Subgroups — C₂×#72240988 #72343803

Order 6144.

At least 14540 conjugacy classes of subgroups; only the 611 normal subgroups are shown.

The subgroup lattice has 203 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×#72236903	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#72236910	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#72236877	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#72236922	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#72236859	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#72236879	C₂	1	2	yes	maximal, complemented
7	C₂×#11714185	C₂²	1	1	yes	characteristic, complemented
8	C₂×#72236866	C₂	1	2	yes	maximal, complemented
9	C₂×#11714186	C₂²	1	1	yes	characteristic, complemented
10	C₂×#72215648	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#11715682	C₂²	1	2	yes	complemented
12	C₂×#72236886	C₂	1	2	yes	maximal, complemented
13	C₂×#11711640	C₂²	1	2	yes	complemented
14	C₂×#11711639	C₂²	1	2	yes	complemented
15	C₂×#11715735	C₂²	1	2	yes	complemented
16	C₂×#11714187	C₂²	1	1	yes	characteristic, complemented
17	C₂×#32947⋊C₃	C₂²	1	2	yes	complemented
18	C₂×#72236850	C₂	1	2	yes	maximal, complemented
19	C₂×#11711638	C₂²	1	2	yes	complemented
20	C₂×#11711638	C₂²	1	2	yes	complemented
21	C₂×#11715736	C₂²	1	2	yes	complemented
22	C₂×#32948⋊C₃	C₂²	1	2	yes	complemented
23	C₂²×#11714184	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂²×#3027⋊C₃	C₂²	1	1	yes	characteristic, complemented
25	C₂²×#87520	C₂²	1	1	yes	characteristic
26	C₂×#11710873	C₂²	1	1	yes	characteristic, complemented
27	C₂²×#87521	C₂²	1	1	yes	characteristic, complemented
28	C₂×#11715667	C₂²	1	1	yes	characteristic, complemented
29	C₂×#11715838	C₂²	1	1	yes	characteristic, complemented
30	C₂×#60398⋊C₃	C₂²	1	1	yes	characteristic, complemented
31	C₂×#6629045	D₃	1	1	yes	characteristic, complemented
32	#72240988	C₂	1	16	yes	maximal, direct product factor
33	#72236922	C₂²	1	8	yes	complemented
34	#72236910	C₂²	1	8	yes	complemented
35	#72236903	C₂²	1	8	yes	complemented
36	#72236886	C₂²	1	16	yes	complemented
37	#72236879	C₂²	1	16	yes	complemented
38	#72236877	C₂²	1	8	yes	complemented
39	#72236866	C₂²	1	16	yes	complemented
40	#72236859	C₂²	1	8	yes	complemented
41	#72236850	C₂²	1	16	yes	complemented
42	C₂×#11714184	C₂²	1	2	yes	—
43	C₂×#11714184	C₂²	1	2	yes	complemented
44	C₂×#11714184	C₂²	1	8	yes	complemented
45	C₂×#11714184	C₂²	1	1	yes	characteristic, complemented
