Subgroups — #72314961

Order 2048.

At least 6013 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

The subgroup lattice has 200 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	#72258094	C₂	1	1	yes	maximal, characteristic, complemented
3	#72255778	C₂	1	1	yes	maximal, characteristic, complemented
4	#72255777	C₂	1	1	yes	maximal, characteristic, complemented
5	#72255774	C₂	1	1	yes	maximal, characteristic, complemented
6	#72255772	C₂	1	1	yes	maximal, characteristic, complemented
7	#72251533	C₂	1	1	yes	maximal, characteristic, complemented
8	#72240132	C₂	1	1	yes	maximal, characteristic, complemented
9	#72238343	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237402	C₂	1	1	yes	maximal, characteristic
11	#72225740	C₂	1	1	yes	maximal, characteristic, complemented
12	#72219158	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218521	C₂	1	1	yes	maximal, characteristic, complemented
14	#72218206	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#139731	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	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	#616961	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	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	#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	#359501	C₂²	1	1	yes	characteristic, complemented
37	#188050	C₂²	1	1	yes	characteristic
38	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
39	#187944	C₂²	1	1	yes	characteristic
40	#187944	C₂²	1	1	yes	characteristic
41	#146479	C₂²	1	1	yes	characteristic, complemented
42	#128033	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	#128009	C₂²	1	1	yes	characteristic, complemented
49	#128009	C₂²	1	1	yes	characteristic, complemented
50	#107114	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	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	#20585	D₄	1	2	yes	complemented
64	C₂×#1781	C₂³	1	1	yes	characteristic
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	#15259	D₄	1	2	yes	—
73	#15258	D₄	1	2	yes	—
74	#14335	C₂³	1	1	yes	characteristic
75	C₂⁷	C₂×D₄	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	C₂³×#41	#41	1	1	yes	characteristic
79	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
80	#3484	C₂×D₄	1	2	yes	—
81	#3484	#41	1	2	yes	—
82	C₂×#534	C₂×D₄	1	2	yes	—
83	C₂²×#127	C₂×D₄	1	1	yes	characteristic
84	C₂²×#127	#41	1	1	yes	characteristic
85	C₂²×#124	C₂×D₄	1	1	yes	characteristic
86	C₂²×#124	#41	1	1	yes	characteristic
87	C₂²×#120	C₂×D₄	1	2	yes	—
88	C₂²×#120	C₂×D₄	1	2	yes	—
89	C₂²×#120	C₂×D₄	1	2	yes	—
90	C₂×#518	C₂×D₄	1	1	yes	characteristic
91	C₂×#518	C₂×D₄	1	1	yes	characteristic
92	C₂×#518	C₂×D₄	1	1	yes	characteristic
93	C₂×#518	#41	1	1	yes	characteristic
94	C₂²×#117	#41	1	1	yes	characteristic
95	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×#99	C₂×D₄	1	2	yes	—
97	C₂²×#99	C₂×D₄	1	2	yes	—
98	C₂²×#99	C₂×D₄	1	2	yes	—
99	C₂×#379	C₂×D₄	1	2	yes	—
100	#1781	C₂×D₄	1	2	yes	—
101	#1438	C₂×D₄	1	2	yes	—
102	C₂⁶	#120	1	2	yes	—
103	C₂⁶	C₂²×D₄	1	1	yes	characteristic
104	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂×H(5,𝔽₂)	#123	1	2	yes	complemented
106	C₂³×Q₈	#123	1	1	yes	characteristic, complemented
107	C₂³×D₄	#120	1	2	yes	—
108	C₂³×D₄	#123	1	1	yes	characteristic, complemented
109	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
110	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
111	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
112	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
113	C₂³×D₄	C₂×#41	1	1	yes	characteristic
114	C₂³×D₄	C₂×#41	1	1	yes	characteristic
115	C₂³×D₄	C₂×#41	1	1	yes	characteristic
116	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
122	#518	#123	1	2	yes	complemented
123	C₂²×C₂²⋊C₄	#120	1	2	yes	—
124	C₂²×C₂²⋊C₄	#120	1	2	yes	—
125	C₂²×C₄²	#123	1	1	yes	characteristic, complemented
126	C₂²×C₄²	#123	1	1	yes	characteristic, complemented
127	C₂⁵	B(4,𝔽₂)	1	2	yes	—
128	C₂⁵	C₂×#120	1	1	yes	characteristic
129	C₂⁵	C₂×#120	1	1	yes	characteristic
