Subgroups — #72262753

Order 2048.

At least 6673 conjugacy classes of subgroups; only the 422 normal subgroups are shown.

The subgroup lattice has 281 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262701	C₂	1	1	yes	maximal, characteristic, complemented
2	#72256193	C₂	1	1	yes	maximal, characteristic, complemented
3	#72255841	C₂	1	1	yes	maximal, characteristic, complemented
4	#72255793	C₂	1	1	yes	maximal, characteristic, complemented
5	#72255772	C₂	1	1	yes	maximal, characteristic, complemented
6	#72251552	C₂	1	1	yes	maximal, characteristic, complemented
7	#72251462	C₂	1	1	yes	maximal, characteristic, complemented
8	#72241271	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237037	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237003	C₂	1	1	yes	maximal, characteristic, complemented
11	#72236948	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#493474	C₂	1	1	yes	maximal, characteristic, complemented
13	#72219157	C₂	1	1	yes	maximal, characteristic
14	#72218855	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#139731	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#62817	C₂²	1	1	yes	characteristic
17	C₂×#23996	C₄	1	2	yes	complemented
18	#6437632	C₄	1	2	yes	complemented
19	C₂×#23895	C₂²	1	1	yes	characteristic
20	C₂×#23900	C₂²	1	1	yes	characteristic
21	#6431795	C₂²	1	1	yes	characteristic
22	C₂×#21778	C₂²	1	1	yes	characteristic
23	C₂×#21776	C₂²	1	1	yes	characteristic
24	C₂×#21746	C₂²	1	1	yes	characteristic
25	#556752	C₂²	1	1	yes	characteristic
26	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
27	C₂×#12705	C₂²	1	1	yes	characteristic
28	C₂×#12705	C₂²	1	1	yes	characteristic
29	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
30	C₂×#12687	C₂²	1	1	yes	characteristic
31	C₂×#12687	C₂²	1	1	yes	characteristic
32	#382434	C₂²	1	1	yes	characteristic
33	#363953	C₄	1	2	yes	complemented
34	#359591	C₄	1	2	yes	complemented
35	#359549	C₂²	1	1	yes	characteristic
36	#359501	C₂²	1	1	yes	characteristic
37	#187944	C₂²	1	1	yes	characteristic
38	#187944	C₂²	1	1	yes	characteristic
39	#187944	C₂²	1	1	yes	characteristic
40	#187944	C₂²	1	1	yes	characteristic
41	#187944	C₂²	1	1	yes	characteristic, complemented
42	#186835	C₂²	1	1	yes	characteristic, complemented
43	#146479	C₂²	1	1	yes	characteristic, complemented
44	#146475	C₂²	1	1	yes	characteristic, complemented
45	#128033	C₂²	1	1	yes	characteristic, complemented
46	#128033	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
51	#128009	C₂²	1	1	yes	characteristic, complemented
52	#128009	C₂²	1	1	yes	characteristic, complemented
53	#108238	C₂²	1	1	yes	characteristic, complemented
54	#107114	C₂²	1	1	yes	characteristic, complemented
55	#62842	C₂×C₄	1	1	yes	characteristic, complemented
56	C₂×#3490	C₂×C₄	1	2	yes	—
57	C₂×#3490	D₄	1	2	yes	—
58	C₂×#3490	D₄	1	2	yes	—
59	C₂×#3484	C₂×C₄	1	2	yes	—
60	C₂×#3484	D₄	1	2	yes	—
61	C₂×#3484	C₂³	1	1	yes	characteristic
62	C₂×#3484	C₂³	1	1	yes	characteristic
63	C₂×#3484	C₂³	1	1	yes	characteristic
64	C₂³×#120	D₄	1	2	yes	—
65	C₂³×#120	C₂³	1	1	yes	characteristic
66	C₂×#3026	C₂×C₄	1	2	yes	complemented
67	C₂²×B(4,𝔽₂)	C₂×C₄	1	2	yes	complemented
68	C₂×#2883	C₂×C₄	1	1	yes	characteristic, complemented
69	C₂×#2883	C₂×C₄	1	1	yes	characteristic, complemented
