Finite Group Databasebeta

← C22×#557×C89 #36440557

Subgroups — C22×#557×C89 #36440557

Order 22784.

7414 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#557×C89C2148yesmaximal, direct product factor
2C22×C4⋊Q8×C89C213yesmaximal
3C22×#125×C89C216yesmaximal
4C22×C4×Q8×C89C212yesmaximal
5C22×C4×Q8×C89C213yesmaximal
6C22×C4×Q8×C89C211yesmaximal, characteristic
7C23×Q8×C89C2213yes—
8C23×Q8×C89C2211yescharacteristic
9#557×C89C221256yesdirect product factor
10C2×C4⋊Q8×C89C22172yes—
11C2×#125×C89C221144yes—
12C2×C4×Q8×C89C22148yes—
13C2×C4×Q8×C89C22172yes—
14C2×C4×Q8×C89C22124yes—
15C22×C4⋊C4×C89C2211yescharacteristic
16C22×C4⋊C4×C89C2216yes—
17C22×C4⋊C4×C89C2213yes—
18C22×C4⋊C4×C89C2216yes—
19C22×C4⋊C4×C89C2216yes—
20C22×C42×C89C2216yes—
21C22×C42×C89C2213yes—
22C22×Q8×C89—224no—
23C22×Q8×C89Q8116yescomplemented
24C22×Q8×C89C23136yes—
25C22×Q8×C89C23112yes—
26C23×C4×C89C2312yes—
27C23×C4×C89C2313yes—
28C23×C4×C89C2316yes—
29C23×C4×C89C2313yes—
30C23×C4×C89C2311yescharacteristic
31C4⋊Q8×C89C231192yes—
32#125×C89C231384yes—
33C4×Q8×C89C231128yes—
34C4×Q8×C89C231192yes—
35C4×Q8×C89C23164yes—
36C2×C4⋊C4×C89C23112yes—
37C2×C4⋊C4×C89C23172yes—
38C2×C4⋊C4×C89C23136yes—
39C2×C4⋊C4×C89C23172yes—
40C2×C4⋊C4×C89C23172yes—
41C2×C42×C89C23172yes—
42C2×C42×C89C23136yes—
43C24×C89C2411yescharacteristic
44C2×Q8×C89—2144no—
45C2×Q8×C89C2×Q8196yescomplemented
46C2×Q8×C89C24148yes—
47C2×Q8×C89C24116yes—
48C22×C4×C89—224no—
49C22×C4×C89—28no—
50C22×C4×C89—212no—
51C22×C4×C89C2×Q8124yes—
52C22×C4×C89#4118yes—
53C22×C4×C89C24112yes—
54C22×C4×C89C24118yes—
55C22×C4×C89C24136yes—
56C22×C4×C89C24118yes—
57C22×C4×C89C2416yes—
58C4⋊C4×C89C24116yes—
59C4⋊C4×C89C24196yes—
60C4⋊C4×C89C24148yes—
61C4⋊C4×C89C24196yes—
62C4⋊C4×C89C24196yes—
63C42×C89C24196yes—
64C42×C89C24148yes—
65C23×C89C22×Q814yes—
66C23×C89C2×#4114yes—
67C23×C892−1+414yes—
68C23×C89C2513yes—
69Q8×C89—296no—
70Q8×C89C22×Q8164yescomplemented
71C2×C4×C89—272no—
72C2×C4×C89—224no—
73C2×C4×C89—236no—
74C2×C4×C89C22×Q8172yes—
75C2×C4×C89C2×#41124yes—
76C2×C4×C89C2518yes—
77C2×C4×C89C25112yes—
78C2×C4×C89C25124yes—
79C2×C4×C89C25112yes—
80C2×C4×C89C2514yes—
81C22×C89#557116yesdirect product factor
82C22×C89C23×Q816yes—
83C22×C89C22×#4116yes—
84C22×C89C2×2−1+416yes—
85C22×C89C2611yescharacteristic
86C4×C89—224no—
87C4×C89—28no—
88C4×C89—212no—
89C4×C89C23×Q8124yes—
90C4×C89C22×#4118yes—
91C22×#557C8911yesmaximal, characteristic, direct product factor, Sylow
92C2×C89C2×#557112yesdirect product factor
93C2×C89C24×Q811yescharacteristic
94C2×C89C23×#4111yescharacteristic
95C2×C89C22×2−1+411yescharacteristic
96C2×#557C2×C89148yesdirect product factor
97C22×C4⋊Q8C2×C8913yes—
98C22×#125C2×C8916yes—
99C22×C4×Q8C2×C8913yes—
100C22×C4×Q8C2×C8911yescharacteristic
101C22×C4×Q8C2×C8912yes—
102C89C22×#55711yescharacteristic, direct product factor, Sylow
103C23×Q8C22×C8913yes—
