Subgroups — #72251580

Order 2048.

At least 6475 conjugacy classes of subgroups; only the 374 normal subgroups are shown.

The subgroup lattice has 250 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72251550	C₂	1	1	yes	maximal, characteristic, complemented
2	#72251528	C₂	1	1	yes	maximal, characteristic, complemented
3	#72251499	C₂	1	1	yes	maximal, characteristic, complemented
4	#72251485	C₂	1	1	yes	maximal, characteristic, complemented
5	#72251473	C₂	1	1	yes	maximal, characteristic, complemented
6	#72251462	C₂	1	1	yes	maximal, characteristic, complemented
7	#72251422	C₂	1	1	yes	maximal, characteristic, complemented
8	#72238343	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237402	C₂	1	1	yes	maximal, characteristic
10	#72235186	C₂	1	1	yes	maximal, characteristic, complemented
11	#72225639	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#139748	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218317	C₂	1	1	yes	maximal, characteristic, complemented
14	#72218206	C₂	1	1	yes	maximal, characteristic, complemented
15	#72218104	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic
17	#7626202	C₂²	1	1	yes	characteristic, complemented
18	C₂×#23900	C₂²	1	1	yes	characteristic
19	C₂×#23900	C₂²	1	1	yes	characteristic
20	C₂×#21778	C₂²	1	1	yes	characteristic
21	#6371558	C₂²	1	1	yes	characteristic
22	#616961	C₂²	1	1	yes	characteristic
23	C₂×#12705	C₂²	1	1	yes	characteristic
24	C₂×#12705	C₂²	1	1	yes	characteristic
25	#493370	C₂²	1	1	yes	characteristic
26	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
27	#373637	C₂²	1	1	yes	characteristic, complemented
28	#359558	C₂²	1	1	yes	characteristic
29	#359558	C₂²	1	1	yes	characteristic
30	#359549	C₂²	1	1	yes	characteristic
31	#359549	C₂²	1	1	yes	characteristic
32	#359509	C₂²	1	1	yes	characteristic
33	#359509	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	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
40	#188050	C₂²	1	1	yes	characteristic
41	#187944	C₂²	1	1	yes	characteristic
42	#187944	C₂²	1	1	yes	characteristic
43	#146479	C₂²	1	1	yes	characteristic, complemented
44	#146475	C₂²	1	1	yes	characteristic, complemented
45	#146475	C₂²	1	1	yes	characteristic, complemented
46	#128033	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
50	#108238	C₂²	1	1	yes	characteristic, complemented
51	#108238	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	#62652	D₄	1	1	yes	characteristic, complemented
60	C₂×#2883	D₄	1	1	yes	characteristic, complemented
61	C₂×#2846	D₄	1	2	yes	—
62	#34171	C₂³	1	1	yes	characteristic
63	#34171	C₂³	1	1	yes	characteristic
64	#34171	C₂³	1	1	yes	characteristic
65	#23900	C₂×C₄	1	4	yes	—
66	#23900	C₂×C₄	1	4	yes	complemented
67	#23900	D₄	1	4	yes	—
68	C₂×#1913	D₄	1	2	yes	complemented
69	C₂×#1898	D₄	1	2	yes	—
70	#20960	D₄	1	2	yes	complemented
71	#20616	D₄	1	2	yes	complemented
72	#20605	D₄	1	2	yes	complemented
