Subgroups — C₂×C₇₁×#1455 #31838204

Order 18176.

994 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₇₁	C₂	1	3	yes	maximal
2	C₇₁×#1455	C₂	1	8	yes	maximal, direct product factor
3	C₂×#322×C₇₁	C₂	1	3	yes	maximal, complemented
4	C₂×C₈²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×M₁₆×C₇₁	C₄	1	6	yes	—
6	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
7	C₄×M₁₆×C₇₁	C₄	1	12	yes	—
8	C₄×M₁₆×C₇₁	C₂²	1	12	yes	—
9	C₂×#97×C₇₁	C₄	1	6	yes	—
10	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
12	#322×C₇₁	C₂²	1	12	yes	complemented
13	C₈²×C₇₁	C₂²	1	4	yes	complemented
14	C₂³×C₄×C₇₁	C₂×C₄	1	3	yes	—
15	C₂×M₁₆×C₇₁	—	2	24	no	—
16	C₂×M₁₆×C₇₁	C₂×C₄	1	24	yes	—
17	C₂²×C₈×C₇₁	C₂×C₄	1	6	yes	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	6	yes	—
19	C₂×C₄²×C₇₁	C₂×C₄	1	3	yes	—
20	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
21	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
22	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
23	#97×C₇₁	C₂×C₄	1	24	yes	—
24	C₄×C₈×C₇₁	C₂×C₄	1	6	yes	—
25	C₄×C₈×C₇₁	C₂×C₄	1	6	yes	—
26	C₄×C₈×C₇₁	C₂³	1	6	yes	—
27	C₄×C₈×C₇₁	C₂³	1	6	yes	—
28	C₂⁴×C₇₁	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	—	2	12	no	—
30	C₂²×C₄×C₇₁	C₄²	1	3	yes	—
31	C₂²×C₄×C₇₁	C₄²	1	6	yes	—
32	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
33	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
34	M₁₆×C₇₁	—	2	48	no	—
35	C₂×C₈×C₇₁	—	2	12	no	—
36	C₂×C₈×C₇₁	M₁₆	1	24	yes	complemented
37	C₂×C₈×C₇₁	C₂²×C₄	1	12	yes	—
38	C₂×C₈×C₇₁	C₂²×C₄	1	12	yes	—
39	C₄²×C₇₁	C₄²	1	6	yes	—
40	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
41	C₄²×C₇₁	C₂²×C₄	1	6	yes	—
42	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
43	C₂³×C₇₁	—	2	6	no	—
44	C₂³×C₇₁	C₂×C₄²	1	2	yes	—
45	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₇₁	—	2	6	no	—
47	C₂×C₄×C₇₁	—	2	12	no	—
48	C₂×C₄×C₇₁	—	2	12	no	—
49	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
50	C₂×C₄×C₇₁	C₂×C₄²	1	3	yes	—
51	C₂×C₄×C₇₁	C₂×M₁₆	1	12	yes	—
52	C₂×C₄×C₇₁	C₂³×C₄	1	3	yes	—
53	C₈×C₇₁	—	2	12	no	—
54	C₈×C₇₁	C₂×M₁₆	1	24	yes	complemented
55	C₂²×C₇₁	—	2	6	no	—
56	C₂²×C₇₁	—	4	4	no	—
57	C₂²×C₇₁	C₄×M₁₆	1	6	yes	—
58	C₂²×C₇₁	C₂²×C₄²	1	1	yes	characteristic
59	C₄×C₇₁	—	2	6	no	—
60	C₄×C₇₁	C₄×M₁₆	1	6	yes	—
61	C₄×C₇₁	C₂²×M₁₆	1	6	yes	—
62	C₂×#1455	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₇₁	—	4	2	no	—
64	C₂×C₇₁	#1455	1	4	yes	direct product factor
65	C₂×C₇₁	C₂×C₄×M₁₆	1	3	yes	—
66	C₂×C₄×M₁₆	C₂×C₇₁	1	3	yes	—
67	#1455	C₂×C₇₁	1	8	yes	direct product factor
68	C₂×#322	C₂×C₇₁	1	3	yes	complemented
69	C₂×C₈²	C₂×C₇₁	1	1	yes	characteristic, complemented
70	C₇₁	C₂×#1455	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²×M₁₆	C₄×C₇₁	1	6	yes	—
72	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
