Subgroups — #72339094

Order 8192.

At least 13578 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

The subgroup lattice has 149 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72339091	C₂	1	1	yes	maximal, characteristic, complemented
2	#72339090	C₂	1	1	yes	maximal, characteristic, complemented
3	#72339086	C₂	1	1	yes	maximal, characteristic, complemented
4	#72339083	C₂	1	1	yes	maximal, characteristic, complemented
5	#72339082	C₂	1	1	yes	maximal, characteristic, complemented
6	#72339066	C₂	1	1	yes	maximal, characteristic, complemented
7	#72339051	C₂	1	1	yes	maximal, characteristic, complemented
8	#72339047	C₂²	1	1	yes	characteristic, complemented
9	#72339044	C₂²	1	1	yes	characteristic, complemented
10	#72339004	C₂	1	1	yes	maximal, characteristic, complemented
11	#72339002	C₂	1	1	yes	maximal, characteristic, complemented
12	#72338987	C₂	1	1	yes	maximal, characteristic, complemented
13	#72338979	C₂²	1	1	yes	characteristic, complemented
14	#72338971	C₂²	1	1	yes	characteristic, complemented
15	#72338966	C₂²	1	1	yes	characteristic, complemented
16	#72338965	C₂²	1	1	yes	characteristic, complemented
17	#72338900	C₂	1	1	yes	maximal, characteristic, complemented
18	#72338874	C₂²	1	1	yes	characteristic, complemented
19	#72338862	C₂²	1	1	yes	characteristic, complemented
20	#72285651	C₂²	1	1	yes	characteristic
21	#72285651	C₂²	1	1	yes	characteristic, complemented
22	#72285651	C₂²	1	1	yes	characteristic, complemented
23	#72285635	C₂³	1	1	yes	characteristic
24	#72283656	D₄	1	2	yes	complemented
25	#72283476	D₄	1	2	yes	complemented
26	#72267058	C₂	1	1	yes	maximal, characteristic, complemented
27	#72252408	C₂²	1	1	yes	characteristic, complemented
28	#72241367	C₂	1	1	yes	maximal, characteristic, complemented
29	#72241353	C₂	1	1	yes	maximal, characteristic, complemented
30	#72241350	C₂²	1	1	yes	characteristic
31	#72241349	C₂²	1	1	yes	characteristic
32	#72241323	C₂²	1	1	yes	characteristic
33	#72241323	C₂²	1	1	yes	characteristic
34	#72237019	C₂	1	1	yes	maximal, characteristic, complemented
35	#72237011	C₂²	1	1	yes	characteristic, complemented
36	#72237011	C₂²	1	1	yes	characteristic, complemented
37	#72237005	C₂²	1	1	yes	characteristic, complemented
38	#72236999	C₂²	1	1	yes	characteristic, complemented
39	#72236986	C₂²	1	1	yes	characteristic
40	#72236979	D₄	1	1	yes	characteristic, complemented
41	#72236975	D₄	1	1	yes	characteristic, complemented
42	#72236975	D₄	1	1	yes	characteristic, complemented
43	#72236936	C₂²	1	1	yes	characteristic
44	#72236936	C₂²	1	1	yes	characteristic, complemented
45	#72236935	C₂²	1	1	yes	characteristic
46	#72236935	C₂²	1	1	yes	characteristic, complemented
47	#72236189	D₄	1	1	yes	characteristic, complemented
48	#72219530	C₂²	1	1	yes	characteristic, complemented
49	#72219524	C₂²	1	1	yes	characteristic, complemented
50	#72219518	C₂²	1	1	yes	characteristic, complemented
51	#72219509	C₂²	1	1	yes	characteristic, complemented
52	#72213834	C₂²	1	1	yes	characteristic
53	#72213834	C₂²	1	1	yes	characteristic, complemented
