Subgroups — #72341591

Order 6144.

At least 36761 conjugacy classes of subgroups; only the 273 normal subgroups are shown.

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	#72341544	C₂	1	1	yes	maximal, characteristic, complemented
2	#72341539	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#72236738	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#72215767	C₂	1	1	yes	maximal, characteristic
5	C₂×#11715831	C₄	1	1	yes	characteristic, complemented
6	C₂×#11715829	C₄	1	1	yes	characteristic, complemented
7	C₂×#11715830	C₄	1	1	yes	characteristic, complemented
8	#72341508	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11715661	C₂²	1	1	yes	characteristic, complemented
10	#72341505	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#20612⋊C₃	C₂²	1	1	yes	characteristic, complemented
12	#72341494	C₂	1	1	yes	maximal, characteristic, complemented
13	#72341492	C₂²	1	1	yes	characteristic, complemented
14	C₂²×#87520	C₂²	1	1	yes	characteristic
15	#72341483	C₂²	1	1	yes	characteristic, complemented
16	#72341463	C₂	1	4	yes	maximal, complemented
17	#72341462	C₂²	1	2	yes	complemented
18	#72341458	C₂	1	4	yes	maximal, complemented
19	#72341455	C₂²	1	4	yes	complemented
20	#72341452	C₂²	1	4	yes	complemented
21	#72341449	C₂²	1	4	yes	complemented
22	#72341447	C₂²	1	4	yes	complemented
23	#72341343	C₂²	1	2	yes	complemented
24	#72280922	D₃	1	1	yes	characteristic, complemented
25	C₂×#11715827	C₂²	1	1	yes	characteristic
26	C₂×#11710937	C₂²	1	1	yes	characteristic
27	#72236738	C₂²	1	2	yes	complemented
28	#72236738	C₂²	1	2	yes	complemented
29	#72215767	C₄	1	2	yes	complemented
30	#72215767	C₄	1	2	yes	complemented
31	#72215767	C₂²	1	2	yes	—
32	#72215767	C₂²	1	2	yes	—
33	C₂×#11715828	C₄	1	1	yes	characteristic, complemented
34	#11715831	C₂×C₄	1	2	yes	complemented
35	#11715831	C₂×C₄	1	2	yes	complemented
36	#11715830	C₂×C₄	1	2	yes	complemented
37	#11715830	C₂×C₄	1	2	yes	complemented
38	#11715829	C₂×C₄	1	2	yes	complemented
39	#11715829	C₂×C₄	1	2	yes	complemented
40	#11715828	C₂×C₄	1	2	yes	complemented
41	#11715828	C₂×C₄	1	2	yes	complemented
42	#11715827	C₂×C₄	1	2	yes	complemented
43	#11715827	C₂³	1	2	yes	—
44	C₂×#87518	C₂×C₄	1	1	yes	characteristic, complemented
45	C₂×#87522	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×#87520	C₂×C₄	1	1	yes	characteristic
47	C₂×#87520	C₂×C₄	1	1	yes	characteristic
48	C₂×#87520	C₂×C₄	1	1	yes	characteristic
49	C₂×#87520	C₂×C₄	1	1	yes	characteristic
50	C₂×#87520	D₄	1	1	yes	characteristic
51	C₂×#87520	D₄	1	1	yes	characteristic
52	C₂×#87520	D₄	1	1	yes	characteristic
53	C₂×#87520	D₄	1	1	yes	characteristic
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	1	yes	characteristic
58	C₂²×#5933	D₄	1	1	yes	characteristic
59	C₂²×#5935	D₄	1	1	yes	characteristic
60	#11715661	C₂³	1	2	yes	complemented
