Subgroups — #72339082

Order 4096.

At least 6391 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

The subgroup lattice has 138 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72339079	C₂	1	2	yes	maximal, complemented
2	#72339077	C₂	1	2	yes	maximal, complemented
3	#72339069	C₂	1	2	yes	maximal, complemented
4	#72339044	C₂	1	1	yes	maximal, characteristic, complemented
5	#72339040	C₂²	1	2	yes	—
6	#72338966	C₂	1	1	yes	maximal, characteristic, complemented
7	#72338947	C₂²	1	2	yes	complemented
8	#72318813	C₂	1	2	yes	maximal, complemented
9	#72318806	C₂²	1	2	yes	—
10	#72318576	C₂²	1	2	yes	complemented
11	#72285651	C₂	1	1	yes	maximal, characteristic, complemented
12	#72285635	C₂²	1	1	yes	characteristic
13	#72285611	C₂²	1	2	yes	complemented
14	#72278795	C₂²	1	2	yes	complemented
15	#72219518	C₂	1	1	yes	maximal, characteristic, complemented
16	#72219509	C₂	1	1	yes	maximal, characteristic, complemented
17	#72213758	C₂²	1	1	yes	characteristic
18	#72213633	C₂²	1	2	yes	—
19	#72213622	C₂²	1	2	yes	complemented
20	#72213565	C₂	1	1	yes	maximal, characteristic, complemented
21	#72213462	C₂²	1	1	yes	characteristic
22	#72213435	C₂²	1	2	yes	complemented
23	#72213423	C₂²	1	1	yes	characteristic
24	#72213387	C₂²	1	2	yes	complemented
25	#72213278	C₂²	1	2	yes	—
26	#72213210	C₂²	1	2	yes	complemented
27	#72210591	C₂	1	1	yes	maximal, characteristic, complemented
28	#72210549	C₂²	1	1	yes	characteristic
29	#72210539	C₂²	1	1	yes	characteristic
30	#72210529	C₂²	1	1	yes	characteristic
31	#72210524	C₂²	1	1	yes	characteristic
32	#72210498	C₂²	1	1	yes	characteristic
33	#72210483	C₂²	1	1	yes	characteristic
34	#72210439	C₂²	1	1	yes	characteristic
35	#10574841	D₄	1	2	yes	complemented
36	#8934366	C₂³	1	1	yes	characteristic
37	#7672751	D₄	1	1	yes	characteristic, complemented
38	#7504292	D₄	1	2	yes	complemented
39	#7501146	D₄	1	2	yes	complemented
40	#6631462	D₄	1	1	yes	characteristic, complemented
41	#6628901	C₂³	1	1	yes	characteristic
42	#6437969	C₂³	1	2	yes	—
43	#6437630	D₄	1	1	yes	characteristic, complemented
44	#496203	C₂³	1	2	yes	—
45	#449492	D₄	1	1	yes	characteristic, complemented
46	#449492	C₂³	1	1	yes	characteristic
47	#449491	D₄	1	2	yes	complemented
48	#438625	D₄	1	2	yes	complemented
49	#432844	D₄	1	2	yes	complemented
50	#430483	D₄	1	2	yes	complemented
51	#417934	C₂³	1	2	yes	—
52	#417907	D₄	1	2	yes	complemented
53	#417807	D₄	1	1	yes	characteristic, complemented
54	#417807	C₂³	1	1	yes	characteristic
55	#391962	D₄	1	2	yes	complemented
56	#374009	D₄	1	1	yes	characteristic, complemented
57	#357286	D₄	1	1	yes	characteristic, complemented
58	#356069	D₄	1	2	yes	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	#62852	C₂×D₄	1	2	yes	—
66	#60619	C₂×D₄	1	2	yes	—
67	C₂×#3021	C₂×D₄	1	2	yes	—
68	C₂×#3019	C₂×D₄	1	2	yes	—
69	C₂×#2888	C₂×D₄	1	2	yes	—
70	C₂×#2867	C₂×D₄	1	1	yes	characteristic
71	#35527	C₂×D₄	1	1	yes	characteristic
