Subgroups — C₂×#72236955 #72341619

Order 6144.

Only the 611 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 205 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×#72236783	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#72236781	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#72236759	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#11714184	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#72236755	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#3027⋊C₃	C₂²	1	1	yes	characteristic, complemented
7	C₂×#72236763	C₂	1	2	yes	maximal, complemented
8	C₂×#72236770	C₂	1	2	yes	maximal, complemented
9	C₂×#11711643	C₂²	1	2	yes	complemented
10	C₂×#11711642	C₂²	1	2	yes	complemented
11	C₂×#72236766	C₂	1	2	yes	maximal, complemented
12	C₂×#11715709	C₂²	1	2	yes	complemented
13	C₂×#32894⋊C₃	C₂²	1	2	yes	complemented
14	C₂×#72236743	C₂	1	2	yes	maximal, complemented
15	C₂×#11711647	C₂²	1	2	yes	complemented
16	C₂×#11715711	C₂²	1	2	yes	complemented
17	C₂×#11711646	C₂²	1	2	yes	complemented
18	C₂×#32895⋊C₃	C₂²	1	2	yes	complemented
19	C₂×#72236738	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂×#11711597	C₂²	1	2	yes	complemented
21	C₂×#11711596	C₂²	1	2	yes	complemented
22	C₂×#72215767	C₂	1	1	yes	maximal, characteristic, complemented
23	C₂×#11715831	C₂²	1	1	yes	characteristic, complemented
24	C₂×#11715829	C₂²	1	1	yes	characteristic, complemented
25	C₂×#11715830	C₂²	1	1	yes	characteristic, complemented
26	C₂×#11715661	C₂²	1	1	yes	characteristic, complemented
27	C₂×#20612⋊C₃	C₂²	1	1	yes	characteristic, complemented
28	C₂²×#87520	C₂²	1	1	yes	characteristic
29	C₂²×#87521	C₂²	1	1	yes	characteristic, complemented
30	C₂×#11715827	C₂²	1	1	yes	characteristic, complemented
31	C₂×#11710937	C₂²	1	1	yes	characteristic, complemented
32	C₂×#6628885	D₃	1	1	yes	characteristic, complemented
33	#72236955	C₂	1	16	yes	maximal, direct product factor
34	#72236783	C₂²	1	8	yes	complemented
35	#72236781	C₂²	1	8	yes	complemented
36	#72236770	C₂²	1	16	yes	complemented
37	#72236766	C₂²	1	16	yes	complemented
38	#72236763	C₂²	1	16	yes	complemented
39	#72236759	C₂²	1	8	yes	complemented
40	#72236755	C₂²	1	8	yes	complemented
41	C₂×#11714184	C₂²	1	2	yes	complemented
42	C₂×#11714184	C₂²	1	2	yes	complemented
43	C₂×#11714184	C₂²	1	8	yes	complemented
44	#72236743	C₂²	1	16	yes	complemented
45	#72236738	C₂²	1	8	yes	complemented
46	#72215767	C₂²	1	8	yes	complemented
47	C₂×#11715828	C₂²	1	1	yes	characteristic, complemented
48	C₂×#87521	C₂³	1	2	yes	—
49	C₂×#87521	C₂³	1	4	yes	complemented
50	#11715831	C₂³	1	4	yes	complemented
51	#11715830	C₂³	1	4	yes	complemented
52	#11715829	C₂³	1	4	yes	complemented
53	#11715828	C₂³	1	4	yes	complemented
54	#11715827	C₂³	1	4	yes	complemented
55	C₂×#87518	C₂³	1	1	yes	characteristic, complemented
56	C₂×#87522	C₂³	1	1	yes	characteristic, complemented
57	C₂×#87520	D₄	1	1	yes	characteristic
58	C₂×#87520	D₄	1	1	yes	characteristic
59	C₂×#87520	D₄	1	4	yes	—
