Subgroups — C₂²×#577 #63085

Order 256.

5559 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#577	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#137	C₂	1	3	yes	maximal, complemented
5	C₂²×AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
6	C₂²×#135	C₂	1	3	yes	maximal, complemented
7	C₂³×SD₁₆	C₂	1	3	yes	maximal, complemented
8	C₂²×#131	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×2₋¹⁺⁴	—	2	12	no	—
10	C₂×2₋¹⁺⁴	C₂²	1	24	yes	complemented
11	C₂×H(5,𝔽₂)	—	2	12	no	—
12	C₂×H(5,𝔽₂)	C₂²	1	24	yes	complemented
13	C₂²×#41	—	2	1	no	—
14	C₂²×#41	—	2	3	no	—
15	C₂²×#41	C₂²	1	3	yes	—
16	C₂²×#41	C₂²	1	1	yes	characteristic
17	C₂²×#41	C₂²	1	3	yes	complemented
18	C₂³×Q₈	—	2	1	no	—
19	C₂³×Q₈	C₂²	1	3	yes	complemented
20	C₂³×D₄	—	2	3	no	—
21	C₂³×D₄	C₂²	1	3	yes	—
22	#577	C₂²	1	256	yes	direct product factor
23	C₂×#137	C₂²	1	72	yes	complemented
24	C₂×AGL(1,ℤ/8)	C₂²	1	72	yes	complemented
25	C₂×#135	C₂²	1	72	yes	complemented
26	C₂²×Q₁₆	C₂²	1	3	yes	complemented
27	C₂²×SD₁₆	C₂²	1	3	yes	complemented
28	C₂²×SD₁₆	C₂²	1	6	yes	complemented
29	C₂²×SD₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₂²×SD₁₆	C₂²	1	72	yes	complemented
31	C₂²×D₈	C₂²	1	3	yes	complemented
32	C₂×#131	C₂²	1	24	yes	complemented
33	C₂²×M₁₆	C₂²	1	3	yes	complemented
34	C₂³×C₈	C₂²	1	3	yes	complemented
35	C₂⁵	—	2	3	no	—
36	2₋¹⁺⁴	—	2	96	no	—
37	2₋¹⁺⁴	C₂³	1	64	yes	complemented
38	H(5,𝔽₂)	—	2	96	no	—
39	H(5,𝔽₂)	C₂³	1	64	yes	complemented
40	C₂×#41	—	2	24	no	—
41	C₂×#41	—	2	72	no	—
42	C₂×#41	—	2	18	no	—
43	C₂×#41	—	2	18	no	—
44	C₂×#41	D₄	1	12	yes	—
45	C₂×#41	C₂³	1	36	yes	—
46	C₂×#41	C₂³	1	12	yes	—
47	C₂×#41	C₂³	1	36	yes	complemented
48	C₂²×Q₈	—	2	24	no	—
49	C₂²×Q₈	—	2	3	no	—
50	C₂²×Q₈	—	2	18	no	—
51	C₂²×Q₈	D₄	1	1	yes	characteristic
52	C₂²×Q₈	C₂³	1	3	yes	—
53	C₂²×Q₈	C₂³	1	1	yes	characteristic
54	C₂²×Q₈	C₂³	1	36	yes	complemented
55	C₂²×D₄	—	2	3	no	—
56	C₂²×D₄	—	2	3	no	—
57	C₂²×D₄	—	2	72	no	—
58	C₂²×D₄	—	2	3	no	—
59	C₂²×D₄	—	2	18	no	—
60	C₂²×D₄	D₄	1	3	yes	—
61	C₂²×D₄	C₂³	1	1	yes	characteristic
62	C₂²×D₄	C₂³	1	3	yes	—
63	C₂²×D₄	C₂³	1	36	yes	—
64	C₂³×C₄	—	2	3	no	—
65	C₂³×C₄	—	2	3	no	—
66	C₂³×C₄	—	2	3	no	—
67	C₂³×C₄	C₂³	1	3	yes	—
68	#137	C₂³	1	192	yes	complemented
69	AGL(1,ℤ/8)	C₂³	1	192	yes	complemented
70	#135	C₂³	1	192	yes	complemented
71	C₂×Q₁₆	C₂³	1	36	yes	complemented
72	C₂×SD₁₆	C₂³	1	36	yes	complemented