73	C₂×#1783	C₂³	1	1	yes	characteristic
74	C₂×#1783	C₂³	1	1	yes	characteristic
75	C₂×#1783	C₂³	1	1	yes	characteristic
76	C₂×#1781	C₂³	1	1	yes	characteristic
77	C₂×#1781	C₂³	1	1	yes	characteristic
78	C₂×#1781	C₂³	1	1	yes	characteristic
79	C₂×#1781	C₂³	1	1	yes	characteristic
80	#20354	D₄	1	1	yes	characteristic, complemented
81	#20354	D₄	1	1	yes	characteristic, complemented
82	#15259	C₂×C₄	1	2	yes	complemented
83	#15258	C₂×C₄	1	2	yes	complemented
84	#14335	C₂³	1	1	yes	characteristic
85	#10740	D₄	1	2	yes	complemented
86	#10716	D₄	1	2	yes	complemented
87	#8546	C₂×C₄	1	2	yes	complemented
88	#8544	C₂×C₄	1	2	yes	complemented
89	#8534	C₂×C₄	1	4	yes	complemented
90	#8534	C₂×C₄	1	4	yes	complemented
91	C₂⁷	C₂×D₄	1	1	yes	characteristic
92	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
93	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
94	C₂³×#41	C₂×D₄	1	1	yes	characteristic
95	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
96	#3484	C₂²×C₄	1	2	yes	—
97	#3484	C₂²×C₄	1	2	yes	—
98	#3484	C₂×D₄	1	2	yes	—
99	#3484	#41	1	2	yes	—
100	#3484	#41	1	2	yes	—
101	#3484	#41	1	2	yes	—
102	C₂×#534	C₂×D₄	1	2	yes	—
103	C₂²×#127	C₂²×C₄	1	1	yes	characteristic
104	C₂²×#127	#41	1	1	yes	characteristic
105	C₂²×#124	C₂²×C₄	1	1	yes	characteristic
106	C₂²×#124	#41	1	1	yes	characteristic
107	C₂²×#120	C₂²×C₄	1	2	yes	—
108	C₂²×#120	C₂×D₄	1	2	yes	—
109	C₂²×#120	C₂×D₄	1	2	yes	—
110	C₂×#518	C₂×D₄	1	1	yes	characteristic
111	C₂×#518	C₂×D₄	1	1	yes	characteristic
112	C₂×#518	C₂×D₄	1	1	yes	characteristic
113	C₂×#518	C₂×D₄	1	1	yes	characteristic
114	C₂×#518	C₂×D₄	1	1	yes	characteristic
115	C₂×#518	C₂×D₄	1	1	yes	characteristic
116	C₂×#518	C₂×D₄	1	1	yes	characteristic
117	C₂×#518	C₂×D₄	1	1	yes	characteristic
118	C₂²×#117	C₂×D₄	1	1	yes	characteristic
119	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
120	C₂²×#100	C₂×D₄	1	2	yes	—
121	C₂²×#99	C₂×D₄	1	2	yes	—
122	C₂×#407	C₂×D₄	1	2	yes	—
123	#2846	C₂²⋊C₄	1	8	yes	complemented
124	#2309	C₂×D₄	1	2	yes	—
125	#2280	C₂×D₄	1	2	yes	—
126	C₂×#379	C₂×D₄	1	2	yes	—
127	#1898	C₂²⋊C₄	1	8	yes	complemented
128	#1781	C₂²×C₄	1	2	yes	—
129	#1781	C₂²×C₄	1	2	yes	—
130	#1781	C₂×D₄	1	2	yes	—
131	#1438	C₂²×C₄	1	2	yes	—
132	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂⁶	C₂²×D₄	1	1	yes	characteristic
134	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
136	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	complemented
137	C₂×H(5,𝔽₂)	#121	1	2	yes	—
138	C₂³×Q₈	C₄×D₄	1	1	yes	characteristic, complemented
139	C₂³×D₄	C₄×D₄	1	1	yes	characteristic, complemented
140	C₂³×D₄	#120	1	2	yes	—
141	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
142	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
143	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
144	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
145	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
146	C₂³×D₄	C₂×#41	1	1	yes	characteristic
