Subgroups — C₂×#72236838 #72344178

Order 3072.

At least 6009 conjugacy classes of subgroups; only the 527 normal subgroups are shown.

The subgroup lattice has 170 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×#11714181	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11714172	C₂	1	2	yes	maximal, complemented
3	C₂×#11714179	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11714177	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#3025⋊C₃	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×#11715827	C₂	1	1	yes	maximal, characteristic, complemented
9	#72236838	C₂	1	16	yes	maximal, direct product factor
10	C₂×#11714171	C₂	1	2	yes	maximal, complemented
11	C₂×#11714162	C₂	1	2	yes	maximal, complemented
12	C₂×#11714175	C₂	1	1	yes	maximal, characteristic, 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	#11715827	C₂²	1	8	yes	complemented
19	C₂×#87518	C₂²	1	1	yes	characteristic, complemented
20	C₂×#87522	C₂²	1	1	yes	characteristic, complemented
21	C₂×#87516	C₂²	1	2	yes	complemented
22	C₂×#87520	C₂²	1	2	yes	complemented
23	#11714181	C₂²	1	8	yes	complemented
24	#11714179	C₂²	1	8	yes	complemented
25	#11714177	C₂²	1	8	yes	complemented
26	#11714175	C₂²	1	8	yes	complemented
27	#11714172	C₂²	1	16	yes	complemented
28	#11714171	C₂²	1	16	yes	complemented
29	#11714169	C₂²	1	16	yes	complemented
30	C₂×#73029	C₂²	1	2	yes	complemented
31	C₂×#73028	C₂²	1	2	yes	complemented
32	C₂×#73032	C₂²	1	2	yes	complemented
33	C₂×#73027	C₂²	1	2	yes	complemented
34	#11714162	C₂²	1	16	yes	complemented
35	C₂×#73031	C₂²	1	2	yes	complemented
36	C₂×#73025	C₂²	1	2	yes	complemented
37	C₂×#73030	C₂²	1	2	yes	complemented
38	C₂×#73026	C₂²	1	2	yes	complemented
39	C₂²×#5950	C₂²	1	1	yes	characteristic, complemented
40	C₂×#3025⋊C₃	C₂²	1	1	yes	characteristic
41	C₂×#3025⋊C₃	C₂²	1	1	yes	characteristic, complemented
42	C₂×#3025⋊C₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×#3025⋊C₃	C₂²	1	1	yes	characteristic, complemented
44	C₂×#3025⋊C₃	C₂²	1	8	yes	complemented
45	C₂²×#458⋊C₃	C₂²	1	1	yes	characteristic, complemented
46	C₂²×B(4,𝔽₂)⋊C₃	C₂²	1	1	yes	characteristic, complemented
47	C₂²×#3025	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	#87522	C₂³	1	4	yes	complemented
53	#87521	C₂³	1	4	yes	complemented
54	#87520	C₂³	1	8	yes	complemented
55	#87519	C₂³	1	8	yes	complemented
56	#87518	C₂³	1	4	yes	complemented
57	#87517	C₂³	1	4	yes	complemented
58	#87516	C₂³	1	8	yes	complemented
59	#87515	C₂³	1	8	yes	complemented
60	C₂×#5935	C₂³	1	2	yes	—
61	C₂×#5934	C₂³	1	2	yes	—
62	C₂×#5933	C₂³	1	2	yes	—
63	C₂×#5932	C₂³	1	2	yes	—
64	#3025⋊C₃	C₂³	1	4	yes	—
65	#3025⋊C₃	C₂³	1	4	yes	complemented
66	#3025⋊C₃	C₂³	1	4	yes	complemented
67	#3025⋊C₃	C₂³	1	4	yes	complemented
68	C₂×#458⋊C₃	C₂³	1	1	yes	characteristic
69	C₂×#458⋊C₃	C₂³	1	1	yes	characteristic
70	C₂×#458⋊C₃	C₂³	1	4	yes	complemented
71	C₂×B(4,𝔽₂)⋊C₃	C₂³	1	1	yes	characteristic
72	C₂×B(4,𝔽₂)⋊C₃	C₂³	1	1	yes	characteristic
