Subgroups — #72324321

Order 4096.

At least 11782 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

The subgroup lattice has 153 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72324320	C₂	1	1	yes	maximal, characteristic, complemented
2	#72324319	C₂	1	1	yes	maximal, characteristic, complemented
3	#72323892	C₂	1	1	yes	maximal, characteristic, complemented
4	#72323800	C₂	1	1	yes	maximal, characteristic, complemented
5	#72323239	C₂	1	1	yes	maximal, characteristic, complemented
6	#72323218	C₂²	1	1	yes	characteristic, complemented
7	#72317192	C₂	1	1	yes	maximal, characteristic, complemented
8	#72316565	C₂	1	1	yes	maximal, characteristic, complemented
9	#72314614	C₂²	1	1	yes	characteristic, complemented
10	#72311054	C₂	1	1	yes	maximal, characteristic, complemented
11	#72310469	C₂²	1	1	yes	characteristic, complemented
12	#72309892	C₂	1	1	yes	maximal, characteristic, complemented
13	#72309784	C₂²	1	1	yes	characteristic, complemented
14	#72309636	C₂²	1	1	yes	characteristic, complemented
15	#72263078	C₂	1	1	yes	maximal, characteristic, complemented
16	#72263077	C₂²	1	1	yes	characteristic, complemented
17	#72262474	C₂	1	1	yes	maximal, characteristic, complemented
18	#72262472	C₂	1	1	yes	maximal, characteristic, complemented
19	#72262471	C₂²	1	1	yes	characteristic, complemented
20	#72237817	C₂	1	1	yes	maximal, characteristic, complemented
21	#72237816	C₂²	1	1	yes	characteristic, complemented
22	#72237816	C₂²	1	1	yes	characteristic, complemented
23	#72237815	C₂²	1	1	yes	characteristic, complemented
24	#72237593	C₂²	1	1	yes	characteristic, complemented
25	#72237578	C₂	1	1	yes	maximal, characteristic, complemented
26	#72237577	C₂²	1	1	yes	characteristic, complemented
27	#72237576	C₂²	1	1	yes	characteristic, complemented
28	#72237576	C₂²	1	1	yes	characteristic, complemented
29	#72237479	C₂²	1	1	yes	characteristic, complemented
30	#72237427	C₂²	1	1	yes	characteristic, complemented
31	#72237377	C₂²	1	1	yes	characteristic, complemented
32	#72237376	C₂²	1	1	yes	characteristic, complemented
33	#72237036	C₂²	1	1	yes	characteristic
34	#72235147	C₂²	1	1	yes	characteristic, complemented
35	#72234642	C₂	1	1	yes	maximal, characteristic, complemented
36	#72234641	C₂²	1	1	yes	characteristic, complemented
37	#72234640	C₂²	1	1	yes	characteristic, complemented
38	#72234637	C₂²	1	1	yes	characteristic
39	#72234637	C₂²	1	1	yes	characteristic, complemented
40	#72234607	C₂²	1	1	yes	characteristic
41	#72234515	C₂²	1	1	yes	characteristic, complemented
42	#72234509	C₂²	1	1	yes	characteristic, complemented
43	#72234108	C₂²	1	1	yes	characteristic
44	#72233792	C₂²	1	1	yes	characteristic, complemented
45	#72233247	C₂²	1	1	yes	characteristic, complemented
46	#72233061	C₂²	1	1	yes	characteristic
47	#72233061	C₂²	1	1	yes	characteristic, complemented
48	#72233061	C₂²	1	1	yes	characteristic, complemented
49	#72226029	C₂²	1	1	yes	characteristic, complemented
50	#72226015	C₂²	1	1	yes	characteristic, complemented
51	#10586472	D₄	1	1	yes	characteristic, complemented
52	#6628892	D₄	1	2	yes	complemented
53	#6499871	C₂³	1	1	yes	characteristic
54	#6499870	C₂³	1	1	yes	characteristic
55	#6499870	C₂³	1	1	yes	characteristic
56	#6499870	C₂³	1	1	yes	characteristic
57	#5024691	C₂³	1	1	yes	characteristic