46	C₂×#11715681	C₂²	1	2	yes	complemented
47	#72215648	C₂²	1	8	yes	complemented
48	#11715838	C₂³	1	4	yes	complemented
49	C₂×#87517	C₂³	1	1	yes	characteristic, complemented
50	C₂×#87521	C₂³	1	2	yes	—
51	C₂×#87521	C₂³	1	4	yes	complemented
52	C₂×#87521	C₂³	1	1	yes	characteristic, complemented
53	C₂×#87520	D₄	1	4	yes	—
54	C₂×#87520	D₄	1	1	yes	characteristic
55	C₂×#87520	D₄	1	1	yes	characteristic
56	C₂×#87520	C₂³	1	1	yes	characteristic
57	C₂×#87520	C₂³	1	4	yes	—
58	C₂×#87520	C₂³	1	1	yes	characteristic
59	C₂²×#5933	D₄	1	1	yes	characteristic
60	C₂²×#5935	D₄	1	1	yes	characteristic
61	#11715736	C₂³	1	8	yes	complemented
62	#11715735	C₂³	1	8	yes	complemented
63	#11715682	C₂³	1	8	yes	complemented
64	#11715681	C₂³	1	8	yes	complemented
65	#11715667	C₂³	1	4	yes	complemented
66	C₂×#85567	C₂³	1	2	yes	complemented
67	#11714187	C₂³	1	4	yes	complemented
68	#11714186	C₂³	1	4	yes	complemented
69	#11714185	C₂³	1	4	yes	complemented
70	#11714184	C₂³	1	8	yes	—
71	#11714184	C₂³	1	8	yes	complemented
72	#11714184	C₂³	1	4	yes	complemented
73	#11711640	C₂³	1	8	yes	complemented
74	#11711639	C₂³	1	8	yes	complemented
75	#11711638	C₂³	1	8	yes	complemented
76	#11711638	C₂³	1	8	yes	complemented
77	C₂²×#5950	C₂³	1	1	yes	characteristic
78	#11710873	C₂³	1	4	yes	complemented
79	C₂×#85666	C₂³	1	2	yes	complemented
80	#60398⋊C₃	C₂³	1	4	yes	complemented
81	C₂×#3028⋊C₃	C₂³	1	1	yes	characteristic, complemented
82	C₂×#3027⋊C₃	C₂³	1	2	yes	—
83	C₂×#3027⋊C₃	C₂³	1	4	yes	complemented
84	C₂×#3027⋊C₃	C₂³	1	1	yes	characteristic, complemented
85	#32948⋊C₃	C₂³	1	8	yes	complemented
86	#32947⋊C₃	C₂³	1	8	yes	complemented
87	C₂×#63076	C₂×D₃	1	1	yes	characteristic, complemented
88	C₂×#60398	C₂×D₃	1	1	yes	characteristic, complemented
89	C₂²×#3027	C₂×D₃	1	1	yes	characteristic, complemented
90	#6629045	C₂×D₃	1	8	yes	complemented
91	C₂×#32948	C₂×D₃	1	2	yes	complemented
92	C₂×#32947	C₂×D₃	1	2	yes	complemented
93	C₂×#5950	C₂×D₄	1	1	yes	characteristic
94	C₂×#5950	C₂×D₄	1	2	yes	—
95	C₂×#5950	C₂⁴	1	1	yes	characteristic
96	C₂×#5950	C₂⁴	1	2	yes	—
97	C₂²×#997	C₂×D₄	1	1	yes	characteristic
98	#87521	C₂⁴	1	4	yes	—
99	#87521	C₂⁴	1	2	yes	complemented
100	#87520	C₂×D₄	1	4	yes	—
101	#87520	C₂×D₄	1	4	yes	—
102	#87520	C₂×D₄	1	2	yes	—
103	#87520	C₂×D₄	1	2	yes	—
104	#87520	C₂⁴	1	2	yes	—
105	#87520	C₂⁴	1	2	yes	—
106	#87517	C₂⁴	1	2	yes	complemented
107	C₂×#5935	C₂×D₄	1	1	yes	characteristic
