Subgroups — #72314858

Order 2048.

At least 6032 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	#72314855	C₂	1	1	yes	maximal, characteristic, complemented
2	#72314452	C₂	1	1	yes	maximal, characteristic, complemented
3	#72310283	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266863	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#139742	C₂	1	1	yes	maximal, characteristic, complemented
6	#72255841	C₂	1	1	yes	maximal, characteristic, complemented
7	#72255778	C₂	1	1	yes	maximal, characteristic, complemented
8	#72251501	C₂	1	1	yes	maximal, characteristic, complemented
9	#72251473	C₂	1	1	yes	maximal, characteristic, complemented
10	#72246450	C₂	1	1	yes	maximal, characteristic, complemented
11	#72246286	C₂	1	1	yes	maximal, characteristic, complemented
12	#72238613	C₂	1	1	yes	maximal, characteristic, complemented
13	#72237599	C₂	1	1	yes	maximal, characteristic, complemented
14	#72237472	C₂	1	1	yes	maximal, characteristic, complemented
15	#72235130	C₂	1	1	yes	maximal, characteristic
16	#10586404	C₂²	1	1	yes	characteristic
17	C₂×#23895	C₂²	1	1	yes	characteristic
18	#6437621	C₂²	1	1	yes	characteristic
19	C₂×#23897	C₂²	1	1	yes	characteristic
20	#6374251	C₂²	1	1	yes	characteristic
21	C₂×#21776	C₂²	1	1	yes	characteristic, complemented
22	#552677	C₂²	1	1	yes	characteristic
23	#511793	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	#493357	C₂²	1	1	yes	characteristic
29	#493357	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	#359442	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, complemented
39	C₂×#8534	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	C₂×#8453	C₂²	1	1	yes	characteristic, complemented
42	#128033	C₂²	1	1	yes	characteristic, complemented
43	C₂×#7304	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	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
50	#108238	C₂²	1	1	yes	characteristic, complemented
51	#108222	C₂²	1	1	yes	characteristic, complemented
52	#62863	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	C₂²×#534	C₂³	1	1	yes	characteristic
57	C₂×#3026	D₄	1	2	yes	—
58	#34176	D₄	1	2	yes	complemented
59	#34171	C₂³	1	1	yes	characteristic
60	#23903	C₂×C₄	1	4	yes	—
61	#23901	D₄	1	2	yes	—
62	#23900	D₄	1	2	yes	—
63	#23897	C₂×C₄	1	4	yes	—
64	#23895	C₂×C₄	1	4	yes	—
65	C₂×#1913	D₄	1	2	yes	complemented
66	C₂×#1913	D₄	1	2	yes	complemented
67	#20960	D₄	1	2	yes	—
68	C₂×#1783	C₂³	1	1	yes	characteristic
69	C₂×#1783	C₂³	1	1	yes	characteristic
70	C₂×#1781	D₄	1	2	yes	complemented
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	C₂×#1438	C₂³	1	1	yes	characteristic
79	#12707	C₂×C₄	1	4	yes	complemented
80	#12705	C₂×C₄	1	4	yes	complemented
81	#12690	C₂×C₄	1	4	yes	complemented
82	#8534	D₄	1	2	yes	complemented
83	#8534	D₄	1	2	yes	complemented
84	#8534	D₄	1	2	yes	complemented
85	#8534	D₄	1	2	yes	complemented
86	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
88	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
89	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
90	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
91	#3484	C₂²×C₄	1	2	yes	—
92	#3484	C₂²×C₄	1	2	yes	—
