Subgroups — #72315059

Order 2048.

At least 6017 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

The subgroup lattice has 208 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72314748	C₂	1	1	yes	maximal, characteristic, complemented
2	#72314729	C₂	1	1	yes	maximal, characteristic, complemented
3	#72310319	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266835	C₂	1	1	yes	maximal, characteristic
5	#72266378	C₂	1	1	yes	maximal, characteristic, complemented
6	#72264765	C₂	1	1	yes	maximal, characteristic, complemented
7	#72264754	C₂	1	1	yes	maximal, characteristic, complemented
8	#72259117	C₂	1	1	yes	maximal, characteristic, complemented
9	#72258094	C₂	1	1	yes	maximal, characteristic, complemented
10	#72251533	C₂	1	1	yes	maximal, characteristic, complemented
11	#72238343	C₂	1	1	yes	maximal, characteristic, complemented
12	#72237900	C₂	1	1	yes	maximal, characteristic, complemented
13	#72237473	C₂	1	1	yes	maximal, characteristic, complemented
14	#72237374	C₂	1	1	yes	maximal, characteristic, complemented
15	#72226366	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic
17	#6437621	C₂²	1	1	yes	characteristic
18	#6437621	C₂²	1	1	yes	characteristic
19	#6374128	C₂²	1	1	yes	characteristic, complemented
20	#763333	C₂²	1	1	yes	characteristic
21	C₂×#12707	C₂²	1	1	yes	characteristic
22	C₂×#12707	C₂²	1	1	yes	characteristic
23	C₂×#12705	C₂²	1	1	yes	characteristic
24	C₂×#12705	C₂²	1	1	yes	characteristic
25	#493370	C₂²	1	1	yes	characteristic
26	#493370	C₂²	1	1	yes	characteristic
27	#493370	C₂²	1	1	yes	characteristic
28	#493357	C₂²	1	1	yes	characteristic, complemented
29	#392030	C₂²	1	1	yes	characteristic, complemented
30	#382544	C₂²	1	1	yes	characteristic, complemented
31	#382434	C₂²	1	1	yes	characteristic
32	#382434	C₂²	1	1	yes	characteristic
33	#359549	C₂²	1	1	yes	characteristic
34	#359549	C₂²	1	1	yes	characteristic
35	#359509	C₂²	1	1	yes	characteristic
36	#359442	C₂²	1	1	yes	characteristic, complemented
37	#188050	C₂²	1	1	yes	characteristic
38	#188049	C₂²	1	1	yes	characteristic, complemented
39	C₂×#8534	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	#146479	C₂²	1	1	yes	characteristic, complemented
42	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
43	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
44	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
50	#108222	C₂²	1	1	yes	characteristic, complemented
51	C₂×#3484	C₂³	1	1	yes	characteristic
52	C₂×#3484	C₂³	1	1	yes	characteristic
53	C₂²×#535	C₂³	1	1	yes	characteristic
54	C₂²×#534	C₂³	1	1	yes	characteristic
55	#34980	D₄	1	2	yes	—
56	#34171	C₂³	1	1	yes	characteristic
57	#23901	D₄	1	2	yes	—
58	#23900	D₄	1	2	yes	—
59	C₂×#1913	D₄	1	2	yes	complemented
60	C₂×#1898	D₄	1	2	yes	—
61	#20960	D₄	1	2	yes	complemented
62	C₂×#1828	D₄	1	2	yes	complemented
63	C₂×#1783	C₂³	1	1	yes	characteristic
64	#20585	D₄	1	2	yes	complemented
65	C₂×#1781	C₂³	1	1	yes	characteristic
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	C₂×#1781	C₂³	1	1	yes	characteristic
73	#15262	D₄	1	1	yes	characteristic
74	#15261	D₄	1	1	yes	characteristic
75	#15259	D₄	1	1	yes	characteristic
76	#15258	D₄	1	1	yes	characteristic
77	C₂×#1438	C₂³	1	1	yes	characteristic