54	#72213834	C₂²	1	1	yes	characteristic, complemented
55	#72213804	D₄	1	2	yes	complemented
56	#72213786	D₄	1	2	yes	complemented
57	#72213758	C₂³	1	1	yes	characteristic
58	#72213758	C₂³	1	1	yes	characteristic
59	#72213758	C₂³	1	1	yes	characteristic
60	#72213758	C₂³	1	1	yes	characteristic
61	#72213565	C₂²	1	1	yes	characteristic, complemented
62	#72213500	C₂³	1	1	yes	characteristic
63	#72213462	C₂³	1	1	yes	characteristic
64	#72213423	C₂³	1	1	yes	characteristic
65	#72210591	C₂²	1	1	yes	characteristic
66	#72210591	C₂²	1	1	yes	characteristic
67	#72210549	D₄	1	1	yes	characteristic
68	#72210549	D₄	1	1	yes	characteristic, complemented
69	#72210539	D₄	1	1	yes	characteristic
70	#72210539	D₄	1	1	yes	characteristic, complemented
71	#72210529	C₂³	1	1	yes	characteristic
72	#72210529	C₂³	1	1	yes	characteristic
73	#72210524	C₂³	1	1	yes	characteristic
74	#72210498	D₄	1	1	yes	characteristic
75	#72210498	D₄	1	1	yes	characteristic, complemented
76	#72210483	D₄	1	1	yes	characteristic
77	#72210483	D₄	1	1	yes	characteristic, complemented
78	#72210439	C₂³	1	1	yes	characteristic
79	#72210439	C₂³	1	1	yes	characteristic
80	#72210439	C₂³	1	1	yes	characteristic
81	#72210439	C₂³	1	1	yes	characteristic
82	#8934366	C₂×D₄	1	1	yes	characteristic
83	#8934366	C₂×D₄	1	1	yes	characteristic
84	#8934366	C₂×D₄	1	1	yes	characteristic
85	#8934366	C₂×D₄	1	1	yes	characteristic
86	#7672751	C₂×D₄	1	1	yes	characteristic
87	#6631462	C₂×D₄	1	1	yes	characteristic, complemented
88	#6628901	C₂⁴	1	1	yes	characteristic
89	#6437630	C₂×D₄	1	1	yes	characteristic, complemented
90	#449492	C₂×D₄	1	1	yes	characteristic
91	#449492	C₂×D₄	1	1	yes	characteristic
92	#417807	C₂×D₄	1	1	yes	characteristic
93	#417807	C₂×D₄	1	1	yes	characteristic
94	#374009	C₂×D₄	1	1	yes	characteristic
95	#357286	C₂×D₄	1	1	yes	characteristic, complemented
96	#356056	C₂×D₄	1	1	yes	characteristic, complemented
97	#251828	C₂×D₄	1	2	yes	—
98	#251828	C₂×D₄	1	2	yes	—
99	#213860	C₂×D₄	1	1	yes	characteristic
100	#213860	C₂×D₄	1	1	yes	characteristic
101	#213860	C₂×D₄	1	1	yes	characteristic
102	#213860	C₂×D₄	1	1	yes	characteristic
103	#149742	C₂×D₄	1	2	yes	complemented
104	#149742	C₂×D₄	1	2	yes	complemented
105	#133835	C₂×D₄	1	1	yes	characteristic
106	#133835	C₂×D₄	1	1	yes	characteristic
107	#97460	C₂×D₄	1	1	yes	characteristic
108	#97460	C₂×D₄	1	1	yes	characteristic
109	C₂×H(7,𝔽₂)	H(5,𝔽₂)	1	1	yes	characteristic
110	#46384	#120	1	2	yes	—
111	C₂×#2867	#120	1	1	yes	characteristic
112	#35527	#120	1	1	yes	characteristic
113	#35527	C₂²×D₄	1	1	yes	characteristic
114	#35527	C₂²×D₄	1	1	yes	characteristic
115	#35527	C₂²×D₄	1	1	yes	characteristic
116	#20586	C₂²×D₄	1	1	yes	characteristic
117	#20586	C₂²×D₄	1	1	yes	characteristic
118	#20586	H(5,𝔽₂)	1	1	yes	characteristic
119	#10981	#120	1	1	yes	characteristic
120	#10981	#120	1	1	yes	characteristic
121	#8748	#120	1	2	yes	—
122	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
123	C₂²×H(5,𝔽₂)	#534	1	1	yes	characteristic
124	C₂²×H(5,𝔽₂)	D₄²	1	1	yes	characteristic