147	C₂³×D₄	C₂×#41	1	1	yes	characteristic
148	C₂³×D₄	C₂×#41	1	1	yes	characteristic
149	C₂³×D₄	C₂×#41	1	1	yes	characteristic
150	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
154	#534	C₄×D₄	1	4	yes	—
155	#534	#121	1	4	yes	—
156	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
157	C₂×#120	#123	1	4	yes	—
158	#518	C₄×D₄	1	2	yes	—
159	#518	C₄×D₄	1	2	yes	complemented
160	#518	#121	1	2	yes	—
161	C₂×#117	#120	1	2	yes	—
162	C₂×#117	#120	1	2	yes	—
163	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
164	C₂²×C₂²⋊C₄	#120	1	2	yes	—
165	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic, complemented
166	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic, complemented
167	#379	C₂×C₂²⋊C₄	1	4	yes	—
168	C₂⁵	B(4,𝔽₂)	1	2	yes	—
169	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
170	C₂⁵	#518	1	1	yes	characteristic
171	C₂⁵	C₂×#120	1	1	yes	characteristic
172	C₂⁵	C₂×#123	1	1	yes	characteristic
173	C₂⁵	#534	1	1	yes	characteristic
174	C₂⁵	#538	1	1	yes	characteristic
175	C₂⁵	D₄²	1	1	yes	characteristic
176	C₂⁵	D₄²	1	1	yes	characteristic
177	C₂⁵	D₄²	1	1	yes	characteristic
178	C₂⁵	#546	1	1	yes	characteristic
179	C₂⁵	#546	1	1	yes	characteristic
180	C₂⁵	#546	1	1	yes	characteristic
181	C₂⁵	#551	1	1	yes	characteristic
182	C₂²×D₄	#386	1	4	yes	—
183	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
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	#518	1	1	yes	characteristic
188	C₂²×D₄	C₂×#121	1	1	yes	characteristic
189	C₂²×D₄	C₂×#121	1	1	yes	characteristic
190	C₂²×D₄	#525	1	1	yes	characteristic
191	C₂²×D₄	#525	1	1	yes	characteristic
192	C₂²×D₄	#535	1	1	yes	characteristic
193	C₂²×D₄	#538	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	#546	1	1	yes	characteristic
198	C₂²×D₄	#546	1	1	yes	characteristic
199	C₂²×D₄	#551	1	1	yes	characteristic
200	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
201	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂³×C₄	C₂×#120	1	1	yes	characteristic
203	C₂³×C₄	#525	1	1	yes	characteristic
204	C₂³×C₄	D₄²	1	1	yes	characteristic
205	C₂³×C₄	D₄²	1	1	yes	characteristic
206	C₂³×C₄	#546	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	#386	1	4	yes	—
208	C₂×C₄²	B(4,𝔽₂)	1	4	yes	complemented
209	C₂⁴	#1889	1	4	yes	—
210	C₂⁴	C₂×#386	1	1	yes	characteristic
211	C₂⁴	#2321	1	1	yes	characteristic
212	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
213	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
214	C₂⁴	#2358	1	1	yes	characteristic
215	C₂⁴	#2403	1	1	yes	characteristic
216	C₂⁴	#2404	1	1	yes	characteristic
217	C₂⁴	#2418	1	1	yes	characteristic
218	C₂⁴	#2418	1	1	yes	characteristic
219	C₂⁴	#2424	1	1	yes	characteristic
220	C₂⁴	#2534	1	1	yes	characteristic
221	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
222	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
223	C₂²×C₄	#1889	1	4	yes	—
224	C₂²×C₄	C₄×#120	1	1	yes	characteristic
