Subgroups — #72235015

Order 2048.

Only the 2294 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 130 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72235014	C₂	1	3	yes	maximal, complemented
2	#72235013	C₂	1	3	yes	maximal, complemented
3	#72235012	C₂	1	12	yes	maximal, complemented
4	#72235011	C₂	1	12	yes	maximal, complemented
5	H(5,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
6	#10586807	C₂²	1	12	yes	—
7	#10586796	C₂²	1	3	yes	—
8	#6516748	C₂²	1	6	yes	complemented
9	#6351730	C₂²	1	4	yes	complemented
10	#6349989	C₂²	1	12	yes	complemented
11	#6347371	C₂²	1	12	yes	complemented
12	#5035253	C₂²	1	1	yes	characteristic, complemented
13	#188167	C₂²	1	3	yes	complemented
14	#187581	C₂²	1	12	yes	complemented
15	#187423	C₂²	1	12	yes	complemented
16	#187276	C₂²	1	24	yes	complemented
17	#187144	C₂²	1	12	yes	complemented
18	#186833	C₂²	1	6	yes	complemented
19	#186167	C₂²	1	24	yes	complemented
20	#185268	C₂²	1	12	yes	complemented
21	#63069	D₄	1	48	yes	complemented
22	#63048	D₄	1	24	yes	complemented
23	#63045	D₄	1	24	yes	complemented
24	#62841	C₂³	1	12	yes	—
25	#62824	D₄	1	12	yes	—
26	#62824	C₂³	1	6	yes	—
27	#62824	C₂³	1	12	yes	—
28	#62817	D₄	1	12	yes	—
29	#62694	C₂³	1	4	yes	—
30	C₂²×H(3,𝔽₄)	C₂³	1	1	yes	characteristic
31	#60798	C₂³	1	12	yes	complemented
32	C₂×#2657	C₂³	1	3	yes	complemented
33	#21798	C₂³	1	6	yes	complemented
34	#21417	C₂³	1	12	yes	complemented
35	#17740	C₂³	1	12	yes	complemented
36	#17673	C₂³	1	12	yes	complemented
37	#17541	C₂³	1	24	yes	complemented
38	#17493	C₂³	1	24	yes	complemented
39	#17393	C₂³	1	12	yes	complemented
40	#14169	C₂³	1	3	yes	complemented
41	#3561	C₂×D₄	1	48	yes	—
42	#3558	C₂×D₄	1	48	yes	—
43	#3550	C₂×D₄	1	24	yes	—
44	#3550	C₂×D₄	1	12	yes	—
45	#3549	C₂×D₄	1	12	yes	—
46	#3531	C₂×D₄	1	24	yes	—
47	#3529	C₂×D₄	1	24	yes	—
48	#3518	C₂×D₄	1	48	yes	—
49	#3514	C₂×D₄	1	48	yes	—
50	#3510	C₂×D₄	1	24	yes	—
51	#3500	C₂×D₄	1	24	yes	—
52	#3490	C₂×D₄	1	48	yes	—
53	#3486	C₂×D₄	1	12	yes	—
54	#3484	C₂×D₄	1	12	yes	—
55	#3444	C₂⁴	1	12	yes	—
56	C₂²×#127	C₂⁴	1	1	yes	characteristic, complemented
57	C₂²×#124	C₂×D₄	1	6	yes	—
58	C₂²×#124	C₂⁴	1	3	yes	—
59	C₂²×#120	C₂×D₄	1	6	yes	—
60	#2920	C₂⁴	1	12	yes	complemented
61	C₂×#376	C₂⁴	1	3	yes	complemented
62	C₂⁶	#120	1	2	yes	—
63	C₂×2₋¹⁺⁴	#120	1	12	yes	—
64	C₂×H(5,𝔽₂)	#120	1	12	yes	—
65	C₂³×Q₈	C₂²×D₄	1	3	yes	—
66	C₂³×D₄	#120	1	6	yes	—
67	C₂³×D₄	C₂²×D₄	1	3	yes	—
68	#564	#120	1	8	yes	complemented
69	#563	#120	1	24	yes	complemented
70	#562	#120	1	24	yes	complemented
71	H(3,𝔽₄)	#120	1	8	yes	complemented
72	#559	#120	1	24	yes	complemented
73	Q₈²	#120	1	24	yes	complemented
74	#556	#120	1	48	yes	complemented
75	#555	#120	1	24	yes	complemented
76	D₄×Q₈	#120	1	48	yes	complemented
77	D₄²	#120	1	24	yes	complemented
78	#543	#120	1	48	yes	—
79	#538	#120	1	96	yes	—
80	#536	#120	1	24	yes	—
81	#535	#120	1	24	yes	—
82	#534	#120	1	48	yes	—
83	#533	C₂²×D₄	1	12	yes	—
84	#533	C₂²×D₄	1	24	yes	—
85	#532	C₂²×D₄	1	24	yes	—
86	#532	C₂²×D₄	1	12	yes	—
87	#529	C₂²×D₄	1	48	yes	—
88	C₂×#124	C₂²×D₄	1	24	yes	—
89	#525	C₂²×D₄	1	24	yes	—
90	#519	#120	1	12	yes	—
91	#518	#120	1	12	yes	—
92	C₄×#41	C₂²×D₄	1	24	yes	—
93	C₂²×C₂²⋊C₄	#120	1	6	yes	—
94	C₂²×C₂²⋊C₄	C₂²×D₄	1	12	yes	—
95	C₂²×C₄²	#120	1	2	yes	—
96	C₂²×C₄²	C₂⁵	1	1	yes	characteristic
97	C₂⁵	C₂×#120	1	6	yes	—
98	C₂×#41	C₂×#120	1	24	yes	—
99	C₂²×Q₈	C₂×#120	1	12	yes	—
100	C₂²×D₄	C₂×#120	1	12	yes	—
101	C₂³×C₄	C₂×#120	1	6	yes	—
102	C₂³×C₄	C₂³×D₄	1	3	yes	—
103	C₄⋊Q₈	C₂×#120	1	24	yes	—
104	C₄⋊Q₈	C₂×#120	1	12	yes	—
105	#127	C₂×#120	1	12	yes	—
106	#125	C₂×#120	1	24	yes	—
107	#124	C₂×#120	1	24	yes	—
108	C₂²⋊Q₈	C₂×#120	1	48	yes	—
109	#121	C₂×#120	1	48	yes	—
110	C₄×Q₈	C₂×#120	1	48	yes	—
111	C₄×D₄	C₂×#120	1	48	yes	—
112	#117	C₂×#120	1	24	yes	—
113	C₂×C₂²⋊C₄	C₂×#120	1	48	yes	—
114	C₂×C₄²	C₂³×D₄	1	12	yes	—
115	C₂⁴	#2846	1	12	yes	—
116	C₂⁴	C₂²×#120	1	1	yes	characteristic
117	C₂×Q₈	#2846	1	24	yes	—
118	C₂×D₄	#2846	1	24	yes	—
119	C₂²×C₄	#2846	1	12	yes	—
120	C₂²×C₄	C₂²×#120	1	12	yes	—
121	C₂²×C₄	C₂²×#120	1	6	yes	—
122	C₄⋊C₄	#2846	1	48	yes	—
123	C₄²	C₂²×#120	1	12	yes	—
124	C₄²	C₂²×#120	1	4	yes	—
125	C₂³	C₂×#2846	1	3	yes	—
126	C₂×C₄	C₂×#2846	1	12	yes	—
127	C₂²	#6342371	1	1	yes	characteristic
128	C₂	#72227588	1	3	yes	—

