Subgroups — #72258416

Order 2048.

At least 6110 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

The subgroup lattice has 207 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72258381	C₂	1	1	yes	maximal, characteristic, complemented
2	#72258026	C₂	1	1	yes	maximal, characteristic, complemented
3	#72240571	C₂	1	1	yes	maximal, characteristic, complemented
4	#72237473	C₂	1	4	yes	maximal, complemented
5	#72236586	C₂	1	1	yes	maximal, characteristic, complemented
6	#72236586	C₂	1	1	yes	maximal, characteristic, complemented
7	#72236508	C₂	1	4	yes	maximal, complemented
8	#72235239	C₂	1	1	yes	maximal, characteristic, complemented
9	#72226123	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#62842	C₂²	1	1	yes	characteristic
11	C₂×#62842	C₂²	1	1	yes	characteristic
12	#7630668	C₂²	1	1	yes	characteristic
13	C₂×#23895	C₂²	1	2	yes	complemented
14	#6437621	C₂²	1	4	yes	—
15	#6437621	C₂²	1	2	yes	complemented
16	C₂×#20585	C₂²	1	1	yes	characteristic
17	C₂×#12707	C₂²	1	2	yes	complemented
18	C₂×#12707	C₂²	1	4	yes	complemented
19	#392111	C₂²	1	1	yes	characteristic
20	#392111	C₂²	1	1	yes	characteristic
21	#392111	C₂²	1	1	yes	characteristic
22	#359549	C₂²	1	2	yes	complemented
23	#359442	C₂²	1	2	yes	complemented
24	#188049	C₂²	1	4	yes	complemented
25	#146479	C₂²	1	2	yes	complemented
26	C₂×#7310	C₂²	1	4	yes	complemented
27	C₂³×H(5,𝔽₂)	D₄	1	1	yes	characteristic, complemented
28	C₂³×H(5,𝔽₂)	D₄	1	1	yes	characteristic, complemented
29	C₂³×H(5,𝔽₂)	C₂³	1	1	yes	characteristic
30	C₂×#3485	D₄	1	2	yes	complemented
31	C₂×#3485	D₄	1	1	yes	characteristic, complemented
32	C₂×#3484	D₄	1	1	yes	characteristic, complemented
33	C₂×#3484	C₂³	1	2	yes	—
34	C₂×#3484	C₂³	1	1	yes	characteristic
35	C₂²×#534	D₄	1	2	yes	complemented
36	C₂²×#534	C₂³	1	2	yes	—
37	C₂²×#518	D₄	1	1	yes	characteristic, complemented
38	C₂²×#518	D₄	1	1	yes	characteristic, complemented
39	C₂³×#99	D₄	1	2	yes	complemented
40	#34172	D₄	1	1	yes	characteristic, complemented
41	#34171	D₄	1	1	yes	characteristic, complemented
42	#34171	C₂³	1	1	yes	characteristic
43	#23901	D₄	1	8	yes	complemented
44	#23900	D₄	1	8	yes	complemented
45	C₂×#1782	D₄	1	2	yes	complemented
46	C₂×#1782	D₄	1	2	yes	complemented
47	C₂×#1781	D₄	1	2	yes	complemented
48	C₂×#1781	D₄	1	2	yes	complemented
49	C₂×#1781	C₂³	1	2	yes	—
50	C₂×#1781	C₂³	1	2	yes	—
51	C₂×#1781	C₂³	1	2	yes	—
52	C₂×#1781	C₂³	1	2	yes	—
53	#14335	D₄	1	2	yes	complemented
54	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
55	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
56	C₂³×#41	C₂×D₄	1	1	yes	characteristic
57	C₂³×#41	C₂×D₄	1	1	yes	characteristic
58	C₂³×#41	C₂×D₄	1	1	yes	characteristic
59	C₂³×#41	C₂×D₄	1	1	yes	characteristic
60	C₂³×#41	#41	1	1	yes	characteristic
61	C₂³×#41	#41	1	1	yes	characteristic
62	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
63	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
64	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
65	C₂⁴×D₄	C₂×D₄	1	2	yes	—
66	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
67	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
68	C₂⁴×D₄	C₂×D₄	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	C₂⁴	1	1	yes	characteristic
72	#3484	C₂×D₄	1	8	yes	—
73	C₂²×#123	C₂×D₄	1	4	yes	—
74	C₂²×#120	C₂×D₄	1	4	yes	—