61	#11715657	C₂³	1	2	yes	complemented
62	#11710937	C₂×C₄	1	2	yes	complemented
63	#11710937	C₂³	1	2	yes	—
64	C₂×#87548	C₂×C₄	1	1	yes	characteristic, complemented
65	C₂×#87547	C₂×C₄	1	1	yes	characteristic, complemented
66	C₂²×#5950	C₂³	1	1	yes	characteristic
67	#20612⋊C₃	C₂³	1	2	yes	complemented
68	#20608⋊C₃	C₂³	1	2	yes	complemented
69	C₂×#63103	C₂×D₃	1	1	yes	characteristic
70	C₂×#20612	C₂×D₃	1	1	yes	characteristic, complemented
71	#6370734	C₂×D₃	1	1	yes	characteristic, complemented
72	#392140	C₂×D₃	1	4	yes	complemented
73	#87548	C₂²×C₄	1	2	yes	—
74	#87547	C₂²×C₄	1	2	yes	—
75	C₂×#5950	C₂²×C₄	1	1	yes	characteristic
76	C₂×#5950	C₂²×C₄	1	1	yes	characteristic
77	C₂×#5950	C₂×D₄	1	1	yes	characteristic
78	C₂×#5950	C₂×D₄	1	1	yes	characteristic
79	C₂×#5950	C₂×D₄	1	1	yes	characteristic
80	C₂×#5950	C₂⁴	1	1	yes	characteristic
81	C₂²×#997	C₂×D₄	1	1	yes	characteristic
82	#87522	C₂²×C₄	1	2	yes	—
83	#87520	C₂²⋊C₄	1	1	yes	characteristic
84	#87520	C₂²⋊C₄	1	1	yes	characteristic
85	#87520	C₂²⋊C₄	1	1	yes	characteristic
86	#87520	C₂²⋊C₄	1	1	yes	characteristic
87	#87520	C₂²⋊C₄	1	1	yes	characteristic
88	#87520	C₂²⋊C₄	1	1	yes	characteristic
89	#87520	C₂²⋊C₄	1	1	yes	characteristic
90	#87520	C₂²⋊C₄	1	1	yes	characteristic
91	#87520	C₂²×C₄	1	1	yes	characteristic
92	#87520	C₂²×C₄	1	1	yes	characteristic
93	#87520	C₂²×C₄	1	1	yes	characteristic
94	#87520	C₂²×C₄	1	1	yes	characteristic
95	#87520	C₂×D₄	1	1	yes	characteristic
96	#87520	C₂×D₄	1	1	yes	characteristic
97	#87520	C₂×D₄	1	1	yes	characteristic
98	#87520	C₂×D₄	1	1	yes	characteristic
99	#87518	C₂²×C₄	1	2	yes	—
100	C₂×#5935	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×#5935	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×#5935	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×#5935	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×#5935	C₂×D₄	1	1	yes	characteristic
105	C₂×#5935	C₂×D₄	1	1	yes	characteristic
106	C₂×#5933	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#5933	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×#5933	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×#5933	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×#5933	C₂×D₄	1	1	yes	characteristic
111	C₂×#5933	C₂×D₄	1	1	yes	characteristic
112	#63103	C₄×D₃	1	2	yes	complemented
113	#63103	C₂²×D₃	1	2	yes	—
114	C₂×2₋¹⁺⁶	C₄×D₃	1	1	yes	characteristic, complemented
115	C₂×H(7,𝔽₂)	C₄×D₃	1	1	yes	characteristic, complemented
116	C₂²×#585	C₂²×D₃	1	1	yes	characteristic
117	C₂²×#421	S₄	1	1	yes	characteristic
118	#20612	C₂²×D₃	1	2	yes	complemented
119	#20608	C₂²×D₃	1	2	yes	complemented
120	#5950	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	#5950	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	#5950	C₂³×C₄	1	1	yes	characteristic
123	#5950	C₂²×D₄	1	1	yes	characteristic
