Subgroups — #72262580

Order 4096.

At least 9123 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

The subgroup lattice has 148 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262579	C₂	1	1	yes	maximal, characteristic, complemented
2	#72262573	C₂	1	1	yes	maximal, characteristic, complemented
3	#72262568	C₂	1	1	yes	maximal, characteristic, complemented
4	#72262565	C₂²	1	1	yes	characteristic, complemented
5	#72262533	C₂	1	1	yes	maximal, characteristic, complemented
6	#72262495	C₂	1	1	yes	maximal, characteristic, complemented
7	#72262493	C₂	1	1	yes	maximal, characteristic, complemented
8	#72262478	C₂²	1	1	yes	characteristic, complemented
9	#72262474	C₂	1	1	yes	maximal, characteristic, complemented
10	#72262472	C₂	1	1	yes	maximal, characteristic
11	#72262471	C₂²	1	1	yes	characteristic
12	#72249352	C₂	1	1	yes	maximal, characteristic, complemented
13	#72238453	C₂²	1	1	yes	characteristic, complemented
14	#72238451	C₂	1	1	yes	maximal, characteristic, complemented
15	#72238385	C₂	1	1	yes	maximal, characteristic, complemented
16	#72238381	C₂	1	1	yes	maximal, characteristic, complemented
17	#72238377	C₂²	1	1	yes	characteristic, complemented
18	#72238374	C₂²	1	1	yes	characteristic, complemented
19	#72238368	C₂²	1	1	yes	characteristic, complemented
20	#72237981	C₂	1	1	yes	maximal, characteristic, complemented
21	#72237980	C₂²	1	1	yes	characteristic, complemented
22	#72237818	C₂	1	1	yes	maximal, characteristic, complemented
23	#72237816	C₂²	1	1	yes	characteristic
24	#72237815	C₂²	1	1	yes	characteristic
25	#72237593	C₂²	1	1	yes	characteristic, complemented
26	#72237580	C₂	1	1	yes	maximal, characteristic, complemented
27	#72237579	C₂²	1	1	yes	characteristic, complemented
28	#72237577	C₂²	1	1	yes	characteristic, complemented
29	#72237576	C₂²	1	1	yes	characteristic, complemented
30	#72237488	C₂²	1	1	yes	characteristic, complemented
31	#72237479	C₂²	1	1	yes	characteristic
32	#72237479	C₂²	1	1	yes	characteristic, complemented
33	#72237427	C₂²	1	1	yes	characteristic, complemented
34	#72237427	C₂²	1	1	yes	characteristic, complemented
35	#72237388	C₂²	1	1	yes	characteristic, complemented
36	#72237377	C₂²	1	1	yes	characteristic
37	#72237376	C₂²	1	1	yes	characteristic
38	#72237364	C₂²	1	1	yes	characteristic
39	#72237364	C₂²	1	1	yes	characteristic, complemented
40	#72237036	C₂²	1	1	yes	characteristic
41	#72237028	C₂²	1	1	yes	characteristic, complemented
42	#72235677	C₂²	1	1	yes	characteristic
43	#72235677	C₂²	1	1	yes	characteristic, complemented
44	#72234744	C₂²	1	1	yes	characteristic, complemented
45	#72234710	C₂²	1	1	yes	characteristic, complemented
46	#72234638	C₂²	1	1	yes	characteristic, complemented
47	#72234607	C₂²	1	1	yes	characteristic
48	#72234515	C₂²	1	1	yes	characteristic, complemented
49	#72234108	C₂²	1	1	yes	characteristic
50	#72226029	C₂²	1	1	yes	characteristic, complemented
51	#10586472	D₄	1	1	yes	characteristic