60	C₂×#87520	C₂³	1	1	yes	characteristic
61	C₂×#87520	C₂³	1	1	yes	characteristic
62	C₂×#87520	C₂³	1	4	yes	—
63	C₂²×#5933	D₄	1	1	yes	characteristic
64	C₂²×#5935	D₄	1	1	yes	characteristic
65	#11715711	C₂³	1	8	yes	complemented
66	#11715709	C₂³	1	8	yes	complemented
67	#11715661	C₂³	1	4	yes	complemented
68	C₂×#85566	C₂³	1	2	yes	complemented
69	#11714184	C₂³	1	8	yes	complemented
70	#11714184	C₂³	1	8	yes	complemented
71	#11711647	C₂³	1	8	yes	complemented
72	#11711646	C₂³	1	8	yes	complemented
73	#11711643	C₂³	1	8	yes	complemented
74	#11711642	C₂³	1	8	yes	complemented
75	#11711597	C₂³	1	8	yes	complemented
76	#11711596	C₂³	1	8	yes	complemented
77	#11710937	C₂³	1	4	yes	complemented
78	C₂×#87548	C₂³	1	1	yes	characteristic, complemented
79	C₂×#87547	C₂³	1	1	yes	characteristic, complemented
80	C₂²×#5950	C₂³	1	1	yes	characteristic
81	C₂×#3027⋊C₃	C₂³	1	2	yes	—
82	C₂×#3027⋊C₃	C₂³	1	4	yes	complemented
83	#32895⋊C₃	C₂³	1	8	yes	complemented
84	#32894⋊C₃	C₂³	1	8	yes	complemented
85	#20612⋊C₃	C₂³	1	4	yes	complemented
86	C₂×#1791⋊C₃	C₂³	1	2	yes	complemented
87	C₂×#63103	C₂×D₃	1	1	yes	characteristic, complemented
88	C₂²×#3027	C₂×D₃	1	1	yes	characteristic, complemented
89	#6628885	C₂×D₃	1	8	yes	complemented
90	C₂×#32895	C₂×D₃	1	2	yes	complemented
91	C₂×#32894	C₂×D₃	1	2	yes	complemented
92	C₂×#20612	C₂×D₃	1	1	yes	characteristic, complemented
93	#87548	C₂⁴	1	2	yes	complemented
94	#87547	C₂⁴	1	2	yes	complemented
95	C₂×#5950	C₂×D₄	1	1	yes	characteristic
96	C₂×#5950	C₂×D₄	1	2	yes	—
97	C₂×#5950	C₂⁴	1	1	yes	characteristic
98	C₂×#5950	C₂⁴	1	2	yes	—
99	C₂²×#997	C₂×D₄	1	1	yes	characteristic
100	#87522	C₂⁴	1	2	yes	complemented
101	#87521	C₂⁴	1	4	yes	—
102	#87520	C₂×D₄	1	2	yes	—
103	#87520	C₂×D₄	1	2	yes	—
104	#87520	C₂×D₄	1	4	yes	—
105	#87520	C₂×D₄	1	4	yes	—
106	#87520	C₂⁴	1	2	yes	—
107	#87520	C₂⁴	1	2	yes	—
108	#87518	C₂⁴	1	2	yes	complemented
109	C₂×#5935	C₂×D₄	1	1	yes	characteristic
110	C₂×#5935	C₂×D₄	1	1	yes	characteristic
111	C₂×#5935	C₂×D₄	1	4	yes	—
112	C₂×#5933	C₂×D₄	1	1	yes	characteristic
113	C₂×#5933	C₂×D₄	1	1	yes	characteristic
114	C₂×#5933	C₂×D₄	1	4	yes	—
115	#85566	C₂⁴	1	4	yes	complemented
116	#3027⋊C₃	C₂⁴	1	4	yes	—
117	#1791⋊C₃	C₂⁴	1	4	yes	complemented
118	#63103	C₂²×D₃	1	4	yes	complemented
119	C₂×2₋¹⁺⁶	C₂²×D₃	1	1	yes	characteristic, complemented
120	C₂×H(7,𝔽₂)	C₂²×D₃	1	1	yes	characteristic, complemented
121	C₂²×#585	C₂²×D₃	1	1	yes	characteristic
122	C₂³×AGL(1,ℤ/8)	S₄	1	1	yes	characteristic, complemented
123	C₂×#3027	C₂²×D₃	1	2	yes	—
124	C₂×#3027	C₂²×D₃	1	4	yes	complemented
125	#32895	C₂²×D₃	1	8	yes	complemented
126	#32894	C₂²×D₃	1	8	yes	complemented
127	#20612	C₂²×D₃	1	4	yes	complemented
128	C₂×#1791	C₂²×D₃	1	2	yes	complemented
129	#5950	C₂²×D₄	1	1	yes	characteristic
130	#5950	C₂²×D₄	1	2	yes	—
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	2₋¹⁺⁶	C₂³×D₃	1	2	yes	complemented
139	H(7,𝔽₂)	C₂³×D₃	1	2	yes	complemented
140	C₂×#585	D₃×D₄	1	2	yes	—
141	C₂×#585	D₃×D₄	1	1	yes	characteristic
