Subgroups — #72219445

Order 2048.

At least 6236 conjugacy classes of subgroups; only the 528 normal subgroups are shown.

The subgroup lattice has 243 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72219418	C₂	1	1	yes	maximal, characteristic, complemented
2	#72219414	C₂	1	1	yes	maximal, characteristic, complemented
3	#72219301	C₂	1	1	yes	maximal, characteristic, complemented
4	#72219158	C₂	1	4	yes	maximal, complemented
5	#72218518	C₂	1	1	yes	maximal, characteristic, complemented
6	#72218306	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#139750	C₂	1	4	yes	maximal, complemented
8	C₂²×#23900	C₂	1	1	yes	maximal, characteristic
9	#72215179	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#3484	C₂²	1	1	yes	characteristic
11	C₂⁴×#120	C₂²	1	1	yes	characteristic
12	C₂²×#2846	C₄	1	2	yes	complemented
13	C₂×#23900	C₄	1	4	yes	complemented
14	C₂×#23900	C₂²	1	2	yes	—
15	C₂×#23900	C₂²	1	2	yes	—
16	C₂×#21776	C₂²	1	2	yes	complemented
17	C₂²×#1898	C₄	1	2	yes	complemented
18	C₂²×#1781	C₂²	1	1	yes	characteristic
19	C₂×#12705	C₂²	1	2	yes	complemented
20	C₂×#12705	C₂²	1	2	yes	complemented
21	C₂×#12687	C₂²	1	4	yes	complemented
22	#392113	C₂²	1	1	yes	characteristic
23	#392109	C₂²	1	1	yes	characteristic
24	#392109	C₂²	1	1	yes	characteristic
25	#392109	C₂²	1	1	yes	characteristic
26	#382434	C₂²	1	2	yes	complemented
27	C₂×#8541	C₂²	1	4	yes	complemented
28	C₂×#7316	C₂²	1	4	yes	complemented
29	C₂×#7310	C₂²	1	4	yes	complemented
30	C₂⁸	C₂×C₄	1	1	yes	characteristic, complemented
31	C₂³×H(5,𝔽₂)	D₄	1	1	yes	characteristic
32	C₂⁵×D₄	C₂³	1	1	yes	characteristic
33	C₂×#3484	C₂×C₄	1	2	yes	—
34	C₂×#3484	D₄	1	2	yes	—
35	C₂×#3484	D₄	1	1	yes	characteristic
36	C₂×#3484	D₄	1	1	yes	characteristic
37	C₂×#3484	C₂³	1	1	yes	characteristic
38	C₂×#3484	C₂³	1	1	yes	characteristic
39	C₂²×#534	D₄	1	2	yes	—
40	C₂³×#120	C₂×C₄	1	2	yes	—
41	C₂³×#120	C₂×C₄	1	2	yes	complemented
42	C₂³×#120	C₂³	1	2	yes	—
43	C₂²×#518	D₄	1	1	yes	characteristic
44	C₂⁴×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic, complemented
45	C₂×#2846	C₂×C₄	1	4	yes	—
46	C₂×#2846	C₂×C₄	1	4	yes	complemented
47	C₂²×#379	C₂×C₄	1	2	yes	complemented
48	#23900	D₄	1	16	yes	—
49	C₂×#1898	C₂×C₄	1	4	yes	—
50	C₂×#1898	C₂×C₄	1	4	yes	complemented
51	C₂×#1783	C₂³	1	2	yes	—
52	C₂×#1781	C₂×C₄	1	2	yes	complemented
53	C₂×#1781	D₄	1	2	yes	complemented
54	C₂×#1781	D₄	1	2	yes	complemented
55	C₂×#1781	D₄	1	2	yes	complemented
56	C₂×#1781	C₂³	1	2	yes	—
57	C₂×#1781	C₂³	1	2	yes	—
58	C₂×#1781	C₂³	1	2	yes	—
59	C₂×#1781	C₂³	1	2	yes	—
60	#14679	D₄	1	2	yes	complemented
61	C₂⁷	C₂²×C₄	1	1	yes	characteristic
62	C₂⁷	C₂²×C₄	1	2	yes	—