93	#3484	C₂²×C₄	1	2	yes	—
94	#3484	C₂×D₄	1	2	yes	—
95	#3484	C₂×D₄	1	2	yes	—
96	#3484	C₂×D₄	1	2	yes	—
97	C₂×#534	C₂²×C₄	1	2	yes	—
98	C₂×#534	C₂×D₄	1	2	yes	—
99	C₂×#534	C₂×D₄	1	2	yes	—
100	C₂×#534	#41	1	2	yes	—
101	C₂²×#123	C₂×D₄	1	1	yes	characteristic
102	C₂²×#123	#41	1	1	yes	characteristic
103	C₂²×#120	C₂×D₄	1	1	yes	characteristic
104	C₂²×#120	#41	1	1	yes	characteristic
105	C₂×#518	C₂×D₄	1	1	yes	characteristic
106	C₂×#518	C₂×D₄	1	1	yes	characteristic
107	C₂×#518	C₂×D₄	1	1	yes	characteristic
108	C₂×#518	#41	1	1	yes	characteristic
109	C₂×#518	#41	1	1	yes	characteristic
110	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
111	C₂×#458	C₂²⋊C₄	1	4	yes	complemented
112	C₂×B(4,𝔽₂)	C₂²⋊C₄	1	4	yes	complemented
113	C₂²×#99	C₂×D₄	1	2	yes	—
114	C₂²×#99	C₂×D₄	1	2	yes	—
115	C₂×#379	C₂×D₄	1	2	yes	—
116	C₂×#379	C₂×D₄	1	2	yes	—
117	#1890	C₂²⋊C₄	1	4	yes	complemented
118	#1889	C₂²⋊C₄	1	4	yes	complemented
119	#1783	C₂²×C₄	1	2	yes	—
120	#1781	C₂²×C₄	1	2	yes	—
121	#1781	C₂²×C₄	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	#1438	C₂×D₄	1	1	yes	characteristic
126	#1438	C₂×D₄	1	1	yes	characteristic
127	#1438	C₂×D₄	1	1	yes	characteristic
128	#1438	C₂×D₄	1	1	yes	characteristic
129	C₂⁶	#121	1	1	yes	characteristic
130	C₂⁶	C₂²×D₄	1	1	yes	characteristic
131	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
134	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
135	C₂³×D₄	C₄×D₄	1	2	yes	—
136	C₂³×D₄	#120	1	2	yes	—
137	C₂³×D₄	C₂³×C₄	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	C₂²×D₄	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	C₂×#41	1	1	yes	characteristic
146	C₂³×D₄	C₂×#41	1	1	yes	characteristic
147	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂⁴×C₄	#121	1	1	yes	characteristic
150	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×#123	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×#123	C₄×D₄	1	2	yes	—
153	C₂×#123	#121	1	2	yes	—
154	C₂×#123	#123	1	2	yes	—
155	C₂×#120	C₂×C₂²⋊C₄	1	2	yes	—
156	C₂×#120	C₄×D₄	1	2	yes	—
157	C₂×#120	#121	1	2	yes	—
158	C₂×#120	#123	1	2	yes	—
159	#518	C₂×C₂²⋊C₄	1	2	yes	—
160	#518	C₄×D₄	1	2	yes	—
161	#518	C₄×D₄	1	2	yes	—
162	C₂²×C₂²⋊C₄	C₄×D₄	1	2	yes	—
163	C₂²×C₂²⋊C₄	#120	1	2	yes	—
164	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
165	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
166	C₂×#99	C₂×C₂²⋊C₄	1	4	yes	—
167	#379	#120	1	4	yes	—
168	C₂⁵	#386	1	2	yes	—
169	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂⁵	#518	1	1	yes	characteristic
171	C₂⁵	C₂×#120	1	1	yes	characteristic
172	C₂⁵	C₂×#120	1	1	yes	characteristic
173	C₂⁵	C₂×#121	1	1	yes	characteristic
174	C₂⁵	#525	1	1	yes	characteristic
175	C₂⁵	#534	1	1	yes	characteristic
176	C₂⁵	#535	1	1	yes	characteristic
177	C₂⁵	#538	1	1	yes	characteristic
178	C₂⁵	D₄²	1	1	yes	characteristic
179	C₂⁵	#546	1	1	yes	characteristic
180	C₂⁵	#546	1	1	yes	characteristic
181	C₂⁵	#546	1	1	yes	characteristic
182	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
183	C₂²×D₄	C₂×C₄×D₄	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×#121	1	1	yes	characteristic
187	C₂²×D₄	C₂×#123	1	1	yes	characteristic
