Subgroups — C₂×#72236922 #72343583

Order 3072.

At least 6247 conjugacy classes of subgroups; only the 527 normal subgroups are shown.

The subgroup lattice has 154 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11714185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11714183	C₂	1	2	yes	maximal, complemented
3	C₂×#11714186	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11714187	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#87520	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11714169	C₂	1	2	yes	maximal, complemented
7	C₂×#11715838	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#60398⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#72236922	C₂	1	16	yes	maximal, direct product factor
10	C₂×#11714182	C₂	1	2	yes	maximal, complemented
11	C₂×#11714184	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11714162	C₂	1	2	yes	maximal, complemented
13	#11715838	C₂²	1	8	yes	complemented
14	C₂×#87517	C₂²	1	1	yes	characteristic, complemented
15	C₂×#87521	C₂²	1	1	yes	characteristic, complemented
16	C₂×#87515	C₂²	1	2	yes	complemented
17	C₂×#87519	C₂²	1	2	yes	complemented
18	C₂×#87520	C₂²	1	1	yes	characteristic
19	C₂×#87520	C₂²	1	1	yes	characteristic
20	C₂×#87520	C₂²	1	1	yes	characteristic, complemented
21	C₂×#87520	C₂²	1	1	yes	characteristic, complemented
22	C₂×#87520	C₂²	1	8	yes	complemented
23	C₂²×#5933	C₂²	1	1	yes	characteristic, complemented
24	C₂²×#5935	C₂²	1	1	yes	characteristic
25	#11714187	C₂²	1	8	yes	complemented
26	#11714186	C₂²	1	8	yes	complemented
27	#11714185	C₂²	1	8	yes	complemented
28	#11714184	C₂²	1	8	yes	complemented
29	#11714183	C₂²	1	16	yes	complemented
30	#11714182	C₂²	1	16	yes	complemented
31	#11714169	C₂²	1	16	yes	complemented
32	C₂×#73029	C₂²	1	2	yes	complemented
33	C₂×#73028	C₂²	1	2	yes	complemented
34	C₂×#73032	C₂²	1	2	yes	complemented
35	C₂×#73027	C₂²	1	2	yes	complemented
36	#11714162	C₂²	1	16	yes	complemented
37	C₂×#73031	C₂²	1	2	yes	complemented
38	C₂×#73025	C₂²	1	2	yes	complemented
39	C₂×#73030	C₂²	1	2	yes	complemented
40	C₂×#73026	C₂²	1	2	yes	complemented
41	C₂²×#5950	C₂²	1	1	yes	characteristic, complemented
42	#60398⋊C₃	C₂²	1	8	yes	complemented
43	C₂×#3028⋊C₃	C₂²	1	1	yes	characteristic, complemented
44	C₂×#3027⋊C₃	C₂²	1	1	yes	characteristic, complemented
45	C₂×#3026⋊C₃	C₂²	1	2	yes	complemented
46	C₂×#3025⋊C₃	C₂²	1	2	yes	complemented
47	C₂×#60398	D₃	1	1	yes	characteristic, complemented
48	C₂×#5950	C₂³	1	1	yes	characteristic
49	C₂×#5950	C₂³	1	1	yes	characteristic
50	C₂×#5950	C₂³	1	4	yes	complemented
51	C₂²×#997	C₂³	1	1	yes	characteristic
52	#87521	C₂³	1	4	yes	complemented
53	#87520	C₂³	1	4	yes	—
54	#87520	C₂³	1	4	yes	—
55	#87520	C₂³	1	4	yes	complemented
56	#87520	C₂³	1	4	yes	complemented
57	#87519	C₂³	1	8	yes	complemented
58	#87517	C₂³	1	4	yes	complemented
59	#87515	C₂³	1	8	yes	complemented
60	C₂×#5935	C₂³	1	1	yes	characteristic
61	C₂×#5935	C₂³	1	1	yes	characteristic
62	C₂×#5935	C₂³	1	4	yes	—
63	C₂×#5934	C₂³	1	2	yes	—
64	C₂×#5933	C₂³	1	1	yes	characteristic
65	C₂×#5933	C₂³	1	1	yes	characteristic
66	C₂×#5933	C₂³	1	4	yes	complemented