52	#6499871	C₂³	1	1	yes	characteristic
53	#6499870	C₂³	1	1	yes	characteristic
54	#6499870	C₂³	1	1	yes	characteristic
55	#6499870	C₂³	1	1	yes	characteristic
56	#5025772	C₂³	1	1	yes	characteristic
57	#4430405	C₂³	1	1	yes	characteristic
58	#4420586	C₂³	1	1	yes	characteristic
59	#485081	D₄	1	2	yes	complemented
60	#485078	D₄	1	2	yes	complemented
61	#347044	D₄	1	1	yes	characteristic
62	#347042	D₄	1	1	yes	characteristic
63	#346447	D₄	1	1	yes	characteristic
64	#346446	D₄	1	1	yes	characteristic
65	#188035	D₄	1	1	yes	characteristic
66	#187949	D₄	1	1	yes	characteristic
67	#187949	D₄	1	1	yes	characteristic
68	#139719	D₄	1	1	yes	characteristic
69	#139718	D₄	1	1	yes	characteristic
70	#139717	D₄	1	1	yes	characteristic
71	#139716	D₄	1	1	yes	characteristic
72	#139625	D₄	1	2	yes	complemented
73	#139622	D₄	1	2	yes	complemented
74	#134037	C₂³	1	1	yes	characteristic
75	#134036	C₂³	1	1	yes	characteristic
76	#134036	C₂³	1	1	yes	characteristic
77	#134036	C₂³	1	1	yes	characteristic
78	#133826	C₂³	1	1	yes	characteristic
79	#133825	C₂³	1	1	yes	characteristic
80	#115633	C₂³	1	1	yes	characteristic
81	#115631	C₂³	1	1	yes	characteristic
82	#62820	C₂×D₄	1	1	yes	characteristic
83	#62817	C₂×D₄	1	1	yes	characteristic
84	#62817	C₂×D₄	1	1	yes	characteristic
85	#62817	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₄²	C₂×D₄	1	1	yes	characteristic
87	C₂×#2840	C₂×D₄	1	1	yes	characteristic, complemented
88	C₂×#2607	C₂×D₄	1	1	yes	characteristic, complemented
89	C₂×#2613	C₂⁴	1	1	yes	characteristic
90	#23985	C₂×D₄	1	2	yes	—
91	#23984	C₂×D₄	1	2	yes	—
92	#21747	C₂×D₄	1	1	yes	characteristic
93	#21746	C₂×D₄	1	1	yes	characteristic
94	#21746	C₂×D₄	1	1	yes	characteristic
95	#21746	C₂×D₄	1	1	yes	characteristic
96	#15948	C₂×D₄	1	1	yes	characteristic
97	#15947	C₂×D₄	1	1	yes	characteristic
98	#15261	C₂×D₄	1	1	yes	characteristic
99	#15258	C₂×D₄	1	1	yes	characteristic
100	#14657	C₂×D₄	1	1	yes	characteristic
101	#13828	C₂×D₄	1	1	yes	characteristic, complemented
102	C₂²⋊C₄²	C₂×D₄	1	1	yes	characteristic, complemented
103	#11500	C₂×D₄	1	2	yes	complemented
104	#11499	C₂×D₄	1	2	yes	complemented
105	#8548	C₂×D₄	1	1	yes	characteristic
106	#8547	C₂×D₄	1	1	yes	characteristic
107	#8543	C₂×D₄	1	1	yes	characteristic
108	#8541	C₂×D₄	1	1	yes	characteristic
109	C₂⁴×D₄	H(5,𝔽₂)	1	1	yes	characteristic
110	#3490	#120	1	2	yes	—
111	C₂²×#127	#120	1	1	yes	characteristic
112	C₂²×#123	C₂²×D₄	1	1	yes	characteristic
113	C₂²×#120	#120	1	1	yes	characteristic
114	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
115	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
116	C₂×#381	C₂²×D₄	1	1	yes	characteristic
117	C₂×#379	C₂²×D₄	1	1	yes	characteristic
118	C₂×#379	H(5,𝔽₂)	1	1	yes	characteristic
119	#1819	#120	1	2	yes	—
120	#1438	#120	1	1	yes	characteristic
121	#1438	#120	1	1	yes	characteristic
122	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