75	C₂×#518	C₂×D₄	1	2	yes	—
76	C₂×#518	C₂×D₄	1	1	yes	characteristic, complemented
77	C₂×#518	C₂×D₄	1	1	yes	characteristic, complemented
78	C₂×#518	C₂×D₄	1	1	yes	characteristic, complemented
79	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
80	C₂²×#99	C₂×D₄	1	4	yes	—
81	C₂²×#99	C₂×D₄	1	4	yes	complemented
82	C₂×#379	C₂×D₄	1	4	yes	complemented
83	#1781	C₂×D₄	1	8	yes	—
84	C₂⁶	#120	1	1	yes	characteristic
85	C₂⁶	#120	1	1	yes	characteristic
86	C₂⁶	#120	1	1	yes	characteristic
87	C₂⁶	#120	1	2	yes	complemented
88	C₂⁶	#127	1	1	yes	characteristic
89	C₂⁶	C₂²×D₄	1	1	yes	characteristic
90	C₂⁶	C₂²×D₄	1	1	yes	characteristic
91	C₂×H(5,𝔽₂)	#120	1	4	yes	—
92	C₂³×Q₈	#121	1	1	yes	characteristic
93	C₂³×Q₈	#121	1	1	yes	characteristic
94	C₂³×D₄	#120	1	1	yes	characteristic
95	C₂³×D₄	#120	1	1	yes	characteristic
96	C₂³×D₄	#120	1	1	yes	characteristic
97	C₂³×D₄	#120	1	1	yes	characteristic
98	C₂³×D₄	#120	1	1	yes	characteristic
99	C₂³×D₄	#120	1	1	yes	characteristic
100	C₂³×D₄	#121	1	1	yes	characteristic
101	C₂³×D₄	#121	1	1	yes	characteristic
102	C₂³×D₄	#121	1	1	yes	characteristic
103	C₂³×D₄	#121	1	1	yes	characteristic
104	C₂³×D₄	#121	1	1	yes	characteristic
105	C₂³×D₄	#121	1	1	yes	characteristic
106	C₂³×D₄	#127	1	1	yes	characteristic
107	C₂³×D₄	#127	1	1	yes	characteristic
108	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
109	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
110	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
111	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
112	C₂³×D₄	C₂²×D₄	1	4	yes	—
113	C₂³×D₄	H(5,𝔽₂)	1	4	yes	—
114	C₂⁴×C₄	#120	1	1	yes	characteristic
115	C₂⁴×C₄	#120	1	1	yes	characteristic
116	C₂⁴×C₄	#120	1	1	yes	characteristic
117	C₂⁴×C₄	#120	1	2	yes	complemented
118	C₂⁴×C₄	#121	1	1	yes	characteristic
119	C₂⁴×C₄	#121	1	1	yes	characteristic
120	C₂⁴×C₄	#121	1	1	yes	characteristic
121	C₂⁴×C₄	#121	1	1	yes	characteristic
122	C₂⁴×C₄	#127	1	1	yes	characteristic
123	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
124	#518	#120	1	4	yes	—
125	C₂²×C₂²⋊C₄	#120	1	4	yes	complemented
126	C₂×#99	#120	1	8	yes	complemented
127	C₂⁵	#392	1	1	yes	characteristic
128	C₂⁵	#392	1	1	yes	characteristic
129	C₂⁵	#392	1	1	yes	characteristic
130	C₂⁵	#392	1	1	yes	characteristic
131	C₂⁵	B(4,𝔽₂)	1	4	yes	—
132	C₂⁵	B(4,𝔽₂)	1	4	yes	—
133	C₂⁵	C₂×#120	1	2	yes	—
134	C₂⁵	C₂×#120	1	1	yes	characteristic
135	C₂⁵	C₂×#120	1	1	yes	characteristic
136	C₂⁵	C₂×#120	1	1	yes	characteristic
137	C₂⁵	C₂×#121	1	1	yes	characteristic
138	C₂⁵	C₂×#127	1	1	yes	characteristic
139	C₂⁵	#534	1	2	yes	—
140	C₂⁵	D₄²	1	4	yes	—
141	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
142	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
143	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
144	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
145	C₂²×D₄	C₂×#120	1	2	yes	—
146	C₂²×D₄	C₂×#120	1	2	yes	—
147	C₂²×D₄	#534	1	2	yes	—
148	C₂²×D₄	#535	1	2	yes	—
149	C₂²×D₄	D₄²	1	4	yes	—
150	C₂²×D₄	D₄²	1	4	yes	—
151	C₂³×C₄	#392	1	1	yes	characteristic
152	C₂³×C₄	#392	1	1	yes	characteristic
153	C₂³×C₄	#392	1	1	yes	characteristic
154	C₂³×C₄	#392	1	1	yes	characteristic
155	C₂³×C₄	C₂×#121	1	1	yes	characteristic
156	C₂³×C₄	C₂×#121	1	1	yes	characteristic
157	C₂³×C₄	#546	1	4	yes	—
158	C₂⁴	#2021	1	2	yes	—
