Subgroups — C₂×#14521816 #16860421

Order 3968.

1460 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	#14521816	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11758525	C₂	1	8	yes	maximal, complemented
3	C₂²×#92474	C₂	1	6	yes	maximal, complemented
4	C₂²×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#41×C₃₁	C₄	1	4	yes	—
6	C₂×#41×C₃₁	C₄	1	8	yes	—
7	C₂×#41×C₃₁	C₂²	1	4	yes	—
8	C₂×#41×C₃₁	C₂²	1	8	yes	—
9	C₂²×Q₈×C₃₁	C₄	1	1	yes	characteristic
10	C₂²×D₄×C₃₁	C₄	1	3	yes	—
11	C₂³×C₄×C₃₁	C₂²	1	3	yes	—
12	#11758525	C₂²	1	64	yes	complemented
13	C₂×#92474	C₂²	1	4	yes	—
14	C₂×#92474	C₂²	1	12	yes	complemented
15	C₂×#92474	C₂²	1	48	yes	complemented
16	C₂²×C₃₁⋊C₈	C₂²	1	12	yes	complemented
17	C₂⁴×C₃₁	C₂×C₄	1	3	yes	—
18	#41×C₃₁	C₂×C₄	1	16	yes	—
19	#41×C₃₁	C₂×C₄	1	32	yes	—
20	#41×C₃₁	C₂³	1	16	yes	—
21	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	—
22	C₂×Q₈×C₃₁	C₂×C₄	1	8	yes	—
23	C₂×D₄×C₃₁	C₂×C₄	1	12	yes	—
24	C₂×D₄×C₃₁	C₂×C₄	1	24	yes	—
25	C₂²×C₄×C₃₁	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₃₁	C₂×C₄	1	6	yes	—
27	C₂²×C₄×C₃₁	C₂×C₄	1	12	yes	—
28	C₂²×C₄×C₃₁	C₂³	1	6	yes	—
29	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	C₂³	1	12	yes	—
31	#92474	C₂³	1	16	yes	—
32	#92474	C₂³	1	48	yes	complemented
33	C₂×C₃₁⋊C₈	C₂³	1	8	yes	—
34	C₂×C₃₁⋊C₈	C₂³	1	48	yes	complemented
35	C₂³×C₃₁	—	4	6	no	—
36	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
37	C₂³×C₃₁	C₂²×C₄	1	6	yes	—
38	C₂³×C₃₁	C₂²×C₄	1	12	yes	—
39	Q₈×C₃₁	C₂²×C₄	1	16	yes	—
40	D₄×C₃₁	C₂²×C₄	1	48	yes	—
41	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₂²×C₄	1	12	yes	—
44	C₂×C₄×C₃₁	C₂²×C₄	1	6	yes	—
45	C₂×C₄×C₃₁	C₂²×C₄	1	12	yes	—
46	C₂×C₄×C₃₁	C₂²×C₄	1	24	yes	—
47	C₂×C₄×C₃₁	C₂⁴	1	12	yes	—
48	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
49	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
50	C₃₁⋊C₈	C₂⁴	1	16	yes	—
51	C₂×#568	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	—	2	2	no	—
53	C₂²×C₃₁	—	2	12	no	—
54	C₂²×C₃₁	—	4	12	no	—
55	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
56	C₂²×C₃₁	C₂³×C₄	1	12	yes	—
57	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₃₁	C₂³×C₄	1	2	yes	—
59	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₃₁	C₂³×C₄	1	12	yes	—
61	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
62	C₂²×#41	D₃₁	1	1	yes	characteristic
63	#568	—	31	16	no	—
64	C₂×#131	—	31	8	no	—
65	C₂²×M₁₆	—	31	6	no	—
66	C₂×C₃₁	—	2	2	no	—
67	C₂×C₃₁	—	2	12	no	—
68	C₂×C₃₁	#568	1	2	yes	complemented
69	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
70	C₂×#41	Dic₃₁	1	4	yes	—
71	C₂×#41	Dic₃₁	1	8	yes	—
72	C₂×#41	C₂×D₃₁	1	4	yes	—
73	C₂×#41	C₂×D₃₁	1	8	yes	—
