Finite Group Databasebeta

← C2×#459×C109 #28360283

Subgroups — C2×#459×C109 #28360283

Order 13952.

648 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 232 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×D8×C109C212yesmaximal, complemented
2C2×#127×C109C211yesmaximal, characteristic, complemented
3C2×C4×D4×C109C211yesmaximal, characteristic, complemented
4#459×C109C218yesmaximal, direct product factor
5C2×C4⋊C8×C109C211yesmaximal, characteristic, complemented
6C2×#102×C109C212yesmaximal, complemented
7C22×D4×C109—21no—
8C22×D4×C109—21no—
9C22×D4×C109C2212yes—
10C22×D4×C109C2211yescharacteristic
11C23×C4×C109—21no—
12C2×D8×C109—28no—
13C2×D8×C109C2218yescomplemented
14C22×C8×C109C2212yes—
15#127×C109—22no—
16#127×C109C2214yescomplemented
17C4×D4×C109—22no—
18C4×D4×C109C2214yescomplemented
19C2×C4⋊C4×C109C2211yescharacteristic
20C2×C22⋊C4×C109—21no—
21C2×C42×C109C2211yescharacteristic
22C4⋊C8×C109C2214yescomplemented
23#102×C109C2218yescomplemented
24C24×C109—21no—
25C24×C109—22no—
26C2×D4×C109—21no—
27C2×D4×C109—24no—
28C2×D4×C109—22no—
29C2×D4×C109—24no—
30C2×D4×C109—24no—
31C2×D4×C109—22no—
32C2×D4×C109—24no—
33C2×D4×C109—44no—
34C2×D4×C109—44no—
35C2×D4×C109D414yescomplemented
36C2×D4×C109C2314yes—
37C2×D4×C109C2312yes—
38C22×C4×C109—21no—
39C22×C4×C109—24no—
40C22×C4×C109—44no—
41C22×C4×C109D411yescharacteristic
42C22×C4×C109D411yescharacteristic
43C22×C4×C109C2311yescharacteristic
44D8×C109—216no—
45C2×C8×C109—24no—
46C2×C8×C109C2314yes—
47C4⋊C4×C109—21no—
48C4⋊C4×C109C2312yes—
49C22⋊C4×C109—24no—
50C42×C109D412yes—
51C42×C109C2312yes—
52C23×C109—24no—
53C23×C109—22no—
54C23×C109—24no—
55C23×C109—42no—
56C23×C109—42no—
57C23×C109—44no—
58C23×C109—44no—
59C23×C109—44no—
60C23×C109C2×D411yescharacteristic
61D4×C109—24no—
62D4×C109—24no—
63D4×C109—22no—
64D4×C109—24no—
65D4×C109—48no—
66D4×C109—44no—
67D4×C109—48no—
68D4×C109C2×D414yescomplemented
69C2×C4×C109—22no—
70C2×C4×C109—22no—
71C2×C4×C109—42no—
72C2×C4×C109—48no—
73C2×C4×C109D814yescomplemented
74C2×C4×C109C2×D412yes—
75C2×C4×C109C2×D411yescharacteristic
76C2×C4×C109C2×D412yes—
77C2×C4×C109C2×D412yes—
78C2×C4×C109#4112yes—
79C2×C4×C109C2411yescharacteristic
80C8×C109—24no—
81C22×C109—24no—
82C22×C109—42no—
83C22×C109—42no—
84C22×C109—48no—
85C22×C109—44no—
86C22×C109—44no—
87C22×C109—44no—
88C22×C109—88no—
89C22×C109#12112yes—
90C22×C109C2×D812yes—
91C22×C109AGL(1,ℤ/8)12yes—
92C22×C109C22×D411yescharacteristic
93C4×C109—22no—
94C4×C109—42no—
95C4×C109#12112yes—
96C4×C109C2×D814yescomplemented
97C4×C109C22×D411yescharacteristic
98C4×C109C2×#4111yescharacteristic
99C2×C109—44no—
100C2×C109—84no—
101C2×C109#45914yesdirect product factor
102C2×C109C2×#12111yescharacteristic
103C2×C109C22×D811yescharacteristic
104C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
105C2×#459C10911yesmaximal, characteristic, direct product factor, Sylow
106C109C2×#45911yescharacteristic, direct product factor, Sylow
107C22×D8C2×C10912yescomplemented
108C2×#127C2×C10911yescharacteristic, complemented
109C2×C4×D4C2×C10911yescharacteristic, complemented
110#459C2×C10918yesdirect product factor
111C2×C4⋊C8C2×C10911yescharacteristic, complemented
112C2×#102C2×C10912yescomplemented
113C22×D4—21no—
114C22×D4—21no—
115C22×D4C22×C10912yes—
