Subgroups — #72251585

Order 2048.

At least 7026 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

The subgroup lattice has 219 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72251533	C₂	1	1	yes	maximal, characteristic, complemented
2	#72251501	C₂	1	1	yes	maximal, characteristic, complemented
3	#72251476	C₂	1	1	yes	maximal, characteristic, complemented
4	#72251463	C₂	1	1	yes	maximal, characteristic, complemented
5	#72238343	C₂	1	1	yes	maximal, characteristic, complemented
6	#72237402	C₂	1	1	yes	maximal, characteristic, complemented
7	#72236948	C₂	1	1	yes	maximal, characteristic, complemented
8	#72236678	C₂	1	1	yes	maximal, characteristic, complemented
9	#72235186	C₂	1	1	yes	maximal, characteristic, complemented
10	#72234100	C₂	1	1	yes	maximal, characteristic, complemented
11	#72225635	C₂	1	1	yes	maximal, characteristic, complemented
12	#72219438	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218215	C₂	1	1	yes	maximal, characteristic, complemented
14	#72218104	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#144382	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×#23900	C₂²	1	1	yes	characteristic, complemented
21	C₂×#23897	C₂²	1	1	yes	characteristic
22	#6437445	C₂²	1	1	yes	characteristic
23	#6431795	C₂²	1	1	yes	characteristic
24	#616961	C₂²	1	1	yes	characteristic
25	C₂×#12705	C₂²	1	1	yes	characteristic
26	C₂×#12705	C₂²	1	1	yes	characteristic
27	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
28	#493370	C₂²	1	1	yes	characteristic
29	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
30	#382544	C₂²	1	1	yes	characteristic, complemented
31	#382436	C₂²	1	1	yes	characteristic
32	#359558	C₂²	1	1	yes	characteristic
33	#359549	C₂²	1	1	yes	characteristic
34	#359549	C₂²	1	1	yes	characteristic
35	#359549	C₂²	1	1	yes	characteristic, complemented
36	#359509	C₂²	1	1	yes	characteristic
37	#359509	C₂²	1	1	yes	characteristic
38	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
39	#188050	C₂²	1	1	yes	characteristic
40	#187944	C₂²	1	1	yes	characteristic
41	#187944	C₂²	1	1	yes	characteristic
42	#187944	C₂²	1	1	yes	characteristic, complemented
43	#146479	C₂²	1	1	yes	characteristic, complemented
44	#146475	C₂²	1	1	yes	characteristic, complemented
45	#128033	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
49	#108238	C₂²	1	1	yes	characteristic, complemented
50	#108238	C₂²	1	1	yes	characteristic, complemented
51	#62842	D₄	1	1	yes	characteristic, complemented
52	C₂×#3484	C₂³	1	1	yes	characteristic
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂²×#535	C₂³	1	1	yes	characteristic
56	C₂³×#120	C₂³	1	1	yes	characteristic
57	C₂×#2883	D₄	1	1	yes	characteristic, complemented
58	C₂×#2883	D₄	1	1	yes	characteristic, complemented
59	C₂×#2846	D₄	1	2	yes	—
60	#34171	C₂³	1	1	yes	characteristic
61	#34171	C₂³	1	1	yes	characteristic
62	#23900	D₄	1	4	yes	complemented
63	C₂×#1913	D₄	1	2	yes	complemented
64	C₂×#1898	D₄	1	2	yes	—
65	#20960	D₄	1	2	yes	complemented
66	#20616	D₄	1	2	yes	complemented
67	#20605	D₄	1	2	yes	complemented
68	C₂×#1783	C₂³	1	1	yes	characteristic
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	#20354	D₄	1	1	yes	characteristic, complemented
76	#15259	D₄	1	2	yes	complemented
77	#15258	D₄	1	2	yes	complemented
78	#14335	C₂³	1	1	yes	characteristic
79	#10740	D₄	1	2	yes	complemented
80	#10716	D₄	1	2	yes	complemented
81	C₂⁷	C₂×D₄	1	1	yes	characteristic
82	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
