Subgroups — #72262450

Order 4096.

At least 9487 conjugacy classes of subgroups; only the 295 normal subgroups are shown.

The subgroup lattice has 155 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262369	C₂	1	2	yes	maximal, complemented
2	#72262257	C₂	1	1	yes	maximal, characteristic, complemented
3	#72256412	C₂	1	1	yes	maximal, characteristic, complemented
4	#72251646	C₂	1	1	yes	maximal, characteristic, complemented
5	#72251605	C₂²	1	1	yes	characteristic
6	#72251577	C₂²	1	1	yes	characteristic, complemented
7	#72249450	C₂	1	2	yes	maximal, complemented
8	#72249444	C₂²	1	1	yes	characteristic
9	#72238445	C₂	1	2	yes	maximal, complemented
10	#72238444	C₂²	1	1	yes	characteristic
11	#72236918	C₂	1	2	yes	maximal, complemented
12	#72236915	C₂	1	2	yes	maximal, complemented
13	#72236901	C₂²	1	2	yes	complemented
14	#72236898	C₂²	1	2	yes	—
15	#72236875	C₂²	1	2	yes	complemented
16	#72236837	C₂²	1	2	yes	—
17	#72236812	C₂²	1	1	yes	characteristic, complemented
18	C₂×#134045	C₂²	1	1	yes	characteristic, complemented
19	#72235872	C₂	1	2	yes	maximal, complemented
20	#72235865	C₂²	1	2	yes	complemented
21	#72234825	C₂²	1	1	yes	characteristic, complemented
22	#72231822	C₂²	1	2	yes	complemented
23	#72231810	C₂²	1	2	yes	—
24	#72231810	C₂²	1	2	yes	—
25	#72226038	C₂²	1	2	yes	—
26	#72225873	C₂²	1	2	yes	complemented
27	#72225733	C₂²	1	2	yes	—
28	#72219034	C₂²	1	2	yes	—
29	C₂×#245225	C₂²	1	2	yes	—
30	C₂×#245225	C₂²	1	2	yes	complemented
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	2	yes	—
35	C₂×#35346	C₂³	1	1	yes	characteristic
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	#515285	C₂³	1	1	yes	characteristic
40	#382422	C₂³	1	2	yes	—
41	#353314	D₄	1	1	yes	characteristic, complemented
42	C₂×#9495	D₄	1	4	yes	complemented
43	C₂×#8547	C₂³	1	2	yes	—
44	C₂×#8547	C₂³	1	2	yes	—
45	C₂×#8541	C₂³	1	2	yes	—
46	#185255	D₄	1	4	yes	complemented
47	#134025	D₄	1	4	yes	complemented
48	#133970	D₄	1	4	yes	complemented
49	#133291	D₄	1	4	yes	complemented
50	#133290	D₄	1	4	yes	complemented
51	#106326	D₄	1	4	yes	complemented
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	C₂×#2403	C₂×D₄	1	4	yes	—
63	#34176	C₂×D₄	1	2	yes	—
64	C₂×#2308	C₂×D₄	1	1	yes	characteristic
65	C₂×#2308	C₂×D₄	1	1	yes	characteristic
66	C₂×#2308	C₂×D₄	1	2	yes	—
67	C₂×#2308	C₂×D₄	1	1	yes	characteristic
68	C₂×#1781	C₂×D₄	1	4	yes	—
69	C₂×#1462	C₂×D₄	1	4	yes	—
70	#13868	C₂×D₄	1	4	yes	—
71	#10965	C₂×D₄	1	4	yes	—
72	#8547	C₂×D₄	1	4	yes	—
73	#8541	C₂×D₄	1	4	yes	—
74	#8541	C₂×D₄	1	4	yes	—
75	C₂⁷	H(5,𝔽₂)	1	1	yes	characteristic
76	C₂⁴×D₄	#120	1	2	yes	—
77	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
78	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
79	C₂×#543	#127	1	2	yes	complemented
80	C₂×#534	#127	1	2	yes	complemented
81	C₂²×#120	C₂²×D₄	1	2	yes	—
82	C₂²×#120	C₂²×D₄	1	2	yes	—
83	C₂²×#120	H(5,𝔽₂)	1	2	yes	—
84	C₂²×#120	H(5,𝔽₂)	1	2	yes	—
85	C₂²×#117	#120	1	2	yes	—
86	C₂³×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
87	C₂³×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂×#381	#120	1	2	yes	—
89	C₂×#379	#120	1	2	yes	—
90	#2273	#127	1	2	yes	complemented
91	#2273	#127	1	2	yes	complemented
92	#1781	#120	1	8	yes	—
93	C₂⁶	B(4,𝔽₂)	1	2	yes	—
94	C₂⁶	C₂×#120	1	1	yes	characteristic
95	C₂⁶	#534	1	1	yes	characteristic