108	C₂×#5935	C₂×D₄	1	1	yes	characteristic
109	C₂×#5935	C₂×D₄	1	4	yes	—
110	C₂×#5933	C₂×D₄	1	1	yes	characteristic
111	C₂×#5933	C₂×D₄	1	1	yes	characteristic
112	C₂×#5933	C₂×D₄	1	4	yes	—
113	#85666	C₂⁴	1	4	yes	complemented
114	#85567	C₂⁴	1	4	yes	complemented
115	#3028⋊C₃	C₂⁴	1	2	yes	complemented
116	#3027⋊C₃	C₂⁴	1	4	yes	—
117	#3027⋊C₃	C₂⁴	1	2	yes	complemented
118	C₂²×#585	C₂²×D₃	1	1	yes	characteristic
119	C₂³×AGL(1,ℤ/8)	S₄	1	1	yes	characteristic, complemented
120	#63076	C₂²×D₃	1	4	yes	complemented
121	C₂×#3573	C₂²×D₃	1	2	yes	complemented
122	#60398	C₂²×D₃	1	4	yes	complemented
123	C₂×#3028	C₂²×D₃	1	1	yes	characteristic, complemented
124	C₂×#3027	C₂²×D₃	1	2	yes	—
125	C₂×#3027	C₂²×D₃	1	4	yes	complemented
126	C₂×#3027	C₂²×D₃	1	1	yes	characteristic, complemented
127	#32948	C₂²×D₃	1	8	yes	complemented
128	#32947	C₂²×D₃	1	8	yes	complemented
129	#5950	C₂²×D₄	1	2	yes	—
130	#5950	C₂²×D₄	1	1	yes	characteristic
131	#5950	C₂⁵	1	1	yes	characteristic
132	C₂×#997	C₂²×D₄	1	1	yes	characteristic
133	C₂×#997	C₂²×D₄	1	2	yes	—
134	#5935	C₂²×D₄	1	2	yes	—
135	#5935	C₂²×D₄	1	2	yes	—
136	#5933	C₂²×D₄	1	2	yes	—
137	#5933	C₂²×D₄	1	2	yes	—
138	C₂×#585	D₃×D₄	1	1	yes	characteristic
139	C₂×#585	D₃×D₄	1	2	yes	—
140	C₂×#585	C₂³×D₃	1	1	yes	characteristic
141	C₂×#585	C₂³×D₃	1	2	yes	—
142	C₂²×H(5,𝔽₂)	D₃×D₄	1	1	yes	characteristic
143	C₂³×#41	C₂×S₄	1	1	yes	characteristic
144	C₂⁴×D₄	C₂×S₄	1	1	yes	characteristic
145	C₂²×AGL(1,ℤ/8)	C₂×S₄	1	8	yes	complemented
146	C₂³×SD₁₆	C₂×S₄	1	2	yes	complemented
147	C₂³×D₈	C₂×S₄	1	2	yes	complemented
148	#3573	C₂³×D₃	1	4	yes	complemented
149	C₂³×M₁₆	C₂×S₄	1	1	yes	characteristic
150	#3028	C₂³×D₃	1	2	yes	complemented
151	#3027	C₂³×D₃	1	4	yes	—
152	#3027	C₂³×D₃	1	2	yes	complemented
153	#997	C₂³×D₄	1	1	yes	characteristic
154	#585	C₂×D₃×D₄	1	2	yes	—
155	#585	C₂×D₃×D₄	1	1	yes	characteristic
156	#585	C₂⁴×D₃	1	1	yes	characteristic
157	C₂×H(5,𝔽₂)	C₂×D₃×D₄	1	1	yes	characteristic
158	C₂×H(5,𝔽₂)	C₂×D₃×D₄	1	2	yes	—
159	C₂²×#41	C₂²×S₄	1	4	yes	—
160	C₂³×Q₈	C₂²×S₄	1	1	yes	characteristic
161	C₂³×D₄	C₂²×S₄	1	2	yes	—
162	C₂³×D₄	C₂²×S₄	1	4	yes	—
163	C₂³×D₄	C₂²×S₄	1	1	yes	characteristic
164	C₂⁴×C₄	C₂²×S₄	1	1	yes	characteristic
165	C₂²×SD₁₆	C₂²×S₄	1	8	yes	complemented
166	C₂²×D₈	C₂²×S₄	1	8	yes	complemented
167	C₂²×M₁₆	C₂²×S₄	1	4	yes	—
168	C₂³×C₈	C₂²×S₄	1	2	yes	—
169	C₂⁵	D₄×S₄	1	1	yes	characteristic
170	H(5,𝔽₂)	C₂²×D₃×D₄	1	1	yes	characteristic
171	C₂²×Q₈	C₂³×S₄	1	2	yes	—
