Subgroups — #72314953

Order 2048.

At least 6047 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

The subgroup lattice has 196 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72314855	C₂	1	1	yes	maximal, characteristic, complemented
2	#72312218	C₂	1	1	yes	maximal, characteristic, complemented
3	#72310358	C₂	1	1	yes	maximal, characteristic, complemented
4	#72310322	C₂	1	1	yes	maximal, characteristic, complemented
5	#72255867	C₂	1	1	yes	maximal, characteristic, complemented
6	#72238976	C₂	1	1	yes	maximal, characteristic, complemented
7	#72225740	C₂	1	1	yes	maximal, characteristic, complemented
8	#72219438	C₂	1	1	yes	maximal, characteristic, complemented
9	#72219393	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#139716	C₂	1	1	yes	maximal, characteristic, complemented
11	#72219162	C₂	1	1	yes	maximal, characteristic, complemented
12	#72218475	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218317	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#139731	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#347042	C₂	1	1	yes	maximal, characteristic
16	C₂×#62817	C₂²	1	1	yes	characteristic
17	C₂×#23900	C₂²	1	1	yes	characteristic
18	C₂×#23900	C₂²	1	1	yes	characteristic
19	C₂×#21746	C₂²	1	1	yes	characteristic, complemented
20	C₂×#15947	C₂²	1	1	yes	characteristic
21	C₂×#12707	C₂²	1	1	yes	characteristic
22	C₂×#12705	C₂²	1	1	yes	characteristic
23	C₂×#12705	C₂²	1	1	yes	characteristic
24	C₂×#12705	C₂²	1	1	yes	characteristic
25	C₂×#12687	C₂²	1	1	yes	characteristic
26	C₂×#12687	C₂²	1	1	yes	characteristic
27	C₂×#12687	C₂²	1	1	yes	characteristic
28	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
29	#392030	C₂²	1	1	yes	characteristic, complemented
30	#382544	C₂²	1	1	yes	characteristic, complemented
31	#382436	C₂²	1	1	yes	characteristic
32	#382434	C₂²	1	1	yes	characteristic
33	#359558	C₂²	1	1	yes	characteristic
34	#359549	C₂²	1	1	yes	characteristic
35	#359501	C₂²	1	1	yes	characteristic
36	#359501	C₂²	1	1	yes	characteristic, complemented
37	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
38	C₂×#8541	C₂²	1	1	yes	characteristic
39	C₂×#8534	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	#146475	C₂²	1	1	yes	characteristic, complemented
42	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
43	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
44	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
50	#107130	C₂²	1	1	yes	characteristic, complemented
51	C₂×#3484	C₂³	1	1	yes	characteristic
52	C₂×#3484	C₂³	1	1	yes	characteristic
53	C₂³×#120	C₂³	1	1	yes	characteristic
54	C₂³×#120	C₂³	1	1	yes	characteristic
55	C₂×#2846	D₄	1	2	yes	—
56	#34171	C₂³	1	1	yes	characteristic
57	#23900	D₄	1	4	yes	—
58	C₂×#1913	D₄	1	2	yes	complemented
59	C₂×#1898	D₄	1	2	yes	—
60	#20960	D₄	1	2	yes	complemented
61	C₂×#1828	D₄	1	2	yes	complemented
62	C₂×#1783	C₂³	1	1	yes	characteristic
63	C₂×#1783	C₂³	1	1	yes	characteristic
64	C₂×#1783	C₂³	1	1	yes	characteristic
65	#20585	D₄	1	2	yes	complemented
66	C₂×#1781	C₂³	1	1	yes	characteristic
67	C₂×#1781	C₂³	1	1	yes	characteristic
68	C₂×#1781	C₂³	1	1	yes	characteristic