116C22×D4C22×C10911yescharacteristic
117C23×C4—21no—
118C2×D8—28no—
119C2×D8C22×C10918yescomplemented
120C22×C8C22×C10912yes—
121#127—22no—
122#127C22×C10914yescomplemented
123C4×D4—22no—
124C4×D4C22×C10914yescomplemented
125C2×C4⋊C4C22×C10911yescharacteristic
126C2×C22⋊C4—21no—
127C2×C42C22×C10911yescharacteristic
128C4⋊C8C22×C10914yescomplemented
129#102C22×C10918yescomplemented
130C24—21no—
131C24—22no—
132C2×D4—21no—
133C2×D4—24no—
134C2×D4—22no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—44no—
140C2×D4—44no—
141C2×D4D4×C10914yescomplemented
142C2×D4C23×C10914yes—
143C2×D4C23×C10912yes—
144C22×C4—21no—
145C22×C4—24no—
146C22×C4—44no—
147C22×C4D4×C10911yescharacteristic
148C22×C4D4×C10911yescharacteristic
149C22×C4C23×C10911yescharacteristic
150D8—216no—
151C2×C8—24no—
152C2×C8C23×C10914yes—
153C4⋊C4—21no—
154C4⋊C4C23×C10912yes—
155C22⋊C4—24no—
156C42D4×C10912yes—
157C42C23×C10912yes—
158C23—24no—
159C23—22no—
160C23—24no—
161C23—42no—
162C23—42no—
163C23—44no—
164C23—44no—
165C23—44no—
166C23C2×D4×C10911yescharacteristic
167D4—24no—
168D4—24no—
169D4—22no—
170D4—24no—
171D4—48no—
172D4—44no—
173D4—48no—
174D4C2×D4×C10914yescomplemented
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—42no—
178C2×C4—48no—
179C2×C4D8×C10914yescomplemented
180C2×C4C2×D4×C10912yes—
181C2×C4C2×D4×C10911yescharacteristic
182C2×C4C2×D4×C10912yes—
183C2×C4C2×D4×C10912yes—
184C2×C4#41×C10912yes—
185C2×C4C24×C10911yescharacteristic
186C8—24no—
187C22—24no—
188C22—42no—
189C22—42no—
190C22—48no—
191C22—44no—
192C22—44no—
193C22—44no—
194C22—88no—
195C22#121×C10912yes—
196C22C2×D8×C10912yes—
197C22AGL(1,ℤ/8)×C10912yes—
198C22C22×D4×C10911yescharacteristic
199C4—22no—
200C4—42no—
201C4#121×C10912yes—
202C4C2×D8×C10914yescomplemented
203C4C22×D4×C10911yescharacteristic
204C4C2×#41×C10911yescharacteristic
205C2—44no—
206C2—84no—
207C2#459×C10914yesdirect product factor
208C2C2×#121×C10911yescharacteristic
209C2C22×D8×C10911yescharacteristic
210C2C2×AGL(1,ℤ/8)×C10911yescharacteristic
subgroupG/H
center, socleC23×C109C2×D4
commutator, Frattini, cosocleC2×C4C24×C109
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×D8×C109C2semidirect
2. C2×#127×C109C2semidirect
3. C2×C4×D4×C109C2semidirect
4. #459×C109C2direct (N × H)
5. C2×C4⋊C8×C109C2semidirect
6. C2×#102×C109C2semidirect
13. C2×D8×C109C22semidirect
16. #127×C109C22semidirect
18. C4×D4×C109C22semidirect
22. C4⋊C8×C109C22semidirect
23. #102×C109C22semidirect
35. C2×D4×C109D4semidirect
68. D4×C109C2×D4semidirect
73. C2×C4×C109D8semidirect
96. C4×C109C2×D8semidirect
101. C2×C109#459direct (N × H)
105. C2×#459C109direct (N × H)
106. C109C2×#459direct (N × H)
107. C22×D8C2×C109semidirect
108. C2×#127C2×C109semidirect
109. C2×C4×D4C2×C109semidirect
110. #459C2×C109direct (N × H)
111. C2×C4⋊C8C2×C109semidirect
112. C2×#102C2×C109semidirect
119. C2×D8C22×C109semidirect
122. #127C22×C109semidirect
124. C4×D4C22×C109semidirect
128. C4⋊C8C22×C109semidirect
129. #102C22×C109semidirect
141. C2×D4D4×C109semidirect
174. D4C2×D4×C109semidirect
179. C2×C4D8×C109semidirect
202. C4C2×D8×C109semidirect
207. C2#459×C109direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D8×C1092
C2×#127×C1092
C2×C4×D4×C1092
#459×C1092
C2×C4⋊C8×C1092
C2×#102×C1092
C2×#459109