130	C₂⁵	#534	1	1	yes	characteristic
131	C₂⁵	#534	1	1	yes	characteristic
132	C₂⁵	#535	1	1	yes	characteristic
133	C₂⁵	#538	1	1	yes	characteristic
134	C₂⁵	#546	1	1	yes	characteristic
135	C₂⁵	#546	1	1	yes	characteristic
136	C₂⁵	#546	1	1	yes	characteristic
137	C₂⁵	#546	1	1	yes	characteristic
138	C₂⁵	#546	1	1	yes	characteristic
139	C₂⁵	#546	1	1	yes	characteristic
140	C₂⁵	#560	1	1	yes	characteristic
141	C₂²×D₄	C₂×#123	1	1	yes	characteristic
142	C₂²×D₄	#525	1	1	yes	characteristic
143	C₂²×D₄	#534	1	1	yes	characteristic
144	C₂²×D₄	#535	1	1	yes	characteristic
145	C₂²×D₄	#538	1	1	yes	characteristic
146	C₂²×D₄	#538	1	1	yes	characteristic
147	C₂²×D₄	D₄²	1	1	yes	characteristic
148	C₂²×D₄	D₄²	1	1	yes	characteristic
149	C₂²×D₄	D₄²	1	1	yes	characteristic
150	C₂²×D₄	D₄²	1	1	yes	characteristic
151	C₂²×D₄	#546	1	1	yes	characteristic
152	C₂²×D₄	#546	1	1	yes	characteristic
153	C₂²×D₄	#546	1	1	yes	characteristic
154	C₂²×D₄	#546	1	1	yes	characteristic
155	C₂²×D₄	#551	1	1	yes	characteristic
156	C₂²×D₄	#560	1	1	yes	characteristic
157	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
158	C₂³×C₄	C₂×#123	1	1	yes	characteristic
159	C₂³×C₄	#525	1	1	yes	characteristic
160	C₂³×C₄	#538	1	1	yes	characteristic
161	C₂³×C₄	#546	1	1	yes	characteristic
162	C₂³×C₄	#546	1	1	yes	characteristic
163	C₂³×C₄	#551	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
165	C₂⁴	#2403	1	1	yes	characteristic
166	C₂⁴	#2404	1	1	yes	characteristic
167	C₂⁴	#2404	1	1	yes	characteristic
168	C₂⁴	#2405	1	1	yes	characteristic
169	C₂⁴	#2405	1	1	yes	characteristic
170	C₂⁴	#2418	1	1	yes	characteristic
171	C₂⁴	#2678	1	1	yes	characteristic
172	C₂⁴	#2678	1	1	yes	characteristic
173	C₂⁴	#2678	1	1	yes	characteristic
174	C₂⁴	#2697	1	1	yes	characteristic
175	C₂⁴	#2697	1	1	yes	characteristic
176	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
177	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
178	C₂²×C₄	#2418	1	1	yes	characteristic
179	C₂²×C₄	#2418	1	1	yes	characteristic
180	C₂²×C₄	#2418	1	1	yes	characteristic
181	C₂²×C₄	#2678	1	1	yes	characteristic
182	C₂³	#15948	1	1	yes	characteristic
183	C₂³	#23900	1	2	yes	—
184	C₂³	#23984	1	2	yes	—
185	C₂³	#23984	1	1	yes	characteristic
186	C₂³	#23984	1	1	yes	characteristic
187	C₂³	#23996	1	1	yes	characteristic
188	C₂³	#23996	1	1	yes	characteristic
189	C₂³	#23996	1	1	yes	characteristic
190	C₂³	#23997	1	1	yes	characteristic
191	C₂²	#251827	1	1	yes	characteristic
192	C₂²	#251838	1	1	yes	characteristic
193	C₂²	#251860	1	1	yes	characteristic
194	C₂²	#251871	1	1	yes	characteristic
195	C₂²	#449521	1	1	yes	characteristic
196	C₂²	#449578	1	1	yes	characteristic
197	C₂²	#449580	1	1	yes	characteristic
198	C₂	#72251429	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72251429
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72237402	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. #72258094	C₂	semidirect
3. #72255778	C₂	semidirect
4. #72255777	C₂	semidirect
5. #72255774	C₂	semidirect
6. #72255772	C₂	semidirect
7. #72251533	C₂	semidirect
8. #72240132	C₂	semidirect
9. #72238343	C₂	semidirect
11. #72225740	C₂	semidirect
12. #72219158	C₂	semidirect
13. #72218521	C₂	semidirect
14. #72218206	C₂	semidirect
15. 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
38. C₂×#8541	C₂²	semidirect
41. #146479	C₂²	semidirect
42. #128033	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. #128009	C₂²	semidirect
49. #128009	C₂²	semidirect
50. #107114	C₂²	semidirect
58. C₂×#1913	D₄	semidirect
60. #20960	D₄	semidirect
61. C₂×#1828	D₄	semidirect
63. #20585	D₄	semidirect
105. C₂×H(5,𝔽₂)	#123	semidirect
106. C₂³×Q₈	#123	semidirect
108. C₂³×D₄	#123	semidirect
122. #518	#123	semidirect
125. C₂²×C₄²	#123	semidirect
126. C₂²×C₄²	#123	semidirect

Maximal subgroups

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

subgroup	index
#72314748	2
#72258094	2
#72255778	2
#72255777	2
#72255774	2
#72255772	2
#72251533	2
#72240132	2
#72238343	2
#72237402	2
#72225740	2
#72219158	2
#72218521	2
#72218206	2
C₂×#139731	2