69	C₂×#1781	C₂³	1	1	yes	characteristic
70	C₂×#1781	C₂³	1	1	yes	characteristic
71	C₂×#1781	C₂³	1	1	yes	characteristic
72	#15947	D₄	1	4	yes	—
73	C₂×#1438	C₂³	1	1	yes	characteristic
74	C₂⁷	C₂×D₄	1	1	yes	characteristic
75	C₂⁷	#41	1	1	yes	characteristic
76	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
77	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
78	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
79	#3484	C₂×D₄	1	2	yes	—
80	#3484	#41	1	2	yes	—
81	C₂×#534	C₂×D₄	1	2	yes	—
82	C₂²×#120	C₂×D₄	1	2	yes	—
83	C₂²×#120	C₂×D₄	1	2	yes	—
84	C₂²×#120	C₂×D₄	1	2	yes	—
85	C₂²×#120	C₂×D₄	1	2	yes	—
86	C₂²×#120	#41	1	2	yes	—
87	C₂×#518	C₂×D₄	1	1	yes	characteristic
88	C₂×#518	C₂×D₄	1	1	yes	characteristic
89	C₂×#518	C₂×D₄	1	1	yes	characteristic
90	C₂×#518	#41	1	1	yes	characteristic
91	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
92	C₂³×C₂²⋊C₄	#41	1	1	yes	characteristic
93	C₂²×#99	C₂×D₄	1	2	yes	—
94	C₂²×#99	C₂×D₄	1	2	yes	—
95	C₂²×#99	C₂×D₄	1	2	yes	—
96	C₂×#379	C₂×D₄	1	2	yes	—
97	#1781	C₂×D₄	1	2	yes	—
98	#1438	C₂×D₄	1	2	yes	—
99	C₂⁶	#120	1	2	yes	—
100	C₂⁶	#123	1	2	yes	complemented
101	C₂⁶	C₂²×D₄	1	1	yes	characteristic
102	C₂⁶	C₂×#41	1	1	yes	characteristic
103	C₂×H(5,𝔽₂)	#123	1	2	yes	complemented
104	C₂³×D₄	#120	1	2	yes	—
105	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
106	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
107	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
108	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
109	C₂³×D₄	C₂×#41	1	1	yes	characteristic
110	C₂³×D₄	C₂×#41	1	1	yes	characteristic
111	C₂³×D₄	C₂×#41	1	1	yes	characteristic
112	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
118	#518	#123	1	2	yes	complemented
119	C₂²×C₂²⋊C₄	#120	1	2	yes	—
120	C₂²×C₂²⋊C₄	#120	1	2	yes	—
121	C₂²×C₂²⋊C₄	#123	1	2	yes	complemented
122	C₂⁵	B(4,𝔽₂)	1	2	yes	—
123	C₂⁵	C₂×#120	1	1	yes	characteristic
124	C₂⁵	C₂×#120	1	1	yes	characteristic
125	C₂⁵	C₂×#123	1	1	yes	characteristic
126	C₂⁵	#525	1	1	yes	characteristic
127	C₂⁵	#534	1	1	yes	characteristic
128	C₂⁵	#538	1	1	yes	characteristic
129	C₂⁵	#546	1	1	yes	characteristic
130	C₂⁵	#546	1	1	yes	characteristic
131	C₂⁵	#546	1	1	yes	characteristic
132	C₂⁵	#546	1	1	yes	characteristic
133	C₂⁵	#546	1	1	yes	characteristic
134	C₂⁵	#546	1	1	yes	characteristic
135	C₂⁵	#551	1	1	yes	characteristic
136	C₂²×D₄	C₂×#123	1	1	yes	characteristic
137	C₂²×D₄	#525	1	1	yes	characteristic
138	C₂²×D₄	#534	1	1	yes	characteristic
139	C₂²×D₄	#535	1	1	yes	characteristic
140	C₂²×D₄	#538	1	1	yes	characteristic
141	C₂²×D₄	#538	1	1	yes	characteristic
142	C₂²×D₄	D₄²	1	1	yes	characteristic
143	C₂²×D₄	D₄²	1	1	yes	characteristic
144	C₂²×D₄	D₄²	1	1	yes	characteristic
145	C₂²×D₄	D₄²	1	1	yes	characteristic
146	C₂²×D₄	#546	1	1	yes	characteristic
147	C₂²×D₄	#546	1	1	yes	characteristic
148	C₂²×D₄	#546	1	1	yes	characteristic
149	C₂²×D₄	#546	1	1	yes	characteristic
150	C₂²×D₄	#551	1	1	yes	characteristic
151	C₂²×D₄	#560	1	1	yes	characteristic
152	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
153	C₂³×C₄	#534	1	1	yes	characteristic
