Subgroups — #72262705

Order 2048.

At least 6624 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

The subgroup lattice has 246 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	#72256195	C₂	1	1	yes	maximal, characteristic, complemented
3	#72255986	C₂	1	1	yes	maximal, characteristic, complemented
4	#72255841	C₂	1	1	yes	maximal, characteristic, complemented
5	#72255778	C₂	1	1	yes	maximal, characteristic, complemented
6	#72251501	C₂	1	1	yes	maximal, characteristic, complemented
7	#72251473	C₂	1	1	yes	maximal, characteristic, complemented
8	#72251471	C₂	1	1	yes	maximal, characteristic, complemented
9	#72241271	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237037	C₂	1	1	yes	maximal, characteristic, complemented
11	#72236636	C₂	1	1	yes	maximal, characteristic, complemented
12	#72225743	C₂	1	1	yes	maximal, characteristic, complemented
13	#72219424	C₂	1	1	yes	maximal, characteristic, complemented
14	#72218475	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#511790	C₂	1	1	yes	maximal, characteristic
16	C₂×#62817	C₂²	1	1	yes	characteristic
17	C₂×#23895	C₂²	1	1	yes	characteristic
18	C₂×#23900	C₂²	1	1	yes	characteristic
19	C₂×#23897	C₂²	1	1	yes	characteristic
20	C₂×#21776	C₂²	1	1	yes	characteristic, complemented
21	C₂×#21746	C₂²	1	1	yes	characteristic
22	#556752	C₂²	1	1	yes	characteristic
23	#511790	C₄	1	8	yes	complemented
24	C₂×#12707	C₂²	1	1	yes	characteristic
25	C₂×#12707	C₂²	1	1	yes	characteristic
26	C₂×#12705	C₂²	1	1	yes	characteristic
27	C₂×#12705	C₂²	1	1	yes	characteristic
28	C₂×#12687	C₂²	1	1	yes	characteristic
29	C₂×#12687	C₂²	1	1	yes	characteristic
30	#392030	C₂²	1	1	yes	characteristic, complemented
31	#359558	C₂²	1	1	yes	characteristic
32	#359549	C₂²	1	1	yes	characteristic
33	#359549	C₂²	1	1	yes	characteristic, complemented
34	#359501	C₂²	1	1	yes	characteristic
35	#355096	C₂²	1	1	yes	characteristic, complemented
36	#187944	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, complemented
41	#186835	C₂²	1	1	yes	characteristic, complemented
42	#128033	C₂²	1	1	yes	characteristic, complemented
43	#128033	C₂²	1	1	yes	characteristic, complemented
44	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7311	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	#108238	C₂²	1	1	yes	characteristic, complemented
51	#107114	C₂²	1	1	yes	characteristic, complemented
52	#62817	C₂×C₄	1	4	yes	complemented
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂×#3484	C₂³	1	1	yes	characteristic
56	C₂³×#120	C₂³	1	1	yes	characteristic
57	C₂²×B(4,𝔽₂)	D₄	1	2	yes	—
58	#34171	C₂³	1	1	yes	characteristic
59	C₂²×#379	D₄	1	2	yes	complemented
60	#23900	C₂×C₄	1	4	yes	—
61	#23900	D₄	1	2	yes	—
62	#23900	D₄	1	2	yes	—
63	#23897	C₂×C₄	1	4	yes	complemented
64	#23895	C₂×C₄	1	4	yes	complemented
65	C₂×#1889	D₄	1	2	yes	—
66	C₂×#1889	D₄	1	2	yes	complemented
