Subgroups — #72314791

Order 2048.

At least 6092 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

The subgroup lattice has 239 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72314734	C₂	1	1	yes	maximal, characteristic, complemented
2	#72258019	C₂	1	1	yes	maximal, characteristic, complemented
3	#72255986	C₂	1	1	yes	maximal, characteristic, complemented
4	#72255977	C₂	1	1	yes	maximal, characteristic, complemented
5	#72255849	C₂	1	1	yes	maximal, characteristic, complemented
6	#72255793	C₂	1	1	yes	maximal, characteristic, complemented
7	#72251528	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	#72225743	C₂	1	1	yes	maximal, characteristic, complemented
12	#72218521	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218206	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#139750	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×#21776	C₂²	1	1	yes	characteristic
20	C₂×#21776	C₂²	1	1	yes	characteristic
21	C₂×#21746	C₂²	1	1	yes	characteristic, complemented
22	#616961	C₂²	1	1	yes	characteristic
23	C₂×#12707	C₂²	1	1	yes	characteristic
24	C₂×#12707	C₂²	1	1	yes	characteristic
25	C₂×#12705	C₂²	1	1	yes	characteristic
26	C₂×#12705	C₂²	1	1	yes	characteristic
27	#493370	C₂²	1	1	yes	characteristic
28	#493370	C₂²	1	1	yes	characteristic
29	#493370	C₂²	1	1	yes	characteristic
30	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
31	#392030	C₂²	1	1	yes	characteristic, complemented
32	#359549	C₂²	1	1	yes	characteristic
33	#359549	C₂²	1	1	yes	characteristic
34	#359509	C₂²	1	1	yes	characteristic
35	#359501	C₂²	1	1	yes	characteristic, complemented
36	#355096	C₂²	1	1	yes	characteristic, complemented
37	#347047	C₄	1	4	yes	complemented
38	#347045	C₄	1	4	yes	complemented
39	#188050	C₂²	1	1	yes	characteristic
40	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
41	#187944	C₂²	1	1	yes	characteristic
42	#187944	C₂²	1	1	yes	characteristic
43	#146479	C₂²	1	1	yes	characteristic, complemented
44	#128033	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
50	#128009	C₂²	1	1	yes	characteristic, complemented
51	#128009	C₂²	1	1	yes	characteristic, complemented
52	#107114	C₂²	1	1	yes	characteristic, complemented
53	#62832	C₂×C₄	1	2	yes	complemented
54	#62824	C₂×C₄	1	2	yes	complemented
55	C₂×#3484	C₂³	1	1	yes	characteristic
56	C₂×#3484	C₂³	1	1	yes	characteristic
57	C₂²×#535	C₂³	1	1	yes	characteristic
58	C₂³×#120	C₂³	1	1	yes	characteristic
59	C₂×#2883	D₄	1	2	yes	complemented
60	C₂×#2846	D₄	1	2	yes	—
61	#34171	C₂³	1	1	yes	characteristic
62	#23900	C₂×C₄	1	4	yes	—
63	#23900	C₂×C₄	1	4	yes	complemented
64	#23900	D₄	1	4	yes	—
65	C₂×#1913	D₄	1	2	yes	complemented
66	C₂×#1898	D₄	1	2	yes	—
67	#20960	D₄	1	2	yes	complemented
68	C₂×#1828	D₄	1	2	yes	complemented
69	C₂×#1783	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	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	C₂×#1781	C₂³	1	1	yes	characteristic
78	#15259	C₂×C₄	1	2	yes	complemented
79	#15258	C₂×C₄	1	2	yes	complemented
80	#14335	C₂³	1	1	yes	characteristic
81	#12707	D₄	1	4	yes	complemented
82	#12694	D₄	1	2	yes	complemented
83	#12693	D₄	1	2	yes	complemented
84	#8546	C₂×C₄	1	2	yes	complemented
85	#8544	C₂×C₄	1	2	yes	complemented
86	#8534	C₂×C₄	1	4	yes	complemented
87	#8534	C₂×C₄	1	4	yes	complemented
88	C₂⁷	C₂×D₄	1	1	yes	characteristic
89	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
90	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
91	C₂³×#41	#41	1	1	yes	characteristic
92	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
93	#3484	C₂²×C₄	1	2	yes	—
94	#3484	C₂²×C₄	1	2	yes	—