154	C₂³×C₄	#535	1	1	yes	characteristic
155	C₂³×C₄	#538	1	1	yes	characteristic
156	C₂³×C₄	#546	1	1	yes	characteristic
157	C₂³×C₄	#546	1	1	yes	characteristic
158	C₂³×C₄	#560	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
160	C₂⁴	#2403	1	1	yes	characteristic
161	C₂⁴	#2404	1	1	yes	characteristic
162	C₂⁴	#2404	1	1	yes	characteristic
163	C₂⁴	#2405	1	1	yes	characteristic
164	C₂⁴	#2405	1	1	yes	characteristic
165	C₂⁴	#2418	1	1	yes	characteristic
166	C₂⁴	#2418	1	1	yes	characteristic
167	C₂⁴	#2418	1	1	yes	characteristic
168	C₂⁴	#2678	1	1	yes	characteristic
169	C₂⁴	#2697	1	1	yes	characteristic
170	C₂⁴	#2697	1	1	yes	characteristic
171	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
172	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
173	C₂²×C₄	#2418	1	1	yes	characteristic
174	C₂²×C₄	#2678	1	1	yes	characteristic
175	C₂²×C₄	#2678	1	1	yes	characteristic
176	C₂²×C₄	#2678	1	1	yes	characteristic
177	C₂³	#15948	1	1	yes	characteristic
178	C₂³	#23900	1	2	yes	—
179	C₂³	#23984	1	1	yes	characteristic
180	C₂³	#23984	1	2	yes	—
181	C₂³	#23984	1	1	yes	characteristic
182	C₂³	#23996	1	1	yes	characteristic
183	C₂³	#23996	1	1	yes	characteristic
184	C₂³	#23996	1	1	yes	characteristic
185	C₂³	#23997	1	1	yes	characteristic
186	C₂²	#251827	1	1	yes	characteristic
187	C₂²	#251838	1	1	yes	characteristic
188	C₂²	#251860	1	1	yes	characteristic
189	C₂²	#251871	1	1	yes	characteristic
190	C₂²	#449521	1	1	yes	characteristic
191	C₂²	#449578	1	1	yes	characteristic
192	C₂²	#449580	1	1	yes	characteristic
193	C₂	#72216344	1	2	yes	—
194	C₂	#72251429	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#251827
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	C₂×#347042	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72314855	C₂	semidirect
2. #72312218	C₂	semidirect
3. #72310358	C₂	semidirect
4. #72310322	C₂	semidirect
5. #72255867	C₂	semidirect
6. #72238976	C₂	semidirect
7. #72225740	C₂	semidirect
8. #72219438	C₂	semidirect
9. #72219393	C₂	semidirect
10. C₂×#139716	C₂	semidirect
11. #72219162	C₂	semidirect
12. #72218475	C₂	semidirect
13. #72218317	C₂	semidirect
14. C₂×#139731	C₂	semidirect
19. C₂×#21746	C₂²	semidirect
28. C₂×#12687	C₂²	semidirect
29. #392030	C₂²	semidirect
30. #382544	C₂²	semidirect
36. #359501	C₂²	semidirect
37. C₂×#8547	C₂²	semidirect
41. #146475	C₂²	semidirect
42. C₂×#7304	C₂²	semidirect
43. C₂×#7304	C₂²	semidirect
44. C₂×#7316	C₂²	semidirect
45. C₂×#7314	C₂²	semidirect
46. C₂×#7314	C₂²	semidirect
47. C₂×#7311	C₂²	semidirect
48. C₂×#7310	C₂²	semidirect
49. C₂×#7307	C₂²	semidirect
50. #107130	C₂²	semidirect
58. C₂×#1913	D₄	semidirect
60. #20960	D₄	semidirect
61. C₂×#1828	D₄	semidirect
65. #20585	D₄	semidirect
100. C₂⁶	#123	semidirect
103. C₂×H(5,𝔽₂)	#123	semidirect
118. #518	#123	semidirect
121. C₂²×C₂²⋊C₄	#123	semidirect

Maximal subgroups

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

subgroup	index
#72314855	2
#72312218	2
#72310358	2
#72310322	2
#72255867	2
#72238976	2
#72225740	2
#72219438	2
#72219393	2
C₂×#139716	2
#72219162	2
#72218475	2
#72218317	2
C₂×#139731	2
C₂×#347042	2