67	C₂×#5932	C₂³	1	2	yes	—
68	#3028⋊C₃	C₂³	1	4	yes	complemented
69	#3027⋊C₃	C₂³	1	4	yes	complemented
70	#3026⋊C₃	C₂³	1	8	yes	complemented
71	#3025⋊C₃	C₂³	1	8	yes	complemented
72	C₂×#458⋊C₃	C₂³	1	2	yes	—
73	C₂×B(4,𝔽₂)⋊C₃	C₂³	1	2	yes	—
74	#73032	C₂³	1	8	yes	complemented
75	#73031	C₂³	1	8	yes	complemented
76	#73030	C₂³	1	8	yes	complemented
77	#73029	C₂³	1	8	yes	complemented
78	#73028	C₂³	1	8	yes	complemented
79	#73027	C₂³	1	8	yes	complemented
80	#73026	C₂³	1	8	yes	complemented
81	#73025	C₂³	1	8	yes	complemented
82	C₂²×#585	C₂×D₃	1	1	yes	characteristic, complemented
83	#60398	C₂×D₃	1	8	yes	complemented
84	C₂×#3028	C₂×D₃	1	1	yes	characteristic, complemented
85	C₂×#3027	C₂×D₃	1	1	yes	characteristic, complemented
86	C₂×#3026	C₂×D₃	1	2	yes	complemented
87	C₂×#3025	C₂×D₃	1	2	yes	complemented
88	#5950	C₂⁴	1	2	yes	—
89	#5950	C₂⁴	1	2	yes	—
90	C₂×#997	C₂⁴	1	1	yes	characteristic
91	C₂×#997	C₂⁴	1	2	yes	—
92	#5935	C₂⁴	1	2	yes	—
93	#5935	C₂⁴	1	2	yes	—
94	#5934	C₂⁴	1	4	yes	—
95	#5933	C₂⁴	1	2	yes	—
96	#5933	C₂⁴	1	2	yes	—
97	#5932	C₂⁴	1	4	yes	—
98	#458⋊C₃	C₂⁴	1	4	yes	—
99	B(4,𝔽₂)⋊C₃	C₂⁴	1	4	yes	—
100	C₂×#585	C₂²×D₃	1	1	yes	characteristic
101	C₂×#585	C₂²×D₃	1	1	yes	characteristic
102	C₂×#585	C₂²×D₃	1	4	yes	complemented
103	C₂²×H(5,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
104	C₂³×#41	S₄	1	1	yes	characteristic, complemented
105	#3028	C₂²×D₃	1	4	yes	complemented
106	#3027	C₂²×D₃	1	4	yes	complemented
107	#3026	C₂²×D₃	1	8	yes	complemented
108	#3025	C₂²×D₃	1	8	yes	complemented
109	C₂×#458	C₂²×D₃	1	2	yes	—
110	C₂×B(4,𝔽₂)	C₂²×D₃	1	2	yes	—
111	#997	C₂⁵	1	1	yes	characteristic
112	#585	C₂³×D₃	1	2	yes	—
113	#585	C₂³×D₃	1	2	yes	—
114	C₂×H(5,𝔽₂)	C₂³×D₃	1	1	yes	characteristic
115	C₂×H(5,𝔽₂)	C₂³×D₃	1	2	yes	—
116	C₂²×#41	C₂×S₄	1	8	yes	complemented
117	C₂³×Q₈	C₂×S₄	1	1	yes	characteristic
118	C₂³×D₄	C₂×S₄	1	2	yes	—
119	C₂³×D₄	C₂×S₄	1	1	yes	characteristic
120	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
121	C₂⁴×C₄	C₂×S₄	1	2	yes	—
122	#458	C₂³×D₃	1	4	yes	—
123	B(4,𝔽₂)	C₂³×D₃	1	4	yes	—
124	C₂⁵	C₂²×S₄	1	1	yes	characteristic
125	C₂⁵	C₂²×S₄	1	2	yes	—
126	H(5,𝔽₂)	C₂⁴×D₃	1	1	yes	characteristic
127	C₂²×Q₈	C₂²×S₄	1	4	yes	—
128	C₂²×D₄	C₂²×S₄	1	8	yes	—
129	C₂²×D₄	C₂²×S₄	1	4	yes	—
130	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
131	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
132	C₂³×C₄	C₂²×S₄	1	4	yes	—
133	C₂³×C₄	C₂²×S₄	1	2	yes	—
134	C₂³×C₄	C₂²×S₄	1	8	yes	—
135	C₂⁴	C₂³×S₄	1	1	yes	characteristic
136	C₂⁴	C₂³×S₄	1	2	yes	—
137	C₂⁴	C₂³×S₄	1	4	yes	—
138	C₂²×C₄	#1020⋊C₂	1	1	yes	characteristic
139	C₂²×C₄	C₂³×S₄	1	2	yes	—
140	C₂²×C₄	C₂³×S₄	1	2	yes	—
141	C₂²×C₄	C₂³×S₄	1	4	yes	—
142	C₂³	C₂×#1020⋊C₂	1	1	yes	characteristic
143	C₂³	C₂⁴×S₄	1	1	yes	characteristic
144	C₂×C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
145	C₂×C₄	C₂×#1020⋊C₂	1	4	yes	—
146	C₂×C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
147	C₂²	C₂²×#1020⋊C₂	1	2	yes	—
148	C₂²	C₂²×#1020⋊C₂	1	1	yes	characteristic
149	C₄	C₂²×#1020⋊C₂	1	2	yes	—
150	C₄	C₂²×#1020⋊C₂	1	2	yes	—
151	C₂	C₂³×#1020⋊C₂	1	1	yes	characteristic
152	C₂	#72236922	1	2	yes	direct product factor