96	C₂⁶	#535	1	2	yes	—
97	C₂⁶	D₄²	1	2	yes	—
98	C₂⁶	D₄²	1	1	yes	characteristic
99	C₂⁶	#560	1	1	yes	characteristic
100	C₂³×D₄	C₂×#120	1	2	yes	—
101	C₂³×D₄	#534	1	2	yes	—
102	C₂³×D₄	D₄²	1	2	yes	—
103	C₂³×D₄	D₄²	1	2	yes	—
104	C₂³×D₄	D₄²	1	2	yes	—
105	C₂³×D₄	D₄²	1	2	yes	—
106	C₂⁴×C₄	B(4,𝔽₂)	1	2	yes	—
107	C₂⁴×C₄	C₂×#120	1	1	yes	characteristic
108	C₂⁴×C₄	D₄²	1	2	yes	—
109	C₂²×C₂²⋊C₄	C₂×#127	1	2	yes	—
110	C₂²×C₂²⋊C₄	#535	1	2	yes	—
111	C₂²×C₂²⋊C₄	#535	1	2	yes	—
112	C₂²×C₂²⋊C₄	D₄²	1	2	yes	—
113	C₂²×C₂²⋊C₄	D₄²	1	2	yes	—
114	C₂²×C₂²⋊C₄	D₄²	1	1	yes	characteristic
115	C₂²×C₂²⋊C₄	#560	1	1	yes	characteristic
116	C₂²×C₄²	B(4,𝔽₂)	1	2	yes	—
117	C₂²×C₄²	B(4,𝔽₂)	1	2	yes	—
118	#379	B(4,𝔽₂)	1	2	yes	—
119	#379	B(4,𝔽₂)	1	2	yes	—
120	C₂⁵	#2403	1	2	yes	—
121	C₂⁵	#2403	1	1	yes	characteristic
122	C₂⁵	#2433	1	2	yes	—
123	C₂⁵	#2669	1	1	yes	characteristic
124	C₂⁵	#2669	1	1	yes	characteristic
125	C₂⁵	#2669	1	2	yes	—
126	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
127	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
128	C₂³×C₄	#2433	1	2	yes	—
129	C₂³×C₄	#2433	1	2	yes	—
130	C₂³×C₄	#2669	1	2	yes	—
131	C₂³×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
132	C₂³×C₄	#3027	1	2	yes	—
133	C₂⁴	#15258	1	1	yes	characteristic
134	C₂⁴	#23895	1	1	yes	characteristic
135	C₂⁴	#23895	1	1	yes	characteristic
136	C₂⁴	#23900	1	1	yes	characteristic
137	C₂⁴	#23900	1	1	yes	characteristic
138	C₂⁴	#23900	1	4	yes	—
139	C₂⁴	#23900	1	1	yes	characteristic
140	C₂⁴	#23904	1	1	yes	characteristic
141	C₂²×C₄	#25771	1	4	yes	—
142	C₂³	#245225	1	1	yes	characteristic
143	C₂³	#245233	1	1	yes	characteristic
144	C₂³	#245249	1	1	yes	characteristic
145	C₂³	#245252	1	1	yes	characteristic
146	C₂³	#417744	1	1	yes	characteristic
147	C₂³	#417821	1	1	yes	characteristic
148	C₂³	#417824	1	1	yes	characteristic
149	C₂²	#72209844	1	1	yes	characteristic
150	C₂²	#72237376	1	1	yes	characteristic
151	C₂²	#72237501	1	1	yes	characteristic
152	C₂	#72251145	1	1	yes	characteristic
153	C₂	#72251610	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#72237501
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. #72262369	C₂	semidirect
2. #72262257	C₂	semidirect
3. #72256412	C₂	semidirect
4. #72251646	C₂	semidirect
6. #72251577	C₂²	semidirect
7. #72249450	C₂	semidirect
9. #72238445	C₂	semidirect
11. #72236918	C₂	semidirect
12. #72236915	C₂	semidirect
13. #72236901	C₂²	semidirect
15. #72236875	C₂²	semidirect
17. #72236812	C₂²	semidirect
18. C₂×#134045	C₂²	semidirect
19. #72235872	C₂	semidirect
20. #72235865	C₂²	semidirect
21. #72234825	C₂²	semidirect
22. #72231822	C₂²	semidirect
26. #72225873	C₂²	semidirect
30. C₂×#245225	C₂²	semidirect
31. C₂×#62842	D₄	semidirect
36. C₂×#32898	D₄	semidirect
37. C₂×#20585	D₄	semidirect
38. C₂×#20585	D₄	semidirect
41. #353314	D₄	semidirect
42. C₂×#9495	D₄	semidirect
46. #185255	D₄	semidirect
47. #134025	D₄	semidirect
48. #133970	D₄	semidirect
49. #133291	D₄	semidirect
50. #133290	D₄	semidirect
51. #106326	D₄	semidirect
79. C₂×#543	#127	semidirect
80. C₂×#534	#127	semidirect
90. #2273	#127	semidirect
91. #2273	#127	semidirect

Maximal subgroups

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

subgroup	index
#72262369	2
#72262257	2
#72256412	2
#72251646	2
#72249450	2
#72238445	2
#72236918	2
#72236915	2
#72235872	2