58	#4930494	C₂³	1	1	yes	characteristic
59	#4420586	C₂³	1	1	yes	characteristic
60	#347044	D₄	1	1	yes	characteristic, complemented
61	#347042	D₄	1	1	yes	characteristic, complemented
62	#346447	D₄	1	1	yes	characteristic, complemented
63	#346446	D₄	1	1	yes	characteristic, complemented
64	#233541	D₄	1	2	yes	complemented
65	#188035	D₄	1	1	yes	characteristic, complemented
66	#187949	D₄	1	1	yes	characteristic, complemented
67	#187949	D₄	1	1	yes	characteristic, complemented
68	#139719	D₄	1	1	yes	characteristic, complemented
69	#139718	D₄	1	1	yes	characteristic, complemented
70	#139717	D₄	1	1	yes	characteristic, complemented
71	#139716	D₄	1	1	yes	characteristic, complemented
72	#134037	C₂³	1	1	yes	characteristic
73	#134036	C₂³	1	1	yes	characteristic
74	#133826	C₂³	1	1	yes	characteristic
75	#133825	C₂³	1	1	yes	characteristic
76	#115633	C₂³	1	1	yes	characteristic
77	#115631	C₂³	1	1	yes	characteristic
78	#115631	C₂³	1	1	yes	characteristic
79	#115631	C₂³	1	1	yes	characteristic
80	#115348	D₄	1	2	yes	complemented
81	#110180	D₄	1	2	yes	complemented
82	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
83	#62820	C₂×D₄	1	1	yes	characteristic, complemented
84	#62817	C₂×D₄	1	1	yes	characteristic
85	#62817	C₂×D₄	1	1	yes	characteristic
86	#62817	C₂×D₄	1	1	yes	characteristic, complemented
87	C₂²×D₄²	C₂×D₄	1	1	yes	characteristic
88	#34980	C₂×D₄	1	2	yes	—
89	C₂×#2403	C₂×D₄	1	1	yes	characteristic, complemented
90	C₂×#2613	C₂⁴	1	1	yes	characteristic
91	#21747	C₂×D₄	1	1	yes	characteristic, complemented
92	#21746	C₂×D₄	1	2	yes	—
93	#21746	C₂×D₄	1	1	yes	characteristic
94	#21746	C₂×D₄	1	1	yes	characteristic
95	#21746	C₂×D₄	1	1	yes	characteristic, complemented
96	C₂×#1781	C₂×D₄	1	2	yes	complemented
97	#15948	C₂×D₄	1	1	yes	characteristic
98	#15947	C₂×D₄	1	1	yes	characteristic
99	#15261	C₂×D₄	1	1	yes	characteristic
100	#15258	C₂×D₄	1	1	yes	characteristic
101	#14657	C₂×D₄	1	1	yes	characteristic
102	C₂×#1438	C₂×D₄	1	1	yes	characteristic, complemented
103	#13827	C₂×D₄	1	1	yes	characteristic, complemented
104	#8548	C₂×D₄	1	1	yes	characteristic, complemented
105	#8547	C₂×D₄	1	1	yes	characteristic, complemented
106	#8543	C₂×D₄	1	1	yes	characteristic, complemented
107	#8541	C₂×D₄	1	1	yes	characteristic, complemented
108	#8101	C₂×D₄	1	2	yes	complemented
109	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
110	C₂²×#127	#120	1	1	yes	characteristic
111	C₂²×#123	C₂²×D₄	1	1	yes	characteristic
112	C₂²×#120	#120	1	2	yes	—
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	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×#379	#120	1	2	yes	—
118	C₂×#379	C₂²×D₄	1	1	yes	characteristic
119	C₂×#379	H(5,𝔽₂)	1	1	yes	characteristic
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	C₂×#120	1	1	yes	characteristic
124	C₂⁶	D₄²	1	1	yes	characteristic
125	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
126	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
127	C₂³×D₄	C₂×#120	1	1	yes	characteristic
128	C₂⁴×C₄	D₄²	1	1	yes	characteristic
129	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
130	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	1	yes	characteristic
131	C₂²×C₂²⋊C₄	#534	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	#2196	1	2	yes	complemented
