Finite Group Databasebeta

← H(5,𝔽2)×C22⋊C9 #11919875

Subgroups — H(5,𝔽2)×C22⋊C9 #11919875

Order 1152.

2574 conjugacy classes of subgroups; only the 340 normal subgroups are shown.

normal subgroups only

This group has 2574 conjugacy classes of subgroups — too many to draw. Only its 340 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#41×C22⋊C9C216yesmaximal, complemented
2C2×D4×C22⋊C9C219yesmaximal, complemented
3C22×C3×H(5,𝔽2)C311yesmaximal, characteristic
4C23×C22⋊C9C2216yescomplemented
5C9×H(5,𝔽2)—41nomaximal
6Q8×C22⋊C9C2212yescomplemented
7D4×C22⋊C9C22118yescomplemented
8C2×C4×C22⋊C9C2219yescomplemented
9C2×C3×H(5,𝔽2)—39no—
10C2×C3×H(5,𝔽2)—36no—
11C2×C3×H(5,𝔽2)—31no—
12C22×C3×#41C2×C316yes—
13C23×C3×D4C2×C319yes—
14C22×C22⋊C9—46no—
15C22×C22⋊C9C2319yes—
16C9×#41—46no—
17C2×D4×C9—49no—
18C4×C22⋊C9C2316yes—
19C22×H(5,𝔽2)C911yescharacteristic, complemented, Sylow
20C25×C3C22×C316yes—
21C3×H(5,𝔽2)—34no—
22C3×H(5,𝔽2)—318no—
23C3×H(5,𝔽2)—336no—
24C3×H(5,𝔽2)—312no—
25C3×H(5,𝔽2)—36no—
26C3×H(5,𝔽2)—39no—
27C3×H(5,𝔽2)A411yescharacteristic
28C2×C3×#41—318no—
29C2×C3×#41—318no—
30C2×C3×#41—36no—
31C2×C3×#41—36no—
32C22×C3×Q8C22×C312yes—
33C22×C3×D4—39no—
34C22×C3×D4—39no—
35C22×C3×D4—318no—
36C22×C3×D4—336no—
37C22×C3×D4C22×C3118yes—
38C23×C3×C4C22×C319yes—
39C23×C9—46no—
40C2×C22⋊C9—29no—
41C2×C22⋊C9C2411yescharacteristic
42Q8×C9—42no—
43D4×C9—418no—
44C2×C4×C9—49no—
45C2×H(5,𝔽2)—39no—
46C2×H(5,𝔽2)—36no—
47C2×H(5,𝔽2)—31no—
48C22×#41C2×C916yescomplemented
49C23×D4C2×C919yescomplemented
50C24×C3—318no—
51C24×C3—36no—
52C24×C3—46no—
53C24×C3—126no—
54C24×C3C23×C319yes—
55C3×#41—336no—
56C3×#41—336no—
57C3×#41—312no—
58C3×#41—36no—
59C3×#41—318no—
60C3×#41—318no—
61C3×#41C2×A416yes—
62C2×C3×Q8—36no—
63C2×C3×Q8—32no—
64C2×C3×D4—372no—
65C2×C3×D4—318no—
66C2×C3×D4—336no—
67C2×C3×D4—318no—
68C2×C3×D4—318no—
69C2×C3×D4—336no—
70C2×C3×D4—318no—
71C2×C3×D4—336no—
72C2×C3×D4—39no—
73C2×C3×D4C2×A419yes—
74C22×C3×C4—39no—
75C22×C3×C4—39no—
76C22×C3×C4—318no—
77C22×C3×C4C23×C316yes—
78C22×C9—49no—
79C22×C9—166no—
80C22⋊C9H(5,𝔽2)11yescharacteristic, direct product factor
81C4×C9—46no—
82C25C22×C916yescomplemented
83H(5,𝔽2)—34no—
84H(5,𝔽2)—318no—
85H(5,𝔽2)—336no—
86H(5,𝔽2)—312no—
87H(5,𝔽2)—36no—
88H(5,𝔽2)—39no—
89H(5,𝔽2)C22⋊C911yescharacteristic, direct product factor
90C2×#41—318no—
91C2×#41—318no—
92C2×#41—36no—
93C2×#41—36no—
94C22×Q8C22×C912yescomplemented
95C22×D4—39no—
96C22×D4—39no—
97C22×D4—318no—
98C22×D4—336no—
99C22×D4C22×C9118yescomplemented
100C23×C4C22×C919yescomplemented
101C23×C3—29no—
102C23×C3—312no—
103C23×C3—39no—
104C23×C3—39no—
105C23×C3—318no—
106C23×C3—69no—
107C23×C3—126no—
108C23×C3—126no—
109C23×C3—1218no—
110C23×C3—1218no—
111C23×C3C22×A416yes—
112C23×C3C24×C311yescharacteristic
113C3×Q8—34no—
114C3×Q8—36no—
115C3×Q8C22×A412yes—
116C3×D4—336no—
117C3×D4—336no—