73	C₂×SD₁₆	C₂³	1	72	yes	complemented
74	C₂×SD₁₆	C₂³	1	12	yes	complemented
75	C₂×SD₁₆	C₂³	1	192	yes	complemented
76	C₂×D₈	C₂³	1	36	yes	complemented
77	#131	C₂³	1	64	yes	complemented
78	C₂×M₁₆	C₂³	1	36	yes	complemented
79	C₂²×C₈	C₂³	1	1	yes	characteristic
80	C₂²×C₈	C₂³	1	3	yes	—
81	C₂²×C₈	C₂³	1	36	yes	complemented
82	C₂⁴	—	2	1	no	—
83	C₂⁴	—	2	3	no	—
84	C₂⁴	—	2	36	no	—
85	C₂⁴	—	8	12	no	—
86	C₂⁴	C₂×D₄	1	3	yes	—
87	#41	—	2	64	no	—
88	#41	—	2	192	no	—
89	#41	—	2	72	no	—
90	#41	—	2	72	no	—
91	#41	C₂×D₄	1	48	yes	—
92	#41	C₂⁴	1	48	yes	—
93	#41	C₂⁴	1	16	yes	—
94	#41	C₂⁴	1	48	yes	complemented
95	C₂×Q₈	—	2	64	no	—
96	C₂×Q₈	—	2	9	no	—
97	C₂×Q₈	—	2	36	no	—
98	C₂×Q₈	—	2	72	no	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×Q₈	C₂×D₄	1	12	yes	—
101	C₂×Q₈	C₂⁴	1	18	yes	—
102	C₂×Q₈	C₂⁴	1	6	yes	—
103	C₂×Q₈	C₂⁴	1	48	yes	complemented
104	C₂×D₄	—	2	36	no	—
105	C₂×D₄	—	2	36	no	—
106	C₂×D₄	—	2	192	no	—
107	C₂×D₄	—	2	36	no	—
108	C₂×D₄	—	2	9	no	—
109	C₂×D₄	—	2	72	no	—
110	C₂×D₄	—	2	3	no	—
111	C₂×D₄	C₂×D₄	1	36	yes	—
112	C₂×D₄	C₂⁴	1	6	yes	—
113	C₂×D₄	C₂⁴	1	18	yes	—
114	C₂×D₄	C₂⁴	1	48	yes	—
115	C₂²×C₄	—	2	36	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	3	no	—
118	C₂²×C₄	—	2	36	no	—
119	C₂²×C₄	—	2	36	no	—
120	C₂²×C₄	C₂×D₄	1	18	yes	—
121	C₂²×C₄	C₂×D₄	1	3	yes	—
122	C₂²×C₄	C₂⁴	1	18	yes	—
123	C₂²×C₄	C₂⁴	1	1	yes	characteristic
124	Q₁₆	C₂⁴	1	48	yes	complemented
125	SD₁₆	C₂⁴	1	48	yes	complemented
126	SD₁₆	C₂⁴	1	96	yes	complemented
127	SD₁₆	C₂⁴	1	16	yes	complemented
128	D₈	C₂⁴	1	48	yes	complemented
129	M₁₆	C₂⁴	1	48	yes	complemented
130	C₂×C₈	C₂⁴	1	6	yes	—
131	C₂×C₈	C₂⁴	1	18	yes	—
132	C₂×C₈	C₂⁴	1	48	yes	complemented
133	C₂³	—	2	12	no	—
134	C₂³	—	2	6	no	—
135	C₂³	—	2	18	no	—
136	C₂³	—	2	48	no	—
137	C₂³	—	4	4	no	—
138	C₂³	—	4	12	no	—
139	C₂³	—	8	72	no	—
140	C₂³	C₂²×D₄	1	18	yes	—
141	C₂³	C₂²×D₄	1	1	yes	characteristic
142	Q₈	—	2	18	no	—
143	Q₈	—	2	48	no	—
144	Q₈	—	2	6	no	—
145	Q₈	C₂²×D₄	1	16	yes	—
146	Q₈	C₂⁵	1	12	yes	—
147	Q₈	C₂⁵	1	4	yes	—
148	D₄	—	2	48	no	—
149	D₄	—	2	48	no	—
150	D₄	—	2	48	no	—
151	D₄	—	2	18	no	—
152	D₄	—	2	6	no	—
153	D₄	C₂²×D₄	1	48	yes	—
154	D₄	C₂⁵	1	4	yes	—
155	D₄	C₂⁵	1	12	yes	—
156	C₂×C₄	—	2	6	no	—
157	C₂×C₄	—	2	18	no	—
158	C₂×C₄	—	2	48	no	—
159	C₂×C₄	—	2	48	no	—
160	C₂×C₄	—	2	48	no	—
161	C₂×C₄	C₂²×D₄	1	18	yes	—
162	C₂×C₄	C₂²×D₄	1	3	yes	—