83	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
84	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
85	C₂³×#41	C₂×D₄	1	1	yes	characteristic
86	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
87	#3484	C₂×D₄	1	2	yes	—
88	#3484	#41	1	2	yes	—
89	C₂×#534	C₂×D₄	1	2	yes	—
90	C₂²×#127	C₂×D₄	1	1	yes	characteristic
91	C₂²×#127	#41	1	1	yes	characteristic
92	C₂²×#124	C₂×D₄	1	1	yes	characteristic
93	C₂²×#124	#41	1	1	yes	characteristic
94	C₂²×#120	C₂×D₄	1	2	yes	—
95	C₂²×#120	C₂×D₄	1	2	yes	—
96	C₂²×#120	C₂×D₄	1	2	yes	—
97	C₂×#518	C₂×D₄	1	1	yes	characteristic
98	C₂×#518	C₂×D₄	1	1	yes	characteristic
99	C₂×#518	C₂×D₄	1	1	yes	characteristic
100	C₂×#518	C₂×D₄	1	1	yes	characteristic
101	C₂×#518	C₂×D₄	1	1	yes	characteristic
102	C₂×#518	C₂×D₄	1	1	yes	characteristic
103	C₂×#518	C₂×D₄	1	1	yes	characteristic
104	C₂²×#117	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
106	C₂²×#100	C₂×D₄	1	2	yes	—
107	C₂²×#99	C₂×D₄	1	2	yes	—
108	C₂×#407	C₂×D₄	1	2	yes	—
109	#2309	C₂×D₄	1	2	yes	—
110	#2280	C₂×D₄	1	2	yes	—
111	C₂×#379	C₂×D₄	1	2	yes	—
112	#1781	C₂×D₄	1	2	yes	—
113	#1438	C₂×D₄	1	2	yes	—
114	C₂⁶	#120	1	2	yes	—
115	C₂⁶	C₂²×D₄	1	1	yes	characteristic
116	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×H(5,𝔽₂)	#121	1	2	yes	complemented
118	C₂³×Q₈	#121	1	1	yes	characteristic, complemented
119	C₂³×D₄	#120	1	2	yes	—
120	C₂³×D₄	#121	1	1	yes	characteristic, complemented
121	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
122	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
123	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
124	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
125	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
126	C₂³×D₄	C₂×#41	1	1	yes	characteristic
127	C₂³×D₄	C₂×#41	1	1	yes	characteristic
128	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
134	#518	#121	1	2	yes	complemented
135	C₂×#117	#120	1	2	yes	—
136	C₂×#117	#120	1	2	yes	—
137	C₂²×C₂²⋊C₄	#120	1	2	yes	—
138	C₂²×C₂²⋊C₄	#120	1	2	yes	—
139	C₂²×C₄²	#121	1	1	yes	characteristic, complemented
140	C₂²×C₄²	#121	1	1	yes	characteristic, complemented
141	C₂⁵	B(4,𝔽₂)	1	2	yes	—
142	C₂⁵	C₂×#120	1	1	yes	characteristic
143	C₂⁵	C₂×#120	1	1	yes	characteristic
144	C₂⁵	#534	1	1	yes	characteristic
145	C₂⁵	#534	1	1	yes	characteristic
146	C₂⁵	#534	1	1	yes	characteristic
147	C₂⁵	#538	1	1	yes	characteristic
148	C₂⁵	D₄²	1	1	yes	characteristic
149	C₂⁵	D₄²	1	1	yes	characteristic
150	C₂⁵	D₄²	1	1	yes	characteristic
151	C₂⁵	#546	1	1	yes	characteristic
152	C₂⁵	#546	1	1	yes	characteristic
153	C₂⁵	#546	1	1	yes	characteristic
154	C₂⁵	#560	1	1	yes	characteristic
155	C₂²×D₄	C₂×#121	1	1	yes	characteristic
156	C₂²×D₄	#525	1	1	yes	characteristic
157	C₂²×D₄	#534	1	1	yes	characteristic
158	C₂²×D₄	#535	1	1	yes	characteristic
159	C₂²×D₄	#535	1	1	yes	characteristic
160	C₂²×D₄	#538	1	1	yes	characteristic
161	C₂²×D₄	D₄²	1	1	yes	characteristic
162	C₂²×D₄	D₄²	1	1	yes	characteristic
163	C₂²×D₄	D₄²	1	1	yes	characteristic
164	C₂²×D₄	D₄²	1	1	yes	characteristic
165	C₂²×D₄	D₄²	1	1	yes	characteristic
166	C₂²×D₄	#546	1	1	yes	characteristic
167	C₂²×D₄	#546	1	1	yes	characteristic
168	C₂²×D₄	#546	1	1	yes	characteristic
169	C₂²×D₄	#546	1	1	yes	characteristic
170	C₂²×D₄	#560	1	1	yes	characteristic
171	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
172	C₂³×C₄	C₂×#120	1	1	yes	characteristic
173	C₂³×C₄	C₂×#121	1	1	yes	characteristic
174	C₂³×C₄	#535	1	1	yes	characteristic