70	#34171	C₂³	1	1	yes	characteristic
71	C₂×#1913	C₂×C₄	1	2	yes	complemented
72	C₂×#1913	C₂×C₄	1	2	yes	complemented
73	#20960	C₂×C₄	1	2	yes	complemented
74	C₂×#1889	C₂×C₄	1	2	yes	complemented
75	C₂×#1889	C₂×C₄	1	2	yes	complemented
76	C₂×#1889	C₂×C₄	1	2	yes	complemented
77	C₂×#1828	C₂×C₄	1	2	yes	complemented
78	C₂×#1783	C₂³	1	1	yes	characteristic
79	C₂×#1783	C₂³	1	1	yes	characteristic
80	#20585	C₂×C₄	1	2	yes	complemented
81	C₂×#1781	C₂³	1	1	yes	characteristic
82	C₂×#1781	C₂³	1	1	yes	characteristic
83	C₂×#1781	C₂³	1	1	yes	characteristic
84	C₂×#1781	C₂³	1	1	yes	characteristic
85	C₂×#1781	C₂³	1	1	yes	characteristic
86	C₂×#1781	C₂³	1	1	yes	characteristic
87	#20354	C₂×C₄	1	1	yes	characteristic, complemented
88	#14335	C₂³	1	1	yes	characteristic
89	#14335	C₂³	1	1	yes	characteristic
90	#12687	D₄	1	2	yes	complemented
91	#12687	D₄	1	2	yes	complemented
92	#12687	D₄	1	4	yes	complemented
93	#8534	D₄	1	4	yes	complemented
94	#8534	D₄	1	4	yes	complemented
95	C₂⁷	C₂×D₄	1	1	yes	characteristic
96	C₂²×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
97	C₂²×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
98	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
99	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
100	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
101	#3484	C₂×D₄	1	2	yes	—
102	#3484	C₂×D₄	1	2	yes	—
103	#3484	#41	1	2	yes	—
104	#3484	#41	1	2	yes	—
105	C₂×#543	C₂²⋊C₄	1	2	yes	—
106	C₂×#534	C₂²⋊C₄	1	2	yes	—
107	C₂×#534	C₂²×C₄	1	2	yes	—
108	C₂×#534	C₂×D₄	1	2	yes	—
109	C₂²×#120	C₂²⋊C₄	1	2	yes	—
110	C₂²×#120	C₂²⋊C₄	1	2	yes	—
111	C₂²×#120	C₂²×C₄	1	2	yes	—
112	C₂²×#120	C₂×D₄	1	2	yes	—
113	C₂²×#120	C₂×D₄	1	2	yes	—
114	C₂²×#120	C₂×D₄	1	2	yes	—
115	C₂²×#120	C₂×D₄	1	1	yes	characteristic
116	C₂²×#120	C₂×D₄	1	1	yes	characteristic
117	C₂²×#120	C₂×D₄	1	2	yes	—
118	C₂²×#120	#41	1	1	yes	characteristic
119	C₂²×#120	#41	1	1	yes	characteristic
120	C₂²×#120	#41	1	2	yes	—
121	C₂×#518	C₂²⋊C₄	1	2	yes	—
122	C₂×#518	C₂²⋊C₄	1	2	yes	—
123	C₂×#518	C₂²×C₄	1	1	yes	characteristic
124	C₂×#518	C₂²×C₄	1	1	yes	characteristic
125	C₂×#518	C₂²×C₄	1	1	yes	characteristic
126	C₂×#518	C₂²×C₄	1	1	yes	characteristic
127	C₂×#518	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
129	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
130	C₂²×#99	C₂²×C₄	1	2	yes	—
131	C₂²×#99	C₂²×C₄	1	2	yes	—
132	#1783	C₂×D₄	1	2	yes	—
133	#1783	C₂×D₄	1	2	yes	—
134	#1781	C₂×D₄	1	2	yes	—
135	#1781	C₂×D₄	1	2	yes	—
136	#1781	C₂×D₄	1	2	yes	—
137	#1781	C₂×D₄	1	2	yes	—
138	#1438	C₂×D₄	1	2	yes	—
139	#1438	C₂×D₄	1	2	yes	—
140	C₂⁶	#99	1	2	yes	complemented
141	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
142	C₂⁶	#121	1	1	yes	characteristic
143	C₂⁶	#121	1	1	yes	characteristic
144	C₂⁶	#121	1	2	yes	—
145	C₂⁶	C₂²×D₄	1	1	yes	characteristic
146	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×H(5,𝔽₂)	#121	1	2	yes	—
148	C₂×H(5,𝔽₂)	#121	1	2	yes	—
149	C₂³×D₄	#99	1	2	yes	complemented
150	C₂³×D₄	#99	1	2	yes	complemented
151	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
154	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
155	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
156	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
157	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