73	C₄×M₁₆	C₄×C₇₁	1	12	yes	—
74	C₄×M₁₆	C₂²×C₇₁	1	12	yes	—
75	C₂×#97	C₄×C₇₁	1	6	yes	—
76	C₂×C₄×C₈	C₂²×C₇₁	1	3	yes	—
77	C₂×C₄×C₈	C₂²×C₇₁	1	3	yes	—
78	#322	C₂²×C₇₁	1	12	yes	complemented
79	C₈²	C₂²×C₇₁	1	4	yes	complemented
80	C₂³×C₄	C₂×C₄×C₇₁	1	3	yes	—
81	C₂×M₁₆	—	2	24	no	—
82	C₂×M₁₆	C₂×C₄×C₇₁	1	24	yes	—
83	C₂²×C₈	C₂×C₄×C₇₁	1	6	yes	—
84	C₂²×C₈	C₂×C₄×C₇₁	1	6	yes	—
85	C₂×C₄²	C₂×C₄×C₇₁	1	3	yes	—
86	C₂×C₄²	C₂×C₄×C₇₁	1	6	yes	—
87	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
88	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
89	#97	C₂×C₄×C₇₁	1	24	yes	—
90	C₄×C₈	C₂×C₄×C₇₁	1	6	yes	—
91	C₄×C₈	C₂×C₄×C₇₁	1	6	yes	—
92	C₄×C₈	C₂³×C₇₁	1	6	yes	—
93	C₄×C₈	C₂³×C₇₁	1	6	yes	—
94	C₂⁴	C₄²×C₇₁	1	1	yes	characteristic
95	C₂²×C₄	—	2	12	no	—
96	C₂²×C₄	C₄²×C₇₁	1	3	yes	—
97	C₂²×C₄	C₄²×C₇₁	1	6	yes	—
98	C₂²×C₄	C₂²×C₄×C₇₁	1	6	yes	—
99	C₂²×C₄	C₂²×C₄×C₇₁	1	3	yes	—
100	M₁₆	—	2	48	no	—
101	C₂×C₈	—	2	12	no	—
102	C₂×C₈	M₁₆×C₇₁	1	24	yes	complemented
103	C₂×C₈	C₂²×C₄×C₇₁	1	12	yes	—
104	C₂×C₈	C₂²×C₄×C₇₁	1	12	yes	—
105	C₄²	C₄²×C₇₁	1	6	yes	—
106	C₄²	C₂²×C₄×C₇₁	1	3	yes	—
107	C₄²	C₂²×C₄×C₇₁	1	6	yes	—
108	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
109	C₂³	—	2	6	no	—
110	C₂³	C₂×C₄²×C₇₁	1	2	yes	—
111	C₂³	C₂×C₄²×C₇₁	1	1	yes	characteristic
112	C₂×C₄	—	2	6	no	—
113	C₂×C₄	—	2	12	no	—
114	C₂×C₄	—	2	12	no	—
115	C₂×C₄	C₂×C₄²×C₇₁	1	6	yes	—
116	C₂×C₄	C₂×C₄²×C₇₁	1	3	yes	—
117	C₂×C₄	C₂×M₁₆×C₇₁	1	12	yes	—
118	C₂×C₄	C₂³×C₄×C₇₁	1	3	yes	—
119	C₈	—	2	12	no	—
120	C₈	C₂×M₁₆×C₇₁	1	24	yes	complemented
121	C₂²	—	2	6	no	—
122	C₂²	—	4	4	no	—
123	C₂²	C₄×M₁₆×C₇₁	1	6	yes	—
124	C₂²	C₂²×C₄²×C₇₁	1	1	yes	characteristic
125	C₄	—	2	6	no	—
126	C₄	C₄×M₁₆×C₇₁	1	6	yes	—
127	C₄	C₂²×M₁₆×C₇₁	1	6	yes	—
128	C₂	—	4	2	no	—
129	C₂	C₇₁×#1455	1	4	yes	direct product factor
130	C₂	C₂×C₄×M₁₆×C₇₁	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²×C₇₁	C₂³
commutator	C₂²	C₂²×C₄²×C₇₁
Frattini, cosocle	C₄²	C₂⁴×C₇₁
socle	C₂³×C₇₁	C₂×C₄²
prime radical	C₂⁴×C₇₁	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#1455	C₂	direct (N × H)
3. C₂×#322×C₇₁	C₂	semidirect
4. C₂×C₈²×C₇₁	C₂	semidirect
12. #322×C₇₁	C₂²	semidirect
13. C₈²×C₇₁	C₂²	semidirect
36. C₂×C₈×C₇₁	M₁₆	semidirect
54. C₈×C₇₁	C₂×M₁₆	semidirect
62. C₂×#1455	C₇₁	direct (N × H)
64. C₂×C₇₁	#1455	direct (N × H)
67. #1455	C₂×C₇₁	direct (N × H)
68. C₂×#322	C₂×C₇₁	semidirect
69. C₂×C₈²	C₂×C₇₁	semidirect
70. C₇₁	C₂×#1455	direct (N × H)
78. #322	C₂²×C₇₁	semidirect
79. C₈²	C₂²×C₇₁	semidirect
102. C₂×C₈	M₁₆×C₇₁	semidirect
120. C₈	C₂×M₁₆×C₇₁	semidirect
129. C₂	C₇₁×#1455	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆×C₇₁	2
C₇₁×#1455	2
C₂×#322×C₇₁	2
C₂×C₈²×C₇₁	2
C₂×#1455	71

Subgroups — C2×C71×#1455 #31838204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₁×#1455 #31838204