Subgroups — #72312569

Order 2048.

At least 6866 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

The subgroup lattice has 226 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72312218	C₂	1	1	yes	maximal, characteristic, complemented
2	#72310358	C₂	1	1	yes	maximal, characteristic, complemented
3	#72309800	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266835	C₂	1	1	yes	maximal, characteristic, complemented
5	#72263490	C₂	1	1	yes	maximal, characteristic, complemented
6	#72256113	C₂	1	1	yes	maximal, characteristic, complemented
7	#72255971	C₂	1	1	yes	maximal, characteristic, complemented
8	#72251533	C₂	1	1	yes	maximal, characteristic, complemented
9	#72238343	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237903	C₂	1	1	yes	maximal, characteristic, complemented
11	#72236678	C₂	1	1	yes	maximal, characteristic, complemented
12	#72236521	C₂	1	1	yes	maximal, characteristic, complemented
13	#72234100	C₂	1	1	yes	maximal, characteristic, complemented
14	#72219163	C₂	1	1	yes	maximal, characteristic, complemented
15	#72219159	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic
17	#6437621	C₂²	1	1	yes	characteristic
18	#6437621	C₂²	1	1	yes	characteristic
19	#6437621	C₂²	1	1	yes	characteristic, complemented
20	C₂×#23897	C₂²	1	1	yes	characteristic
21	#6437445	C₂²	1	1	yes	characteristic
22	#6431795	C₂²	1	1	yes	characteristic
23	#6374128	C₂²	1	1	yes	characteristic, complemented
24	#763333	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	#493357	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	#188109	C₂²	1	1	yes	characteristic, complemented
39	#188050	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	C₂×#8534	C₂²	1	1	yes	characteristic
42	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
43	#146479	C₂²	1	1	yes	characteristic, complemented
44	#146475	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7304	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	#107130	C₂²	1	1	yes	characteristic, complemented
50	#107130	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	C₂²×#534	C₂³	1	1	yes	characteristic
57	C₂×#2883	D₄	1	1	yes	characteristic, complemented
58	C₂×#2883	D₄	1	1	yes	characteristic, complemented
59	#34980	D₄	1	2	yes	—
60	#34171	C₂³	1	1	yes	characteristic
61	#34171	C₂³	1	1	yes	characteristic
62	#23901	D₄	1	2	yes	complemented
63	#23900	D₄	1	2	yes	complemented
64	C₂×#1913	D₄	1	2	yes	complemented
65	C₂×#1898	D₄	1	2	yes	—
66	#20960	D₄	1	2	yes	complemented
67	#20616	D₄	1	2	yes	complemented
68	#20605	D₄	1	2	yes	complemented
69	C₂×#1783	C₂³	1	1	yes	characteristic
70	C₂×#1783	C₂³	1	1	yes	characteristic
71	C₂×#1783	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	#20354	D₄	1	1	yes	characteristic, complemented
77	#15262	D₄	1	1	yes	characteristic, complemented
78	#15261	D₄	1	1	yes	characteristic, complemented
79	#15259	D₄	1	1	yes	characteristic, complemented
80	#15258	D₄	1	1	yes	characteristic, complemented
81	C₂×#1438	C₂³	1	1	yes	characteristic
82	#10740	D₄	1	2	yes	complemented
83	#10716	D₄	1	2	yes	complemented
84	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
85	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
86	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
87	C₂³×#41	C₂×D₄	1	1	yes	characteristic
88	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