142	C₂×#585	C₂³×D₃	1	1	yes	characteristic
143	C₂×#585	C₂³×D₃	1	2	yes	—
144	C₂²×H(5,𝔽₂)	D₃×D₄	1	1	yes	characteristic
145	C₂³×#41	C₂×S₄	1	1	yes	characteristic
146	C₂⁴×D₄	C₂×S₄	1	1	yes	characteristic
147	C₂²×AGL(1,ℤ/8)	C₂×S₄	1	8	yes	complemented
148	C₂³×SD₁₆	C₂×S₄	1	2	yes	complemented
149	C₂³×D₈	C₂×S₄	1	2	yes	complemented
150	C₂³×M₁₆	C₂×S₄	1	1	yes	characteristic
151	#3027	C₂³×D₃	1	4	yes	—
152	#1791	C₂³×D₃	1	4	yes	complemented
153	#997	C₂³×D₄	1	1	yes	characteristic
154	#585	C₂×D₃×D₄	1	1	yes	characteristic
155	#585	C₂×D₃×D₄	1	2	yes	—
156	#585	C₂⁴×D₃	1	1	yes	characteristic
157	C₂×H(5,𝔽₂)	C₂×D₃×D₄	1	2	yes	—
158	C₂×H(5,𝔽₂)	C₂×D₃×D₄	1	1	yes	characteristic
159	C₂²×#41	C₂²×S₄	1	4	yes	—
160	C₂³×Q₈	C₂²×S₄	1	1	yes	characteristic
161	C₂³×D₄	C₂²×S₄	1	1	yes	characteristic
162	C₂³×D₄	C₂²×S₄	1	2	yes	—
163	C₂³×D₄	C₂²×S₄	1	4	yes	—
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	C₂×#41	#1020⋊C₂	1	1	yes	characteristic
172	C₂²×Q₈	C₂³×S₄	1	2	yes	—
173	C₂²×D₄	C₂³×S₄	1	2	yes	—
174	C₂²×D₄	C₂³×S₄	1	4	yes	—
175	C₂³×C₄	D₄×S₄	1	2	yes	—
176	C₂³×C₄	D₄×S₄	1	1	yes	characteristic
177	C₂³×C₄	C₂³×S₄	1	1	yes	characteristic
178	C₂³×C₄	C₂³×S₄	1	2	yes	—
179	C₂²×C₈	C₂³×S₄	1	4	yes	—
180	C₂⁴	C₂×D₄×S₄	1	2	yes	—
181	C₂⁴	C₂×D₄×S₄	1	1	yes	characteristic
182	#41	C₂×#1020⋊C₂	1	4	yes	—
183	C₂×Q₈	C₂×#1020⋊C₂	1	1	yes	characteristic
184	C₂×D₄	C₂×#1020⋊C₂	1	1	yes	characteristic
185	C₂²×C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×S₄	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×S₄	1	2	yes	—
188	C₂²×C₄	C₂⁴×S₄	1	1	yes	characteristic
189	C₂³	#11713200	1	1	yes	characteristic
190	C₂³	C₂²×D₄×S₄	1	1	yes	characteristic
191	Q₈	C₂²×#1020⋊C₂	1	2	yes	—
192	D₄	C₂²×#1020⋊C₂	1	2	yes	—
193	C₂×C₄	#11713200	1	2	yes	—
194	C₂×C₄	#11713200	1	1	yes	characteristic
195	C₂×C₄	C₂²×#1020⋊C₂	1	2	yes	—
196	C₂×C₄	C₂²×#1020⋊C₂	1	1	yes	characteristic
197	C₂²	C₂×#11713200	1	2	yes	—
198	C₂²	C₂×#11713200	1	1	yes	characteristic
199	C₄	C₂×#11713200	1	2	yes	—
200	C₄	C₂×#11713200	1	1	yes	characteristic
201	C₄	C₂³×#1020⋊C₂	1	1	yes	characteristic
202	C₂	#72236955	1	2	yes	direct product factor
203	C₂	C₂²×#11713200	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#11713200
commutator, cosocle	#5950	C₂⁵
Frattini	C₂²×C₄	C₂⁴×S₄
Fitting	C₂×#6628885	D₃
hypercenter	C₂×#41	#1020⋊C₂
hypocenter	#997	C₂³×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#72236783	C₂	semidirect
2. C₂×#72236781	C₂	semidirect
3. C₂×#72236759	C₂	semidirect
4. C₂²×#11714184	C₂	semidirect
5. C₂×#72236755	C₂	semidirect
6. C₂²×#3027⋊C₃	C₂²	semidirect
7. C₂×#72236763	C₂	semidirect
8. C₂×#72236770	C₂	semidirect
9. C₂×#11711643	C₂²	semidirect
10. C₂×#11711642	C₂²	semidirect
11. C₂×#72236766	C₂	semidirect
12. C₂×#11715709	C₂²	semidirect
13. C₂×#32894⋊C₃	C₂²	semidirect
14. C₂×#72236743	C₂	semidirect
15. C₂×#11711647	C₂²	semidirect
16. C₂×#11715711	C₂²	semidirect
17. C₂×#11711646	C₂²	semidirect
18. C₂×#32895⋊C₃	C₂²	semidirect
19. C₂×#72236738	C₂	semidirect
