Subgroups — C₂²×C₅×#347 #11956047

Order 1280.

782 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	2	yes	maximal
2	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#347	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₂×C₅×M₃₂	C₄	1	12	yes	complemented
8	C₂×C₅×M₃₂	C₂²	1	12	yes	—
9	C₂²×C₅×C₁₆	C₄	1	4	yes	complemented
10	C₂×C₅×#97	C₄	1	6	yes	—
11	C₂×C₅×#97	C₂²	1	6	yes	—
12	C₅×#347	C₂²	1	16	yes	direct product factor
13	C₂²×#347	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₄×C₅	—	2	1	no	—
15	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₅×C₈	—	2	1	no	—
17	C₂²×C₅×C₈	—	2	3	no	—
18	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
19	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
20	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	C₂³	1	3	yes	—
23	C₂×C₄²×C₅	—	2	3	no	—
24	C₂×C₄²×C₅	C₂×C₄	1	6	yes	—
25	C₅×M₃₂	C₂×C₄	1	24	yes	complemented
26	C₅×M₃₂	C₂³	1	8	yes	—
27	C₂×C₅×C₁₆	C₂×C₄	1	24	yes	complemented
28	C₅×#97	C₂×C₄	1	12	yes	—
29	C₅×#97	C₂³	1	4	yes	—
30	C₂²×M₃₂	C₂×C₅	1	2	yes	—
31	C₂²×#97	C₂×C₅	1	1	yes	characteristic
32	C₂×#347	C₂×C₅	1	12	yes	direct product factor
33	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₅	—	2	6	no	—
35	C₂²×C₄×C₅	—	4	4	no	—
36	C₂²×C₄×C₅	C₄²	1	3	yes	—
37	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₅	M₁₆	1	3	yes	—
39	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₅	M₁₆	1	3	yes	—
41	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
42	C₂×C₅×C₈	—	2	6	no	—
43	C₂×C₅×C₈	—	2	6	no	—
44	C₂×C₅×C₈	C₄²	1	6	yes	—
45	C₂×C₅×C₈	C₄²	1	6	yes	—
46	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
47	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
48	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
49	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
50	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
51	C₄²×C₅	—	2	6	no	—
52	C₄²×C₅	C₂²×C₄	1	4	yes	—
53	C₅×C₁₆	C₂²×C₄	1	16	yes	complemented
54	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
55	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
56	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
57	C₂×M₃₂	C₄×C₅	1	12	yes	complemented
58	C₂×M₃₂	C₂²×C₅	1	12	yes	—
59	C₂²×C₁₆	C₄×C₅	1	4	yes	complemented
60	C₂×#97	C₄×C₅	1	6	yes	—
61	C₂×#97	C₂²×C₅	1	6	yes	—
62	#347	C₂²×C₅	1	16	yes	direct product factor
63	C₂³×C₅	—	2	4	no	—
64	C₂³×C₅	#97	1	3	yes	—
65	C₂³×C₅	#97	1	1	yes	characteristic
66	C₂³×C₅	C₂×M₁₆	1	3	yes	—
67	C₂×C₄×C₅	—	2	4	no	—
68	C₂×C₄×C₅	—	4	12	no	—
69	C₂×C₄×C₅	#97	1	6	yes	—
70	C₂×C₄×C₅	#97	1	3	yes	—
71	C₂×C₄×C₅	#97	1	3	yes	—
72	C₂×C₄×C₅	C₂×C₄²	1	3	yes	—
73	C₂×C₄×C₅	C₂×C₄²	1	3	yes	—
74	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
75	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
76	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
77	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
78	C₅×C₈	—	2	4	no	—
79	C₅×C₈	C₂×C₄²	1	6	yes	—
80	C₅×C₈	C₂³×C₄	1	2	yes	—
81	C₂³×C₄	—	2	1	no	—
82	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
83	C₂²×C₈	—	2	3	no	—
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	C₂×C₄×C₅	1	6	yes	—
86	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
87	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
88	C₂²×C₈	C₂×C₄×C₅	1	6	yes	—
89	C₂²×C₈	C₂³×C₅	1	3	yes	—
90	C₂×C₄²	—	2	3	no	—
91	C₂×C₄²	C₂×C₄×C₅	1	6	yes	—
92	M₃₂	C₂×C₄×C₅	1	24	yes	complemented
93	M₃₂	C₂³×C₅	1	8	yes	—
94	C₂×C₁₆	C₂×C₄×C₅	1	24	yes	complemented