73	C₂×B(4,𝔽₂)⋊C₃	C₂³	1	4	yes	complemented
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	C₂×#3025	C₂×D₃	1	1	yes	characteristic
84	C₂×#3025	C₂×D₃	1	1	yes	characteristic
85	C₂×#3025	C₂×D₃	1	1	yes	characteristic, complemented
86	C₂×#3025	C₂×D₃	1	1	yes	characteristic, complemented
87	C₂×#3025	C₂×D₃	1	8	yes	complemented
88	C₂²×#458	C₂×D₃	1	1	yes	characteristic, complemented
89	C₂²×B(4,𝔽₂)	C₂×D₃	1	1	yes	characteristic
90	#5950	C₂⁴	1	2	yes	—
91	#5950	C₂⁴	1	2	yes	—
92	C₂×#997	C₂⁴	1	1	yes	characteristic
93	C₂×#997	C₂⁴	1	2	yes	—
94	#5935	C₂⁴	1	4	yes	—
95	#5934	C₂⁴	1	4	yes	—
96	#5933	C₂⁴	1	4	yes	—
97	#5932	C₂⁴	1	4	yes	—
98	#458⋊C₃	C₂⁴	1	2	yes	—
99	#458⋊C₃	C₂⁴	1	2	yes	—
100	B(4,𝔽₂)⋊C₃	C₂⁴	1	2	yes	—
101	B(4,𝔽₂)⋊C₃	C₂⁴	1	2	yes	—
102	C₂×#585	C₂²×D₃	1	1	yes	characteristic
103	C₂×#585	C₂²×D₃	1	1	yes	characteristic
104	C₂×#585	C₂²×D₃	1	4	yes	complemented
105	C₂²×H(5,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
106	C₂⁵×C₄	S₄	1	1	yes	characteristic, complemented
107	#3025	C₂²×D₃	1	4	yes	—
108	#3025	C₂²×D₃	1	4	yes	—
109	#3025	C₂²×D₃	1	4	yes	complemented
110	#3025	C₂²×D₃	1	4	yes	complemented
111	C₂×#458	C₂²×D₃	1	1	yes	characteristic
112	C₂×#458	C₂²×D₃	1	1	yes	characteristic
113	C₂×#458	C₂²×D₃	1	4	yes	complemented
114	C₂×B(4,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
115	C₂×B(4,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
116	C₂×B(4,𝔽₂)	C₂²×D₃	1	4	yes	—
117	#997	C₂⁵	1	1	yes	characteristic
118	C₂⁶	C₂×S₄	1	1	yes	characteristic
119	#585	C₂³×D₃	1	2	yes	—
120	#585	C₂³×D₃	1	2	yes	—
121	C₂×H(5,𝔽₂)	C₂³×D₃	1	1	yes	characteristic
122	C₂×H(5,𝔽₂)	C₂³×D₃	1	2	yes	—
123	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
124	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
125	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
126	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
127	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
128	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
129	C₂⁴×C₄	C₂×S₄	1	8	yes	complemented
130	#458	C₂³×D₃	1	2	yes	—
131	#458	C₂³×D₃	1	2	yes	—
132	B(4,𝔽₂)	C₂³×D₃	1	2	yes	—
133	B(4,𝔽₂)	C₂³×D₃	1	2	yes	—
134	C₂⁵	C₂²×S₄	1	1	yes	characteristic
135	C₂⁵	C₂²×S₄	1	1	yes	characteristic
136	C₂⁵	C₂²×S₄	1	1	yes	characteristic
137	C₂⁵	C₂²×S₄	1	4	yes	—
138	H(5,𝔽₂)	C₂⁴×D₃	1	1	yes	characteristic
139	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
140	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
141	C₂³×C₄	C₂²×S₄	1	4	yes	—
142	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
143	C₂³×C₄	C₂²×S₄	1	4	yes	—
144	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
145	C₂³×C₄	C₂²×S₄	1	4	yes	—
146	C₂³×C₄	C₂²×S₄	1	4	yes	—
147	C₂³×C₄	C₂²×S₄	1	4	yes	—
148	C₂³×C₄	C₂²×S₄	1	4	yes	—
149	C₂⁴	C₂³×S₄	1	1	yes	characteristic
150	C₂⁴	C₂³×S₄	1	2	yes	—
151	C₂⁴	C₂³×S₄	1	2	yes	—
152	C₂⁴	C₂³×S₄	1	2	yes	—
153	C₂²×C₄	#1020⋊C₂	1	1	yes	characteristic
154	C₂²×C₄	C₂³×S₄	1	2	yes	—
155	C₂²×C₄	C₂³×S₄	1	2	yes	—
156	C₂²×C₄	C₂³×S₄	1	2	yes	—
157	C₂²×C₄	C₂³×S₄	1	2	yes	—