172	C₂²×D₄	C₂³×S₄	1	4	yes	—
173	C₂²×D₄	C₂³×S₄	1	2	yes	—
174	C₂³×C₄	D₄×S₄	1	1	yes	characteristic
175	C₂³×C₄	D₄×S₄	1	2	yes	—
176	C₂³×C₄	C₂³×S₄	1	1	yes	characteristic
177	C₂³×C₄	C₂³×S₄	1	2	yes	—
178	C₂×M₁₆	#1020⋊C₂	1	1	yes	characteristic
179	C₂²×C₈	C₂³×S₄	1	4	yes	—
180	C₂⁴	C₂×D₄×S₄	1	1	yes	characteristic
181	C₂⁴	C₂×D₄×S₄	1	2	yes	—
182	C₂²×C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×S₄	1	2	yes	—
184	C₂²×C₄	C₂×D₄×S₄	1	1	yes	characteristic
185	C₂²×C₄	C₂⁴×S₄	1	1	yes	characteristic
186	M₁₆	C₂×#1020⋊C₂	1	4	yes	—
187	C₂×C₈	C₂×#1020⋊C₂	1	2	yes	—
188	C₂³	#11713271	1	1	yes	characteristic
189	C₂³	C₂²×D₄×S₄	1	1	yes	characteristic
190	C₂×C₄	#11713271	1	2	yes	—
191	C₂×C₄	#11713271	1	1	yes	characteristic
192	C₂×C₄	C₂²×#1020⋊C₂	1	2	yes	—
193	C₂×C₄	C₂²×#1020⋊C₂	1	1	yes	characteristic
194	C₈	C₂²×#1020⋊C₂	1	4	yes	—
195	C₂²	C₂×#11713271	1	2	yes	—
196	C₂²	C₂×#11713271	1	1	yes	characteristic
197	C₄	C₂³×#1020⋊C₂	1	1	yes	characteristic
198	C₄	C₂×#11713271	1	2	yes	—
199	C₄	C₂×#11713271	1	1	yes	characteristic
200	C₂	#72240988	1	2	yes	direct product factor
201	C₂	C₂²×#11713271	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#11713271
commutator, cosocle	#5950	C₂⁵
Frattini	C₂²×C₄	C₂⁴×S₄
Fitting	C₂×#6629045	D₃
hypercenter	C₂×M₁₆	#1020⋊C₂
hypocenter	#997	C₂³×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#72236903	C₂	semidirect
2. C₂×#72236910	C₂	semidirect
3. C₂×#72236877	C₂	semidirect
4. C₂×#72236922	C₂	semidirect
5. C₂×#72236859	C₂	semidirect
6. C₂×#72236879	C₂	semidirect
7. C₂×#11714185	C₂²	semidirect
8. C₂×#72236866	C₂	semidirect
9. C₂×#11714186	C₂²	semidirect
10. C₂×#72215648	C₂	semidirect
11. C₂×#11715682	C₂²	semidirect
12. C₂×#72236886	C₂	semidirect
13. C₂×#11711640	C₂²	semidirect
14. C₂×#11711639	C₂²	semidirect
15. C₂×#11715735	C₂²	semidirect
16. C₂×#11714187	C₂²	semidirect
17. C₂×#32947⋊C₃	C₂²	semidirect
18. C₂×#72236850	C₂	semidirect
19. C₂×#11711638	C₂²	semidirect
20. C₂×#11711638	C₂²	semidirect
21. C₂×#11715736	C₂²	semidirect
22. C₂×#32948⋊C₃	C₂²	semidirect
23. C₂²×#11714184	C₂	semidirect
24. C₂²×#3027⋊C₃	C₂²	semidirect
26. C₂×#11710873	C₂²	semidirect
27. C₂²×#87521	C₂²	semidirect
28. C₂×#11715667	C₂²	semidirect
29. C₂×#11715838	C₂²	semidirect
30. C₂×#60398⋊C₃	C₂²	semidirect
31. C₂×#6629045	D₃	semidirect
32. #72240988	C₂	direct (N × H)
33. #72236922	C₂²	semidirect
34. #72236910	C₂²	semidirect
35. #72236903	C₂²	semidirect
