Subgroups — #72262443

Order 4096.

At least 8760 conjugacy classes of subgroups; only the 299 normal subgroups are shown.

The subgroup lattice has 157 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262371	C₂	1	2	yes	maximal, complemented
2	#72262256	C₂	1	1	yes	maximal, characteristic, complemented
3	#72256412	C₂	1	1	yes	maximal, characteristic, complemented
4	#72251612	C₂	1	1	yes	maximal, characteristic, complemented
5	#72251607	C₂²	1	1	yes	characteristic, complemented
6	#72251605	C₂²	1	1	yes	characteristic
7	#72249450	C₂	1	2	yes	maximal, complemented
8	#72249444	C₂²	1	1	yes	characteristic, complemented
9	#72238445	C₂	1	2	yes	maximal, complemented
10	#72238444	C₂²	1	1	yes	characteristic, complemented
11	#72236923	C₂	1	2	yes	maximal, complemented
12	#72236921	C₂	1	2	yes	maximal, complemented
13	#72236901	C₂²	1	2	yes	complemented
14	#72236898	C₂²	1	2	yes	complemented
15	#72236875	C₂²	1	2	yes	complemented
16	#72236837	C₂²	1	2	yes	complemented
17	#72236816	C₂²	1	1	yes	characteristic, complemented
18	#72236814	C₂²	1	1	yes	characteristic, complemented
19	#72236754	C₂²	1	2	yes	complemented
20	#72231951	C₂	1	2	yes	maximal, complemented
21	#72231822	C₂²	1	2	yes	complemented
22	#72231822	C₂²	1	2	yes	complemented
23	C₂×#8932885	C₂²	1	1	yes	characteristic, complemented
24	#72231810	C₂²	1	2	yes	—
25	#72231810	C₂²	1	2	yes	—
26	#72226038	C₂²	1	2	yes	complemented
27	#72225873	C₂²	1	2	yes	complemented
28	#72225733	C₂²	1	2	yes	—
29	#72219034	C₂²	1	2	yes	complemented
30	C₂×#245225	C₂²	1	2	yes	—
31	C₂×#62842	D₄	1	1	yes	characteristic, complemented
32	#7642273	C₂³	1	2	yes	—
33	C₂×#35346	C₂³	1	1	yes	characteristic
34	C₂×#35346	C₂³	1	1	yes	characteristic
35	C₂×#35346	C₂³	1	2	yes	—
36	C₂×#32898	D₄	1	4	yes	complemented
37	C₂×#20585	D₄	1	1	yes	characteristic, complemented
38	C₂×#20585	D₄	1	1	yes	characteristic, complemented
39	#6346594	D₄	1	4	yes	complemented
40	#515285	C₂³	1	1	yes	characteristic
41	#417896	D₄	1	4	yes	complemented
42	#417824	D₄	1	4	yes	complemented
43	#382422	C₂³	1	2	yes	—
44	#353314	D₄	1	1	yes	characteristic, complemented
45	#249331	D₄	1	4	yes	complemented
46	#249319	D₄	1	4	yes	complemented
47	C₂×#9495	D₄	1	4	yes	complemented
48	#201462	D₄	1	4	yes	complemented
49	C₂×#8547	C₂³	1	2	yes	—
50	C₂×#8547	C₂³	1	2	yes	—
51	C₂×#8541	C₂³	1	2	yes	—
52	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
53	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
54	C₂³×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
55	C₂×#3490	C₂×D₄	1	2	yes	—
56	C₂³×#120	C₂⁴	1	1	yes	characteristic
57	C₂²×#518	C₂×D₄	1	1	yes	characteristic
58	#35390	C₂×D₄	1	2	yes	—
59	#35390	C₂×D₄	1	2	yes	—
60	#35367	C₂×D₄	1	2	yes	—
61	#35367	C₂×D₄	1	2	yes	—
62	#35346	C₂×D₄	1	4	yes	—
63	#35346	C₂×D₄	1	4	yes	—
64	C₂×#2404	C₂×D₄	1	4	yes	—
65	C₂×#2403	C₂×D₄	1	4	yes	—
66	#34176	C₂×D₄	1	2	yes	—
67	C₂×#2308	C₂×D₄	1	1	yes	characteristic
68	C₂×#2308	C₂×D₄	1	1	yes	characteristic
69	C₂×#2308	C₂×D₄	1	2	yes	—
70	C₂×#2308	C₂×D₄	1	1	yes	characteristic
71	C₂×#1781	C₂×D₄	1	4	yes	—
72	C₂×#1462	C₂×D₄	1	4	yes	—
73	#10965	C₂×D₄	1	4	yes	—
74	#8547	C₂×D₄	1	4	yes	—
75	C₂⁷	H(5,𝔽₂)	1	1	yes	characteristic
76	C₂²×H(5,𝔽₂)	#120	1	4	yes	—
77	C₂³×#41	#120	1	2	yes	—
78	C₂⁴×D₄	#120	1	2	yes	—
79	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
80	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
81	C₂×#543	#120	1	2	yes	complemented
82	C₂×#534	#120	1	2	yes	complemented
83	C₂²×#120	C₂²×D₄	1	2	yes	—
84	C₂²×#120	C₂²×D₄	1	2	yes	—
85	C₂²×#120	H(5,𝔽₂)	1	2	yes	—
86	C₂²×#120	H(5,𝔽₂)	1	2	yes	—
87	C₂³×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
88	C₂³×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
89	#2308	#120	1	4	yes	—
90	C₂×#381	#120	1	2	yes	—
91	C₂×#379	#120	1	2	yes	—
92	#2273	#120	1	2	yes	complemented
