Subgroups — C₄×#1747 #353172

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1747	C₂	1	4	yes	maximal
2	C₄×#405	C₂	1	3	yes	maximal
3	#10699	C₂	1	4	yes	maximal
4	C₄×#338	C₂	1	1	yes	maximal, characteristic
5	C₄×H(3,ℤ/4)	C₂	1	3	yes	maximal
6	C₂×#405	C₂²	1	6	yes	—
7	C₂×#405	C₂²	1	6	yes	—
8	C₂×C₄×M₁₆	C₂²	1	3	yes	—
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄×#117	C₂²	1	3	yes	—
11	#1747	C₄	1	32	yes	direct product factor
12	C₂×#338	C₂²	1	4	yes	—
13	C₂×H(3,ℤ/4)	C₂²	1	12	yes	—
14	#1726	C₄	1	6	yes	complemented
15	#1726	C₄	1	6	yes	complemented
16	C₄×#97	C₄	1	6	yes	complemented
17	C₄²×C₈	C₄	1	6	yes	complemented
18	C₂²×M₁₆	C₂³	1	6	yes	—
19	C₂³×C₈	C₂³	1	1	yes	characteristic
20	C₂³×C₈	C₂³	1	1	yes	characteristic
21	C₂×#117	C₂³	1	6	yes	—
22	C₂²×C₄²	C₂³	1	1	yes	characteristic
23	#405	—	2	24	no	—
24	#405	C₂×C₄	1	48	yes	—
25	C₄×M₁₆	—	2	12	no	—
26	C₂×#97	C₂×C₄	1	6	yes	—
27	C₂×#97	C₂×C₄	1	12	yes	—
28	C₂×#97	C₂×C₄	1	12	yes	—
29	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
30	C₂×C₄×C₈	C₂×C₄	1	12	yes	—
31	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
32	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
33	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
34	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
35	C₂×C₄×C₈	C₂×C₄	1	12	yes	—
36	C₄×C₄⋊C₄	—	2	6	no	—
37	C₄×C₂²⋊C₄	—	2	6	no	—
38	#376	C₂×C₄	1	6	yes	—
39	C₄³	C₂×C₄	1	6	yes	—
40	#338	C₂×C₄	1	16	yes	—
41	H(3,ℤ/4)	C₂×C₄	1	48	yes	—
42	C₂³×C₄	#41	1	2	yes	—
43	C₂³×C₄	C₂⁴	1	1	yes	characteristic
44	C₂×M₁₆	—	2	12	no	—
45	C₂×M₁₆	—	2	12	no	—
46	C₂×M₁₆	C₂²×C₄	1	24	yes	—
47	C₂²×C₈	C₂²×C₄	1	12	yes	—
48	C₂²×C₈	C₂²×C₄	1	3	yes	—
49	C₂²×C₈	C₂²×C₄	1	3	yes	—
50	C₂²×C₈	C₂²×C₄	1	8	yes	—
51	C₂²×C₈	C₂²×C₄	1	3	yes	—
52	C₂²×C₈	C₂²×C₄	1	3	yes	—
53	C₂²×C₈	#41	1	8	yes	—
54	#117	—	2	12	no	—
55	#117	C₂²×C₄	1	24	yes	—
56	C₂×C₄⋊C₄	—	2	6	no	—
57	C₂×C₂²⋊C₄	—	2	6	no	—
58	C₂×C₄²	—	2	12	no	—
59	C₂×C₄²	C₂²×C₄	1	12	yes	—
60	C₂×C₄²	C₂²×C₄	1	3	yes	—
61	C₂×C₄²	C₂²×C₄	1	3	yes	—
62	C₂×C₄²	#41	1	6	yes	—
63	#97	—	2	24	no	—
64	#97	—	2	12	no	—
65	#97	C₄²	1	48	yes	complemented
66	C₄×C₈	—	2	24	no	—
67	C₄×C₈	—	2	12	no	—
68	C₄×C₈	C₄²	1	6	yes	—
69	C₄×C₈	C₄²	1	3	yes	—
70	C₄×C₈	C₄²	1	3	yes	—
71	C₄×C₈	C₄²	1	3	yes	—
72	C₄×C₈	C₄²	1	1	yes	characteristic