63	C₂⁷	C₂×D₄	1	1	yes	characteristic
64	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	2	yes	—
65	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
66	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
67	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
68	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
69	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
70	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
71	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
72	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
73	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
74	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
75	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
76	C₂⁴×D₄	#41	1	1	yes	characteristic
77	C₂⁴×D₄	#41	1	1	yes	characteristic
78	C₂⁴×D₄	#41	1	1	yes	characteristic
79	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
80	C₂⁵×C₄	#41	1	1	yes	characteristic
81	#3484	C₂×D₄	1	8	yes	—
82	C₂×#534	C₂²⋊C₄	1	2	yes	—
83	C₂×#534	C₂²⋊C₄	1	2	yes	—
84	C₂×#534	C₂×D₄	1	2	yes	—
85	C₂×#534	C₂×D₄	1	2	yes	—
86	C₂²×#120	C₂²⋊C₄	1	2	yes	—
87	C₂²×#120	C₂²⋊C₄	1	2	yes	—
88	C₂²×#120	C₂²×C₄	1	2	yes	—
89	C₂²×#120	C₂²×C₄	1	2	yes	—
90	C₂²×#120	C₂×D₄	1	8	yes	—
91	C₂×#518	C₂²⋊C₄	1	2	yes	—
92	C₂×#518	C₂×D₄	1	1	yes	characteristic
93	C₂×#518	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
95	C₂³×C₂²⋊C₄	C₂²×C₄	1	2	yes	—
96	C₂²×#99	C₂×D₄	1	4	yes	complemented
97	#2846	C₂²⋊C₄	1	16	yes	complemented
98	C₂×#379	C₂²×C₄	1	4	yes	—
99	C₂×#379	C₂×D₄	1	4	yes	complemented
100	#1898	C₂²⋊C₄	1	16	yes	complemented
101	#1781	C₂×D₄	1	8	yes	—
102	C₂⁶	#99	1	2	yes	complemented
103	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂⁶	C₂×C₂²⋊C₄	1	4	yes	complemented
106	C₂⁶	C₄×D₄	1	1	yes	characteristic
107	C₂⁶	C₄×D₄	1	1	yes	characteristic
108	C₂⁶	C₄×D₄	1	1	yes	characteristic
109	C₂⁶	C₄×D₄	1	1	yes	characteristic
110	C₂⁶	C₄×D₄	1	1	yes	characteristic
111	C₂⁶	#120	1	1	yes	characteristic
112	C₂⁶	#120	1	1	yes	characteristic
113	C₂⁶	#121	1	1	yes	characteristic
114	C₂⁶	#121	1	1	yes	characteristic
115	C₂⁶	C₂³×C₄	1	1	yes	characteristic
116	C₂⁶	C₂²×D₄	1	1	yes	characteristic
117	C₂×H(5,𝔽₂)	#120	1	4	yes	—
118	C₂³×D₄	#99	1	2	yes	complemented
119	C₂³×D₄	#99	1	2	yes	complemented
120	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
125	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
126	C₂³×D₄	#120	1	1	yes	characteristic
127	C₂³×D₄	#120	1	1	yes	characteristic
128	C₂³×D₄	#121	1	1	yes	characteristic
129	C₂³×D₄	#121	1	1	yes	characteristic
130	C₂³×D₄	#121	1	1	yes	characteristic
131	C₂³×D₄	#121	1	1	yes	characteristic