20. C₂×#11711597	C₂²	semidirect
21. C₂×#11711596	C₂²	semidirect
22. C₂×#72215767	C₂	semidirect
23. C₂×#11715831	C₂²	semidirect
24. C₂×#11715829	C₂²	semidirect
25. C₂×#11715830	C₂²	semidirect
26. C₂×#11715661	C₂²	semidirect
27. C₂×#20612⋊C₃	C₂²	semidirect
29. C₂²×#87521	C₂²	semidirect
30. C₂×#11715827	C₂²	semidirect
31. C₂×#11710937	C₂²	semidirect
32. C₂×#6628885	D₃	semidirect
33. #72236955	C₂	direct (N × H)
34. #72236783	C₂²	semidirect
35. #72236781	C₂²	semidirect
36. #72236770	C₂²	semidirect
37. #72236766	C₂²	semidirect
38. #72236763	C₂²	semidirect
39. #72236759	C₂²	semidirect
40. #72236755	C₂²	semidirect
41. C₂×#11714184	C₂²	semidirect
42. C₂×#11714184	C₂²	semidirect
43. C₂×#11714184	C₂²	semidirect
44. #72236743	C₂²	semidirect
45. #72236738	C₂²	semidirect
46. #72215767	C₂²	semidirect
47. C₂×#11715828	C₂²	semidirect
49. C₂×#87521	C₂³	semidirect
50. #11715831	C₂³	semidirect
51. #11715830	C₂³	semidirect
52. #11715829	C₂³	semidirect
53. #11715828	C₂³	semidirect
54. #11715827	C₂³	semidirect
55. C₂×#87518	C₂³	semidirect
56. C₂×#87522	C₂³	semidirect
65. #11715711	C₂³	semidirect
66. #11715709	C₂³	semidirect
67. #11715661	C₂³	semidirect
68. C₂×#85566	C₂³	semidirect
69. #11714184	C₂³	semidirect
70. #11714184	C₂³	semidirect
71. #11711647	C₂³	semidirect
72. #11711646	C₂³	semidirect
73. #11711643	C₂³	semidirect
74. #11711642	C₂³	semidirect
75. #11711597	C₂³	semidirect
76. #11711596	C₂³	semidirect
77. #11710937	C₂³	semidirect
78. C₂×#87548	C₂³	semidirect
79. C₂×#87547	C₂³	semidirect
82. C₂×#3027⋊C₃	C₂³	semidirect
83. #32895⋊C₃	C₂³	semidirect
84. #32894⋊C₃	C₂³	semidirect
85. #20612⋊C₃	C₂³	semidirect
86. C₂×#1791⋊C₃	C₂³	semidirect
87. C₂×#63103	C₂×D₃	semidirect
88. C₂²×#3027	C₂×D₃	semidirect
89. #6628885	C₂×D₃	semidirect
90. C₂×#32895	C₂×D₃	semidirect
91. C₂×#32894	C₂×D₃	semidirect
92. C₂×#20612	C₂×D₃	semidirect
93. #87548	C₂⁴	semidirect
94. #87547	C₂⁴	semidirect
100. #87522	C₂⁴	semidirect
108. #87518	C₂⁴	semidirect
115. #85566	C₂⁴	semidirect
117. #1791⋊C₃	C₂⁴	semidirect
118. #63103	C₂²×D₃	semidirect
119. C₂×2₋¹⁺⁶	C₂²×D₃	semidirect
120. C₂×H(7,𝔽₂)	C₂²×D₃	semidirect
122. C₂³×AGL(1,ℤ/8)	S₄	semidirect
124. C₂×#3027	C₂²×D₃	semidirect
125. #32895	C₂²×D₃	semidirect
126. #32894	C₂²×D₃	semidirect
127. #20612	C₂²×D₃	semidirect
128. C₂×#1791	C₂²×D₃	semidirect
138. 2₋¹⁺⁶	C₂³×D₃	semidirect
139. H(7,𝔽₂)	C₂³×D₃	semidirect
147. C₂²×AGL(1,ℤ/8)	C₂×S₄	semidirect
148. C₂³×SD₁₆	C₂×S₄	semidirect
149. C₂³×D₈	C₂×S₄	semidirect
152. #1791	C₂³×D₃	semidirect
165. C₂²×SD₁₆	C₂²×S₄	semidirect
166. C₂²×D₈	C₂²×S₄	semidirect
202. C₂	#72236955	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×#72236783	2
C₂×#72236781	2
C₂×#72236759	2
C₂²×#11714184	2
C₂×#72236755	2
C₂×#72236763	2
C₂×#72236770	2
C₂×#72236766	2
C₂×#72236743	2
C₂×#72236738	2
C₂×#72215767	2
C₂×#72236949	3
#72236955	2
C₂×S₄×AGL(1,ℤ/8)	4

Subgroups — C2×#72236955 #72341619

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72236955 #72341619