36. #72236886	C₂²	semidirect
37. #72236879	C₂²	semidirect
38. #72236877	C₂²	semidirect
39. #72236866	C₂²	semidirect
40. #72236859	C₂²	semidirect
41. #72236850	C₂²	semidirect
43. C₂×#11714184	C₂²	semidirect
44. C₂×#11714184	C₂²	semidirect
45. C₂×#11714184	C₂²	semidirect
46. C₂×#11715681	C₂²	semidirect
47. #72215648	C₂²	semidirect
48. #11715838	C₂³	semidirect
49. C₂×#87517	C₂³	semidirect
51. C₂×#87521	C₂³	semidirect
52. C₂×#87521	C₂³	semidirect
61. #11715736	C₂³	semidirect
62. #11715735	C₂³	semidirect
63. #11715682	C₂³	semidirect
64. #11715681	C₂³	semidirect
65. #11715667	C₂³	semidirect
66. C₂×#85567	C₂³	semidirect
67. #11714187	C₂³	semidirect
68. #11714186	C₂³	semidirect
69. #11714185	C₂³	semidirect
71. #11714184	C₂³	semidirect
72. #11714184	C₂³	semidirect
73. #11711640	C₂³	semidirect
74. #11711639	C₂³	semidirect
75. #11711638	C₂³	semidirect
76. #11711638	C₂³	semidirect
78. #11710873	C₂³	semidirect
79. C₂×#85666	C₂³	semidirect
80. #60398⋊C₃	C₂³	semidirect
81. C₂×#3028⋊C₃	C₂³	semidirect
83. C₂×#3027⋊C₃	C₂³	semidirect
84. C₂×#3027⋊C₃	C₂³	semidirect
85. #32948⋊C₃	C₂³	semidirect
86. #32947⋊C₃	C₂³	semidirect
87. C₂×#63076	C₂×D₃	semidirect
88. C₂×#60398	C₂×D₃	semidirect
89. C₂²×#3027	C₂×D₃	semidirect
90. #6629045	C₂×D₃	semidirect
91. C₂×#32948	C₂×D₃	semidirect
92. C₂×#32947	C₂×D₃	semidirect
99. #87521	C₂⁴	semidirect
106. #87517	C₂⁴	semidirect
113. #85666	C₂⁴	semidirect
114. #85567	C₂⁴	semidirect
115. #3028⋊C₃	C₂⁴	semidirect
117. #3027⋊C₃	C₂⁴	semidirect
119. C₂³×AGL(1,ℤ/8)	S₄	semidirect
120. #63076	C₂²×D₃	semidirect
121. C₂×#3573	C₂²×D₃	semidirect
122. #60398	C₂²×D₃	semidirect
123. C₂×#3028	C₂²×D₃	semidirect
125. C₂×#3027	C₂²×D₃	semidirect
126. C₂×#3027	C₂²×D₃	semidirect
127. #32948	C₂²×D₃	semidirect
128. #32947	C₂²×D₃	semidirect
145. C₂²×AGL(1,ℤ/8)	C₂×S₄	semidirect
146. C₂³×SD₁₆	C₂×S₄	semidirect
147. C₂³×D₈	C₂×S₄	semidirect
148. #3573	C₂³×D₃	semidirect
150. #3028	C₂³×D₃	semidirect
152. #3027	C₂³×D₃	semidirect
165. C₂²×SD₁₆	C₂²×S₄	semidirect
166. C₂²×D₈	C₂²×S₄	semidirect
200. C₂	#72240988	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×#72236903	2
C₂×#72236910	2
C₂×#72236877	2
C₂×#72236922	2
C₂×#72236859	2
C₂×#72236879	2
C₂×#72236866	2
C₂×#72215648	2
C₂×#72236886	2
C₂×#72236850	2
C₂²×#11714184	2
C₂×#72236900	3
#72240988	2
C₂×S₄×AGL(1,ℤ/8)	4

Subgroups — C2×#72240988 #72343803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72240988 #72343803