	subgroup	G/H
center	C₂×C₄	C₂×#1020⋊C₂
commutator, cosocle, hypocenter	#997	C₂⁵
Frattini	C₂³	C₂⁴×S₄
Fitting	C₂×#60398	D₃
socle	C₂²	C₂²×#1020⋊C₂
hypercenter	C₂²×C₄	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#11714185	C₂	semidirect
2. C₂×#11714183	C₂	semidirect
3. C₂×#11714186	C₂	semidirect
4. C₂×#11714187	C₂	semidirect
5. C₂²×#87520	C₂	semidirect
6. C₂×#11714169	C₂	semidirect
7. C₂×#11715838	C₂	semidirect
8. C₂×#60398⋊C₃	C₂	semidirect
9. #72236922	C₂	direct (N × H)
10. C₂×#11714182	C₂	semidirect
11. C₂×#11714184	C₂	semidirect
12. C₂×#11714162	C₂	semidirect
13. #11715838	C₂²	semidirect
14. C₂×#87517	C₂²	semidirect
15. C₂×#87521	C₂²	semidirect
16. C₂×#87515	C₂²	semidirect
17. C₂×#87519	C₂²	semidirect
20. C₂×#87520	C₂²	semidirect
21. C₂×#87520	C₂²	semidirect
22. C₂×#87520	C₂²	semidirect
23. C₂²×#5933	C₂²	semidirect
25. #11714187	C₂²	semidirect
26. #11714186	C₂²	semidirect
27. #11714185	C₂²	semidirect
28. #11714184	C₂²	semidirect
29. #11714183	C₂²	semidirect
30. #11714182	C₂²	semidirect
31. #11714169	C₂²	semidirect
32. C₂×#73029	C₂²	semidirect
33. C₂×#73028	C₂²	semidirect
34. C₂×#73032	C₂²	semidirect
35. C₂×#73027	C₂²	semidirect
36. #11714162	C₂²	semidirect
37. C₂×#73031	C₂²	semidirect
38. C₂×#73025	C₂²	semidirect
39. C₂×#73030	C₂²	semidirect
40. C₂×#73026	C₂²	semidirect
41. C₂²×#5950	C₂²	semidirect
42. #60398⋊C₃	C₂²	semidirect
43. C₂×#3028⋊C₃	C₂²	semidirect
44. C₂×#3027⋊C₃	C₂²	semidirect
45. C₂×#3026⋊C₃	C₂²	semidirect
46. C₂×#3025⋊C₃	C₂²	semidirect
47. C₂×#60398	D₃	semidirect
50. C₂×#5950	C₂³	semidirect
52. #87521	C₂³	semidirect
55. #87520	C₂³	semidirect
56. #87520	C₂³	semidirect
57. #87519	C₂³	semidirect
58. #87517	C₂³	semidirect
59. #87515	C₂³	semidirect
66. C₂×#5933	C₂³	semidirect
68. #3028⋊C₃	C₂³	semidirect
69. #3027⋊C₃	C₂³	semidirect
70. #3026⋊C₃	C₂³	semidirect
71. #3025⋊C₃	C₂³	semidirect
74. #73032	C₂³	semidirect
75. #73031	C₂³	semidirect
76. #73030	C₂³	semidirect
77. #73029	C₂³	semidirect
78. #73028	C₂³	semidirect
79. #73027	C₂³	semidirect
80. #73026	C₂³	semidirect
81. #73025	C₂³	semidirect
82. C₂²×#585	C₂×D₃	semidirect
83. #60398	C₂×D₃	semidirect
84. C₂×#3028	C₂×D₃	semidirect
85. C₂×#3027	C₂×D₃	semidirect
86. C₂×#3026	C₂×D₃	semidirect
87. C₂×#3025	C₂×D₃	semidirect
102. C₂×#585	C₂²×D₃	semidirect
104. C₂³×#41	S₄	semidirect
105. #3028	C₂²×D₃	semidirect
106. #3027	C₂²×D₃	semidirect
107. #3026	C₂²×D₃	semidirect
108. #3025	C₂²×D₃	semidirect
116. C₂²×#41	C₂×S₄	semidirect
152. C₂	#72236922	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11714185	2
C₂×#11714183	2
C₂×#11714186	2
C₂×#11714187	2
C₂²×#87520	2
C₂×#7622874	3
C₂×#11714169	2
C₂×#11715838	2
C₂×#60398⋊C₃	2
#72236922	2
C₂×#11714182	2
C₂×#11714184	2
C₂×#11714162	2
C₂×#41×S₄	4

Subgroups — C2×#72236922 #72343583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72236922 #72343583