72	#35527	C₂×D₄	1	1	yes	characteristic
73	#35527	C₂×D₄	1	1	yes	characteristic
74	#35527	C₂⁴	1	1	yes	characteristic
75	#23469	C₂×D₄	1	2	yes	—
76	#23461	C₂×D₄	1	2	yes	—
77	#23430	C₂×D₄	1	2	yes	—
78	#20956	C₂×D₄	1	2	yes	—
79	#20855	C₂×D₄	1	2	yes	—
80	#20615	C₂×D₄	1	2	yes	—
81	#20586	C₂×D₄	1	1	yes	characteristic
82	#20586	C₂×D₄	1	1	yes	characteristic
83	#20586	C₂×D₄	1	1	yes	characteristic
84	#11006	C₂×D₄	1	2	yes	—
85	#10989	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	C₂×#585	#120	1	2	yes	—
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)	#120	1	1	yes	characteristic
95	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
96	#3458	#120	1	2	yes	—
97	C₂×#535	C₂²×D₄	1	2	yes	—
98	C₂²×#127	#120	1	1	yes	characteristic
99	C₂²×#127	C₂²×D₄	1	1	yes	characteristic
100	#3428	#120	1	2	yes	—
101	C₂×#413	#120	1	2	yes	—
102	C₂×#413	#120	1	2	yes	—
103	C₂²×#100	#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	C₂×#100	B(4,𝔽₂)	1	4	yes	—
120	#379	B(4,𝔽₂)	1	2	yes	—
121	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
122	C₂⁵	#3026	1	1	yes	characteristic
123	C₂²×Q₈	#3026	1	2	yes	—
124	C₂²×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
125	C₂³×C₄	#2846	1	1	yes	characteristic
126	C₂⁴	#32898	1	1	yes	characteristic
127	C₂×D₄	#33559	1	4	yes	—
128	C₂²×C₄	#32898	1	2	yes	—
129	C₂³	#347047	1	1	yes	characteristic
130	C₂³	#511790	1	1	yes	characteristic
131	C₂³	#511812	1	1	yes	characteristic
132	C₂×C₄	#511787	1	2	yes	—
133	C₂×C₄	#511790	1	2	yes	—
134	C₂²	#72233368	1	2	yes	—
135	C₂²	#72249397	1	1	yes	characteristic
136	C₂	#72262873	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72262873
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. #72339079	C₂	semidirect
2. #72339077	C₂	semidirect
3. #72339069	C₂	semidirect
4. #72339044	C₂	semidirect
6. #72338966	C₂	semidirect
7. #72338947	C₂²	semidirect
8. #72318813	C₂	semidirect
10. #72318576	C₂²	semidirect
11. #72285651	C₂	semidirect
13. #72285611	C₂²	semidirect
14. #72278795	C₂²	semidirect
15. #72219518	C₂	semidirect
16. #72219509	C₂	semidirect
19. #72213622	C₂²	semidirect
20. #72213565	C₂	semidirect
22. #72213435	C₂²	semidirect
24. #72213387	C₂²	semidirect
26. #72213210	C₂²	semidirect
27. #72210591	C₂	semidirect
35. #10574841	D₄	semidirect
37. #7672751	D₄	semidirect
38. #7504292	D₄	semidirect
39. #7501146	D₄	semidirect
40. #6631462	D₄	semidirect
43. #6437630	D₄	semidirect
45. #449492	D₄	semidirect
47. #449491	D₄	semidirect
48. #438625	D₄	semidirect
49. #432844	D₄	semidirect
50. #430483	D₄	semidirect
52. #417907	D₄	semidirect
53. #417807	D₄	semidirect
55. #391962	D₄	semidirect
56. #374009	D₄	semidirect
57. #357286	D₄	semidirect
58. #356069	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
#72339079	2
#72339077	2
#72339069	2
#72339044	2
#72338966	2
#72318813	2
#72285651	2
#72219518	2
#72219509	2
#72213565	2
#72210591	2