124	C₂×#997	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×#997	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×#997	C₂²×D₄	1	1	yes	characteristic
127	#5935	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	#5935	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	#5935	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	#5935	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	#5933	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	#5933	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	#5933	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	#5933	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	2₋¹⁺⁶	C₂×C₄×D₃	1	2	yes	—
136	H(7,𝔽₂)	C₂×C₄×D₃	1	2	yes	—
137	C₂×#585	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂×#585	C₂×C₄×D₃	1	1	yes	characteristic
139	C₂×#585	D₃×D₄	1	1	yes	characteristic
140	C₂×#585	D₃×D₄	1	1	yes	characteristic
141	C₂×#585	D₃×D₄	1	1	yes	characteristic
142	C₂×#585	C₂³×D₃	1	1	yes	characteristic
143	C₂²×H(5,𝔽₂)	D₃×D₄	1	1	yes	characteristic
144	C₂³×#41	C₂×S₄	1	1	yes	characteristic
145	C₂³×M₁₆	C₂×S₄	1	1	yes	characteristic
146	C₂²×#117	C₂×S₄	1	1	yes	characteristic
147	C₂²×#104	C₂×S₄	1	4	yes	—
148	#997	C₂²×C₂²⋊C₄	1	1	yes	characteristic
149	#585	D₃×C₂²⋊C₄	1	1	yes	characteristic
150	#585	D₃×C₂²⋊C₄	1	1	yes	characteristic
151	#585	C₂²×C₄×D₃	1	1	yes	characteristic
152	#585	C₂×D₃×D₄	1	1	yes	characteristic
153	C₂×H(5,𝔽₂)	D₃×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×H(5,𝔽₂)	D₃×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×H(5,𝔽₂)	C₂×D₃×D₄	1	1	yes	characteristic
156	C₂²×#41	C₄×S₄	1	2	yes	—
157	C₂²×#41	C₂²×S₄	1	2	yes	—
158	C₂³×Q₈	C₄×S₄	1	1	yes	characteristic
159	C₂³×D₄	C₄×S₄	1	1	yes	characteristic
160	C₂⁴×C₄	C₂²×S₄	1	1	yes	characteristic
161	C₂²×M₁₆	C₂²×S₄	1	2	yes	—
162	C₂²×C₄²	C₂²×S₄	1	2	yes	—
163	C₂⁵	D₄×S₄	1	1	yes	characteristic
164	H(5,𝔽₂)	C₂×D₃×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×#41	#1020⋊C₂	1	1	yes	characteristic
166	C₂²×Q₈	C₂×C₄×S₄	1	2	yes	—
167	C₂²×D₄	C₂×C₄×S₄	1	2	yes	—
168	C₂³×C₄	C₂×C₄×S₄	1	1	yes	characteristic
169	C₂³×C₄	C₂×C₄×S₄	1	1	yes	characteristic
170	C₂³×C₄	D₄×S₄	1	1	yes	characteristic
171	C₂³×C₄	D₄×S₄	1	1	yes	characteristic
172	C₂³×C₄	D₄×S₄	1	1	yes	characteristic
173	C₂³×C₄	C₂³×S₄	1	1	yes	characteristic
174	C₂⁴	C₂²⋊C₄×S₄	1	1	yes	characteristic
175	C₂⁴	C₂²⋊C₄×S₄	1	1	yes	characteristic
176	C₂⁴	C₂×D₄×S₄	1	1	yes	characteristic
177	#41	#1020⋊C₄	1	2	yes	—
178	#41	C₂×#1020⋊C₂	1	2	yes	—
179	C₂×Q₈	#1020⋊C₄	1	1	yes	characteristic
180	C₂×D₄	#1020⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊C₄×S₄	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊C₄×S₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
184	C₂²×C₄	C₂²×C₄×S₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×S₄	1	1	yes	characteristic
186	C₂³	#11713200	1	1	yes	characteristic