67	C₂×#1889	D₄	1	2	yes	complemented
68	C₂×#1819	D₄	1	2	yes	complemented
69	C₂×#1783	C₂³	1	1	yes	characteristic
70	C₂×#1783	C₂³	1	1	yes	characteristic
71	C₂×#1781	C₂³	1	1	yes	characteristic
72	C₂×#1781	C₂³	1	1	yes	characteristic
73	C₂×#1781	C₂³	1	1	yes	characteristic
74	C₂×#1781	C₂³	1	1	yes	characteristic
75	C₂×#1781	C₂³	1	1	yes	characteristic
76	C₂×#1781	C₂³	1	1	yes	characteristic
77	#14335	C₂³	1	1	yes	characteristic
78	#14335	C₂³	1	1	yes	characteristic
79	#12707	C₂×C₄	1	4	yes	complemented
80	#12705	C₂×C₄	1	4	yes	complemented
81	#12687	C₂×C₄	1	4	yes	complemented
82	#8534	D₄	1	4	yes	complemented
83	#8534	D₄	1	4	yes	complemented
84	C₂⁷	C₂×D₄	1	1	yes	characteristic
85	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
86	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
88	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
89	#3484	C₂²×C₄	1	2	yes	—
90	#3484	C₂²×C₄	1	2	yes	—
91	#3484	C₂²×C₄	1	2	yes	—
92	#3484	C₂×D₄	1	2	yes	—
93	#3484	C₂×D₄	1	2	yes	—
94	#3484	C₂×D₄	1	2	yes	—
95	C₂²×#120	C₂²×C₄	1	2	yes	—
96	C₂²×#120	C₂×D₄	1	2	yes	—
97	C₂²×#120	C₂×D₄	1	2	yes	—
98	C₂²×#120	C₂×D₄	1	1	yes	characteristic
99	C₂²×#120	C₂×D₄	1	1	yes	characteristic
100	C₂²×#120	#41	1	2	yes	—
101	C₂²×#120	#41	1	1	yes	characteristic
102	C₂²×#120	#41	1	1	yes	characteristic
103	C₂×#518	C₂×D₄	1	1	yes	characteristic
104	C₂×#518	C₂×D₄	1	1	yes	characteristic
105	C₂×#518	C₂×D₄	1	1	yes	characteristic
106	C₂×#518	#41	1	1	yes	characteristic
107	C₂×#518	#41	1	1	yes	characteristic
108	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
109	C₂×B(4,𝔽₂)	C₂²⋊C₄	1	4	yes	complemented
110	C₂×B(4,𝔽₂)	C₂²⋊C₄	1	4	yes	complemented
111	C₂²×#99	C₂×D₄	1	2	yes	—
112	C₂²×#99	C₂×D₄	1	2	yes	—
113	C₂×#379	C₂×D₄	1	2	yes	—
114	C₂×#379	C₂×D₄	1	2	yes	—
115	#1889	C₂²⋊C₄	1	4	yes	complemented
116	#1889	C₂²⋊C₄	1	4	yes	complemented
117	#1783	C₂²×C₄	1	2	yes	—
118	#1781	C₂²×C₄	1	2	yes	—
119	#1781	C₂²×C₄	1	2	yes	—
120	#1781	C₂×D₄	1	2	yes	—
121	#1781	C₂×D₄	1	2	yes	—
122	#1781	C₂×D₄	1	2	yes	—
123	#1438	C₂×D₄	1	2	yes	—
124	#1438	C₂×D₄	1	2	yes	—
125	C₂⁶	C₄×D₄	1	2	yes	—
126	C₂⁶	#120	1	2	yes	—
127	C₂⁶	#121	1	1	yes	characteristic
128	C₂⁶	#121	1	1	yes	characteristic
129	C₂⁶	C₂²×D₄	1	1	yes	characteristic
130	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	complemented
133	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	complemented
134	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
135	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
136	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
137	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
138	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
139	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
140	C₂³×D₄	C₂×#41	1	1	yes	characteristic
141	C₂³×D₄	C₂×#41	1	1	yes	characteristic
142	C₂³×D₄	C₂×#41	1	1	yes	characteristic