78	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
79	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
80	C₂³×#41	#41	1	1	yes	characteristic
81	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
82	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
83	#3484	C₂×D₄	1	2	yes	—
84	#3484	#41	1	2	yes	—
85	C₂×#534	C₂×D₄	1	2	yes	—
86	C₂²×#127	C₂×D₄	1	1	yes	characteristic
87	C₂²×#127	#41	1	1	yes	characteristic
88	C₂²×#124	C₂×D₄	1	1	yes	characteristic
89	C₂²×#124	#41	1	1	yes	characteristic
90	C₂²×#123	C₂×D₄	1	1	yes	characteristic
91	C₂²×#123	C₂×D₄	1	1	yes	characteristic
92	C₂²×#120	C₂×D₄	1	2	yes	—
93	C₂²×#120	C₂×D₄	1	1	yes	characteristic
94	C₂²×#120	C₂×D₄	1	1	yes	characteristic
95	C₂×#518	C₂×D₄	1	1	yes	characteristic
96	C₂×#518	C₂×D₄	1	1	yes	characteristic
97	C₂×#518	C₂×D₄	1	1	yes	characteristic
98	C₂×#518	#41	1	1	yes	characteristic
99	C₂²×#117	#41	1	1	yes	characteristic
100	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
101	C₂²×#99	C₂×D₄	1	2	yes	—
102	C₂²×#99	C₂×D₄	1	2	yes	—
103	C₂²×#99	C₂×D₄	1	2	yes	—
104	C₂×#379	C₂×D₄	1	2	yes	—
105	#1781	C₂×D₄	1	2	yes	—
106	#1438	C₂×D₄	1	1	yes	characteristic
107	#1438	C₂×D₄	1	1	yes	characteristic
108	C₂⁶	#120	1	1	yes	characteristic
109	C₂⁶	C₂²×D₄	1	1	yes	characteristic
110	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×H(5,𝔽₂)	#123	1	2	yes	—
112	C₂³×Q₈	#123	1	1	yes	characteristic
113	C₂³×D₄	#120	1	2	yes	—
114	C₂³×D₄	#123	1	1	yes	characteristic
115	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
116	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
117	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
118	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
119	C₂³×D₄	C₂×#41	1	1	yes	characteristic
120	C₂³×D₄	C₂×#41	1	1	yes	characteristic
121	C₂³×D₄	C₂×#41	1	1	yes	characteristic
122	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂⁴×C₄	#120	1	1	yes	characteristic
128	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
129	#518	#123	1	2	yes	—
130	C₂²×C₂²⋊C₄	#120	1	2	yes	—
131	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
132	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
133	C₂²×C₄²	#123	1	1	yes	characteristic
134	C₂²×C₄²	#123	1	1	yes	characteristic
135	C₂⁵	B(4,𝔽₂)	1	2	yes	—
136	C₂⁵	C₂×#120	1	1	yes	characteristic
137	C₂⁵	C₂×#120	1	1	yes	characteristic
138	C₂⁵	#534	1	1	yes	characteristic
139	C₂⁵	#534	1	1	yes	characteristic
140	C₂⁵	#535	1	1	yes	characteristic
141	C₂⁵	#538	1	1	yes	characteristic
142	C₂⁵	#546	1	1	yes	characteristic
143	C₂⁵	#546	1	1	yes	characteristic
144	C₂⁵	#546	1	1	yes	characteristic
145	C₂⁵	#546	1	1	yes	characteristic
146	C₂⁵	#546	1	1	yes	characteristic
147	C₂⁵	#546	1	1	yes	characteristic
148	C₂⁵	#560	1	1	yes	characteristic
149	C₂²×D₄	C₂×#123	1	1	yes	characteristic
150	C₂²×D₄	#525	1	1	yes	characteristic
151	C₂²×D₄	#534	1	1	yes	characteristic
152	C₂²×D₄	#535	1	1	yes	characteristic
153	C₂²×D₄	#538	1	1	yes	characteristic
154	C₂²×D₄	#538	1	1	yes	characteristic
155	C₂²×D₄	D₄²	1	1	yes	characteristic
156	C₂²×D₄	D₄²	1	1	yes	characteristic
157	C₂²×D₄	D₄²	1	1	yes	characteristic
158	C₂²×D₄	D₄²	1	1	yes	characteristic
159	C₂²×D₄	#546	1	1	yes	characteristic
160	C₂²×D₄	#546	1	1	yes	characteristic