187	C₂³	C₂×C₂²⋊C₄×S₄	1	1	yes	characteristic
188	Q₈	C₂×#1020⋊C₄	1	2	yes	—
189	D₄	C₂×#1020⋊C₄	1	2	yes	—
190	C₂×C₄	C₂×#1020⋊C₄	1	1	yes	characteristic
191	C₂×C₄	C₂×#1020⋊C₄	1	1	yes	characteristic
192	C₂×C₄	#11713200	1	1	yes	characteristic
193	C₂×C₄	#11713200	1	1	yes	characteristic
194	C₂×C₄	#11713200	1	1	yes	characteristic
195	C₂×C₄	C₂²×#1020⋊C₂	1	1	yes	characteristic
196	C₂²	#72215370	1	1	yes	characteristic
197	C₂²	#72215370	1	1	yes	characteristic
198	C₂²	C₂×#11713200	1	1	yes	characteristic
199	C₄	#72215370	1	1	yes	characteristic
200	C₄	#72215370	1	1	yes	characteristic
201	C₄	C₂×#11713200	1	1	yes	characteristic
202	C₄	C₂²×#1020⋊C₄	1	1	yes	characteristic
203	C₂	C₂×#72215370	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#11713200
commutator	#5950	C₂³×C₄
Frattini	C₂³×C₄	C₂³×S₄
Fitting	#72280922	D₃
socle	C₂	C₂×#72215370
prime radical	C₂×#72215767	C₂
cosocle	C₂×#5950	C₂⁴
hypercenter	C₂×#41	#1020⋊C₂
hypocenter	#997	C₂²×C₂²⋊C₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #72341544	C₂	semidirect
2. #72341539	C₂	semidirect
3. C₂×#72236738	C₂	semidirect
5. C₂×#11715831	C₄	semidirect
6. C₂×#11715829	C₄	semidirect
7. C₂×#11715830	C₄	semidirect
8. #72341508	C₂	semidirect
9. C₂×#11715661	C₂²	semidirect
10. #72341505	C₂	semidirect
11. C₂×#20612⋊C₃	C₂²	semidirect
12. #72341494	C₂	semidirect
13. #72341492	C₂²	semidirect
15. #72341483	C₂²	semidirect
16. #72341463	C₂	semidirect
17. #72341462	C₂²	semidirect
18. #72341458	C₂	semidirect
19. #72341455	C₂²	semidirect
20. #72341452	C₂²	semidirect
21. #72341449	C₂²	semidirect
22. #72341447	C₂²	semidirect
23. #72341343	C₂²	semidirect
24. #72280922	D₃	semidirect
27. #72236738	C₂²	semidirect
28. #72236738	C₂²	semidirect
29. #72215767	C₄	semidirect
30. #72215767	C₄	semidirect
33. C₂×#11715828	C₄	semidirect
34. #11715831	C₂×C₄	semidirect
35. #11715831	C₂×C₄	semidirect
36. #11715830	C₂×C₄	semidirect
37. #11715830	C₂×C₄	semidirect
38. #11715829	C₂×C₄	semidirect
39. #11715829	C₂×C₄	semidirect
40. #11715828	C₂×C₄	semidirect
41. #11715828	C₂×C₄	semidirect
42. #11715827	C₂×C₄	semidirect
44. C₂×#87518	C₂×C₄	semidirect
45. C₂×#87522	C₂×C₄	semidirect
60. #11715661	C₂³	semidirect
61. #11715657	C₂³	semidirect
62. #11710937	C₂×C₄	semidirect
64. C₂×#87548	C₂×C₄	semidirect
65. C₂×#87547	C₂×C₄	semidirect
67. #20612⋊C₃	C₂³	semidirect
68. #20608⋊C₃	C₂³	semidirect
70. C₂×#20612	C₂×D₃	semidirect
71. #6370734	C₂×D₃	semidirect
72. #392140	C₂×D₃	semidirect
112. #63103	C₄×D₃	semidirect
114. C₂×2₋¹⁺⁶	C₄×D₃	semidirect
115. C₂×H(7,𝔽₂)	C₄×D₃	semidirect
118. #20612	C₂²×D₃	semidirect
119. #20608	C₂²×D₃	semidirect

Maximal subgroups

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

subgroup	index
#72341544	2
#72341539	2
C₂×#72236738	2
C₂×#72215767	2
#72341508	2
#72341505	2
#72341494	2
#72341477	3
#72341463	2
#72341458	2
S₄×#421	4