74	C₂²×Q₈	Dic₃₁	1	1	yes	characteristic
75	C₂²×D₄	Dic₃₁	1	3	yes	—
76	C₂³×C₄	C₂×D₃₁	1	3	yes	—
77	#131	—	31	64	no	—
78	C₂×M₁₆	—	31	4	no	—
79	C₂×M₁₆	—	31	12	no	—
80	C₂×M₁₆	—	31	48	no	—
81	C₂²×C₈	—	31	12	no	—
82	C₃₁	C₂×#568	1	1	yes	characteristic, complemented, Sylow
83	C₂⁴	C₂×Dic₃₁	1	3	yes	—
84	#41	C₂×Dic₃₁	1	16	yes	—
85	#41	C₂×Dic₃₁	1	32	yes	—
86	#41	C₂²×D₃₁	1	16	yes	—
87	C₂×Q₈	C₂×Dic₃₁	1	4	yes	—
88	C₂×Q₈	C₂×Dic₃₁	1	8	yes	—
89	C₂×D₄	C₂×Dic₃₁	1	12	yes	—
90	C₂×D₄	C₂×Dic₃₁	1	24	yes	—
91	C₂²×C₄	C₂×Dic₃₁	1	3	yes	—
92	C₂²×C₄	C₂×Dic₃₁	1	6	yes	—
93	C₂²×C₄	C₂×Dic₃₁	1	12	yes	—
94	C₂²×C₄	C₂²×D₃₁	1	12	yes	—
95	C₂²×C₄	C₂²×D₃₁	1	6	yes	—
96	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
97	M₁₆	—	31	16	no	—
98	M₁₆	—	31	48	no	—
99	C₂×C₈	—	31	48	no	—
100	C₂×C₈	—	31	8	no	—
101	C₂³	—	4	6	no	—
102	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
103	C₂³	C₂²×Dic₃₁	1	6	yes	—
104	C₂³	C₂²×Dic₃₁	1	12	yes	—
105	Q₈	C₂²×Dic₃₁	1	16	yes	—
106	D₄	C₂²×Dic₃₁	1	48	yes	—
107	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
108	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
109	C₂×C₄	C₂²×Dic₃₁	1	12	yes	—
110	C₂×C₄	C₂²×Dic₃₁	1	6	yes	—
111	C₂×C₄	C₂²×Dic₃₁	1	24	yes	—
112	C₂×C₄	C₂²×Dic₃₁	1	12	yes	—
113	C₂×C₄	C₂³×D₃₁	1	12	yes	—
114	C₂×C₄	C₂³×D₃₁	1	2	yes	—
115	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
116	C₈	—	31	16	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	12	no	—
119	C₂²	—	4	12	no	—
120	C₂²	C₂³×Dic₃₁	1	2	yes	—
121	C₂²	C₂³×Dic₃₁	1	12	yes	—
122	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
123	C₄	C₂³×Dic₃₁	1	2	yes	—
124	C₄	C₂³×Dic₃₁	1	1	yes	characteristic
125	C₄	C₂³×Dic₃₁	1	12	yes	—
126	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	12	no	—
129	C₂	#14521816	1	2	yes	direct product factor
130	C₂	C₂⁴×Dic₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₃₁
Fitting, prime radical	C₂²×#41×C₃₁	C₂
socle	C₂²×C₃₁	C₂³×C₄
cosocle	C₄×C₃₁	C₂⁵
hypercenter	C₂²×#41	D₃₁
hypocenter	C₃₁	C₂×#568
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #14521816	C₂	direct (N × H)
2. C₂×#11758525	C₂	semidirect
3. C₂²×#92474	C₂	semidirect
12. #11758525	C₂²	semidirect
14. C₂×#92474	C₂²	semidirect
15. C₂×#92474	C₂²	semidirect
16. C₂²×C₃₁⋊C₈	C₂²	semidirect
32. #92474	C₂³	semidirect
34. C₂×C₃₁⋊C₈	C₂³	semidirect
68. C₂×C₃₁	#568	semidirect
82. C₃₁	C₂×#568	semidirect
129. C₂	#14521816	direct (N × H)

Maximal subgroups

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

subgroup	index
#14521816	2
C₂×#11758525	2
C₂²×#92474	2
C₂²×#41×C₃₁	2
C₂×#568	31

Subgroups — C2×#14521816 #16860421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521816 #16860421