Subgroups — #72310367

Order 2048.

At least 6081 conjugacy classes of subgroups; only the 422 normal subgroups are shown.

The subgroup lattice has 279 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72310366	C₂	1	1	yes	maximal, characteristic, complemented
2	#72310319	C₂	1	1	yes	maximal, characteristic, complemented
3	#72309552	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266863	C₂	1	1	yes	maximal, characteristic, complemented
5	#72260537	C₂	1	1	yes	maximal, characteristic, complemented
6	#72258028	C₂	1	1	yes	maximal, characteristic, complemented
7	#72256193	C₂	1	1	yes	maximal, characteristic, complemented
8	#72255793	C₂	1	1	yes	maximal, characteristic, complemented
9	#72251552	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237638	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237637	C₂	1	1	yes	maximal, characteristic, complemented
12	#72237003	C₂	1	1	yes	maximal, characteristic, complemented
13	#72234953	C₂	1	1	yes	maximal, characteristic
14	#72219159	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#139743	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586404	C₂²	1	1	yes	characteristic
17	#6451863	C₄	1	2	yes	complemented
18	#6437629	C₄	1	2	yes	complemented
19	C₂×#23895	C₂²	1	1	yes	characteristic
20	#6437621	C₂²	1	1	yes	characteristic
21	#6431795	C₂²	1	1	yes	characteristic
22	#6374251	C₂²	1	1	yes	characteristic
23	C₂×#21778	C₂²	1	1	yes	characteristic
24	C₂×#21776	C₂²	1	1	yes	characteristic
25	#552677	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	#493357	C₂²	1	1	yes	characteristic
31	#493357	C₂²	1	1	yes	characteristic
32	#382434	C₂²	1	1	yes	characteristic
33	#363969	C₄	1	2	yes	complemented
34	#359549	C₂²	1	1	yes	characteristic
35	#359528	C₄	1	2	yes	complemented
36	#359442	C₂²	1	1	yes	characteristic
37	#187944	C₂²	1	1	yes	characteristic
38	#187944	C₂²	1	1	yes	characteristic
39	C₂×#8534	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	C₂×#8534	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	C₂×#7304	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	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
53	#107130	C₂²	1	1	yes	characteristic, complemented
54	#107114	C₂²	1	1	yes	characteristic, complemented
55	#62844	C₂×C₄	1	2	yes	—
56	#62844	D₄	1	2	yes	—
57	#62842	C₂×C₄	1	2	yes	—
58	#62842	C₂×C₄	1	1	yes	characteristic, complemented
59	#62842	D₄	1	2	yes	—
60	C₂×#3490	D₄	1	2	yes	—
61	C₂×#3484	D₄	1	2	yes	—
62	C₂×#3484	C₂³	1	1	yes	characteristic
63	C₂×#3484	C₂³	1	1	yes	characteristic
64	C₂×#3484	C₂³	1	1	yes	characteristic
65	C₂²×#534	C₂³	1	1	yes	characteristic
66	C₂×#3026	C₂×C₄	1	2	yes	—
67	C₂×#3026	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	—
72	C₂×#1913	C₂×C₄	1	2	yes	—
73	C₂×#1913	C₂×C₄	1	2	yes	complemented
74	C₂×#1913	C₂×C₄	1	2	yes	complemented
75	#20960	C₂×C₄	1	2	yes	—
76	#20960	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	C₂×#1438	C₂³	1	1	yes	characteristic
90	#12690	D₄	1	4	yes	complemented
91	#12688	D₄	1	2	yes	complemented
92	#12687	D₄	1	2	yes	complemented
93	#8534	D₄	1	2	yes	complemented
94	#8534	D₄	1	2	yes	complemented
95	#8534	D₄	1	2	yes	complemented
96	#8534	D₄	1	2	yes	complemented
97	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	2	yes	—
98	C₂²×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
99	C₂²×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
100	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
101	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
102	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
103	#3484	C₂×D₄	1	2	yes	—
104	#3484	C₂×D₄	1	2	yes	—
105	#3484	#41	1	2	yes	—