161	C₂²×D₄	#546	1	1	yes	characteristic
162	C₂²×D₄	#546	1	1	yes	characteristic
163	C₂²×D₄	#551	1	1	yes	characteristic
164	C₂²×D₄	#560	1	1	yes	characteristic
165	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
166	C₂³×C₄	C₂×#123	1	1	yes	characteristic
167	C₂³×C₄	#525	1	1	yes	characteristic
168	C₂³×C₄	#538	1	1	yes	characteristic
169	C₂³×C₄	#546	1	1	yes	characteristic
170	C₂³×C₄	#546	1	1	yes	characteristic
171	C₂³×C₄	#551	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
173	C₂⁴	#2403	1	1	yes	characteristic
174	C₂⁴	#2404	1	1	yes	characteristic
175	C₂⁴	#2404	1	1	yes	characteristic
176	C₂⁴	#2405	1	1	yes	characteristic
177	C₂⁴	#2405	1	1	yes	characteristic
178	C₂⁴	#2418	1	1	yes	characteristic
179	C₂⁴	#2678	1	1	yes	characteristic
180	C₂⁴	#2678	1	1	yes	characteristic
181	C₂⁴	#2678	1	1	yes	characteristic
182	C₂⁴	#2697	1	1	yes	characteristic
183	C₂⁴	#2697	1	1	yes	characteristic
184	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
185	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
186	C₂²×C₄	#2418	1	1	yes	characteristic
187	C₂²×C₄	#2418	1	1	yes	characteristic
188	C₂²×C₄	#2418	1	1	yes	characteristic
189	C₂²×C₄	#2678	1	1	yes	characteristic
190	C₂³	#15948	1	1	yes	characteristic
191	C₂³	#23900	1	2	yes	—
192	C₂³	#23984	1	2	yes	—
193	C₂³	#23984	1	1	yes	characteristic
194	C₂³	#23984	1	1	yes	characteristic
195	C₂³	#23996	1	1	yes	characteristic
196	C₂³	#23996	1	1	yes	characteristic
197	C₂³	#23996	1	1	yes	characteristic
198	C₂³	#23997	1	1	yes	characteristic
199	C₂²	#251828	1	1	yes	characteristic
200	C₂²	#251837	1	1	yes	characteristic
201	C₂²	#251859	1	1	yes	characteristic
202	C₂²	#251872	1	1	yes	characteristic
203	C₂²	#449521	1	1	yes	characteristic
204	C₂²	#449578	1	1	yes	characteristic
205	C₂²	#449580	1	1	yes	characteristic
206	C₂	#72251438	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72251438
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72266835	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ 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. #72314748	C₂	semidirect
2. #72314729	C₂	semidirect
3. #72310319	C₂	semidirect
5. #72266378	C₂	semidirect
6. #72264765	C₂	semidirect
7. #72264754	C₂	semidirect
8. #72259117	C₂	semidirect
9. #72258094	C₂	semidirect
10. #72251533	C₂	semidirect
11. #72238343	C₂	semidirect
12. #72237900	C₂	semidirect
13. #72237473	C₂	semidirect
14. #72237374	C₂	semidirect
15. #72226366	C₂	semidirect
19. #6374128	C₂²	semidirect
28. #493357	C₂²	semidirect
29. #392030	C₂²	semidirect
30. #382544	C₂²	semidirect
36. #359442	C₂²	semidirect
38. #188049	C₂²	semidirect
41. #146479	C₂²	semidirect
42. C₂×#7304	C₂²	semidirect
43. C₂×#7316	C₂²	semidirect
44. C₂×#7314	C₂²	semidirect
45. C₂×#7310	C₂²	semidirect
46. C₂×#7310	C₂²	semidirect
47. C₂×#7310	C₂²	semidirect
48. C₂×#7307	C₂²	semidirect
49. C₂×#7307	C₂²	semidirect
50. #108222	C₂²	semidirect
59. C₂×#1913	D₄	semidirect
61. #20960	D₄	semidirect
62. C₂×#1828	D₄	semidirect
64. #20585	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72314748	2
#72314729	2
#72310319	2
#72266835	2
#72266378	2
#72264765	2
#72264754	2
#72259117	2
#72258094	2
#72251533	2
#72238343	2
#72237900	2
#72237473	2
#72237374	2
#72226366	2