118C3×D4—318no—
119C3×D4C22×A4118yes—
120C2×C3×C4—318no—
121C2×C3×C4—39no—
122C2×C3×C4—36no—
123C2×C3×C4—318no—
124C2×C3×C4—36no—
125C2×C3×C4C22×A419yes—
126C2×C9—41no—
127C2×C9—89no—
128C24—318no—
129C24—36no—
130C24—46no—
131C24—126no—
132C24C23×C919yes—
133#41—336no—
134#41—336no—
135#41—312no—
136#41—36no—
137#41—318no—
138#41—318no—
139#41C2×C22⋊C916yescomplemented
140C2×Q8—36no—
141C2×Q8—32no—
142C2×D4—372no—
143C2×D4—318no—
144C2×D4—336no—
145C2×D4—318no—
146C2×D4—318no—
147C2×D4—336no—
148C2×D4—318no—
149C2×D4—336no—
150C2×D4—39no—
151C2×D4C2×C22⋊C919yescomplemented
152C22×C4—39no—
153C22×C4—39no—
154C22×C4—318no—
155C22×C4C23×C916yes—
156C22×C3—31no—
157C22×C3—31no—
158C22×C3—39no—
159C22×C3—46no—
160C22×C3—69no—
161C22×C3—69no—
162C22×C3—69no—
163C22×C3—69no—
164C22×C3—1218no—
165C22×C3—1212no—
166C22×C3C3×H(5,𝔽2)11yescharacteristic
167C22×C3C23×A419yes—
168C3×C4—36no—
169C3×C4C23×A416yes—
170C9—41noSylow
171C23—29no—
172C23—312no—
173C23—39no—
174C23—39no—
175C23—318no—
176C23—69no—
177C23—126no—
178C23—126no—
179C23—1218no—
180C23—1218no—
181C23C22×C22⋊C916yescomplemented
182C23C24×C911yescharacteristic
183Q8—34no—
184Q8—36no—
185Q8C22×C22⋊C912yescomplemented
186D4—336no—
187D4—336no—
188D4—318no—
189D4C22×C22⋊C9118yescomplemented
190C2×C4—318no—
191C2×C4—39no—
192C2×C4—36no—
193C2×C4—318no—
194C2×C4—36no—
195C2×C4C22×C22⋊C919yescomplemented
196C2×C3—29no—
197C2×C3—31no—
198C2×C3—31no—
199C2×C3—69no—
200C2×C3C24×A411yescharacteristic
201C22—31no—
202C22—31no—
203C22—39no—
204C22—46no—
205C22—69no—
206C22—69no—
207C22—69no—
208C22—69no—
209C22—1218no—
210C22—1212no—
211C22C9×H(5,𝔽2)11yescharacteristic, complemented
212C22C23×C22⋊C919yes—
213C4—36no—
214C4C23×C22⋊C916yes—
215C3A4×H(5,𝔽2)11yescharacteristic
216C2—29no—
217C2—31no—
218C2—31no—
219C2—69no—
220C2C24×C22⋊C911yescharacteristic
subgroupG/H
center, FrattiniC2×C3C24×A4
commutatorC23C24×C9
Fitting, prime radicalC22×C3×H(5,𝔽2)C3
socle, cosocleC23×C3C24×C3
hypercenterC3×H(5,𝔽2)A4
hypocenterC22C9×H(5,𝔽2)
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C3 ◁ C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #41×C22⋊C9C2semidirect
2. C2×D4×C22⋊C9C2semidirect
4. C23×C22⋊C9C22semidirect
6. Q8×C22⋊C9C22semidirect
7. D4×C22⋊C9C22semidirect
8. C2×C4×C22⋊C9C22semidirect
19. C22×H(5,𝔽2)C9semidirect
48. C22×#41C2×C9semidirect
49. C23×D4C2×C9semidirect
80. C22⋊C9H(5,𝔽2)direct (N × H)
82. C25C22×C9semidirect
89. H(5,𝔽2)C22⋊C9direct (N × H)
94. C22×Q8C22×C9semidirect
99. C22×D4C22×C9semidirect
100. C23×C4C22×C9semidirect
139. #41C2×C22⋊C9semidirect
151. C2×D4C2×C22⋊C9semidirect
181. C23C22×C22⋊C9semidirect
185. Q8C22×C22⋊C9semidirect
189. D4C22×C22⋊C9semidirect
195. C2×C4C22×C22⋊C9semidirect
211. C22C9×H(5,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
#41×C22⋊C92
C2×D4×C22⋊C92
C22×C3×H(5,𝔽2)3
C9×H(5,𝔽2)4