158	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
159	C₂³×D₄	C₂×#41	1	1	yes	characteristic
160	C₂³×D₄	C₂×#41	1	1	yes	characteristic
161	C₂³×D₄	C₂×#41	1	1	yes	characteristic
162	C₂³×D₄	C₂×#41	1	1	yes	characteristic
163	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂⁴×C₄	#99	1	2	yes	complemented
166	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
167	#534	C₄×D₄	1	4	yes	—
168	#534	C₄×D₄	1	4	yes	—
169	C₂×#120	#99	1	4	yes	complemented
170	C₂×#120	#99	1	4	yes	complemented
171	C₂×#120	C₄×D₄	1	2	yes	—
172	C₂×#120	C₄×D₄	1	2	yes	—
173	C₂×#120	C₄×D₄	1	2	yes	—
174	C₂×#120	C₄×D₄	1	2	yes	—
175	C₂×#120	#120	1	2	yes	—
176	C₂×#120	#120	1	2	yes	—
177	C₂×#120	#120	1	4	yes	—
178	C₂×#120	#123	1	2	yes	—
179	C₂×#120	#123	1	2	yes	—
180	C₂×#120	#123	1	4	yes	—
181	#518	#121	1	2	yes	—
182	#518	#121	1	2	yes	—
183	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
184	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
185	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂²×C₂²⋊C₄	#121	1	2	yes	—
187	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
188	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
189	C₂⁵	#386	1	2	yes	—
190	C₂⁵	#386	1	2	yes	—
191	C₂⁵	C₂×#99	1	2	yes	—
192	C₂⁵	C₂×#99	1	2	yes	—
193	C₂⁵	C₂×#99	1	2	yes	—
194	C₂⁵	C₂×#99	1	2	yes	—
195	C₂⁵	C₂×#99	1	2	yes	—
196	C₂⁵	C₂×#99	1	2	yes	—
197	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
198	C₂⁵	#518	1	1	yes	characteristic
199	C₂⁵	C₂×#120	1	1	yes	characteristic
200	C₂⁵	C₂×#121	1	1	yes	characteristic
201	C₂⁵	C₂×#121	1	1	yes	characteristic
202	C₂⁵	C₂×#123	1	1	yes	characteristic
203	C₂⁵	#534	1	1	yes	characteristic
204	C₂⁵	#534	1	1	yes	characteristic
205	C₂⁵	#534	1	1	yes	characteristic
206	C₂⁵	D₄²	1	1	yes	characteristic
207	C₂⁵	#546	1	1	yes	characteristic
208	C₂⁵	#546	1	1	yes	characteristic
209	C₂⁵	#551	1	1	yes	characteristic
210	C₂²×D₄	#386	1	4	yes	—
211	C₂²×D₄	C₂×#99	1	4	yes	—
212	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
213	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
216	C₂²×D₄	C₂×#120	1	1	yes	characteristic
217	C₂²×D₄	C₂×#121	1	1	yes	characteristic
218	C₂²×D₄	C₂×#121	1	1	yes	characteristic
219	C₂²×D₄	C₂×#123	1	1	yes	characteristic
220	C₂²×D₄	D₄²	1	1	yes	characteristic
221	C₂²×D₄	D₄²	1	1	yes	characteristic
222	C₂²×D₄	#546	1	1	yes	characteristic
223	C₂²×D₄	#546	1	1	yes	characteristic
224	C₂²×D₄	#546	1	1	yes	characteristic
225	C₂²×D₄	#546	1	1	yes	characteristic
226	C₂²×D₄	#551	1	1	yes	characteristic
227	C₂²×D₄	#551	1	1	yes	characteristic
228	C₂³×C₄	C₂×#99	1	2	yes	—
229	C₂³×C₄	C₂×#99	1	2	yes	—
230	C₂³×C₄	#518	1	1	yes	characteristic
231	C₂³×C₄	#534	1	1	yes	characteristic
232	C₂³×C₄	D₄²	1	1	yes	characteristic
233	C₂³×C₄	#546	1	1	yes	characteristic
234	C₂³×C₄	#546	1	1	yes	characteristic
235	C₂³×C₄	#551	1	1	yes	characteristic
236	C₂×C₂²⋊C₄	#386	1	4	yes	—
237	C₂×C₂²⋊C₄	#386	1	2	yes	—
238	C₂×C₂²⋊C₄	#386	1	2	yes	—
239	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
240	C₂⁴	#1898	1	2	yes	—
241	C₂⁴	#1898	1	2	yes	—
242	C₂⁴	#1898	1	2	yes	—
243	C₂⁴	#1898	1	4	yes	—
244	C₂⁴	#1898	1	2	yes	—
245	C₂⁴	C₂×#386	1	1	yes	characteristic
246	C₂⁴	C₂×#386	1	1	yes	characteristic
247	C₂⁴	#2308	1	1	yes	characteristic
248	C₂⁴	#2358	1	1	yes	characteristic