188	C₂²×D₄	#525	1	1	yes	characteristic
189	C₂²×D₄	#525	1	1	yes	characteristic
190	C₂²×D₄	D₄²	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	#546	1	1	yes	characteristic
195	C₂²×D₄	#546	1	1	yes	characteristic
196	C₂²×D₄	#551	1	1	yes	characteristic
197	C₂²×D₄	#551	1	1	yes	characteristic
198	C₂³×C₄	#386	1	2	yes	—
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	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
208	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
209	C₂⁴	C₂×#386	1	1	yes	characteristic
210	C₂⁴	C₄×#120	1	1	yes	characteristic
211	C₂⁴	#2321	1	1	yes	characteristic
212	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
213	C₂⁴	#2358	1	1	yes	characteristic
214	C₂⁴	#2403	1	1	yes	characteristic
215	C₂⁴	#2404	1	1	yes	characteristic
216	C₂⁴	#2408	1	1	yes	characteristic
217	C₂⁴	#2418	1	1	yes	characteristic
218	C₂⁴	#2424	1	1	yes	characteristic
219	C₂⁴	#2534	1	1	yes	characteristic
220	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
221	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
222	C₂²×C₄	D₄×C₂²⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	#2358	1	1	yes	characteristic
224	C₂²×C₄	#2418	1	1	yes	characteristic
225	C₂²×C₄	#2424	1	1	yes	characteristic
226	C₂²⋊C₄	#1889	1	4	yes	—
227	C₂²⋊C₄	#1889	1	4	yes	—
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	#251828	1	1	yes	characteristic
238	C₂²	#251838	1	1	yes	characteristic
239	C₂²	#251860	1	1	yes	characteristic
240	C₂²	#251872	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	#72241178	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241178
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72235130	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. #72314855	C₂	semidirect
2. #72314452	C₂	semidirect
3. #72310283	C₂	semidirect
4. #72266863	C₂	semidirect
5. C₂×#139742	C₂	semidirect
6. #72255841	C₂	semidirect
7. #72255778	C₂	semidirect
8. #72251501	C₂	semidirect
9. #72251473	C₂	semidirect
10. #72246450	C₂	semidirect
11. #72246286	C₂	semidirect
12. #72238613	C₂	semidirect
13. #72237599	C₂	semidirect
14. #72237472	C₂	semidirect
21. C₂×#21776	C₂²	semidirect
23. #511793	C₄	semidirect
30. #392030	C₂²	semidirect
33. #359549	C₂²	semidirect
35. #355096	C₂²	semidirect
38. #187944	C₂²	semidirect
41. C₂×#8453	C₂²	semidirect
42. #128033	C₂²	semidirect
43. C₂×#7304	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. C₂×#7307	C₂²	semidirect
50. #108238	C₂²	semidirect
51. #108222	C₂²	semidirect
52. #62863	C₂×C₄	semidirect
58. #34176	D₄	semidirect
65. C₂×#1913	D₄	semidirect
66. C₂×#1913	D₄	semidirect
70. C₂×#1781	D₄	semidirect
79. #12707	C₂×C₄	semidirect
80. #12705	C₂×C₄	semidirect
81. #12690	C₂×C₄	semidirect
82. #8534	D₄	semidirect
83. #8534	D₄	semidirect
84. #8534	D₄	semidirect
85. #8534	D₄	semidirect
111. C₂×#458	C₂²⋊C₄	semidirect
112. C₂×B(4,𝔽₂)	C₂²⋊C₄	semidirect
117. #1890	C₂²⋊C₄	semidirect
118. #1889	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72314855	2
#72314452	2
#72310283	2
#72266863	2
C₂×#139742	2
#72255841	2
#72255778	2
#72251501	2
#72251473	2
#72246450	2
#72246286	2
#72238613	2
#72237599	2
#72237472	2
#72235130	2