89	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
90	#3484	C₂×D₄	1	2	yes	—
91	#3484	#41	1	2	yes	—
92	C₂×#534	C₂×D₄	1	2	yes	—
93	C₂²×#127	C₂×D₄	1	1	yes	characteristic
94	C₂²×#127	#41	1	1	yes	characteristic
95	C₂²×#124	C₂×D₄	1	1	yes	characteristic
96	C₂²×#124	#41	1	1	yes	characteristic
97	C₂²×#123	C₂×D₄	1	1	yes	characteristic
98	C₂²×#123	C₂×D₄	1	1	yes	characteristic
99	C₂²×#120	C₂×D₄	1	2	yes	—
100	C₂²×#120	C₂×D₄	1	1	yes	characteristic
101	C₂²×#120	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	C₂×#518	C₂×D₄	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	C₂×D₄	1	1	yes	characteristic
109	C₂²×#117	C₂×D₄	1	1	yes	characteristic
110	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
111	C₂²×#100	C₂×D₄	1	2	yes	—
112	C₂²×#99	C₂×D₄	1	2	yes	—
113	C₂×#407	C₂×D₄	1	2	yes	—
114	#2309	C₂×D₄	1	2	yes	—
115	#2280	C₂×D₄	1	2	yes	—
116	C₂×#379	C₂×D₄	1	2	yes	—
117	#1781	C₂×D₄	1	2	yes	—
118	#1438	C₂×D₄	1	1	yes	characteristic
119	#1438	C₂×D₄	1	1	yes	characteristic
120	C₂⁶	#120	1	1	yes	characteristic
121	C₂⁶	C₂²×D₄	1	1	yes	characteristic
122	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×H(5,𝔽₂)	#121	1	2	yes	—
124	C₂³×Q₈	#121	1	1	yes	characteristic
125	C₂³×D₄	#120	1	2	yes	—
126	C₂³×D₄	#121	1	1	yes	characteristic
127	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
128	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
129	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
130	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
131	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
132	C₂³×D₄	C₂×#41	1	1	yes	characteristic
133	C₂³×D₄	C₂×#41	1	1	yes	characteristic
134	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂⁴×C₄	#120	1	1	yes	characteristic
140	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
141	#518	#121	1	2	yes	—
142	C₂×#117	#120	1	2	yes	—
143	C₂×#117	#120	1	2	yes	—
144	C₂²×C₂²⋊C₄	#120	1	2	yes	—
145	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
146	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
147	C₂²×C₄²	#121	1	1	yes	characteristic
148	C₂²×C₄²	#121	1	1	yes	characteristic
149	C₂⁵	B(4,𝔽₂)	1	2	yes	—
150	C₂⁵	C₂×#120	1	1	yes	characteristic
151	C₂⁵	C₂×#120	1	1	yes	characteristic
152	C₂⁵	#534	1	1	yes	characteristic
153	C₂⁵	#534	1	1	yes	characteristic
154	C₂⁵	#534	1	1	yes	characteristic
155	C₂⁵	#538	1	1	yes	characteristic
156	C₂⁵	D₄²	1	1	yes	characteristic
157	C₂⁵	D₄²	1	1	yes	characteristic
158	C₂⁵	D₄²	1	1	yes	characteristic
159	C₂⁵	#546	1	1	yes	characteristic
160	C₂⁵	#546	1	1	yes	characteristic
161	C₂⁵	#546	1	1	yes	characteristic
162	C₂⁵	#560	1	1	yes	characteristic
163	C₂²×D₄	C₂×#121	1	1	yes	characteristic
164	C₂²×D₄	#525	1	1	yes	characteristic
165	C₂²×D₄	#534	1	1	yes	characteristic
166	C₂²×D₄	#535	1	1	yes	characteristic
167	C₂²×D₄	#535	1	1	yes	characteristic
168	C₂²×D₄	#538	1	1	yes	characteristic
169	C₂²×D₄	D₄²	1	1	yes	characteristic
170	C₂²×D₄	D₄²	1	1	yes	characteristic
171	C₂²×D₄	D₄²	1	1	yes	characteristic
172	C₂²×D₄	D₄²	1	1	yes	characteristic
173	C₂²×D₄	D₄²	1	1	yes	characteristic
174	C₂²×D₄	#546	1	1	yes	characteristic
175	C₂²×D₄	#546	1	1	yes	characteristic
176	C₂²×D₄	#546	1	1	yes	characteristic
177	C₂²×D₄	#546	1	1	yes	characteristic
178	C₂²×D₄	#560	1	1	yes	characteristic
179	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