158	C₂³	C₂×#1020⋊C₂	1	1	yes	characteristic
159	C₂³	C₂⁴×S₄	1	1	yes	characteristic
160	C₂×C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
161	C₂×C₄	C₂×#1020⋊C₂	1	4	yes	—
162	C₂×C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
163	C₂²	C₂²×#1020⋊C₂	1	2	yes	—
164	C₂²	C₂²×#1020⋊C₂	1	1	yes	characteristic
165	C₄	C₂²×#1020⋊C₂	1	2	yes	—
166	C₄	C₂²×#1020⋊C₂	1	2	yes	—
167	C₂	C₂³×#1020⋊C₂	1	1	yes	characteristic
168	C₂	#72236838	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	C₂²×#3025	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×#11714181	C₂	semidirect
2. C₂×#11714172	C₂	semidirect
3. C₂×#11714179	C₂	semidirect
4. C₂×#11714177	C₂	semidirect
5. C₂²×#3025⋊C₃	C₂	semidirect
6. C₂×#11714169	C₂	semidirect
7. C₂×#11715838	C₂	semidirect
8. C₂×#11715827	C₂	semidirect
9. #72236838	C₂	direct (N × H)
10. C₂×#11714171	C₂	semidirect
11. C₂×#11714162	C₂	semidirect
12. C₂×#11714175	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
18. #11715827	C₂²	semidirect
19. C₂×#87518	C₂²	semidirect
20. C₂×#87522	C₂²	semidirect
21. C₂×#87516	C₂²	semidirect
22. C₂×#87520	C₂²	semidirect
23. #11714181	C₂²	semidirect
24. #11714179	C₂²	semidirect
25. #11714177	C₂²	semidirect
26. #11714175	C₂²	semidirect
27. #11714172	C₂²	semidirect
28. #11714171	C₂²	semidirect
29. #11714169	C₂²	semidirect
30. C₂×#73029	C₂²	semidirect
31. C₂×#73028	C₂²	semidirect
32. C₂×#73032	C₂²	semidirect
33. C₂×#73027	C₂²	semidirect
34. #11714162	C₂²	semidirect
35. C₂×#73031	C₂²	semidirect
36. C₂×#73025	C₂²	semidirect
37. C₂×#73030	C₂²	semidirect
38. C₂×#73026	C₂²	semidirect
39. C₂²×#5950	C₂²	semidirect
41. C₂×#3025⋊C₃	C₂²	semidirect
42. C₂×#3025⋊C₃	C₂²	semidirect
43. C₂×#3025⋊C₃	C₂²	semidirect
44. C₂×#3025⋊C₃	C₂²	semidirect
45. C₂²×#458⋊C₃	C₂²	semidirect
46. C₂²×B(4,𝔽₂)⋊C₃	C₂²	semidirect
47. C₂²×#3025	D₃	semidirect
50. C₂×#5950	C₂³	semidirect
52. #87522	C₂³	semidirect
53. #87521	C₂³	semidirect
54. #87520	C₂³	semidirect
55. #87519	C₂³	semidirect
56. #87518	C₂³	semidirect
57. #87517	C₂³	semidirect
58. #87516	C₂³	semidirect
59. #87515	C₂³	semidirect
65. #3025⋊C₃	C₂³	semidirect
66. #3025⋊C₃	C₂³	semidirect
67. #3025⋊C₃	C₂³	semidirect
70. C₂×#458⋊C₃	C₂³	semidirect
73. C₂×B(4,𝔽₂)⋊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
85. C₂×#3025	C₂×D₃	semidirect
86. C₂×#3025	C₂×D₃	semidirect
87. C₂×#3025	C₂×D₃	semidirect
88. C₂²×#458	C₂×D₃	semidirect
104. C₂×#585	C₂²×D₃	semidirect
106. C₂⁵×C₄	S₄	semidirect
109. #3025	C₂²×D₃	semidirect
110. #3025	C₂²×D₃	semidirect
113. C₂×#458	C₂²×D₃	semidirect
129. C₂⁴×C₄	C₂×S₄	semidirect
168. C₂	#72236838	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×#11714181	2
C₂×#11714172	2
C₂×#11714179	2
C₂×#11714177	2
C₂²×#3025⋊C₃	2
C₂×#11714169	2
C₂×#11715838	2
C₂×#11715827	2
C₂×#7622868	3
#72236838	2
C₂×#11714171	2
C₂×#11714162	2
C₂×#11714175	2
C₂²×#5912	4

Subgroups — C2×#72236838 #72344178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72236838 #72344178