Subgroups — #72337733

Order 4096.

At least 6041 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

The subgroup lattice has 137 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72337581	C₂	1	1	yes	maximal, characteristic, complemented
2	#72337580	C₂	1	2	yes	maximal, complemented
3	#72337417	C₂	1	1	yes	maximal, characteristic, complemented
4	#72334268	C₂	1	1	yes	maximal, characteristic, complemented
5	#72334267	C₂	1	2	yes	maximal, complemented
6	#72327248	C₂²	1	2	yes	—
7	#72327231	C₂²	1	1	yes	characteristic
8	#72325871	C₂	1	1	yes	maximal, characteristic, complemented
9	#72310520	C₂	1	1	yes	maximal, characteristic, complemented
10	#72310519	C₂	1	2	yes	maximal, complemented
11	#72310516	C₂²	1	1	yes	characteristic
12	#72310515	C₂²	1	2	yes	—
13	#72309390	C₂²	1	1	yes	characteristic
14	#72309389	C₂²	1	2	yes	—
15	#72265989	C₂	1	1	yes	maximal, characteristic, complemented
16	#72265913	C₂²	1	2	yes	—
17	#72265913	C₂²	1	1	yes	characteristic
18	#72237741	C₂	1	1	yes	maximal, characteristic, complemented
19	#72237740	C₂²	1	1	yes	characteristic
20	#72237735	C₂	1	2	yes	maximal, complemented
21	#72237734	C₂²	1	2	yes	—
22	#72237734	C₂²	1	2	yes	—
23	#72237733	C₂²	1	2	yes	—
24	#72237732	C₂²	1	2	yes	—
25	#72237726	C₂²	1	1	yes	characteristic
26	#72237725	C₂²	1	2	yes	—
27	#72237724	C₂²	1	1	yes	characteristic
28	#72234627	C₂²	1	2	yes	—
29	#72234627	C₂²	1	1	yes	characteristic
30	#72227772	C₂²	1	2	yes	—
31	#72226351	C₂²	1	1	yes	characteristic
32	#72226351	C₂²	1	2	yes	—
33	#72216464	C₂²	1	1	yes	characteristic
34	#72216432	C₂²	1	1	yes	characteristic
35	#8932885	C₂³	1	1	yes	characteristic
36	#8093108	D₄	1	2	yes	complemented
37	#8093107	D₄	1	1	yes	characteristic, complemented
38	#7513948	D₄	1	2	yes	complemented
39	#6631461	D₄	1	1	yes	characteristic, complemented
40	#6437629	D₄	1	1	yes	characteristic, complemented
41	#6437629	D₄	1	2	yes	complemented
42	#6432953	C₂³	1	1	yes	characteristic
43	#6432953	C₂³	1	2	yes	—
44	#449521	D₄	1	2	yes	complemented
45	#449475	D₄	1	1	yes	characteristic, complemented
46	#449475	C₂³	1	1	yes	characteristic
47	#449090	D₄	1	2	yes	complemented
48	#432839	D₄	1	2	yes	complemented
49	#418911	D₄	1	2	yes	complemented
50	#418213	D₄	1	2	yes	complemented
51	#417896	C₂³	1	2	yes	—
52	#417824	D₄	1	1	yes	characteristic, complemented
53	#417824	C₂³	1	1	yes	characteristic
54	#388001	D₄	1	2	yes	complemented
55	#375666	D₄	1	1	yes	characteristic, complemented
56	#374004	D₄	1	2	yes	complemented
57	#357285	D₄	1	1	yes	characteristic, complemented
58	#356055	D₄	1	1	yes	characteristic, complemented
59	#213829	C₂³	1	2	yes	—
60	#213829	C₂³	1	1	yes	characteristic
61	#134045	C₂³	1	2	yes	—
62	#134045	C₂³	1	1	yes	characteristic
63	#97550	C₂³	1	1	yes	characteristic
64	#62842	C₂×D₄	1	1	yes	characteristic
65	#62842	C₂×D₄	1	2	yes	—
66	C₂×#2846	C₂×D₄	1	1	yes	characteristic
67	C₂×#2846	C₂×D₄	1	2	yes	—
68	#35390	C₂×D₄	1	1	yes	characteristic
69	#35367	C₂×D₄	1	2	yes	—
70	#35350	C₂×D₄	1	1	yes	characteristic
71	#35350	C₂×D₄	1	2	yes	—
72	#35346	C₂×D₄	1	2	yes	—
73	#35346	C₂×D₄	1	1	yes	characteristic
74	#35346	C₂⁴	1	1	yes	characteristic
75	#23468	C₂×D₄	1	2	yes	—
76	#23468	C₂×D₄	1	2	yes	—
77	#23444	C₂×D₄	1	2	yes	—
78	#20689	C₂×D₄	1	2	yes	—
79	#20689	C₂×D₄	1	1	yes	characteristic
