Subgroups — #72339066

Order 4096.

At least 6274 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

The subgroup lattice has 136 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72339064	C₂	1	2	yes	maximal, complemented
2	#72339062	C₂	1	2	yes	maximal, complemented
3	#72339059	C₂	1	2	yes	maximal, complemented
4	#72339056	C₂	1	2	yes	maximal, complemented
5	#72339044	C₂	1	1	yes	maximal, characteristic, complemented
6	#72339041	C₂²	1	2	yes	—
7	#72339039	C₂²	1	2	yes	—
8	#72338971	C₂	1	1	yes	maximal, characteristic, complemented
9	#72338965	C₂	1	1	yes	maximal, characteristic, complemented
10	#72338952	C₂²	1	2	yes	complemented
11	#72338951	C₂²	1	2	yes	complemented
12	#72338949	C₂²	1	2	yes	complemented
13	#72285635	C₂²	1	1	yes	characteristic
14	#72235221	C₂²	1	2	yes	complemented
15	#72219530	C₂	1	1	yes	maximal, characteristic, complemented
16	#72219524	C₂	1	1	yes	maximal, characteristic, complemented
17	#72213834	C₂	1	1	yes	maximal, characteristic, complemented
18	#72213758	C₂²	1	1	yes	characteristic
19	#72213734	C₂²	1	2	yes	complemented
20	#72213651	C₂²	1	2	yes	complemented
21	#72213638	C₂²	1	2	yes	complemented
22	#72213500	C₂²	1	1	yes	characteristic
23	#72213423	C₂²	1	1	yes	characteristic
24	#72213406	C₂²	1	2	yes	complemented
25	#72213400	C₂²	1	2	yes	complemented
26	#72213224	C₂²	1	2	yes	complemented
27	#72210591	C₂	1	1	yes	maximal, characteristic, complemented
28	#72210549	C₂²	1	1	yes	characteristic, complemented
29	#72210539	C₂²	1	1	yes	characteristic, complemented
30	#72210529	C₂²	1	1	yes	characteristic
31	#72210524	C₂²	1	1	yes	characteristic
32	#72210498	C₂²	1	1	yes	characteristic, complemented
33	#72210483	C₂²	1	1	yes	characteristic, complemented
34	#72210439	C₂²	1	1	yes	characteristic
35	#8934366	C₂³	1	1	yes	characteristic
36	#7672751	D₄	1	1	yes	characteristic
37	#6631462	D₄	1	1	yes	characteristic, complemented
38	#6628901	C₂³	1	1	yes	characteristic
39	#6437630	D₄	1	1	yes	characteristic, complemented
40	#496222	C₂³	1	2	yes	—
41	#479834	D₄	1	2	yes	complemented
42	#477023	D₄	1	2	yes	complemented
43	#453908	D₄	1	2	yes	complemented
44	#449492	D₄	1	1	yes	characteristic
45	#449492	C₂³	1	1	yes	characteristic
46	#418884	D₄	1	2	yes	complemented
47	#417807	D₄	1	1	yes	characteristic
48	#417807	C₂³	1	1	yes	characteristic
49	#417798	C₂³	1	2	yes	—
50	#417790	D₄	1	2	yes	complemented
51	#392138	D₄	1	2	yes	complemented
52	#391720	C₂³	1	2	yes	—
53	#391655	D₄	1	2	yes	complemented
54	#386041	D₄	1	2	yes	complemented
55	#383184	D₄	1	2	yes	complemented
56	#378105	D₄	1	2	yes	complemented
57	#374009	D₄	1	1	yes	characteristic
58	#357286	D₄	1	1	yes	characteristic, complemented
59	#356056	D₄	1	1	yes	characteristic, complemented
60	#213860	C₂³	1	1	yes	characteristic
61	#134056	C₂³	1	2	yes	—
62	#133835	C₂³	1	1	yes	characteristic
63	#97460	C₂³	1	1	yes	characteristic
64	C₂×H(7,𝔽₂)	C₂×D₄	1	1	yes	characteristic
65	#60644	C₂×D₄	1	2	yes	—
66	C₂×#3019	C₂×D₄	1	2	yes	—
67	C₂×#2867	C₂×D₄	1	1	yes	characteristic
68	#35527	C₂×D₄	1	1	yes	characteristic
69	#35527	C₂×D₄	1	1	yes	characteristic
70	#35527	C₂×D₄	1	1	yes	characteristic
71	#35527	C₂⁴	1	1	yes	characteristic
72	#23445	C₂×D₄	1	2	yes	—
73	#23430	C₂×D₄	1	2	yes	—
74	#23406	C₂×D₄	1	2	yes	—
75	#20592	C₂×D₄	1	2	yes	—
76	#20586	C₂×D₄	1	2	yes	—
77	#20586	C₂×D₄	1	1	yes	characteristic
78	#20586	C₂×D₄	1	1	yes	characteristic
79	#20586	C₂×D₄	1	1	yes	characteristic