163	C₂×C₄	C₂²×D₄	1	36	yes	—
164	C₂×C₄	C₂⁵	1	3	yes	—
165	C₂×C₄	C₂⁵	1	12	yes	—
166	C₈	C₂⁵	1	4	yes	—
167	C₈	C₂⁵	1	12	yes	—
168	C₂²	—	2	36	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	12	no	—
171	C₂²	—	4	12	no	—
172	C₂²	—	4	36	no	—
173	C₂²	—	8	48	no	—
174	C₂²	#577	1	4	yes	direct product factor
175	C₂²	C₂³×D₄	1	12	yes	—
176	C₂²	C₂³×D₄	1	3	yes	—
177	C₄	—	2	4	no	—
178	C₄	—	2	12	no	—
179	C₄	C₂³×D₄	1	12	yes	—
180	C₄	C₂³×D₄	1	3	yes	—
181	C₄	C₂⁶	1	1	yes	characteristic
182	C₂	—	2	12	no	—
183	C₂	—	4	4	no	—
184	C₂	—	4	12	no	—
185	C₂	C₂×#577	1	6	yes	direct product factor
186	C₂	C₂⁴×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁶
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×2₋¹⁺⁴	C₂	semidirect
2. C₂²×H(5,𝔽₂)	C₂	semidirect
3. C₂×#577	C₂	direct (N × H)
4. C₂²×#137	C₂	semidirect
5. C₂²×AGL(1,ℤ/8)	C₂	semidirect
6. C₂²×#135	C₂	semidirect
7. C₂³×SD₁₆	C₂	semidirect
8. C₂²×#131	C₂	semidirect
10. C₂×2₋¹⁺⁴	C₂²	semidirect
12. C₂×H(5,𝔽₂)	C₂²	semidirect
17. C₂²×#41	C₂²	semidirect
19. C₂³×Q₈	C₂²	semidirect
22. #577	C₂²	direct (N × H)
23. C₂×#137	C₂²	semidirect
24. C₂×AGL(1,ℤ/8)	C₂²	semidirect
25. C₂×#135	C₂²	semidirect
26. C₂²×Q₁₆	C₂²	semidirect
27. C₂²×SD₁₆	C₂²	semidirect
28. C₂²×SD₁₆	C₂²	semidirect
29. C₂²×SD₁₆	C₂²	semidirect
30. C₂²×SD₁₆	C₂²	semidirect
31. C₂²×D₈	C₂²	semidirect
32. C₂×#131	C₂²	semidirect
33. C₂²×M₁₆	C₂²	semidirect
34. C₂³×C₈	C₂²	semidirect
37. 2₋¹⁺⁴	C₂³	semidirect
39. H(5,𝔽₂)	C₂³	semidirect
47. C₂×#41	C₂³	semidirect
54. C₂²×Q₈	C₂³	semidirect
68. #137	C₂³	semidirect
69. AGL(1,ℤ/8)	C₂³	semidirect
70. #135	C₂³	semidirect
71. C₂×Q₁₆	C₂³	semidirect
72. C₂×SD₁₆	C₂³	semidirect
73. C₂×SD₁₆	C₂³	semidirect
74. C₂×SD₁₆	C₂³	semidirect
75. C₂×SD₁₆	C₂³	semidirect
76. C₂×D₈	C₂³	semidirect
77. #131	C₂³	semidirect
78. C₂×M₁₆	C₂³	semidirect
81. C₂²×C₈	C₂³	semidirect
94. #41	C₂⁴	semidirect
103. C₂×Q₈	C₂⁴	semidirect
124. Q₁₆	C₂⁴	semidirect
125. SD₁₆	C₂⁴	semidirect
126. SD₁₆	C₂⁴	semidirect
127. SD₁₆	C₂⁴	semidirect
128. D₈	C₂⁴	semidirect
129. M₁₆	C₂⁴	semidirect
132. C₂×C₈	C₂⁴	semidirect
174. C₂²	#577	direct (N × H)
185. C₂	C₂×#577	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴	2
C₂²×H(5,𝔽₂)	2
C₂×#577	2
C₂²×#137	2
C₂²×AGL(1,ℤ/8)	2
C₂²×#135	2
C₂³×SD₁₆	2
C₂²×#131	2

Subgroups — C22×#577 #63085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#577 #63085