123	C₂⁶	#534	1	1	yes	characteristic
124	C₂⁶	D₄²	1	1	yes	characteristic
125	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
126	C₂³×D₄	C₂×#120	1	1	yes	characteristic
127	C₂⁴×C₄	D₄²	1	1	yes	characteristic
128	#543	B(4,𝔽₂)	1	2	yes	—
129	#534	B(4,𝔽₂)	1	2	yes	—
130	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	1	yes	characteristic
131	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
132	C₂²×C₂²⋊C₄	D₄²	1	1	yes	characteristic
133	C₂²×C₂²⋊C₄	D₄²	1	1	yes	characteristic
134	C₂²×C₄²	B(4,𝔽₂)	1	1	yes	characteristic
135	C₂⁵	#2403	1	1	yes	characteristic
136	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
137	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
138	C₂×C₂²⋊C₄	C₂×B(4,𝔽₂)	1	2	yes	—
139	C₂⁴	#23895	1	1	yes	characteristic
140	C₂⁴	#23900	1	1	yes	characteristic
141	C₂⁴	#23900	1	1	yes	characteristic
142	C₂³	#245225	1	1	yes	characteristic
143	C₂²	#72209844	1	1	yes	characteristic
144	C₂²	#72251567	1	1	yes	characteristic
145	C₂²	#72251567	1	1	yes	characteristic
146	C₂	#72262420	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72262420
commutator, Frattini, cosocle	C₂×#2613	C₂⁴
prime radical	#72262472	C₂
LCS	G ▷ C₂×#2613 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁵ ◁ C₂×#2613 ◁ G
LECS	G ▷ C₂×#2613 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72262579	C₂	semidirect
2. #72262573	C₂	semidirect
3. #72262568	C₂	semidirect
4. #72262565	C₂²	semidirect
5. #72262533	C₂	semidirect
6. #72262495	C₂	semidirect
7. #72262493	C₂	semidirect
8. #72262478	C₂²	semidirect
9. #72262474	C₂	semidirect
12. #72249352	C₂	semidirect
13. #72238453	C₂²	semidirect
14. #72238451	C₂	semidirect
15. #72238385	C₂	semidirect
16. #72238381	C₂	semidirect
17. #72238377	C₂²	semidirect
18. #72238374	C₂²	semidirect
19. #72238368	C₂²	semidirect
20. #72237981	C₂	semidirect
21. #72237980	C₂²	semidirect
22. #72237818	C₂	semidirect
25. #72237593	C₂²	semidirect
26. #72237580	C₂	semidirect
27. #72237579	C₂²	semidirect
28. #72237577	C₂²	semidirect
29. #72237576	C₂²	semidirect
30. #72237488	C₂²	semidirect
32. #72237479	C₂²	semidirect
33. #72237427	C₂²	semidirect
34. #72237427	C₂²	semidirect
35. #72237388	C₂²	semidirect
39. #72237364	C₂²	semidirect
41. #72237028	C₂²	semidirect
43. #72235677	C₂²	semidirect
44. #72234744	C₂²	semidirect
45. #72234710	C₂²	semidirect
46. #72234638	C₂²	semidirect
48. #72234515	C₂²	semidirect
50. #72226029	C₂²	semidirect
59. #485081	D₄	semidirect
60. #485078	D₄	semidirect
72. #139625	D₄	semidirect
73. #139622	D₄	semidirect
87. C₂×#2840	C₂×D₄	semidirect
88. C₂×#2607	C₂×D₄	semidirect
101. #13828	C₂×D₄	semidirect
102. C₂²⋊C₄²	C₂×D₄	semidirect
103. #11500	C₂×D₄	semidirect
104. #11499	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72262579	2
#72262573	2
#72262568	2
#72262533	2
#72262495	2
#72262493	2
#72262474	2
#72262472	2
#72249352	2
#72238451	2
#72238385	2
#72238381	2
#72237981	2
#72237818	2
#72237580	2