	subgroup	G/H
center, socle	C₂²	#6342371
commutator, Frattini, cosocle	C₂²×C₄²	C₂⁵
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72235014	C₂	semidirect
2. #72235013	C₂	semidirect
3. #72235012	C₂	semidirect
4. #72235011	C₂	semidirect
5. H(5,𝔽₄)	C₂	semidirect
8. #6516748	C₂²	semidirect
9. #6351730	C₂²	semidirect
10. #6349989	C₂²	semidirect
11. #6347371	C₂²	semidirect
12. #5035253	C₂²	semidirect
13. #188167	C₂²	semidirect
14. #187581	C₂²	semidirect
15. #187423	C₂²	semidirect
16. #187276	C₂²	semidirect
17. #187144	C₂²	semidirect
18. #186833	C₂²	semidirect
19. #186167	C₂²	semidirect
20. #185268	C₂²	semidirect
21. #63069	D₄	semidirect
22. #63048	D₄	semidirect
23. #63045	D₄	semidirect
31. #60798	C₂³	semidirect
32. C₂×#2657	C₂³	semidirect
33. #21798	C₂³	semidirect
34. #21417	C₂³	semidirect
35. #17740	C₂³	semidirect
36. #17673	C₂³	semidirect
37. #17541	C₂³	semidirect
38. #17493	C₂³	semidirect
39. #17393	C₂³	semidirect
40. #14169	C₂³	semidirect
56. C₂²×#127	C₂⁴	semidirect
60. #2920	C₂⁴	semidirect
61. C₂×#376	C₂⁴	semidirect
68. #564	#120	semidirect
69. #563	#120	semidirect
70. #562	#120	semidirect
71. H(3,𝔽₄)	#120	semidirect
72. #559	#120	semidirect
73. Q₈²	#120	semidirect
74. #556	#120	semidirect
75. #555	#120	semidirect
76. D₄×Q₈	#120	semidirect
77. D₄²	#120	semidirect

Maximal subgroups

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

subgroup	index
#72235014	2
#72235013	2
#72235012	2
#72235011	2
H(5,𝔽₄)	2