Subgroups — C₂×#72227646 #72271668

Order 3072.

Only the 176 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 123 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11715667	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11710937	C₂	1	1	yes	maximal, characteristic
3	#72227646	C₂	1	4	yes	maximal, direct product factor
4	C₂×#11715663	C₂	1	1	yes	maximal, characteristic, complemented
5	#11715667	C₂²	1	2	yes	complemented
6	#11715663	C₂²	1	2	yes	complemented
7	#11710937	C₄	1	2	yes	complemented
8	#11710937	C₂²	1	2	yes	—
9	C₂×#87548	C₄	1	1	yes	characteristic, complemented
10	C₂×#87547	C₄	1	1	yes	characteristic, complemented
11	C₂²×#5950	C₂²	1	1	yes	characteristic
12	C₂×#63103	D₃	1	1	yes	characteristic
13	#87548	C₂×C₄	1	2	yes	complemented
14	#87547	C₂×C₄	1	2	yes	complemented
15	C₂×#5950	C₂×C₄	1	1	yes	characteristic
16	C₂×#5950	C₂×C₄	1	1	yes	characteristic
17	C₂×#5950	D₄	1	1	yes	characteristic
18	C₂×#5950	D₄	1	1	yes	characteristic
19	C₂×#5950	D₄	1	1	yes	characteristic
20	C₂×#5950	C₂³	1	1	yes	characteristic
21	C₂²×#997	D₄	1	1	yes	characteristic
22	#63103	Dic₃	1	2	yes	complemented
23	#63103	C₂×D₃	1	2	yes	—
24	C₂×2₋¹⁺⁶	Dic₃	1	1	yes	characteristic, complemented
25	C₂×H(7,𝔽₂)	Dic₃	1	1	yes	characteristic, complemented
26	C₂²×#585	C₂×D₃	1	1	yes	characteristic
27	#5950	C₂²⋊C₄	1	1	yes	characteristic, complemented
28	#5950	C₂²⋊C₄	1	1	yes	characteristic, complemented
29	#5950	C₂²×C₄	1	1	yes	characteristic
30	#5950	C₂×D₄	1	1	yes	characteristic
31	C₂×#997	C₂²⋊C₄	1	1	yes	characteristic, complemented
32	C₂×#997	C₂²⋊C₄	1	1	yes	characteristic, complemented
33	C₂×#997	C₂×D₄	1	1	yes	characteristic
34	2₋¹⁺⁶	C₂×Dic₃	1	2	yes	complemented
35	H(7,𝔽₂)	C₂×Dic₃	1	2	yes	complemented
36	C₂×#585	C₂×Dic₃	1	1	yes	characteristic
37	C₂×#585	C₂×Dic₃	1	1	yes	characteristic
38	C₂×#585	#67	1	1	yes	characteristic
39	C₂×#585	#67	1	1	yes	characteristic
40	C₂×#585	#67	1	1	yes	characteristic
41	C₂×#585	S₄	1	2	yes	—
42	C₂×#585	C₂²×D₃	1	1	yes	characteristic
43	C₂²×H(5,𝔽₂)	#67	1	1	yes	characteristic
44	C₂³×#41	S₄	1	1	yes	characteristic
45	#997	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
46	#585	#217	1	1	yes	characteristic
47	#585	#217	1	1	yes	characteristic
48	#585	SL(2,ℤ/4)	1	4	yes	complemented
49	#585	C₂²×Dic₃	1	1	yes	characteristic
50	#585	C₂×#67	1	1	yes	characteristic
51	#585	C₂×S₄	1	4	yes	—
52	C₂×2₋¹⁺⁴	SL(2,ℤ/4)	1	2	yes	complemented
53	C₂×H(5,𝔽₂)	#217	1	1	yes	characteristic
54	C₂×H(5,𝔽₂)	#217	1	1	yes	characteristic
55	C₂×H(5,𝔽₂)	SL(2,ℤ/4)	1	2	yes	complemented
56	C₂×H(5,𝔽₂)	C₂×#67	1	1	yes	characteristic
57	C₂²×#41	SL(2,ℤ/4)	1	2	yes	—
58	C₂²×#41	C₂×S₄	1	2	yes	—
59	C₂²×#41	C₂×S₄	1	2	yes	—
60	C₂³×Q₈	SL(2,ℤ/4)	1	1	yes	characteristic
61	C₂³×D₄	SL(2,ℤ/4)	1	1	yes	characteristic
62	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
63	C₂⁵	GL(2,ℤ/4)	1	1	yes	characteristic
64	2₋¹⁺⁴	C₂×SL(2,ℤ/4)	1	4	yes	complemented
65	H(5,𝔽₂)	C₂×#217	1	1	yes	characteristic
66	H(5,𝔽₂)	C₂×SL(2,ℤ/4)	1	4	yes	complemented
67	C₂×#41	C₂×SL(2,ℤ/4)	1	2	yes	—
68	C₂×#41	C₂×SL(2,ℤ/4)	1	2	yes	—
69	C₂×#41	GL(2,ℤ/4)	1	2	yes	—
70	C₂×#41	GL(2,ℤ/4)	1	2	yes	—
71	C₂×#41	GL(2,ℤ/4)	1	2	yes	—
72	C₂×#41	C₂²×S₄	1	2	yes	—
73	C₂×#41	#1020	1	1	yes	characteristic
74	C₂²×Q₈	C₂×SL(2,ℤ/4)	1	2	yes	—
75	C₂²×Q₈	GL(2,ℤ/4)	1	2	yes	—
76	C₂²×D₄	C₂×SL(2,ℤ/4)	1	2	yes	—
77	C₂³×C₄	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
78	C₂³×C₄	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
79	C₂³×C₄	GL(2,ℤ/4)	1	1	yes	characteristic
80	C₂³×C₄	GL(2,ℤ/4)	1	1	yes	characteristic
81	C₂³×C₄	GL(2,ℤ/4)	1	1	yes	characteristic
82	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
83	C₂⁴	#5432	1	1	yes	characteristic
84	C₂⁴	#5432	1	1	yes	characteristic
85	C₂⁴	C₂×GL(2,ℤ/4)	1	1	yes	characteristic
86	#41	#5432	1	2	yes	—
87	#41	#5432	1	2	yes	—
88	#41	C₂²×SL(2,ℤ/4)	1	2	yes	—
89	#41	C₂×GL(2,ℤ/4)	1	2	yes	—
90	#41	#5936	1	2	yes	—
91	#41	C₂×#1020	1	2	yes	—
92	C₂×Q₈	#5432	1	2	yes	—
93	C₂×Q₈	#5432	1	2	yes	—
94	C₂×Q₈	C₂×GL(2,ℤ/4)	1	2	yes	—
95	C₂×Q₈	#5936	1	1	yes	characteristic
96	C₂×D₄	#5936	1	1	yes	characteristic
97	C₂²×C₄	#5432	1	1	yes	characteristic
98	C₂²×C₄	#5432	1	1	yes	characteristic
99	C₂²×C₄	C₂²×SL(2,ℤ/4)	1	1	yes	characteristic
100	C₂²×C₄	C₂×GL(2,ℤ/4)	1	1	yes	characteristic
101	C₂²×C₄	C₂×#1020	1	1	yes	characteristic
102	C₂³	C₂×#5432	1	1	yes	characteristic
103	C₂³	#87524	1	1	yes	characteristic
104	Q₈	C₂×#5432	1	2	yes	—
105	Q₈	C₂×#5936	1	2	yes	—
106	D₄	C₂×#5936	1	2	yes	—
107	C₂×C₄	C₂×#5936	1	1	yes	characteristic
108	C₂×C₄	C₂×#5936	1	1	yes	characteristic
109	C₂×C₄	#87524	1	1	yes	characteristic
110	C₂×C₄	#87524	1	1	yes	characteristic
111	C₂×C₄	#87524	1	1	yes	characteristic
112	C₂×C₄	C₂²×#1020	1	1	yes	characteristic
113	C₂²	#11715569	1	1	yes	characteristic
114	C₂²	#11715569	1	1	yes	characteristic
115	C₂²	C₂×#87524	1	1	yes	characteristic
116	C₄	#11715569	1	1	yes	characteristic
117	C₄	#11715569	1	1	yes	characteristic
118	C₄	C₂²×#5936	1	1	yes	characteristic
119	C₄	C₂×#87524	1	1	yes	characteristic
120	C₂	C₂×#11715569	1	1	yes	characteristic
121	C₂	#72227646	1	2	yes	direct product factor