180	C₂³×C₄	C₂×#120	1	1	yes	characteristic
181	C₂³×C₄	C₂×#121	1	1	yes	characteristic
182	C₂³×C₄	#535	1	1	yes	characteristic
183	C₂³×C₄	D₄²	1	1	yes	characteristic
184	C₂³×C₄	D₄²	1	1	yes	characteristic
185	C₂³×C₄	#546	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
187	C₂⁴	#2403	1	1	yes	characteristic
188	C₂⁴	#2403	1	1	yes	characteristic
189	C₂⁴	#2404	1	1	yes	characteristic
190	C₂⁴	#2410	1	1	yes	characteristic
191	C₂⁴	#2410	1	1	yes	characteristic
192	C₂⁴	#2424	1	1	yes	characteristic
193	C₂⁴	#2668	1	1	yes	characteristic
194	C₂⁴	#2668	1	1	yes	characteristic
195	C₂⁴	#2669	1	1	yes	characteristic
196	C₂⁴	#2678	1	1	yes	characteristic
197	C₂⁴	#2678	1	1	yes	characteristic
198	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
199	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
200	C₂²×C₄	#2408	1	1	yes	characteristic
201	C₂²×C₄	#2408	1	1	yes	characteristic
202	C₂²×C₄	#2424	1	1	yes	characteristic
203	C₂²×C₄	#2669	1	1	yes	characteristic
204	C₂²×C₄	#3026	1	2	yes	—
205	C₂²×C₄	#3027	1	2	yes	—
206	C₂³	#15257	1	1	yes	characteristic
207	C₂³	#23895	1	1	yes	characteristic
208	C₂³	#23895	1	1	yes	characteristic
209	C₂³	#23900	1	2	yes	—
210	C₂³	#23900	1	1	yes	characteristic
211	C₂³	#23900	1	1	yes	characteristic
212	C₂³	#23903	1	1	yes	characteristic
213	C₂³	#23904	1	1	yes	characteristic
214	C₂³	#23984	1	2	yes	—
215	C₂×C₄	#24262	1	2	yes	—
216	C₂×C₄	#24262	1	2	yes	—
217	C₂²	#245246	1	1	yes	characteristic
218	C₂²	#245249	1	1	yes	characteristic
219	C₂²	#245252	1	1	yes	characteristic
220	C₂²	#245255	1	1	yes	characteristic
221	C₂²	#417493	1	1	yes	characteristic
222	C₂²	#417528	1	1	yes	characteristic
223	C₂²	#417532	1	1	yes	characteristic
224	C₂	#72251507	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72251507
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. #72312218	C₂	semidirect
2. #72310358	C₂	semidirect
3. #72309800	C₂	semidirect
4. #72266835	C₂	semidirect
5. #72263490	C₂	semidirect
6. #72256113	C₂	semidirect
7. #72255971	C₂	semidirect
8. #72251533	C₂	semidirect
9. #72238343	C₂	semidirect
10. #72237903	C₂	semidirect
11. #72236678	C₂	semidirect
12. #72236521	C₂	semidirect
13. #72234100	C₂	semidirect
14. #72219163	C₂	semidirect
15. #72219159	C₂	semidirect
19. #6437621	C₂²	semidirect
23. #6374128	C₂²	semidirect
27. C₂×#12705	C₂²	semidirect
29. #493357	C₂²	semidirect
30. #382544	C₂²	semidirect
35. #359549	C₂²	semidirect
38. #188109	C₂²	semidirect
42. C₂×#8534	C₂²	semidirect
43. #146479	C₂²	semidirect
44. #146475	C₂²	semidirect
45. C₂×#7304	C₂²	semidirect
46. C₂×#7316	C₂²	semidirect
47. C₂×#7314	C₂²	semidirect
48. C₂×#7311	C₂²	semidirect
49. #107130	C₂²	semidirect
50. #107130	C₂²	semidirect
51. #62842	D₄	semidirect
57. C₂×#2883	D₄	semidirect
58. C₂×#2883	D₄	semidirect
62. #23901	D₄	semidirect
63. #23900	D₄	semidirect
64. C₂×#1913	D₄	semidirect
66. #20960	D₄	semidirect
67. #20616	D₄	semidirect
68. #20605	D₄	semidirect
76. #20354	D₄	semidirect
77. #15262	D₄	semidirect
78. #15261	D₄	semidirect
79. #15259	D₄	semidirect
80. #15258	D₄	semidirect
82. #10740	D₄	semidirect
83. #10716	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72312218	2
#72310358	2
#72309800	2
#72266835	2
#72263490	2
#72256113	2
#72255971	2
#72251533	2
#72238343	2
#72237903	2
#72236678	2
#72236521	2
#72234100	2
#72219163	2
#72219159	2