Subgroups — #72315031

Order 2048.

At least 6642 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

The subgroup lattice has 216 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	#72314748	C₂	1	1	yes	maximal, characteristic, complemented
3	#72310322	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	#72260562	C₂	1	1	yes	maximal, characteristic, complemented
7	#72258094	C₂	1	1	yes	maximal, characteristic, complemented
8	#72251530	C₂	1	1	yes	maximal, characteristic, complemented
9	#72238342	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237899	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237638	C₂	1	1	yes	maximal, characteristic, complemented
12	#72236508	C₂	1	1	yes	maximal, characteristic, complemented
13	#72226366	C₂	1	1	yes	maximal, characteristic, complemented
14	#72219160	C₂	1	1	yes	maximal, characteristic, complemented
15	#72219160	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic
17	C₂×#23895	C₂²	1	1	yes	characteristic
18	C₂×#23895	C₂²	1	1	yes	characteristic
19	#6437621	C₂²	1	1	yes	characteristic
20	#6437621	C₂²	1	1	yes	characteristic
21	#6437621	C₂²	1	1	yes	characteristic, complemented
22	#6437445	C₂²	1	1	yes	characteristic
23	#6374251	C₂²	1	1	yes	characteristic, complemented
24	#763333	C₂²	1	1	yes	characteristic
25	C₂×#12707	C₂²	1	1	yes	characteristic
26	C₂×#12707	C₂²	1	1	yes	characteristic
27	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
28	C₂×#12705	C₂²	1	1	yes	characteristic
29	C₂×#12705	C₂²	1	1	yes	characteristic
30	#493370	C₂²	1	1	yes	characteristic
31	#493370	C₂²	1	1	yes	characteristic
32	#493370	C₂²	1	1	yes	characteristic
33	#493357	C₂²	1	1	yes	characteristic, complemented
34	#392030	C₂²	1	1	yes	characteristic, complemented
35	#382434	C₂²	1	1	yes	characteristic
36	#382434	C₂²	1	1	yes	characteristic
37	#382434	C₂²	1	1	yes	characteristic, complemented
38	#188050	C₂²	1	1	yes	characteristic
39	#188049	C₂²	1	1	yes	characteristic, complemented
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	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
44	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
51	C₂×#3484	C₂³	1	1	yes	characteristic
52	C₂×#3484	C₂³	1	1	yes	characteristic
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂²×#535	C₂³	1	1	yes	characteristic
55	C₂²×#534	C₂³	1	1	yes	characteristic
56	C₂×#3026	D₄	1	2	yes	complemented
57	#34980	D₄	1	2	yes	complemented
58	#23901	D₄	1	2	yes	complemented
59	#23900	D₄	1	2	yes	complemented
60	C₂×#1913	D₄	1	2	yes	complemented
61	C₂×#1898	D₄	1	2	yes	complemented
62	C₂×#1828	D₄	1	2	yes	complemented
63	C₂×#1783	C₂³	1	1	yes	characteristic
64	#20585	D₄	1	2	yes	complemented
65	C₂×#1781	C₂³	1	1	yes	characteristic
66	C₂×#1781	C₂³	1	1	yes	characteristic
67	C₂×#1781	C₂³	1	1	yes	characteristic
68	C₂×#1781	C₂³	1	1	yes	characteristic
69	C₂×#1781	C₂³	1	1	yes	characteristic
70	C₂×#1781	C₂³	1	1	yes	characteristic
71	C₂×#1781	C₂³	1	1	yes	characteristic
72	C₂×#1781	C₂³	1	1	yes	characteristic
73	#15262	D₄	1	1	yes	characteristic, complemented
74	#15261	D₄	1	1	yes	characteristic, complemented
75	#15259	D₄	1	1	yes	characteristic, complemented
76	#15258	D₄	1	1	yes	characteristic, complemented
77	C₂×#1438	C₂³	1	1	yes	characteristic
78	#12705	D₄	1	4	yes	complemented