106	#3484	#41	1	2	yes	—
107	C₂×#543	C₂²⋊C₄	1	2	yes	—
108	C₂×#534	C₂²⋊C₄	1	2	yes	—
109	C₂×#534	C₂²⋊C₄	1	2	yes	—
110	C₂×#534	C₂²⋊C₄	1	2	yes	—
111	C₂×#534	C₂²×C₄	1	2	yes	—
112	C₂×#534	C₂²×C₄	1	2	yes	—
113	C₂×#534	C₂×D₄	1	2	yes	—
114	C₂×#534	C₂×D₄	1	2	yes	—
115	C₂×#534	C₂×D₄	1	2	yes	—
116	C₂×#534	#41	1	2	yes	—
117	C₂²×#123	C₂×D₄	1	1	yes	characteristic
118	C₂²×#123	#41	1	1	yes	characteristic
119	C₂²×#120	C₂×D₄	1	2	yes	—
120	C₂²×#120	C₂×D₄	1	2	yes	—
121	C₂²×#120	C₂×D₄	1	1	yes	characteristic
122	C₂²×#120	#41	1	1	yes	characteristic
123	C₂×#518	C₂²⋊C₄	1	2	yes	—
124	C₂×#518	C₂²⋊C₄	1	2	yes	—
125	C₂×#518	C₂²⋊C₄	1	2	yes	—
126	C₂×#518	C₂²×C₄	1	1	yes	characteristic
127	C₂×#518	C₂²×C₄	1	1	yes	characteristic
128	C₂×#518	C₂²×C₄	1	1	yes	characteristic
129	C₂×#518	C₂²×C₄	1	1	yes	characteristic
130	C₂×#518	C₂×D₄	1	1	yes	characteristic
131	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
132	C₂²×#99	C₂²×C₄	1	2	yes	—
133	C₂²×#99	C₂²×C₄	1	2	yes	—
134	#1783	C₂×D₄	1	2	yes	—
135	#1783	C₂×D₄	1	2	yes	—
136	#1781	C₂×D₄	1	2	yes	—
137	#1781	C₂×D₄	1	2	yes	—
138	#1781	C₂×D₄	1	2	yes	—
139	#1781	C₂×D₄	1	2	yes	—
140	#1438	C₂×D₄	1	1	yes	characteristic
141	#1438	C₂×D₄	1	1	yes	characteristic
142	#1438	C₂×D₄	1	1	yes	characteristic
143	#1438	C₂×D₄	1	1	yes	characteristic
144	C₂⁶	#121	1	1	yes	characteristic
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	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂³×D₄	#121	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	#121	1	1	yes	characteristic
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×#123	#99	1	4	yes	complemented
170	C₂×#123	C₄×D₄	1	2	yes	—
171	C₂×#123	C₄×D₄	1	2	yes	—
172	C₂×#123	#120	1	2	yes	—
173	C₂×#123	#123	1	2	yes	—
174	C₂×#120	#99	1	4	yes	complemented
175	C₂×#120	C₄×D₄	1	2	yes	—
176	C₂×#120	C₄×D₄	1	2	yes	—
177	C₂×#120	#120	1	2	yes	—
178	C₂×#120	#120	1	4	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	C₂×#117	#99	1	4	yes	complemented
184	C₂²×C₂²⋊C₄	#99	1	4	yes	complemented
185	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
188	C₂²×C₂²⋊C₄	#121	1	2	yes	—
189	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
190	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
191	C₂⁵	#386	1	2	yes	—
192	C₂⁵	C₂×#99	1	2	yes	—
193	C₂⁵	C₂×#99	1	2	yes	—
194	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
195	C₂⁵	#518	1	1	yes	characteristic
196	C₂⁵	C₂×#120	1	1	yes	characteristic
197	C₂⁵	C₂×#121	1	1	yes	characteristic
198	C₂⁵	C₂×#121	1	1	yes	characteristic
199	C₂⁵	C₂×#123	1	1	yes	characteristic
200	C₂⁵	#534	1	1	yes	characteristic
201	C₂⁵	#534	1	1	yes	characteristic
202	C₂⁵	#534	1	1	yes	characteristic
203	C₂⁵	D₄²	1	1	yes	characteristic
204	C₂⁵	#546	1	1	yes	characteristic
205	C₂⁵	#546	1	1	yes	characteristic
206	C₂⁵	#551	1	1	yes	characteristic
207	C₂²×D₄	#386	1	4	yes	—
208	C₂²×D₄	C₂×#99	1	4	yes	—
209	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
210	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
213	C₂²×D₄	C₂×#120	1	1	yes	characteristic
214	C₂²×D₄	C₂×#121	1	1	yes	characteristic
215	C₂²×D₄	C₂×#121	1	1	yes	characteristic
216	C₂²×D₄	C₂×#123	1	1	yes	characteristic
217	C₂²×D₄	D₄²	1	1	yes	characteristic
218	C₂²×D₄	D₄²	1	1	yes	characteristic
219	C₂²×D₄	#546	1	1	yes	characteristic
220	C₂²×D₄	#546	1	1	yes	characteristic
221	C₂²×D₄	#546	1	1	yes	characteristic
222	C₂²×D₄	#546	1	1	yes	characteristic
223	C₂²×D₄	#551	1	1	yes	characteristic
224	C₂²×D₄	#551	1	1	yes	characteristic
225	C₂³×C₄	#386	1	2	yes	—
226	C₂³×C₄	C₂×#99	1	2	yes	—
227	C₂³×C₄	C₂×#99	1	2	yes	—
228	C₂³×C₄	#518	1	1	yes	characteristic
229	C₂³×C₄	#534	1	1	yes	characteristic
230	C₂³×C₄	D₄²	1	1	yes	characteristic