175	C₂³×C₄	D₄²	1	1	yes	characteristic
176	C₂³×C₄	D₄²	1	1	yes	characteristic
177	C₂³×C₄	#546	1	1	yes	characteristic
178	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
179	C₂×C₄²	B(4,𝔽₂)	1	4	yes	complemented
180	C₂⁴	#2403	1	1	yes	characteristic
181	C₂⁴	#2403	1	1	yes	characteristic
182	C₂⁴	#2404	1	1	yes	characteristic
183	C₂⁴	#2410	1	1	yes	characteristic
184	C₂⁴	#2410	1	1	yes	characteristic
185	C₂⁴	#2424	1	1	yes	characteristic
186	C₂⁴	#2668	1	1	yes	characteristic
187	C₂⁴	#2668	1	1	yes	characteristic
188	C₂⁴	#2669	1	1	yes	characteristic
189	C₂⁴	#2678	1	1	yes	characteristic
190	C₂⁴	#2678	1	1	yes	characteristic
191	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
192	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
193	C₂²×C₄	#2408	1	1	yes	characteristic
194	C₂²×C₄	#2408	1	1	yes	characteristic
195	C₂²×C₄	#2424	1	1	yes	characteristic
196	C₂²×C₄	#2669	1	1	yes	characteristic
197	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
198	C₂²×C₄	#3025	1	2	yes	—
199	C₂³	#15257	1	1	yes	characteristic
200	C₂³	#23895	1	1	yes	characteristic
201	C₂³	#23895	1	1	yes	characteristic
202	C₂³	#23900	1	2	yes	—
203	C₂³	#23900	1	1	yes	characteristic
204	C₂³	#23900	1	1	yes	characteristic
205	C₂³	#23900	1	1	yes	characteristic
206	C₂³	#23902	1	1	yes	characteristic
207	C₂³	#23984	1	2	yes	—
208	C₂×C₄	#24260	1	2	yes	—
209	C₂×C₄	#24260	1	2	yes	—
210	C₂²	#245225	1	1	yes	characteristic
211	C₂²	#245228	1	1	yes	characteristic
212	C₂²	#245231	1	1	yes	characteristic
213	C₂²	#245233	1	1	yes	characteristic
214	C₂²	#417493	1	1	yes	characteristic
215	C₂²	#417526	1	1	yes	characteristic
216	C₂²	#417532	1	1	yes	characteristic
217	C₂	#72237498	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72237498
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×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. #72251533	C₂	semidirect
2. #72251501	C₂	semidirect
3. #72251476	C₂	semidirect
4. #72251463	C₂	semidirect
5. #72238343	C₂	semidirect
6. #72237402	C₂	semidirect
7. #72236948	C₂	semidirect
8. #72236678	C₂	semidirect
9. #72235186	C₂	semidirect
10. #72234100	C₂	semidirect
11. #72225635	C₂	semidirect
12. #72219438	C₂	semidirect
13. #72218215	C₂	semidirect
14. #72218104	C₂	semidirect
15. C₂×#144382	C₂	semidirect
17. #7626202	C₂²	semidirect
20. C₂×#23900	C₂²	semidirect
27. C₂×#12705	C₂²	semidirect
29. C₂×#12687	C₂²	semidirect
30. #382544	C₂²	semidirect
35. #359549	C₂²	semidirect
38. C₂×#8547	C₂²	semidirect
42. #187944	C₂²	semidirect
43. #146479	C₂²	semidirect
44. #146475	C₂²	semidirect
45. #128033	C₂²	semidirect
46. C₂×#7316	C₂²	semidirect
47. C₂×#7314	C₂²	semidirect
48. C₂×#7311	C₂²	semidirect
49. #108238	C₂²	semidirect
50. #108238	C₂²	semidirect
51. #62842	D₄	semidirect
57. C₂×#2883	D₄	semidirect
58. C₂×#2883	D₄	semidirect
62. #23900	D₄	semidirect
63. C₂×#1913	D₄	semidirect
65. #20960	D₄	semidirect
66. #20616	D₄	semidirect
67. #20605	D₄	semidirect
75. #20354	D₄	semidirect
76. #15259	D₄	semidirect
77. #15258	D₄	semidirect
79. #10740	D₄	semidirect
80. #10716	D₄	semidirect
117. C₂×H(5,𝔽₂)	#121	semidirect
118. C₂³×Q₈	#121	semidirect
120. C₂³×D₄	#121	semidirect
134. #518	#121	semidirect
139. C₂²×C₄²	#121	semidirect
140. C₂²×C₄²	#121	semidirect
179. 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
#72251533	2
#72251501	2
#72251476	2
#72251463	2
#72238343	2
#72237402	2
#72236948	2
#72236678	2
#72235186	2
#72234100	2
#72225635	2
#72219438	2
#72218215	2
#72218104	2
C₂×#144382	2