73	C₄×C₈	C₄²	1	48	yes	complemented
74	#95	—	2	12	no	—
75	C₂⁴	C₂×#41	1	1	yes	characteristic
76	C₂²×C₄	—	2	12	no	—
77	C₂²×C₄	—	4	4	no	—
78	C₂²×C₄	#117	1	6	yes	—
79	C₂²×C₄	#117	1	6	yes	—
80	C₂²×C₄	C₂³×C₄	1	3	yes	—
81	C₂²×C₄	C₂³×C₄	1	4	yes	—
82	C₂²×C₄	C₂×#41	1	4	yes	—
83	C₂²×C₄	C₂×#41	1	3	yes	—
84	M₁₆	—	2	48	no	—
85	C₂×C₈	—	2	12	no	—
86	C₂×C₈	—	2	12	no	—
87	C₂×C₈	C₂×C₄²	1	24	yes	—
88	C₂×C₈	C₂×C₄²	1	3	yes	—
89	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₈	C₂×C₄²	1	12	yes	—
91	C₂×C₈	C₂×C₄²	1	12	yes	—
92	C₂×C₈	C₂×C₄²	1	3	yes	—
93	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₈	#117	1	12	yes	—
95	C₂×C₈	#117	1	12	yes	—
96	C₄⋊C₄	—	2	24	no	—
97	C₂²⋊C₄	—	2	24	no	—
98	C₄²	—	2	12	no	—
99	C₄²	—	4	24	no	—
100	C₄²	C₂×C₄²	1	24	yes	—
101	C₄²	C₂×C₄²	1	3	yes	—
102	C₄²	C₂×C₄²	1	1	yes	characteristic
103	C₄²	#117	1	6	yes	—
104	C₄²	#117	1	6	yes	—
105	C₂³	—	4	2	no	—
106	C₂³	C₂×#117	1	3	yes	—
107	C₂³	C₄×#41	1	4	yes	—
108	C₂×C₄	—	2	12	no	—
109	C₂×C₄	—	2	24	no	—
110	C₂×C₄	—	4	6	no	—
111	C₂×C₄	—	4	24	no	—
112	C₂×C₄	#376	1	6	yes	—
113	C₂×C₄	#376	1	2	yes	—
114	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
115	C₂×C₄	C₂²×C₄²	1	6	yes	—
116	C₂×C₄	C₂×#117	1	6	yes	—
117	C₂×C₄	C₂×#117	1	3	yes	—
118	C₂×C₄	C₄×#41	1	12	yes	—
119	C₈	—	2	24	no	—
120	C₈	C₄³	1	6	yes	—
121	C₈	C₄³	1	2	yes	—
122	C₈	#376	1	6	yes	—
123	C₈	#376	1	2	yes	—
124	C₂²	—	2	6	no	—
125	C₂²	—	4	4	no	—
126	C₂²	C₂×#376	1	1	yes	characteristic
127	C₂²	C₄×#117	1	6	yes	—
128	C₄	—	2	6	no	—
129	C₄	—	4	24	no	—
130	C₄	#1747	1	4	yes	direct product factor
131	C₄	C₂×C₄³	1	1	yes	characteristic
132	C₄	C₂×#376	1	1	yes	characteristic
133	C₄	C₄×#117	1	6	yes	—
134	C₂	—	2	6	no	—
135	C₂	C₄×#376	1	1	yes	characteristic
136	C₂	C₂×#1747	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₄²
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#376
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #1747	C₄	direct (N × H)
14. #1726	C₄	semidirect
15. #1726	C₄	semidirect
16. C₄×#97	C₄	semidirect
17. C₄²×C₈	C₄	semidirect
65. #97	C₄²	semidirect
73. C₄×C₈	C₄²	semidirect
130. C₄	#1747	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×#1747	2
C₄×#405	2
#10699	2
C₄×#338	2
C₄×H(3,ℤ/4)	2

Subgroups — C4×#1747 #353172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#1747 #353172