132	C₂³×D₄	#123	1	1	yes	characteristic
133	C₂³×D₄	#123	1	1	yes	characteristic
134	C₂³×D₄	C₂²×D₄	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	4	yes	—
138	C₂³×D₄	C₂×#41	1	1	yes	characteristic
139	C₂³×D₄	H(5,𝔽₂)	1	4	yes	—
140	C₂⁴×C₄	#99	1	2	yes	complemented
141	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
142	C₂⁴×C₄	#121	1	1	yes	characteristic
143	C₂⁴×C₄	#121	1	1	yes	characteristic
144	C₂⁴×C₄	#123	1	1	yes	characteristic
145	C₂⁴×C₄	#123	1	1	yes	characteristic
146	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
147	#534	#120	1	8	yes	—
148	C₂×#120	#99	1	8	yes	complemented
149	C₂×#120	C₂×C₂²⋊C₄	1	8	yes	—
150	#518	#120	1	4	yes	—
151	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	complemented
154	#379	C₂×C₂²⋊C₄	1	8	yes	complemented
155	C₂⁵	#386	1	1	yes	characteristic
156	C₂⁵	#386	1	1	yes	characteristic
157	C₂⁵	#386	1	1	yes	characteristic
158	C₂⁵	#386	1	1	yes	characteristic
159	C₂⁵	#386	1	1	yes	characteristic
160	C₂⁵	#386	1	1	yes	characteristic
161	C₂⁵	C₂×#99	1	2	yes	—
162	C₂⁵	C₂×#99	1	2	yes	—
163	C₂⁵	C₂×#99	1	2	yes	—
164	C₂⁵	C₂×#99	1	4	yes	—
165	C₂⁵	C₂×#99	1	2	yes	—
166	C₂⁵	B(4,𝔽₂)	1	4	yes	—
167	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
168	C₂⁵	C₂²×C₂²⋊C₄	1	2	yes	—
169	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂⁵	#518	1	2	yes	—
172	C₂⁵	C₂×#120	1	1	yes	characteristic
173	C₂⁵	C₂×#121	1	1	yes	characteristic
174	C₂⁵	D₄²	1	4	yes	—
175	C₂²×D₄	#379	1	4	yes	—
176	C₂²×D₄	C₂×#99	1	4	yes	—
177	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
178	C₂²×D₄	C₂×#120	1	2	yes	—
179	C₂²×D₄	C₂×#120	1	2	yes	—
180	C₂²×D₄	#534	1	2	yes	—
181	C₂²×D₄	#534	1	2	yes	—
182	C₂²×D₄	D₄²	1	4	yes	—
183	C₂²×D₄	#546	1	4	yes	—
184	C₂³×C₄	#386	1	1	yes	characteristic
185	C₂³×C₄	#386	1	1	yes	characteristic
186	C₂³×C₄	C₂×#99	1	2	yes	—
187	C₂³×C₄	C₂×#99	1	2	yes	—
188	C₂³×C₄	C₂×#121	1	1	yes	characteristic
189	C₂³×C₄	C₂×#123	1	1	yes	characteristic
190	C₂³×C₄	#546	1	4	yes	—
191	C₂×C₂²⋊C₄	#379	1	4	yes	—
192	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
193	C₂⁴	#1781	1	4	yes	complemented
194	C₂⁴	#1889	1	2	yes	—
195	C₂⁴	#1889	1	2	yes	—
196	C₂⁴	#1889	1	2	yes	—
197	C₂⁴	#1898	1	1	yes	characteristic
198	C₂⁴	#1898	1	2	yes	—
199	C₂⁴	#1898	1	2	yes	—
200	C₂⁴	#1898	1	1	yes	characteristic
201	C₂⁴	#1898	1	2	yes	—
202	C₂⁴	#1913	1	2	yes	—
203	C₂⁴	#1913	1	2	yes	—
204	C₂⁴	C₂×#379	1	1	yes	characteristic
205	C₂⁴	C₂×#386	1	1	yes	characteristic
206	C₂⁴	#2308	1	1	yes	characteristic
207	C₂⁴	D₄×C₂²⋊C₄	1	2	yes	—
208	C₂⁴	D₄×C₂²⋊C₄	1	2	yes	—
209	C₂⁴	#2403	1	2	yes	—
210	C₂⁴	#2410	1	2	yes	—
211	C₂⁴	C₂²×#99	1	2	yes	—
212	C₂⁴	C₂²×#99	1	2	yes	—