95	#3484	C₂×D₄	1	2	yes	—
96	#3484	C₂×D₄	1	2	yes	—
97	#3484	#41	1	2	yes	—
98	#3484	#41	1	2	yes	—
99	C₂×#534	C₂×D₄	1	2	yes	—
100	C₂²×#127	C₂²×C₄	1	1	yes	characteristic
101	C₂²×#127	C₂×D₄	1	1	yes	characteristic
102	C₂²×#124	C₂²×C₄	1	1	yes	characteristic
103	C₂²×#124	C₂×D₄	1	1	yes	characteristic
104	C₂²×#120	C₂²×C₄	1	2	yes	—
105	C₂²×#120	C₂×D₄	1	2	yes	—
106	C₂²×#120	C₂×D₄	1	2	yes	—
107	C₂×#518	C₂×D₄	1	1	yes	characteristic
108	C₂×#518	C₂×D₄	1	1	yes	characteristic
109	C₂×#518	C₂×D₄	1	1	yes	characteristic
110	C₂×#518	#41	1	1	yes	characteristic
111	C₂²×#117	#41	1	1	yes	characteristic
112	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
113	C₂²×#99	C₂×D₄	1	2	yes	—
114	C₂²×#99	C₂×D₄	1	2	yes	—
115	C₂²×#99	C₂×D₄	1	2	yes	—
116	#2846	C₂²⋊C₄	1	8	yes	complemented
117	C₂×#379	C₂×D₄	1	2	yes	—
118	#1898	C₂²⋊C₄	1	8	yes	complemented
119	#1781	C₂²×C₄	1	2	yes	—
120	#1781	C₂²×C₄	1	2	yes	—
121	#1781	C₂×D₄	1	2	yes	—
122	#1781	C₂×D₄	1	2	yes	—
123	#1781	C₂×D₄	1	2	yes	—
124	#1781	C₂×D₄	1	2	yes	—
125	#1781	C₂×D₄	1	2	yes	—
126	#1438	C₂²×C₄	1	2	yes	—
127	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂⁶	C₂²×D₄	1	1	yes	characteristic
129	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
131	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	complemented
132	C₂×H(5,𝔽₂)	#121	1	2	yes	—
133	C₂³×Q₈	C₄×D₄	1	1	yes	characteristic, complemented
134	C₂³×D₄	C₄×D₄	1	1	yes	characteristic, complemented
135	C₂³×D₄	#120	1	2	yes	—
136	C₂³×D₄	C₂³×C₄	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	C₂²×D₄	1	1	yes	characteristic
141	C₂³×D₄	C₂²×D₄	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	C₂×#41	1	1	yes	characteristic
145	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
149	#534	C₄×D₄	1	4	yes	—
150	#534	#121	1	4	yes	—
151	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
152	C₂×#120	#123	1	4	yes	—
153	#518	C₄×D₄	1	2	yes	—
154	#518	C₄×D₄	1	2	yes	complemented
155	#518	#121	1	2	yes	—
156	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
157	C₂²×C₂²⋊C₄	#120	1	2	yes	—
158	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic, complemented
159	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic, complemented
160	C₂×#99	#120	1	4	yes	—
161	#379	C₂×C₂²⋊C₄	1	4	yes	—
162	C₂⁵	B(4,𝔽₂)	1	2	yes	—
163	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
164	C₂⁵	#518	1	1	yes	characteristic
165	C₂⁵	C₂×#120	1	1	yes	characteristic
166	C₂⁵	C₂×#123	1	1	yes	characteristic
167	C₂⁵	#535	1	1	yes	characteristic
168	C₂⁵	#538	1	1	yes	characteristic
169	C₂⁵	D₄²	1	1	yes	characteristic
170	C₂⁵	D₄²	1	1	yes	characteristic
171	C₂⁵	D₄²	1	1	yes	characteristic
172	C₂⁵	#546	1	1	yes	characteristic
173	C₂⁵	#546	1	1	yes	characteristic
174	C₂⁵	#546	1	1	yes	characteristic
175	C₂⁵	#546	1	1	yes	characteristic
176	C₂²×D₄	#386	1	4	yes	—
177	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
178	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
179	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
181	C₂²×D₄	#518	1	1	yes	characteristic
182	C₂²×D₄	C₂×#120	1	1	yes	characteristic
183	C₂²×D₄	C₂×#121	1	1	yes	characteristic
184	C₂²×D₄	C₂×#121	1	1	yes	characteristic
185	C₂²×D₄	#525	1	1	yes	characteristic
186	C₂²×D₄	#534	1	1	yes	characteristic
187	C₂²×D₄	#538	1	1	yes	characteristic
188	C₂²×D₄	D₄²	1	1	yes	characteristic
189	C₂²×D₄	D₄²	1	1	yes	characteristic
190	C₂²×D₄	D₄²	1	1	yes	characteristic
191	C₂²×D₄	#546	1	1	yes	characteristic
192	C₂²×D₄	#546	1	1	yes	characteristic
193	C₂²×D₄	#551	1	1	yes	characteristic
194	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