79	#12694	D₄	1	2	yes	complemented
80	#12693	D₄	1	2	yes	complemented
81	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
82	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
83	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
84	C₂³×#41	#41	1	1	yes	characteristic
85	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
86	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
87	#3484	C₂×D₄	1	2	yes	—
88	#3484	C₂×D₄	1	2	yes	—
89	C₂×#534	C₂×D₄	1	2	yes	—
90	C₂²×#127	C₂×D₄	1	1	yes	characteristic
91	C₂²×#127	C₂×D₄	1	1	yes	characteristic
92	C₂²×#124	C₂×D₄	1	1	yes	characteristic
93	C₂²×#124	C₂×D₄	1	1	yes	characteristic
94	C₂²×#123	C₂×D₄	1	1	yes	characteristic
95	C₂²×#123	C₂×D₄	1	1	yes	characteristic
96	C₂²×#120	C₂×D₄	1	2	yes	—
97	C₂²×#120	C₂×D₄	1	1	yes	characteristic
98	C₂²×#120	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	#41	1	1	yes	characteristic
102	C₂²×#117	#41	1	1	yes	characteristic
103	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
104	C₂²×#99	C₂×D₄	1	2	yes	—
105	C₂²×#99	C₂×D₄	1	2	yes	—
106	C₂²×#99	C₂×D₄	1	2	yes	—
107	C₂×#379	C₂×D₄	1	2	yes	—
108	#1783	C₂×D₄	1	2	yes	—
109	#1781	C₂×D₄	1	2	yes	—
110	#1781	C₂×D₄	1	2	yes	—
111	#1781	C₂×D₄	1	2	yes	—
112	#1781	C₂×D₄	1	2	yes	—
113	#1438	C₂×D₄	1	1	yes	characteristic
114	#1438	C₂×D₄	1	1	yes	characteristic
115	C₂⁶	#120	1	1	yes	characteristic
116	C₂⁶	C₂²×D₄	1	1	yes	characteristic
117	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×H(5,𝔽₂)	#121	1	2	yes	—
119	C₂³×Q₈	#121	1	1	yes	characteristic
120	C₂³×D₄	#120	1	2	yes	—
121	C₂³×D₄	#121	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	C₂²×D₄	1	1	yes	characteristic
127	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
128	C₂³×D₄	C₂×#41	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	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂⁴×C₄	#120	1	1	yes	characteristic
135	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
136	#518	#121	1	2	yes	—
137	C₂²×C₂²⋊C₄	#120	1	2	yes	—
138	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
139	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
140	C₂²×C₄²	#121	1	1	yes	characteristic
141	C₂²×C₄²	#121	1	1	yes	characteristic
142	C₂×#99	#120	1	4	yes	—
143	C₂⁵	B(4,𝔽₂)	1	2	yes	—
144	C₂⁵	C₂×#120	1	1	yes	characteristic
145	C₂⁵	C₂×#120	1	1	yes	characteristic
146	C₂⁵	#534	1	1	yes	characteristic
147	C₂⁵	#534	1	1	yes	characteristic
148	C₂⁵	#535	1	1	yes	characteristic
149	C₂⁵	#535	1	1	yes	characteristic
150	C₂⁵	D₄²	1	1	yes	characteristic
151	C₂⁵	D₄²	1	1	yes	characteristic
152	C₂⁵	D₄²	1	1	yes	characteristic
153	C₂⁵	#546	1	1	yes	characteristic
154	C₂⁵	#546	1	1	yes	characteristic
155	C₂⁵	#546	1	1	yes	characteristic
156	C₂⁵	#560	1	1	yes	characteristic
157	C₂²×D₄	C₂×#120	1	1	yes	characteristic
158	C₂²×D₄	C₂×#121	1	1	yes	characteristic
159	C₂²×D₄	#534	1	1	yes	characteristic
160	C₂²×D₄	#534	1	1	yes	characteristic
161	C₂²×D₄	#535	1	1	yes	characteristic
162	C₂²×D₄	#538	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	D₄²	1	1	yes	characteristic
167	C₂²×D₄	D₄²	1	1	yes	characteristic
168	C₂²×D₄	D₄²	1	1	yes	characteristic
169	C₂²×D₄	D₄²	1	1	yes	characteristic
170	C₂²×D₄	#546	1	1	yes	characteristic
171	C₂²×D₄	#546	1	1	yes	characteristic
172	C₂²×D₄	#560	1	1	yes	characteristic