213	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
214	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
215	C₂²×C₄	#1781	1	4	yes	complemented
216	C₂²×C₄	#1889	1	2	yes	—
217	C₂²×C₄	#2405	1	2	yes	—
218	C₂²×C₄	#2410	1	2	yes	—
219	C₂³	#9495	1	2	yes	—
220	C₂³	#9495	1	2	yes	—
221	C₂³	#9495	1	4	yes	—
222	C₂³	#13867	1	1	yes	characteristic
223	C₂³	C₂×#1781	1	2	yes	—
224	C₂³	C₂×#1889	1	1	yes	characteristic
225	C₂³	C₂×#1889	1	1	yes	characteristic
226	C₂³	C₂×#1898	1	1	yes	characteristic
227	C₂³	C₂×#1898	1	1	yes	characteristic
228	C₂³	C₂×#1913	1	1	yes	characteristic
229	C₂³	C₂×#1913	1	1	yes	characteristic
230	C₂³	#21746	1	2	yes	—
231	C₂³	#23900	1	4	yes	—
232	C₂²	#139750	1	4	yes	complemented
233	C₂²	C₂×#9495	1	1	yes	characteristic
234	C₂²	C₂×#9495	1	1	yes	characteristic
235	C₂²	#201664	1	1	yes	characteristic
236	C₂²	#201664	1	1	yes	characteristic
237	C₂²	#359428	1	1	yes	characteristic
238	C₂²	#359442	1	1	yes	characteristic
239	C₂²	#359501	1	1	yes	characteristic
240	C₂	C₂×#139750	1	2	yes	—
241	C₂	#72218846	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#9495
commutator	C₂⁶	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	C₂²×#23900	C₂
LCS	G ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁵×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. #72219418	C₂	semidirect
2. #72219414	C₂	semidirect
3. #72219301	C₂	semidirect
4. #72219158	C₂	semidirect
5. #72218518	C₂	semidirect
6. #72218306	C₂	semidirect
7. C₂×#139750	C₂	semidirect
9. #72215179	C₂	semidirect
12. C₂²×#2846	C₄	semidirect
13. C₂×#23900	C₄	semidirect
16. C₂×#21776	C₂²	semidirect
17. C₂²×#1898	C₄	semidirect
19. C₂×#12705	C₂²	semidirect
20. C₂×#12705	C₂²	semidirect
21. C₂×#12687	C₂²	semidirect
26. #382434	C₂²	semidirect
27. C₂×#8541	C₂²	semidirect
28. C₂×#7316	C₂²	semidirect
29. C₂×#7310	C₂²	semidirect
30. C₂⁸	C₂×C₄	semidirect
41. C₂³×#120	C₂×C₄	semidirect
44. C₂⁴×C₂²⋊C₄	C₂×C₄	semidirect
46. C₂×#2846	C₂×C₄	semidirect
47. C₂²×#379	C₂×C₄	semidirect
50. C₂×#1898	C₂×C₄	semidirect
52. C₂×#1781	C₂×C₄	semidirect
53. C₂×#1781	D₄	semidirect
54. C₂×#1781	D₄	semidirect
55. C₂×#1781	D₄	semidirect
60. #14679	D₄	semidirect
96. C₂²×#99	C₂×D₄	semidirect
97. #2846	C₂²⋊C₄	semidirect
99. C₂×#379	C₂×D₄	semidirect
100. #1898	C₂²⋊C₄	semidirect
102. C₂⁶	#99	semidirect
105. C₂⁶	C₂×C₂²⋊C₄	semidirect
118. C₂³×D₄	#99	semidirect
119. C₂³×D₄	#99	semidirect
140. C₂⁴×C₄	#99	semidirect
148. C₂×#120	#99	semidirect
153. C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	semidirect
154. #379	C₂×C₂²⋊C₄	semidirect
193. C₂⁴	#1781	semidirect
215. C₂²×C₄	#1781	semidirect
232. C₂²	#139750	semidirect