143	C₂³×D₄	C₂×#41	1	1	yes	characteristic
144	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×#120	C₂×C₂²⋊C₄	1	2	yes	—
148	C₂×#120	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂×#120	C₄×D₄	1	2	yes	—
150	C₂×#120	C₄×D₄	1	2	yes	—
151	C₂×#120	#121	1	2	yes	—
152	C₂×#120	#121	1	2	yes	—
153	C₂×#120	#123	1	2	yes	—
154	C₂×#120	#123	1	2	yes	—
155	#518	C₂×C₂²⋊C₄	1	2	yes	—
156	#518	C₄×D₄	1	2	yes	complemented
157	#518	C₄×D₄	1	2	yes	complemented
158	C₂²×C₂²⋊C₄	C₄×D₄	1	2	yes	—
159	C₂²×C₂²⋊C₄	#120	1	2	yes	—
160	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
161	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
162	C₂×#99	C₂×C₂²⋊C₄	1	4	yes	—
163	#379	#120	1	4	yes	—
164	C₂⁵	#386	1	2	yes	—
165	C₂⁵	#386	1	2	yes	—
166	C₂⁵	B(4,𝔽₂)	1	2	yes	—
167	C₂⁵	B(4,𝔽₂)	1	2	yes	—
168	C₂⁵	B(4,𝔽₂)	1	2	yes	—
169	C₂⁵	B(4,𝔽₂)	1	2	yes	—
170	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂⁵	#518	1	1	yes	characteristic
172	C₂⁵	C₂×#120	1	1	yes	characteristic
173	C₂⁵	C₂×#120	1	1	yes	characteristic
174	C₂⁵	C₂×#121	1	1	yes	characteristic
175	C₂⁵	#525	1	1	yes	characteristic
176	C₂⁵	#534	1	1	yes	characteristic
177	C₂⁵	#535	1	1	yes	characteristic
178	C₂⁵	#538	1	1	yes	characteristic
179	C₂⁵	D₄²	1	1	yes	characteristic
180	C₂⁵	#546	1	1	yes	characteristic
181	C₂⁵	#546	1	1	yes	characteristic
182	C₂⁵	#546	1	1	yes	characteristic
183	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
184	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
185	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
186	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂²×D₄	C₂×#121	1	1	yes	characteristic
188	C₂²×D₄	C₂×#123	1	1	yes	characteristic
189	C₂²×D₄	#525	1	1	yes	characteristic
190	C₂²×D₄	#525	1	1	yes	characteristic
191	C₂²×D₄	D₄²	1	1	yes	characteristic
192	C₂²×D₄	D₄²	1	1	yes	characteristic
193	C₂²×D₄	D₄²	1	1	yes	characteristic
194	C₂²×D₄	D₄²	1	1	yes	characteristic
195	C₂²×D₄	#546	1	1	yes	characteristic
196	C₂²×D₄	#546	1	1	yes	characteristic
197	C₂²×D₄	#551	1	1	yes	characteristic
198	C₂²×D₄	#551	1	1	yes	characteristic
199	C₂³×C₄	#518	1	1	yes	characteristic
200	C₂³×C₄	#538	1	1	yes	characteristic
201	C₂³×C₄	D₄²	1	1	yes	characteristic
202	C₂³×C₄	#546	1	1	yes	characteristic
203	C₂³×C₄	#546	1	1	yes	characteristic
204	C₂³×C₄	#546	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	#386	1	2	yes	—
206	C₂×C₂²⋊C₄	#386	1	2	yes	—
207	C₂⁴	#1889	1	2	yes	—
208	C₂⁴	#1889	1	2	yes	—
209	C₂⁴	#1889	1	2	yes	—
210	C₂⁴	#1889	1	2	yes	—
211	C₂⁴	C₂×#386	1	1	yes	characteristic
212	C₂⁴	C₄×#120	1	1	yes	characteristic
213	C₂⁴	#2321	1	1	yes	characteristic
214	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
215	C₂⁴	#2358	1	1	yes	characteristic
216	C₂⁴	#2403	1	1	yes	characteristic
217	C₂⁴	#2404	1	1	yes	characteristic
218	C₂⁴	#2408	1	1	yes	characteristic
219	C₂⁴	#2418	1	1	yes	characteristic