173	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
174	C₂³×C₄	C₂×#121	1	1	yes	characteristic
175	C₂³×C₄	#525	1	1	yes	characteristic
176	C₂³×C₄	#538	1	1	yes	characteristic
177	C₂³×C₄	#546	1	1	yes	characteristic
178	C₂³×C₄	#546	1	1	yes	characteristic
179	C₂³×C₄	#546	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
181	C₂⁴	#2403	1	1	yes	characteristic
182	C₂⁴	#2403	1	1	yes	characteristic
183	C₂⁴	#2404	1	1	yes	characteristic
184	C₂⁴	#2408	1	1	yes	characteristic
185	C₂⁴	#2410	1	1	yes	characteristic
186	C₂⁴	#2410	1	1	yes	characteristic
187	C₂⁴	#2668	1	1	yes	characteristic
188	C₂⁴	#2669	1	1	yes	characteristic
189	C₂⁴	#2669	1	1	yes	characteristic
190	C₂⁴	#2678	1	1	yes	characteristic
191	C₂⁴	#2678	1	1	yes	characteristic
192	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
193	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
194	C₂²×C₄	#2408	1	1	yes	characteristic
195	C₂²×C₄	#2424	1	1	yes	characteristic
196	C₂²×C₄	#2424	1	1	yes	characteristic
197	C₂²×C₄	#2668	1	1	yes	characteristic
198	C₂³	#15257	1	1	yes	characteristic
199	C₂³	#23900	1	2	yes	—
200	C₂³	#23900	1	2	yes	—
201	C₂³	#23984	1	1	yes	characteristic
202	C₂³	#23984	1	1	yes	characteristic
203	C₂³	#23996	1	1	yes	characteristic
204	C₂³	#23996	1	1	yes	characteristic
205	C₂³	#23996	1	1	yes	characteristic
206	C₂³	#23996	1	1	yes	characteristic
207	C₂²	#251854	1	1	yes	characteristic
208	C₂²	#251855	1	1	yes	characteristic
209	C₂²	#251859	1	1	yes	characteristic
210	C₂²	#251860	1	1	yes	characteristic
211	C₂²	#417493	1	1	yes	characteristic
212	C₂²	#417538	1	1	yes	characteristic
213	C₂²	#417542	1	1	yes	characteristic
214	C₂	#72241285	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241285
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. #72314855	C₂	semidirect
2. #72314748	C₂	semidirect
3. #72310322	C₂	semidirect
4. #72266835	C₂	semidirect
5. #72263490	C₂	semidirect
6. #72260562	C₂	semidirect
7. #72258094	C₂	semidirect
8. #72251530	C₂	semidirect
9. #72238342	C₂	semidirect
10. #72237899	C₂	semidirect
11. #72237638	C₂	semidirect
12. #72236508	C₂	semidirect
13. #72226366	C₂	semidirect
14. #72219160	C₂	semidirect
15. #72219160	C₂	semidirect
21. #6437621	C₂²	semidirect
23. #6374251	C₂²	semidirect
27. C₂×#12707	C₂²	semidirect
33. #493357	C₂²	semidirect
34. #392030	C₂²	semidirect
37. #382434	C₂²	semidirect
39. #188049	C₂²	semidirect
42. C₂×#8534	C₂²	semidirect
43. C₂×#7304	C₂²	semidirect
44. C₂×#7316	C₂²	semidirect
45. C₂×#7314	C₂²	semidirect
46. C₂×#7310	C₂²	semidirect
47. C₂×#7310	C₂²	semidirect
48. C₂×#7310	C₂²	semidirect
49. C₂×#7307	C₂²	semidirect
50. C₂×#7307	C₂²	semidirect
56. C₂×#3026	D₄	semidirect
57. #34980	D₄	semidirect
58. #23901	D₄	semidirect
59. #23900	D₄	semidirect
60. C₂×#1913	D₄	semidirect
61. C₂×#1898	D₄	semidirect
62. C₂×#1828	D₄	semidirect
64. #20585	D₄	semidirect
73. #15262	D₄	semidirect
74. #15261	D₄	semidirect
75. #15259	D₄	semidirect
76. #15258	D₄	semidirect
78. #12705	D₄	semidirect
79. #12694	D₄	semidirect
80. #12693	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72314855	2
#72314748	2
#72310322	2
#72266835	2
#72263490	2
#72260562	2
#72258094	2
#72251530	2
#72238342	2
#72237899	2
#72237638	2
#72236508	2
#72226366	2
#72219160	2