195	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂³×C₄	#525	1	1	yes	characteristic
197	C₂³×C₄	#525	1	1	yes	characteristic
198	C₂³×C₄	#546	1	1	yes	characteristic
199	C₂³×C₄	#546	1	1	yes	characteristic
200	C₂³×C₄	#551	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	#386	1	4	yes	—
202	C₂⁴	#1889	1	4	yes	—
203	C₂⁴	C₂×#386	1	1	yes	characteristic
204	C₂⁴	#2321	1	1	yes	characteristic
205	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
206	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
207	C₂⁴	#2358	1	1	yes	characteristic
208	C₂⁴	#2403	1	1	yes	characteristic
209	C₂⁴	#2404	1	1	yes	characteristic
210	C₂⁴	#2408	1	1	yes	characteristic
211	C₂⁴	#2418	1	1	yes	characteristic
212	C₂⁴	#2418	1	1	yes	characteristic
213	C₂⁴	#2424	1	1	yes	characteristic
214	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
215	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
216	C₂²×C₄	#1889	1	4	yes	—
217	C₂²×C₄	C₄×#120	1	1	yes	characteristic
218	C₂²×C₄	#2358	1	1	yes	characteristic
219	C₂²×C₄	#2424	1	1	yes	characteristic
220	C₂²×C₄	#2534	1	1	yes	characteristic
221	C₂³	#13967	1	1	yes	characteristic
222	C₂³	C₂×#1889	1	2	yes	—
223	C₂³	#23900	1	2	yes	—
224	C₂³	#23984	1	1	yes	characteristic
225	C₂³	#23984	1	1	yes	characteristic
226	C₂³	#23996	1	1	yes	characteristic
227	C₂³	#23996	1	1	yes	characteristic
228	C₂³	#23996	1	1	yes	characteristic
229	C₂³	#23997	1	1	yes	characteristic
230	C₂²	#251827	1	1	yes	characteristic
231	C₂²	#251837	1	1	yes	characteristic
232	C₂²	#251859	1	1	yes	characteristic
233	C₂²	#251871	1	1	yes	characteristic
234	C₂²	#364410	1	1	yes	characteristic
235	C₂²	#364532	1	1	yes	characteristic
236	C₂²	#364545	1	1	yes	characteristic
237	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	#72237402	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. #72314734	C₂	semidirect
2. #72258019	C₂	semidirect
3. #72255986	C₂	semidirect
4. #72255977	C₂	semidirect
5. #72255849	C₂	semidirect
6. #72255793	C₂	semidirect
7. #72251528	C₂	semidirect
8. #72240132	C₂	semidirect
9. #72238343	C₂	semidirect
11. #72225743	C₂	semidirect
12. #72218521	C₂	semidirect
13. #72218206	C₂	semidirect
14. C₂×#139750	C₂	semidirect
15. C₂×#139731	C₂	semidirect
21. C₂×#21746	C₂²	semidirect
30. C₂×#12687	C₂²	semidirect
31. #392030	C₂²	semidirect
35. #359501	C₂²	semidirect
36. #355096	C₂²	semidirect
37. #347047	C₄	semidirect
38. #347045	C₄	semidirect
40. C₂×#8541	C₂²	semidirect
43. #146479	C₂²	semidirect
44. #128033	C₂²	semidirect
45. C₂×#7316	C₂²	semidirect
46. C₂×#7314	C₂²	semidirect
47. C₂×#7310	C₂²	semidirect
48. C₂×#7310	C₂²	semidirect
49. C₂×#7310	C₂²	semidirect
50. #128009	C₂²	semidirect
51. #128009	C₂²	semidirect
52. #107114	C₂²	semidirect
53. #62832	C₂×C₄	semidirect
54. #62824	C₂×C₄	semidirect
59. C₂×#2883	D₄	semidirect
63. #23900	C₂×C₄	semidirect
65. C₂×#1913	D₄	semidirect
67. #20960	D₄	semidirect
68. C₂×#1828	D₄	semidirect
78. #15259	C₂×C₄	semidirect
79. #15258	C₂×C₄	semidirect
81. #12707	D₄	semidirect
82. #12694	D₄	semidirect
83. #12693	D₄	semidirect
84. #8546	C₂×C₄	semidirect
85. #8544	C₂×C₄	semidirect
86. #8534	C₂×C₄	semidirect
87. #8534	C₂×C₄	semidirect
116. #2846	C₂²⋊C₄	semidirect
118. #1898	C₂²⋊C₄	semidirect
131. C₂×H(5,𝔽₂)	C₄×D₄	semidirect
133. C₂³×Q₈	C₄×D₄	semidirect
134. C₂³×D₄	C₄×D₄	semidirect
154. #518	C₄×D₄	semidirect
158. C₂²×C₄²	C₄×D₄	semidirect
159. C₂²×C₄²	C₄×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72314734	2
#72258019	2
#72255986	2
#72255977	2
#72255849	2
#72255793	2
#72251528	2
#72240132	2
#72238343	2
#72237402	2
#72225743	2
#72218521	2
#72218206	2
C₂×#139750	2
C₂×#139731	2