104C23×Q8C22×C8911yescharacteristic
105#557C22×C891256yesdirect product factor
106C2×C4⋊Q8C22×C89172yes—
107C2×#125C22×C891144yes—
108C2×C4×Q8C22×C89148yes—
109C2×C4×Q8C22×C89172yes—
110C2×C4×Q8C22×C89124yes—
111C22×C4⋊C4C22×C8911yescharacteristic
112C22×C4⋊C4C22×C8913yes—
113C22×C4⋊C4C22×C8916yes—
114C22×C4⋊C4C22×C8916yes—
115C22×C4⋊C4C22×C8916yes—
116C22×C42C22×C8916yes—
117C22×C42C22×C8913yes—
118C22×Q8—224no—
119C22×Q8Q8×C89116yescomplemented
120C22×Q8C23×C89136yes—
121C22×Q8C23×C89112yes—
122C23×C4C23×C8913yes—
123C23×C4C23×C8916yes—
124C23×C4C23×C8912yes—
125C23×C4C23×C8913yes—
126C23×C4C23×C8911yescharacteristic
127C4⋊Q8C23×C891192yes—
128#125C23×C891384yes—
129C4×Q8C23×C891128yes—
130C4×Q8C23×C891192yes—
131C4×Q8C23×C89164yes—
132C2×C4⋊C4C23×C89112yes—
133C2×C4⋊C4C23×C89172yes—
134C2×C4⋊C4C23×C89136yes—
135C2×C4⋊C4C23×C89172yes—
136C2×C4⋊C4C23×C89172yes—
137C2×C42C23×C89172yes—
138C2×C42C23×C89136yes—
139C24C24×C8911yescharacteristic
140C2×Q8—2144no—
141C2×Q8C2×Q8×C89196yescomplemented
142C2×Q8C24×C89148yes—
143C2×Q8C24×C89116yes—
144C22×C4—224no—
145C22×C4—212no—
146C22×C4—28no—
147C22×C4C2×Q8×C89124yes—
148C22×C4#41×C8918yes—
149C22×C4C24×C89112yes—
150C22×C4C24×C89118yes—
151C22×C4C24×C89136yes—
152C22×C4C24×C89118yes—
153C22×C4C24×C8916yes—
154C4⋊C4C24×C89116yes—
155C4⋊C4C24×C89196yes—
156C4⋊C4C24×C89148yes—
157C4⋊C4C24×C89196yes—
158C4⋊C4C24×C89196yes—
159C42C24×C89196yes—
160C42C24×C89148yes—
161C23C22×Q8×C8914yes—
162C23C2×#41×C8914yes—
163C232−1+4×C8914yes—
164C23C25×C8913yes—
165Q8—296no—
166Q8C22×Q8×C89164yescomplemented
167C2×C4—272no—
168C2×C4—224no—
169C2×C4—236no—
170C2×C4C22×Q8×C89172yes—
171C2×C4C2×#41×C89124yes—
172C2×C4C25×C8918yes—
173C2×C4C25×C89112yes—
174C2×C4C25×C89124yes—
175C2×C4C25×C89112yes—
176C2×C4C25×C8914yes—
177C22#557×C89116yesdirect product factor
178C22C23×Q8×C8916yes—
179C22C22×#41×C8916yes—
180C22C2×2−1+4×C8916yes—
181C22C26×C8911yescharacteristic
182C4—224no—
183C4—28no—
184C4—212no—
185C4C23×Q8×C89124yes—
186C4C22×#41×C8918yes—
187C2C2×#557×C89112yesdirect product factor
188C2C24×Q8×C8911yescharacteristic
189C2C23×#41×C8911yescharacteristic
190C2C22×2−1+4×C8911yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C89C24
commutator, Frattini, cosocleC22C26×C89
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#557×C89C2direct (N × H)
9. #557×C89C22direct (N × H)
23. C22×Q8×C89Q8semidirect
45. C2×Q8×C89C2×Q8semidirect
70. Q8×C89C22×Q8semidirect
81. C22×C89#557direct (N × H)
91. C22×#557C89direct (N × H)
92. C2×C89C2×#557direct (N × H)
96. C2×#557C2×C89direct (N × H)
102. C89C22×#557direct (N × H)
105. #557C22×C89direct (N × H)
119. C22×Q8Q8×C89semidirect
141. C2×Q8C2×Q8×C89semidirect
166. Q8C22×Q8×C89semidirect
177. C22#557×C89direct (N × H)
187. C2C2×#557×C89direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#557×C892
C22×C4⋊Q8×C892
C22×#125×C892
C22×C4×Q8×C892
C22×#55789