125	C₂²×H(5,𝔽₂)	D₄²	1	1	yes	characteristic
126	C₂²×#127	B(4,𝔽₂)	1	1	yes	characteristic
127	C₂²×#127	C₂×#120	1	1	yes	characteristic
128	#2842	B(4,𝔽₂)	1	2	yes	—
129	#2403	B(4,𝔽₂)	1	2	yes	—
130	#2309	B(4,𝔽₂)	1	1	yes	characteristic
131	#2309	C₂×#120	1	1	yes	characteristic
132	#2309	D₄²	1	1	yes	characteristic
133	#2309	D₄²	1	1	yes	characteristic
134	C₂×C₄³	B(4,𝔽₂)	1	1	yes	characteristic
135	C₂³×D₄	#2403	1	1	yes	characteristic
136	C₂³×D₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
137	C₂²×C₄²	C₂×B(4,𝔽₂)	1	1	yes	characteristic
138	#379	C₂×B(4,𝔽₂)	1	2	yes	—
139	C₂⁵	#23895	1	1	yes	characteristic
140	C₂⁵	#23900	1	1	yes	characteristic
141	C₂³×C₄	#23900	1	1	yes	characteristic
142	C₂⁴	#245225	1	1	yes	characteristic
143	C₂³	#72209844	1	1	yes	characteristic
144	C₂³	#72210342	1	1	yes	characteristic
145	C₂³	#72251567	1	1	yes	characteristic
146	C₂²	#72262416	1	1	yes	characteristic
147	C₂	#72338818	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72338818
commutator, Frattini, cosocle	#6628901	C₂⁴
LCS	G ▷ #6628901 ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #6628901 ◁ G
LECS	G ▷ #6628901 ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72339091	C₂	semidirect
2. #72339090	C₂	semidirect
3. #72339086	C₂	semidirect
4. #72339083	C₂	semidirect
5. #72339082	C₂	semidirect
6. #72339066	C₂	semidirect
7. #72339051	C₂	semidirect
8. #72339047	C₂²	semidirect
9. #72339044	C₂²	semidirect
10. #72339004	C₂	semidirect
11. #72339002	C₂	semidirect
12. #72338987	C₂	semidirect
13. #72338979	C₂²	semidirect
14. #72338971	C₂²	semidirect
15. #72338966	C₂²	semidirect
16. #72338965	C₂²	semidirect
17. #72338900	C₂	semidirect
18. #72338874	C₂²	semidirect
19. #72338862	C₂²	semidirect
21. #72285651	C₂²	semidirect
22. #72285651	C₂²	semidirect
24. #72283656	D₄	semidirect
25. #72283476	D₄	semidirect
26. #72267058	C₂	semidirect
27. #72252408	C₂²	semidirect
28. #72241367	C₂	semidirect
29. #72241353	C₂	semidirect
34. #72237019	C₂	semidirect
35. #72237011	C₂²	semidirect
36. #72237011	C₂²	semidirect
37. #72237005	C₂²	semidirect
38. #72236999	C₂²	semidirect
40. #72236979	D₄	semidirect
41. #72236975	D₄	semidirect
42. #72236975	D₄	semidirect
44. #72236936	C₂²	semidirect
46. #72236935	C₂²	semidirect
47. #72236189	D₄	semidirect
48. #72219530	C₂²	semidirect
49. #72219524	C₂²	semidirect
50. #72219518	C₂²	semidirect
51. #72219509	C₂²	semidirect
53. #72213834	C₂²	semidirect
54. #72213834	C₂²	semidirect
55. #72213804	D₄	semidirect
56. #72213786	D₄	semidirect
61. #72213565	C₂²	semidirect
68. #72210549	D₄	semidirect
70. #72210539	D₄	semidirect
75. #72210498	D₄	semidirect
77. #72210483	D₄	semidirect
87. #6631462	C₂×D₄	semidirect
89. #6437630	C₂×D₄	semidirect
95. #357286	C₂×D₄	semidirect
96. #356056	C₂×D₄	semidirect
103. #149742	C₂×D₄	semidirect
104. #149742	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72339091	2
#72339090	2
#72339086	2
#72339083	2
#72339082	2
#72339066	2
#72339051	2
#72339004	2
#72339002	2
#72338987	2
#72338900	2
#72267058	2
#72241367	2
#72241353	2
#72237019	2