	subgroup	G/H
center	C₂×C₄	#87524
commutator	#5950	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1020
Fitting	C₂×#63103	D₃
socle	C₂²	#11715569
prime radical	C₂×#11710937	C₂
cosocle	C₂×#5950	C₂³
hypercenter	C₂×#41	#1020
hypocenter	#997	C₂×C₂²⋊C₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#11715667	C₂	semidirect
3. #72227646	C₂	direct (N × H)
4. C₂×#11715663	C₂	semidirect
5. #11715667	C₂²	semidirect
6. #11715663	C₂²	semidirect
7. #11710937	C₄	semidirect
9. C₂×#87548	C₄	semidirect
10. C₂×#87547	C₄	semidirect
13. #87548	C₂×C₄	semidirect
14. #87547	C₂×C₄	semidirect
22. #63103	Dic₃	semidirect
24. C₂×2₋¹⁺⁶	Dic₃	semidirect
25. C₂×H(7,𝔽₂)	Dic₃	semidirect
27. #5950	C₂²⋊C₄	semidirect
28. #5950	C₂²⋊C₄	semidirect
31. C₂×#997	C₂²⋊C₄	semidirect
32. C₂×#997	C₂²⋊C₄	semidirect
34. 2₋¹⁺⁶	C₂×Dic₃	semidirect
35. H(7,𝔽₂)	C₂×Dic₃	semidirect
45. #997	C₂×C₂²⋊C₄	semidirect
48. #585	SL(2,ℤ/4)	semidirect
52. C₂×2₋¹⁺⁴	SL(2,ℤ/4)	semidirect
55. C₂×H(5,𝔽₂)	SL(2,ℤ/4)	semidirect
64. 2₋¹⁺⁴	C₂×SL(2,ℤ/4)	semidirect
66. H(5,𝔽₂)	C₂×SL(2,ℤ/4)	semidirect
121. C₂	#72227646	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×#11715667	2
C₂×#11710937	2
C₂×#392140	3
#72227646	2
C₂×#11715663	2
C₂×#73157	4
C₂×#73150	4

Subgroups — C2×#72227646 #72271668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72227646 #72271668