80	#11160	C₂×D₄	1	2	yes	—
81	#11160	C₂×D₄	1	2	yes	—
82	#11006	C₂×D₄	1	2	yes	—
83	#10989	C₂×D₄	1	2	yes	—
84	#10987	C₂×D₄	1	2	yes	—
85	#10987	C₂×D₄	1	2	yes	—
86	#10981	C₂×D₄	1	1	yes	characteristic
87	#10981	C₂×D₄	1	1	yes	characteristic
88	#10959	C₂×D₄	1	2	yes	—
89	#10939	C₂×D₄	1	2	yes	—
90	#10923	C₂×D₄	1	2	yes	—
91	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
92	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
93	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
94	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
95	#3449	#120	1	2	yes	—
96	C₂×#535	C₂²×D₄	1	2	yes	—
97	C₂²×#127	#120	1	1	yes	characteristic
98	C₂²×#127	C₂²×D₄	1	1	yes	characteristic
99	C₂²×#100	#120	1	2	yes	—
100	C₂×#410	#120	1	2	yes	—
101	C₂×#410	#120	1	2	yes	—
102	#2879	#120	1	2	yes	—
103	#2879	#120	1	2	yes	—
104	#2873	#120	1	2	yes	—
105	#2309	#120	1	1	yes	characteristic
106	#2309	#120	1	1	yes	characteristic
107	#2309	#120	1	1	yes	characteristic
108	#2309	C₂²×D₄	1	1	yes	characteristic
109	#2297	C₂²×D₄	1	2	yes	—
110	C₂×C₄³	#120	1	1	yes	characteristic
111	#1742	#120	1	2	yes	—
112	#1740	#120	1	2	yes	—
113	#1728	#120	1	2	yes	—
114	C₂²×#41	C₂×#120	1	2	yes	—
115	C₂³×D₄	C₂×#120	1	1	yes	characteristic
116	C₂³×D₄	C₂×#120	1	1	yes	characteristic
117	C₂×#117	C₂×#120	1	2	yes	—
118	C₂²×C₄²	C₂×#120	1	1	yes	characteristic
119	#379	B(4,𝔽₂)	1	2	yes	—
120	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
121	C₂⁵	#3026	1	1	yes	characteristic
122	C₂²×Q₈	#3027	1	2	yes	—
123	C₂²×D₄	#3025	1	2	yes	—
124	C₂³×C₄	#2846	1	1	yes	characteristic
125	C₂⁴	#32898	1	1	yes	characteristic
126	C₂²×C₄	#32899	1	2	yes	—
127	C₂³	#347045	1	1	yes	characteristic
128	C₂³	#511790	1	1	yes	characteristic
129	C₂³	#511811	1	1	yes	characteristic
130	C₂×C₄	#511853	1	2	yes	—
131	C₂×C₄	#511853	1	2	yes	—
132	C₂²	#72249407	1	1	yes	characteristic
133	C₂²	#72255837	1	2	yes	—
134	C₂	#72262878	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72262878
commutator, Frattini, cosocle	#35527	C₂⁴
LCS	G ▷ #35527 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35527 ◁ G
LECS	G ▷ #35527 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72339064	C₂	semidirect
2. #72339062	C₂	semidirect
3. #72339059	C₂	semidirect
4. #72339056	C₂	semidirect
5. #72339044	C₂	semidirect
8. #72338971	C₂	semidirect
9. #72338965	C₂	semidirect
10. #72338952	C₂²	semidirect
11. #72338951	C₂²	semidirect
12. #72338949	C₂²	semidirect
14. #72235221	C₂²	semidirect
15. #72219530	C₂	semidirect
16. #72219524	C₂	semidirect
17. #72213834	C₂	semidirect
19. #72213734	C₂²	semidirect
20. #72213651	C₂²	semidirect
21. #72213638	C₂²	semidirect
24. #72213406	C₂²	semidirect
25. #72213400	C₂²	semidirect
26. #72213224	C₂²	semidirect
27. #72210591	C₂	semidirect
28. #72210549	C₂²	semidirect
29. #72210539	C₂²	semidirect
32. #72210498	C₂²	semidirect
33. #72210483	C₂²	semidirect
37. #6631462	D₄	semidirect
39. #6437630	D₄	semidirect
41. #479834	D₄	semidirect
42. #477023	D₄	semidirect
43. #453908	D₄	semidirect
46. #418884	D₄	semidirect
50. #417790	D₄	semidirect
51. #392138	D₄	semidirect
53. #391655	D₄	semidirect
54. #386041	D₄	semidirect
55. #383184	D₄	semidirect
56. #378105	D₄	semidirect
58. #357286	D₄	semidirect
59. #356056	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72339064	2
#72339062	2
#72339059	2
#72339056	2
#72339044	2
#72338971	2
#72338965	2
#72219530	2
#72219524	2
#72213834	2
#72210591	2