80	#20585	C₂×D₄	1	1	yes	characteristic
81	#20585	C₂×D₄	1	2	yes	—
82	#20585	C₂×D₄	1	1	yes	characteristic
83	#20585	C₂×D₄	1	2	yes	—
84	#10965	C₂×D₄	1	1	yes	characteristic
85	#10965	C₂×D₄	1	2	yes	—
86	#10965	C₂×D₄	1	1	yes	characteristic
87	#10965	C₂×D₄	1	2	yes	—
88	#10938	C₂×D₄	1	2	yes	—
89	#10938	C₂×D₄	1	2	yes	—
90	#10938	C₂×D₄	1	2	yes	—
91	C₂⁷	#120	1	1	yes	characteristic
92	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
93	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
94	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
95	C₂×#543	#120	1	2	yes	—
96	C₂×#534	#120	1	1	yes	characteristic
97	C₂²×#120	C₂²×D₄	1	2	yes	—
98	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
99	C₂×#518	#120	1	1	yes	characteristic
100	C₂×#518	#120	1	2	yes	—
101	#2308	#120	1	1	yes	characteristic
102	#2308	#120	1	2	yes	—
103	#2308	#120	1	2	yes	—
104	#2308	#120	1	1	yes	characteristic
105	#2308	C₂²×D₄	1	2	yes	—
106	#2308	C₂²×D₄	1	1	yes	characteristic
107	C₂×#379	#120	1	1	yes	characteristic
108	C₂×#379	#120	1	2	yes	—
109	#2273	#120	1	2	yes	—
110	#2273	#120	1	1	yes	characteristic
111	C₂×#342	#120	1	2	yes	—
112	C₂×#342	#120	1	2	yes	—
113	C₂×#342	#120	1	2	yes	—
114	C₂⁶	C₂×#120	1	1	yes	characteristic
115	C₂³×D₄	C₂×#120	1	1	yes	characteristic
116	C₂³×D₄	C₂×#120	1	2	yes	—
117	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
118	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
119	#379	B(4,𝔽₂)	1	4	yes	—
120	#379	B(4,𝔽₂)	1	2	yes	—
121	C₂⁵	#2846	1	1	yes	characteristic
122	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
123	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
124	C₂³×C₄	#3026	1	2	yes	—
125	C₂³×C₄	#3026	1	1	yes	characteristic
126	C₂⁴	#32898	1	2	yes	—
127	C₂⁴	#32898	1	1	yes	characteristic
128	C₂³	#347047	1	1	yes	characteristic
129	C₂³	#511790	1	1	yes	characteristic
130	C₂³	#511790	1	2	yes	—
131	C₂³	#511809	1	2	yes	—
132	C₂³	#511812	1	1	yes	characteristic
133	C₂²	#72249397	1	1	yes	characteristic
134	C₂²	#72265781	1	2	yes	—
135	C₂	#72266703	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266703
commutator, Frattini, cosocle	#35346	C₂⁴
LCS	G ▷ #35346 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁵ ◁ #35346 ◁ G
LECS	G ▷ #35346 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72337581	C₂	semidirect
2. #72337580	C₂	semidirect
3. #72337417	C₂	semidirect
4. #72334268	C₂	semidirect
5. #72334267	C₂	semidirect
8. #72325871	C₂	semidirect
9. #72310520	C₂	semidirect
10. #72310519	C₂	semidirect
15. #72265989	C₂	semidirect
18. #72237741	C₂	semidirect
20. #72237735	C₂	semidirect
36. #8093108	D₄	semidirect
37. #8093107	D₄	semidirect
38. #7513948	D₄	semidirect
39. #6631461	D₄	semidirect
40. #6437629	D₄	semidirect
41. #6437629	D₄	semidirect
44. #449521	D₄	semidirect
45. #449475	D₄	semidirect
47. #449090	D₄	semidirect
48. #432839	D₄	semidirect
49. #418911	D₄	semidirect
50. #418213	D₄	semidirect
52. #417824	D₄	semidirect
54. #388001	D₄	semidirect
55. #375666	D₄	semidirect
56. #374004	D₄	semidirect
57. #357285	D₄	semidirect
58. #356055	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72337581	2
#72337580	2
#72337417	2
#72334268	2
#72334267	2
#72325871	2
#72310520	2
#72310519	2
#72265989	2
#72237741	2
#72237735	2