231	C₂³×C₄	#546	1	1	yes	characteristic
232	C₂³×C₄	#546	1	1	yes	characteristic
233	C₂³×C₄	#551	1	1	yes	characteristic
234	C₂×C₂²⋊C₄	#386	1	4	yes	—
235	C₂×C₂²⋊C₄	#386	1	2	yes	—
236	C₂×C₂²⋊C₄	#386	1	2	yes	—
237	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
238	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
239	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
240	C₂⁴	#1898	1	4	yes	—
241	C₂⁴	C₂×#386	1	1	yes	characteristic
242	C₂⁴	C₂×#386	1	1	yes	characteristic
243	C₂⁴	#2308	1	1	yes	characteristic
244	C₂⁴	#2358	1	1	yes	characteristic
245	C₂⁴	#2358	1	1	yes	characteristic
246	C₂⁴	#2410	1	1	yes	characteristic
247	C₂⁴	#2410	1	1	yes	characteristic
248	C₂⁴	#2410	1	1	yes	characteristic
249	C₂⁴	#2412	1	1	yes	characteristic
250	C₂⁴	#2412	1	1	yes	characteristic
251	C₂⁴	#2412	1	1	yes	characteristic
252	C₂⁴	C₂²×#99	1	2	yes	—
253	C₂⁴	C₂²×#99	1	2	yes	—
254	C₂²×C₄	#1898	1	4	yes	—
255	C₂²×C₄	#2358	1	1	yes	characteristic
256	C₂²×C₄	#2358	1	1	yes	characteristic
257	C₂²×C₄	#2410	1	1	yes	characteristic
258	C₂²×C₄	#2412	1	1	yes	characteristic
259	C₂²⋊C₄	#1898	1	4	yes	—
260	C₂²⋊C₄	#1898	1	4	yes	—
261	C₂³	#14134	1	1	yes	characteristic
262	C₂³	C₂×#1898	1	2	yes	—
263	C₂³	C₂×#1898	1	2	yes	—
264	C₂³	#21746	1	1	yes	characteristic
265	C₂³	#21746	1	1	yes	characteristic
266	C₂³	#21776	1	1	yes	characteristic
267	C₂³	#21776	1	1	yes	characteristic
268	C₂³	#21776	1	1	yes	characteristic
269	C₂³	#21777	1	1	yes	characteristic
270	C₂²	#213832	1	1	yes	characteristic
271	C₂²	#213838	1	1	yes	characteristic
272	C₂²	#213854	1	1	yes	characteristic
273	C₂²	#213864	1	1	yes	characteristic
274	C₂²	#374004	1	1	yes	characteristic
275	C₂²	#374008	1	1	yes	characteristic
276	C₂²	#374022	1	1	yes	characteristic
277	C₂	#72241175	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241175
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72234953	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. #72310366	C₂	semidirect
2. #72310319	C₂	semidirect
3. #72309552	C₂	semidirect
4. #72266863	C₂	semidirect
5. #72260537	C₂	semidirect
6. #72258028	C₂	semidirect
7. #72256193	C₂	semidirect
8. #72255793	C₂	semidirect
9. #72251552	C₂	semidirect
10. #72237638	C₂	semidirect
11. #72237637	C₂	semidirect
12. #72237003	C₂	semidirect
14. #72219159	C₂	semidirect
15. C₂×#139743	C₂	semidirect
17. #6451863	C₄	semidirect
18. #6437629	C₄	semidirect
26. C₂×#12707	C₂²	semidirect
29. C₂×#12705	C₂²	semidirect
33. #363969	C₄	semidirect
35. #359528	C₄	semidirect
41. C₂×#8534	C₂²	semidirect
42. #186835	C₂²	semidirect
43. #146479	C₂²	semidirect
44. #146475	C₂²	semidirect
45. #128033	C₂²	semidirect
46. C₂×#7304	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. C₂×#7307	C₂²	semidirect
53. #107130	C₂²	semidirect
54. #107114	C₂²	semidirect
58. #62842	C₂×C₄	semidirect
67. C₂×#3026	C₂×C₄	semidirect
68. C₂×#2883	C₂×C₄	semidirect
69. C₂×#2883	C₂×C₄	semidirect
73. C₂×#1913	C₂×C₄	semidirect
74. C₂×#1913	C₂×C₄	semidirect
76. #20960	C₂×C₄	semidirect
77. C₂×#1828	C₂×C₄	semidirect
80. #20585	C₂×C₄	semidirect
87. #20354	C₂×C₄	semidirect
90. #12690	D₄	semidirect
91. #12688	D₄	semidirect
92. #12687	D₄	semidirect
93. #8534	D₄	semidirect
94. #8534	D₄	semidirect
95. #8534	D₄	semidirect
96. #8534	D₄	semidirect
169. C₂×#123	#99	semidirect
174. C₂×#120	#99	semidirect
183. C₂×#117	#99	semidirect
184. C₂²×C₂²⋊C₄	#99	semidirect

Maximal subgroups

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

subgroup	index
#72310366	2
#72310319	2
#72309552	2
#72266863	2
#72260537	2
#72258028	2
#72256193	2
#72255793	2
#72251552	2
#72237638	2
#72237637	2
#72237003	2
#72234953	2
#72219159	2
C₂×#139743	2