93	#2273	#120	1	2	yes	complemented
94	C₂⁶	B(4,𝔽₂)	1	2	yes	—
95	C₂⁶	C₂×#120	1	1	yes	characteristic
96	C₂⁶	#534	1	1	yes	characteristic
97	C₂⁶	#534	1	2	yes	—
98	C₂⁶	D₄²	1	2	yes	—
99	C₂⁶	D₄²	1	1	yes	characteristic
100	C₂⁶	H(3,𝔽₄)	1	1	yes	characteristic
101	C₂³×Q₈	B(4,𝔽₂)	1	2	yes	—
102	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
103	C₂³×D₄	C₂×#120	1	2	yes	—
104	C₂³×D₄	#534	1	2	yes	—
105	C₂³×D₄	D₄²	1	2	yes	—
106	C₂³×D₄	D₄²	1	2	yes	—
107	C₂³×D₄	D₄²	1	2	yes	—
108	C₂³×D₄	D₄²	1	2	yes	—
109	C₂⁴×C₄	B(4,𝔽₂)	1	2	yes	—
110	C₂⁴×C₄	C₂×#120	1	1	yes	characteristic
111	C₂⁴×C₄	D₄²	1	2	yes	—
112	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
113	C₂²×C₂²⋊C₄	#534	1	2	yes	—
114	C₂²×C₂²⋊C₄	#534	1	2	yes	—
115	C₂²×C₂²⋊C₄	D₄²	1	2	yes	—
116	C₂²×C₂²⋊C₄	D₄²	1	2	yes	—
117	C₂²×C₂²⋊C₄	D₄²	1	1	yes	characteristic
118	C₂²×C₂²⋊C₄	H(3,𝔽₄)	1	1	yes	characteristic
119	#381	B(4,𝔽₂)	1	4	yes	—
120	#379	B(4,𝔽₂)	1	2	yes	—
121	#379	B(4,𝔽₂)	1	2	yes	—
122	C₂⁵	#2403	1	2	yes	—
123	C₂⁵	#2403	1	2	yes	—
124	C₂⁵	#2403	1	1	yes	characteristic
125	C₂⁵	#2679	1	1	yes	characteristic
126	C₂⁵	#2679	1	1	yes	characteristic
127	C₂⁵	#2679	1	2	yes	—
128	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
129	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
130	C₂³×C₄	#2403	1	2	yes	—
131	C₂³×C₄	#2403	1	2	yes	—
132	C₂³×C₄	#2679	1	2	yes	—
133	C₂³×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
134	C₂³×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
135	C₂⁴	#15947	1	1	yes	characteristic
136	C₂⁴	#23895	1	1	yes	characteristic
137	C₂⁴	#23896	1	1	yes	characteristic
138	C₂⁴	#23900	1	1	yes	characteristic
139	C₂⁴	#23900	1	1	yes	characteristic
140	C₂⁴	#23900	1	4	yes	—
141	C₂⁴	#23900	1	1	yes	characteristic
142	C₂⁴	#23900	1	1	yes	characteristic
143	C₂²×C₄	#23900	1	4	yes	—
144	C₂³	#245225	1	1	yes	characteristic
145	C₂³	#245229	1	1	yes	characteristic
146	C₂³	#245229	1	1	yes	characteristic
147	C₂³	#245233	1	1	yes	characteristic
148	C₂³	#449469	1	1	yes	characteristic
149	C₂³	#449475	1	1	yes	characteristic
150	C₂³	#449475	1	1	yes	characteristic
151	C₂²	#72209844	1	1	yes	characteristic
152	C₂²	#72219165	1	1	yes	characteristic
153	C₂²	#72237377	1	1	yes	characteristic
154	C₂	#72251115	1	1	yes	characteristic
155	C₂	#72251583	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#72219165
commutator, Frattini, cosocle	C₂³×#120	C₂⁴
LCS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72262371	C₂	semidirect
2. #72262256	C₂	semidirect
3. #72256412	C₂	semidirect
4. #72251612	C₂	semidirect
5. #72251607	C₂²	semidirect
7. #72249450	C₂	semidirect
8. #72249444	C₂²	semidirect
9. #72238445	C₂	semidirect
10. #72238444	C₂²	semidirect
11. #72236923	C₂	semidirect
12. #72236921	C₂	semidirect
13. #72236901	C₂²	semidirect
14. #72236898	C₂²	semidirect
15. #72236875	C₂²	semidirect
16. #72236837	C₂²	semidirect
17. #72236816	C₂²	semidirect
18. #72236814	C₂²	semidirect
19. #72236754	C₂²	semidirect
20. #72231951	C₂	semidirect
21. #72231822	C₂²	semidirect
22. #72231822	C₂²	semidirect
23. C₂×#8932885	C₂²	semidirect
26. #72226038	C₂²	semidirect
27. #72225873	C₂²	semidirect
29. #72219034	C₂²	semidirect
31. C₂×#62842	D₄	semidirect
36. C₂×#32898	D₄	semidirect
37. C₂×#20585	D₄	semidirect
38. C₂×#20585	D₄	semidirect
39. #6346594	D₄	semidirect
41. #417896	D₄	semidirect
42. #417824	D₄	semidirect
44. #353314	D₄	semidirect
45. #249331	D₄	semidirect
46. #249319	D₄	semidirect
47. C₂×#9495	D₄	semidirect
48. #201462	D₄	semidirect
81. C₂×#543	#120	semidirect
82. C₂×#534	#120	semidirect
92. #2273	#120	semidirect
93. #2273	#120	semidirect

Maximal subgroups

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

subgroup	index
#72262371	2
#72262256	2
#72256412	2
#72251612	2
#72249450	2
#72238445	2
#72236923	2
#72236921	2
#72231951	2