220	C₂⁴	#2424	1	1	yes	characteristic
221	C₂⁴	#2534	1	1	yes	characteristic
222	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
223	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
224	C₂²×C₄	D₄×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	#2358	1	1	yes	characteristic
226	C₂²×C₄	#2418	1	1	yes	characteristic
227	C₂²×C₄	#2424	1	1	yes	characteristic
228	C₂³	#13967	1	1	yes	characteristic
229	C₂³	C₂×#1889	1	2	yes	—
230	C₂³	#23900	1	2	yes	—
231	C₂³	#23984	1	1	yes	characteristic
232	C₂³	#23984	1	1	yes	characteristic
233	C₂³	#23996	1	1	yes	characteristic
234	C₂³	#23996	1	1	yes	characteristic
235	C₂³	#23996	1	1	yes	characteristic
236	C₂³	#23997	1	1	yes	characteristic
237	C₂²	#251827	1	1	yes	characteristic
238	C₂²	#251837	1	1	yes	characteristic
239	C₂²	#251859	1	1	yes	characteristic
240	C₂²	#251871	1	1	yes	characteristic
241	C₂²	#364410	1	1	yes	characteristic
242	C₂²	#364532	1	1	yes	characteristic
243	C₂²	#364545	1	1	yes	characteristic
244	C₂	#72241183	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241183
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	C₂×#511790	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. #72256195	C₂	semidirect
3. #72255986	C₂	semidirect
4. #72255841	C₂	semidirect
5. #72255778	C₂	semidirect
6. #72251501	C₂	semidirect
7. #72251473	C₂	semidirect
8. #72251471	C₂	semidirect
9. #72241271	C₂	semidirect
10. #72237037	C₂	semidirect
11. #72236636	C₂	semidirect
12. #72225743	C₂	semidirect
13. #72219424	C₂	semidirect
14. #72218475	C₂	semidirect
20. C₂×#21776	C₂²	semidirect
23. #511790	C₄	semidirect
30. #392030	C₂²	semidirect
33. #359549	C₂²	semidirect
35. #355096	C₂²	semidirect
40. #187944	C₂²	semidirect
41. #186835	C₂²	semidirect
42. #128033	C₂²	semidirect
43. #128033	C₂²	semidirect
44. C₂×#7316	C₂²	semidirect
45. C₂×#7316	C₂²	semidirect
46. C₂×#7311	C₂²	semidirect
47. C₂×#7310	C₂²	semidirect
48. #128009	C₂²	semidirect
49. #128009	C₂²	semidirect
50. #108238	C₂²	semidirect
51. #107114	C₂²	semidirect
52. #62817	C₂×C₄	semidirect
59. C₂²×#379	D₄	semidirect
63. #23897	C₂×C₄	semidirect
64. #23895	C₂×C₄	semidirect
66. C₂×#1889	D₄	semidirect
67. C₂×#1889	D₄	semidirect
68. C₂×#1819	D₄	semidirect
79. #12707	C₂×C₄	semidirect
80. #12705	C₂×C₄	semidirect
81. #12687	C₂×C₄	semidirect
82. #8534	D₄	semidirect
83. #8534	D₄	semidirect
109. C₂×B(4,𝔽₂)	C₂²⋊C₄	semidirect
110. C₂×B(4,𝔽₂)	C₂²⋊C₄	semidirect
115. #1889	C₂²⋊C₄	semidirect
116. #1889	C₂²⋊C₄	semidirect
132. C₂×H(5,𝔽₂)	C₄×D₄	semidirect
133. C₂×H(5,𝔽₂)	C₄×D₄	semidirect
156. #518	C₄×D₄	semidirect
157. #518	C₄×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72262701	2
#72256195	2
#72255986	2
#72255841	2
#72255778	2
#72251501	2
#72251473	2
#72251471	2
#72241271	2
#72237037	2
#72236636	2
#72225743	2
#72219424	2
#72218475	2
C₂×#511790	2