—212no—
140C2×D4—26no—
141C2×D4—34612no—
142C2×D4—34696no—
143C2×D4—34612no—
144C2×D4—34612no—
145C2×D4—34612no—
146C2×D4—3466no—
147C22×C4—34612no—
148C22×C4—3466no—
149C22×C4—1731no—
150C22×C4—1733no—
151C22×C4—17312no—
152C22×C4C22×D17311yescharacteristic
153C22×C4C22×D17313yes—
154C22⋊C4—26no—
155C22⋊C4—3466no—
156C22⋊C4—3466no—
157C22⋊C4—3466no—
158C23—23no—
159C23—26no—
160C23—26no—
161C23—23no—
162C23—34648no—
163C23—34624no—
164C23—3463no—
165C23—3466no—
166C23—3466no—
167C23—3463no—
168C23—1736no—
169C23—1736no—
170C23—1736no—
171C23—1732no—
172C23—69212no—
173C23—69224no—
174C23—69212no—
175C23D4×D17316yescomplemented
176C23C23×D17311yescharacteristic
177D4—224no—
178D4—34624no—
179D4—34648no—
180C2×C4—26no—
181C2×C4—23no—
182C2×C4—34624no—
183C2×C4—3466no—
184C2×C4—3463no—
185C2×C4—1736no—
186C2×C4—6924no—
187C2×C4D4×D17316yescomplemented
188C22—212no—
189C22—26no—
190C22—43no—
191C22—46no—
192C22—43no—
193C22—34612no—
194C22—3466no—
195C22—17312no—
196C22—17312no—
197C22—1734no—
198C22—69248no—
199C22—6923no—
200C22—6926no—
201C22—6923no—
202C22C2×D4×D17313yes—
203C22C2×D4×D17313yes—
204C22#41×D17311yescharacteristic
205C4—26no—
206C4—41no—
207C4—3466no—
208C4—6921no—
209C2—46no—
210C2—1738no—
211C2—6926no—
212C2#120×D17313yes—
213C2#121×D17313yes—
214C2#127×D17311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#392×C173C2
hypercenter#392D173
hypocenterC173C2×#392
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D173C2semidirect
2. C2×C22⋊C4×D173C2semidirect
3. #26011401C2semidirect
4. #26011400C2semidirect
5. #26011399C2semidirect
6. #95×D173C2semidirect
7. #392×C173C2semidirect
9. C22×D692C22semidirect
10. C22×C22⋊D173C22semidirect
15. C2×C22⋊Dic173C22semidirect
16. C2×#14583803C22semidirect
17. C2×#14583803C22semidirect
19. #19149586C22semidirect
20. C22×D4×C173C22semidirect
21. C2×C22⋊C4×C173C22semidirect
22. #95×C173C22semidirect
27. C23×D173D4semidirect
40. C2×C4×D173D4semidirect
61. C22×D173C2×D4semidirect
67. C2×Dic173C2×D4semidirect
74. C23×C173C2×D4semidirect
79. C2×C4×C173C2×D4semidirect
106. D173#392direct (N × H)
107. C173C2×#392semidirect
114. #392D173direct (N × H)
121. C22×D4C2×D173semidirect
128. C2×C22⋊C4C2×D173semidirect
130. #95C2×D173semidirect
175. C23D4×D173semidirect
187. C2×C4D4×D173semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1732
C2×C22⋊C4×D1732
#260114012
#260114002
#260113992
#95×D1732
#392×C1732
C2×#392173