225	C₂²×C₄	#2358	1	1	yes	characteristic
226	C₂²×C₄	#2408	1	1	yes	characteristic
227	C₂²×C₄	#2424	1	1	yes	characteristic
228	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
229	C₂²×C₄	#3025	1	2	yes	—
230	C₂³	#13967	1	1	yes	characteristic
231	C₂³	C₂×#1889	1	2	yes	—
232	C₂³	#23895	1	1	yes	characteristic
233	C₂³	#23896	1	1	yes	characteristic
234	C₂³	#23900	1	1	yes	characteristic
235	C₂³	#23900	1	1	yes	characteristic
236	C₂³	#23900	1	1	yes	characteristic
237	C₂³	#23902	1	1	yes	characteristic
238	C₂³	#23984	1	2	yes	—
239	C₂×C₄	C₄×B(4,𝔽₂)	1	2	yes	—
240	C₂×C₄	#24260	1	2	yes	—
241	C₂²	#245225	1	1	yes	characteristic
242	C₂²	#245227	1	1	yes	characteristic
243	C₂²	#245229	1	1	yes	characteristic
244	C₂²	#245231	1	1	yes	characteristic
245	C₂²	#364410	1	1	yes	characteristic
246	C₂²	#364500	1	1	yes	characteristic
247	C₂²	#364513	1	1	yes	characteristic
248	C₂	#72237497	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72237497
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. #72251550	C₂	semidirect
2. #72251528	C₂	semidirect
3. #72251499	C₂	semidirect
4. #72251485	C₂	semidirect
5. #72251473	C₂	semidirect
6. #72251462	C₂	semidirect
7. #72251422	C₂	semidirect
8. #72238343	C₂	semidirect
10. #72235186	C₂	semidirect
11. #72225639	C₂	semidirect
12. C₂×#139748	C₂	semidirect
13. #72218317	C₂	semidirect
14. #72218206	C₂	semidirect
15. #72218104	C₂	semidirect
17. #7626202	C₂²	semidirect
26. C₂×#12687	C₂²	semidirect
27. #373637	C₂²	semidirect
35. #359501	C₂²	semidirect
36. #355096	C₂²	semidirect
37. #347047	C₄	semidirect
38. #347045	C₄	semidirect
39. C₂×#8547	C₂²	semidirect
43. #146479	C₂²	semidirect
44. #146475	C₂²	semidirect
45. #146475	C₂²	semidirect
46. #128033	C₂²	semidirect
47. C₂×#7316	C₂²	semidirect
48. C₂×#7314	C₂²	semidirect
49. C₂×#7311	C₂²	semidirect
50. #108238	C₂²	semidirect
51. #108238	C₂²	semidirect
52. #107114	C₂²	semidirect
53. #62832	C₂×C₄	semidirect
54. #62824	C₂×C₄	semidirect
59. #62652	D₄	semidirect
60. C₂×#2883	D₄	semidirect
66. #23900	C₂×C₄	semidirect
68. C₂×#1913	D₄	semidirect
70. #20960	D₄	semidirect
71. #20616	D₄	semidirect
72. #20605	D₄	semidirect
80. #20354	D₄	semidirect
81. #20354	D₄	semidirect
82. #15259	C₂×C₄	semidirect
83. #15258	C₂×C₄	semidirect
85. #10740	D₄	semidirect
86. #10716	D₄	semidirect
87. #8546	C₂×C₄	semidirect
88. #8544	C₂×C₄	semidirect
89. #8534	C₂×C₄	semidirect
90. #8534	C₂×C₄	semidirect
123. #2846	C₂²⋊C₄	semidirect
127. #1898	C₂²⋊C₄	semidirect
136. C₂×H(5,𝔽₂)	C₄×D₄	semidirect
138. C₂³×Q₈	C₄×D₄	semidirect
139. C₂³×D₄	C₄×D₄	semidirect
159. #518	C₄×D₄	semidirect
165. C₂²×C₄²	C₄×D₄	semidirect
166. C₂²×C₄²	C₄×D₄	semidirect
208. C₂×C₄²	B(4,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#72251550	2
#72251528	2
#72251499	2
#72251485	2
#72251473	2
#72251462	2
#72251422	2
#72238343	2
#72237402	2
#72235186	2
#72225639	2
C₂×#139748	2
#72218317	2
#72218206	2
#72218104	2