249	C₂⁴	#2358	1	1	yes	characteristic
250	C₂⁴	#2410	1	1	yes	characteristic
251	C₂⁴	#2410	1	1	yes	characteristic
252	C₂⁴	#2410	1	1	yes	characteristic
253	C₂⁴	#2412	1	1	yes	characteristic
254	C₂⁴	#2412	1	1	yes	characteristic
255	C₂⁴	#2412	1	1	yes	characteristic
256	C₂⁴	C₂²×#99	1	2	yes	—
257	C₂⁴	C₂²×#99	1	2	yes	—
258	C₂²×C₄	#1898	1	4	yes	—
259	C₂²×C₄	#2358	1	1	yes	characteristic
260	C₂²×C₄	#2358	1	1	yes	characteristic
261	C₂²×C₄	#2410	1	1	yes	characteristic
262	C₂²×C₄	#2412	1	1	yes	characteristic
263	C₂³	#14134	1	1	yes	characteristic
264	C₂³	C₂×#1898	1	2	yes	—
265	C₂³	C₂×#1898	1	2	yes	—
266	C₂³	#21746	1	1	yes	characteristic
267	C₂³	#21746	1	1	yes	characteristic
268	C₂³	#21776	1	1	yes	characteristic
269	C₂³	#21776	1	1	yes	characteristic
270	C₂³	#21776	1	1	yes	characteristic
271	C₂³	#21777	1	1	yes	characteristic
272	C₂²	#213828	1	1	yes	characteristic
273	C₂²	#213836	1	1	yes	characteristic
274	C₂²	#213852	1	1	yes	characteristic
275	C₂²	#213860	1	1	yes	characteristic
276	C₂²	#374004	1	1	yes	characteristic
277	C₂²	#374008	1	1	yes	characteristic
278	C₂²	#374022	1	1	yes	characteristic
279	C₂	#72241293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241293
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72219157	C₂
LCS	G ▷ C₂³×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. #72262701	C₂	semidirect
2. #72256193	C₂	semidirect
3. #72255841	C₂	semidirect
4. #72255793	C₂	semidirect
5. #72255772	C₂	semidirect
6. #72251552	C₂	semidirect
7. #72251462	C₂	semidirect
8. #72241271	C₂	semidirect
9. #72237037	C₂	semidirect
10. #72237003	C₂	semidirect
11. #72236948	C₂	semidirect
12. C₂×#493474	C₂	semidirect
14. #72218855	C₂	semidirect
15. C₂×#139731	C₂	semidirect
17. C₂×#23996	C₄	semidirect
18. #6437632	C₄	semidirect
26. C₂×#12707	C₂²	semidirect
29. C₂×#12705	C₂²	semidirect
33. #363953	C₄	semidirect
34. #359591	C₄	semidirect
41. #187944	C₂²	semidirect
42. #186835	C₂²	semidirect
43. #146479	C₂²	semidirect
44. #146475	C₂²	semidirect
45. #128033	C₂²	semidirect
46. #128033	C₂²	semidirect
47. C₂×#7314	C₂²	semidirect
48. C₂×#7314	C₂²	semidirect
49. C₂×#7310	C₂²	semidirect
50. C₂×#7310	C₂²	semidirect
51. #128009	C₂²	semidirect
52. #128009	C₂²	semidirect
53. #108238	C₂²	semidirect
54. #107114	C₂²	semidirect
55. #62842	C₂×C₄	semidirect
66. C₂×#3026	C₂×C₄	semidirect
67. C₂²×B(4,𝔽₂)	C₂×C₄	semidirect
68. C₂×#2883	C₂×C₄	semidirect
69. C₂×#2883	C₂×C₄	semidirect
71. C₂×#1913	C₂×C₄	semidirect
72. C₂×#1913	C₂×C₄	semidirect
73. #20960	C₂×C₄	semidirect
74. C₂×#1889	C₂×C₄	semidirect
75. C₂×#1889	C₂×C₄	semidirect
76. C₂×#1889	C₂×C₄	semidirect
77. C₂×#1828	C₂×C₄	semidirect
80. #20585	C₂×C₄	semidirect
87. #20354	C₂×C₄	semidirect
90. #12687	D₄	semidirect
91. #12687	D₄	semidirect
92. #12687	D₄	semidirect
93. #8534	D₄	semidirect
94. #8534	D₄	semidirect
140. C₂⁶	#99	semidirect
149. C₂³×D₄	#99	semidirect
150. C₂³×D₄	#99	semidirect
165. C₂⁴×C₄	#99	semidirect
169. C₂×#120	#99	semidirect
170. C₂×#120	#99	semidirect

Maximal subgroups

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

subgroup	index
#72262701	2
#72256193	2
#72255841	2
#72255793	2
#72255772	2
#72251552	2
#72251462	2
#72241271	2
#72237037	2
#72237003	2
#72236948	2
C₂×#493474	2
#72219157	2
#72218855	2
C₂×#139731	2