136	C₂⁵	#2403	1	1	yes	characteristic
137	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
138	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
139	C₂³×C₄	#2196	1	2	yes	complemented
140	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
141	C₂⁴	#23895	1	1	yes	characteristic
142	C₂⁴	#23900	1	1	yes	characteristic
143	C₂⁴	#23900	1	1	yes	characteristic
144	C₂⁴	C₂×#2196	1	2	yes	—
145	C₂⁴	#33551	1	2	yes	—
146	C₂³	#245225	1	1	yes	characteristic
147	C₂³	#499787	1	2	yes	—
148	C₂²	B(5,𝔽₂)	1	1	yes	characteristic
149	C₂²	B(5,𝔽₂)	1	1	yes	characteristic
150	C₂²	#72210587	1	1	yes	characteristic
151	C₂	#72216737	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72216737
commutator, Frattini, cosocle	C₂×#2613	C₂⁴
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. #72324320	C₂	semidirect
2. #72324319	C₂	semidirect
3. #72323892	C₂	semidirect
4. #72323800	C₂	semidirect
5. #72323239	C₂	semidirect
6. #72323218	C₂²	semidirect
7. #72317192	C₂	semidirect
8. #72316565	C₂	semidirect
9. #72314614	C₂²	semidirect
10. #72311054	C₂	semidirect
11. #72310469	C₂²	semidirect
12. #72309892	C₂	semidirect
13. #72309784	C₂²	semidirect
14. #72309636	C₂²	semidirect
15. #72263078	C₂	semidirect
16. #72263077	C₂²	semidirect
17. #72262474	C₂	semidirect
18. #72262472	C₂	semidirect
19. #72262471	C₂²	semidirect
20. #72237817	C₂	semidirect
21. #72237816	C₂²	semidirect
22. #72237816	C₂²	semidirect
23. #72237815	C₂²	semidirect
24. #72237593	C₂²	semidirect
25. #72237578	C₂	semidirect
26. #72237577	C₂²	semidirect
27. #72237576	C₂²	semidirect
28. #72237576	C₂²	semidirect
29. #72237479	C₂²	semidirect
30. #72237427	C₂²	semidirect
31. #72237377	C₂²	semidirect
32. #72237376	C₂²	semidirect
34. #72235147	C₂²	semidirect
35. #72234642	C₂	semidirect
36. #72234641	C₂²	semidirect
37. #72234640	C₂²	semidirect
39. #72234637	C₂²	semidirect
41. #72234515	C₂²	semidirect
42. #72234509	C₂²	semidirect
44. #72233792	C₂²	semidirect
45. #72233247	C₂²	semidirect
47. #72233061	C₂²	semidirect
48. #72233061	C₂²	semidirect
49. #72226029	C₂²	semidirect
50. #72226015	C₂²	semidirect
51. #10586472	D₄	semidirect
52. #6628892	D₄	semidirect
60. #347044	D₄	semidirect
61. #347042	D₄	semidirect
62. #346447	D₄	semidirect
63. #346446	D₄	semidirect
64. #233541	D₄	semidirect
65. #188035	D₄	semidirect
66. #187949	D₄	semidirect
67. #187949	D₄	semidirect
68. #139719	D₄	semidirect
69. #139718	D₄	semidirect
70. #139717	D₄	semidirect
71. #139716	D₄	semidirect
80. #115348	D₄	semidirect
81. #110180	D₄	semidirect
82. C₂³×H(5,𝔽₂)	C₂×D₄	semidirect
83. #62820	C₂×D₄	semidirect
86. #62817	C₂×D₄	semidirect
89. C₂×#2403	C₂×D₄	semidirect
91. #21747	C₂×D₄	semidirect
95. #21746	C₂×D₄	semidirect
96. C₂×#1781	C₂×D₄	semidirect
102. C₂×#1438	C₂×D₄	semidirect
103. #13827	C₂×D₄	semidirect
104. #8548	C₂×D₄	semidirect
105. #8547	C₂×D₄	semidirect
106. #8543	C₂×D₄	semidirect
107. #8541	C₂×D₄	semidirect
108. #8101	C₂×D₄	semidirect
135. C₂⁵	#2196	semidirect
139. C₂³×C₄	#2196	semidirect

Maximal subgroups

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

subgroup	index
#72324320	2
#72324319	2
#72323892	2
#72323800	2
#72323239	2
#72317192	2
#72316565	2
#72311054	2
#72309892	2
#72263078	2
#72262474	2
#72262472	2
#72237817	2
#72237578	2
#72234642	2