159	C₂⁴	#2021	1	2	yes	—
160	C₂⁴	#2021	1	2	yes	—
161	C₂⁴	#2021	1	1	yes	characteristic
162	C₂⁴	#2021	1	2	yes	—
163	C₂⁴	#2021	1	2	yes	—
164	C₂⁴	#2021	1	2	yes	—
165	C₂⁴	#2021	1	1	yes	characteristic
166	C₂⁴	C₂×#392	1	1	yes	characteristic
167	C₂⁴	#2403	1	2	yes	—
168	C₂⁴	#2403	1	2	yes	—
169	C₂⁴	#2410	1	2	yes	—
170	C₂⁴	#2433	1	2	yes	—
171	C₂⁴	#2613	1	1	yes	characteristic
172	C₂⁴	#2846	1	1	yes	characteristic
173	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
174	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
175	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
176	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
177	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
178	C₂⁴	#3026	1	2	yes	—
179	C₂²×C₄	#2021	1	2	yes	—
180	C₂²×C₄	#2021	1	2	yes	—
181	C₂²×C₄	#2021	1	2	yes	—
182	C₂²×C₄	#2408	1	2	yes	—
183	C₂²×C₄	#2410	1	2	yes	—
184	C₂³	#10336	1	2	yes	—
185	C₂³	#10336	1	4	yes	—
186	C₂³	#10336	1	2	yes	—
187	C₂³	#14715	1	1	yes	characteristic
188	C₂³	C₂×#2021	1	1	yes	characteristic
189	C₂³	C₂×#2021	1	1	yes	characteristic
190	C₂³	C₂×#2021	1	1	yes	characteristic
191	C₂³	C₂×#2021	1	1	yes	characteristic
192	C₂³	C₂×#2021	1	1	yes	characteristic
193	C₂³	#23444	1	1	yes	characteristic
194	C₂³	#23900	1	4	yes	—
195	C₂³	#23900	1	4	yes	—
196	C₂³	#29634	1	2	yes	—
197	C₂³	#32898	1	2	yes	—
198	C₂²	C₂×#10336	1	1	yes	characteristic
199	C₂²	C₂×#10336	1	1	yes	characteristic
200	C₂²	#233541	1	1	yes	characteristic
201	C₂²	#233541	1	1	yes	characteristic
202	C₂²	#393929	1	1	yes	characteristic
203	C₂²	#394058	1	1	yes	characteristic
204	C₂²	#394058	1	1	yes	characteristic
205	C₂	#72214549	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72214549
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂³×H(5,𝔽₂) ◁ 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. #72258381	C₂	semidirect
2. #72258026	C₂	semidirect
3. #72240571	C₂	semidirect
4. #72237473	C₂	semidirect
5. #72236586	C₂	semidirect
6. #72236586	C₂	semidirect
7. #72236508	C₂	semidirect
8. #72235239	C₂	semidirect
9. #72226123	C₂	semidirect
13. C₂×#23895	C₂²	semidirect
15. #6437621	C₂²	semidirect
17. C₂×#12707	C₂²	semidirect
18. C₂×#12707	C₂²	semidirect
22. #359549	C₂²	semidirect
23. #359442	C₂²	semidirect
24. #188049	C₂²	semidirect
25. #146479	C₂²	semidirect
26. C₂×#7310	C₂²	semidirect
27. C₂³×H(5,𝔽₂)	D₄	semidirect
28. C₂³×H(5,𝔽₂)	D₄	semidirect
30. C₂×#3485	D₄	semidirect
31. C₂×#3485	D₄	semidirect
32. C₂×#3484	D₄	semidirect
35. C₂²×#534	D₄	semidirect
37. C₂²×#518	D₄	semidirect
38. C₂²×#518	D₄	semidirect
39. C₂³×#99	D₄	semidirect
40. #34172	D₄	semidirect
41. #34171	D₄	semidirect
43. #23901	D₄	semidirect
44. #23900	D₄	semidirect
45. C₂×#1782	D₄	semidirect
46. C₂×#1782	D₄	semidirect
47. C₂×#1781	D₄	semidirect
48. C₂×#1781	D₄	semidirect
53. #14335	D₄	semidirect
55. C₂²×H(5,𝔽₂)	C₂×D₄	semidirect
76. C₂×#518	C₂×D₄	semidirect
77. C₂×#518	C₂×D₄	semidirect
78. C₂×#518	C₂×D₄	semidirect
81. C₂²×#99	C₂×D₄	semidirect
82. C₂×#379	C₂×D₄	semidirect
87. C₂⁶	#120	semidirect
117. C₂⁴×C₄	#120	semidirect
125. C₂²×C₂²⋊C₄	#120	semidirect
126. C₂×#99	#120	semidirect

Maximal subgroups

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

subgroup	index
#72258381	2
#72258026	2
#72240571	2
#72237473	2
#72236586	2
#72236508	2
#72235239	2
#72226123	2