95	#97	C₂×C₄×C₅	1	12	yes	—
96	#97	C₂³×C₅	1	4	yes	—
97	C₂²×C₅	—	2	6	no	—
98	C₂²×C₅	—	2	6	no	—
99	C₂²×C₅	#347	1	4	yes	direct product factor
100	C₂²×C₅	C₂×#97	1	3	yes	—
101	C₂²×C₅	C₂×#97	1	3	yes	—
102	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
103	C₄×C₅	—	4	4	no	—
104	C₄×C₅	C₂×#97	1	3	yes	—
105	C₄×C₅	C₂×#97	1	3	yes	—
106	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
107	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
108	C₂⁴	C₅×M₁₆	1	1	yes	characteristic
109	C₂²×C₄	—	2	6	no	—
110	C₂²×C₄	—	4	4	no	—
111	C₂²×C₄	C₄²×C₅	1	3	yes	—
112	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
113	C₂²×C₄	C₅×M₁₆	1	3	yes	—
114	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
115	C₂²×C₄	C₅×M₁₆	1	3	yes	—
116	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
117	C₂×C₈	—	2	6	no	—
118	C₂×C₈	—	2	6	no	—
119	C₂×C₈	C₄²×C₅	1	6	yes	—
120	C₂×C₈	C₄²×C₅	1	6	yes	—
121	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
122	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
123	C₂×C₈	C₂²×C₄×C₅	1	6	yes	—
124	C₂×C₈	C₂²×C₄×C₅	1	6	yes	—
125	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
126	C₄²	—	2	6	no	—
127	C₄²	C₂²×C₄×C₅	1	4	yes	—
128	C₁₆	C₂²×C₄×C₅	1	16	yes	complemented
129	C₂×C₅	—	2	1	no	—
130	C₂×C₅	—	2	3	no	—
131	C₂×C₅	C₂×#347	1	6	yes	direct product factor
132	C₂×C₅	C₂²×#97	1	1	yes	characteristic
133	C₂³	—	2	4	no	—
134	C₂³	C₅×#97	1	3	yes	—
135	C₂³	C₅×#97	1	1	yes	characteristic
136	C₂³	C₂×C₅×M₁₆	1	3	yes	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	4	12	no	—
139	C₂×C₄	C₅×#97	1	6	yes	—
140	C₂×C₄	C₅×#97	1	3	yes	—
141	C₂×C₄	C₅×#97	1	3	yes	—
142	C₂×C₄	C₂×C₄²×C₅	1	3	yes	—
143	C₂×C₄	C₂×C₄²×C₅	1	3	yes	—
144	C₂×C₄	C₂×C₅×M₁₆	1	3	yes	—
145	C₂×C₄	C₂×C₅×M₁₆	1	3	yes	—
146	C₂×C₄	C₂×C₅×M₁₆	1	3	yes	—
147	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
148	C₈	—	2	4	no	—
149	C₈	C₂×C₄²×C₅	1	6	yes	—
150	C₈	C₂³×C₄×C₅	1	2	yes	—
151	C₅	C₂²×#347	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²	—	2	6	no	—
153	C₂²	—	2	6	no	—
154	C₂²	C₅×#347	1	4	yes	direct product factor
155	C₂²	C₂×C₅×#97	1	3	yes	—
156	C₂²	C₂×C₅×#97	1	3	yes	—
157	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
158	C₄	—	4	4	no	—
159	C₄	C₂×C₅×#97	1	3	yes	—
160	C₄	C₂×C₅×#97	1	3	yes	—
161	C₄	C₂²×C₄²×C₅	1	1	yes	characteristic
162	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	3	no	—
165	C₂	C₂×C₅×#347	1	6	yes	direct product factor
166	C₂	C₂²×C₅×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₄²
commutator	C₄	C₂²×C₄²×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂³×C₅	#97
prime radical	C₂⁴×C₅	M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#347	C₂	direct (N × H)
7. C₂×C₅×M₃₂	C₄	semidirect
9. C₂²×C₅×C₁₆	C₄	semidirect
12. C₅×#347	C₂²	direct (N × H)
13. C₂²×#347	C₅	direct (N × H)
25. C₅×M₃₂	C₂×C₄	semidirect
27. C₂×C₅×C₁₆	C₂×C₄	semidirect
32. C₂×#347	C₂×C₅	direct (N × H)
53. C₅×C₁₆	C₂²×C₄	semidirect
57. C₂×M₃₂	C₄×C₅	semidirect
59. C₂²×C₁₆	C₄×C₅	semidirect
62. #347	C₂²×C₅	direct (N × H)
92. M₃₂	C₂×C₄×C₅	semidirect
94. C₂×C₁₆	C₂×C₄×C₅	semidirect
99. C₂²×C₅	#347	direct (N × H)
128. C₁₆	C₂²×C₄×C₅	semidirect
131. C₂×C₅	C₂×#347	direct (N × H)
151. C₅	C₂²×#347	direct (N × H)
154. C₂²	C₅×#347	direct (N × H)
165. C₂	C₂×C₅×#347	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×M₃₂	2
C₂²×C₅×#97	2
C₂×C₅×#347	2
C₂²×#347	5

Subgroups — C22×C5×#347 #11956047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#347 #11956047