Subgroups — C₂²×#575 #63083

Order 256.

5647 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41	C₂	1	2	yes	maximal, complemented
2	C₂×#575	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#137	C₂	1	4	yes	maximal, complemented
4	C₂²×AGL(1,ℤ/8)	C₂	1	4	yes	maximal, complemented
5	C₂²×#135	C₂	1	4	yes	maximal, complemented
6	C₂³×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41	—	2	2	no	—
8	C₂²×#41	—	2	24	no	—
9	C₂²×#41	—	2	2	no	—
10	C₂²×#41	C₂²	1	4	yes	—
11	C₂²×#41	C₂²	1	4	yes	complemented
12	C₂²×#41	C₂²	1	48	yes	complemented
13	C₂³×Q₈	C₂²	1	2	yes	complemented
14	C₂³×D₄	—	2	2	no	—
15	C₂³×D₄	C₂²	1	2	yes	complemented
16	C₂⁴×C₄	—	2	2	no	—
17	C₂⁴×C₄	C₂²	1	1	yes	characteristic
18	#575	C₂²	1	256	yes	direct product factor
19	C₂×#137	C₂²	1	96	yes	complemented
20	C₂×AGL(1,ℤ/8)	C₂²	1	96	yes	complemented
21	C₂×#135	C₂²	1	96	yes	complemented
22	C₂²×Q₁₆	C₂²	1	4	yes	complemented
23	C₂²×SD₁₆	C₂²	1	8	yes	complemented
24	C₂²×D₈	C₂²	1	4	yes	complemented
25	C₂²×M₁₆	C₂²	1	4	yes	complemented
26	C₂²×M₁₆	C₂²	1	24	yes	complemented
27	C₂³×C₈	C₂²	1	2	yes	complemented
28	C₂⁵	—	2	2	no	—
29	C₂⁵	D₄	1	1	yes	characteristic
30	C₂×#41	—	2	24	no	—
31	C₂×#41	—	2	48	no	—
32	C₂×#41	—	2	48	no	—
33	C₂×#41	—	2	24	no	—
34	C₂×#41	—	2	192	no	—
35	C₂×#41	C₂³	1	48	yes	—
36	C₂×#41	C₂³	1	48	yes	complemented
37	C₂×#41	C₂³	1	128	yes	complemented
38	C₂²×Q₈	—	2	2	no	—
39	C₂²×Q₈	—	2	12	no	—
40	C₂²×Q₈	C₂³	1	4	yes	—
41	C₂²×Q₈	C₂³	1	24	yes	complemented
42	C₂²×D₄	—	2	4	no	—
43	C₂²×D₄	—	2	4	no	—
44	C₂²×D₄	—	2	48	no	—
45	C₂²×D₄	—	2	2	no	—
46	C₂²×D₄	—	2	12	no	—
47	C₂²×D₄	C₂³	1	4	yes	—
48	C₂²×D₄	C₂³	1	24	yes	complemented
49	C₂³×C₄	—	2	2	no	—
50	C₂³×C₄	—	2	4	no	—
51	C₂³×C₄	—	2	48	no	—
52	C₂³×C₄	—	2	4	no	—
53	C₂³×C₄	D₄	1	1	yes	characteristic
54	C₂³×C₄	D₄	1	12	yes	—
55	C₂³×C₄	D₄	1	2	yes	—
56	C₂³×C₄	C₂³	1	2	yes	—
57	C₂³×C₄	C₂³	1	12	yes	—
58	C₂³×C₄	C₂³	1	1	yes	characteristic
59	#137	C₂³	1	256	yes	complemented
60	AGL(1,ℤ/8)	C₂³	1	256	yes	complemented
61	#135	C₂³	1	256	yes	complemented
62	C₂×Q₁₆	C₂³	1	48	yes	complemented
63	C₂×SD₁₆	C₂³	1	96	yes	complemented
64	C₂×D₈	C₂³	1	48	yes	complemented
65	C₂×M₁₆	C₂³	1	48	yes	complemented
66	C₂×M₁₆	C₂³	1	64	yes	complemented
67	C₂²×C₈	C₂³	1	4	yes	—
68	C₂²×C₈	C₂³	1	24	yes	complemented
69	C₂⁴	—	2	4	no	—
70	C₂⁴	—	2	24	no	—
71	C₂⁴	—	4	4	no	—
72	C₂⁴	—	4	16	no	—
73	C₂⁴	C₂×D₄	1	2	yes	—
74	C₂⁴	C₂×D₄	1	1	yes	characteristic
75	C₂⁴	C₂×D₄	1	12	yes	—
76	#41	—	2	96	no	—
77	#41	—	2	128	no	—
78	#41	—	2	128	no	—
79	#41	—	2	96	no	—
80	#41	C₂⁴	1	64	yes	—
81	#41	C₂⁴	1	64	yes	complemented
82	C₂×Q₈	—	2	12	no	—
83	C₂×Q₈	—	2	24	no	—
84	C₂×Q₈	—	2	48	no	—
85	C₂×Q₈	C₂⁴	1	24	yes	—
86	C₂×Q₈	C₂⁴	1	32	yes	complemented
87	C₂×D₄	—	2	48	no	—
88	C₂×D₄	—	2	48	no	—
89	C₂×D₄	—	2	128	no	—
90	C₂×D₄	—	2	12	no	—
91	C₂×D₄	—	2	24	no	—
92	C₂×D₄	—	2	48	no	—
93	C₂×D₄	C₂⁴	1	24	yes	—
94	C₂×D₄	C₂⁴	1	32	yes	complemented
95	C₂²×C₄	—	2	4	no	—
96	C₂²×C₄	—	2	24	no	—
97	C₂²×C₄	—	2	48	no	—
98	C₂²×C₄	—	2	48	no	—
99	C₂²×C₄	—	2	128	no	—
100	C₂²×C₄	C₂×D₄	1	2	yes	—
101	C₂²×C₄	C₂×D₄	1	24	yes	—
102	C₂²×C₄	C₂×D₄	1	12	yes	—
103	C₂²×C₄	C₂×D₄	1	6	yes	—
104	C₂²×C₄	C₂×D₄	1	12	yes	—
105	C₂²×C₄	C₂×D₄	1	48	yes	—
106	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄	C₂⁴	1	1	yes	characteristic
108	C₂²×C₄	C₂⁴	1	6	yes	—
109	C₂²×C₄	C₂⁴	1	12	yes	—
110	C₂²×C₄	C₂⁴	1	16	yes	—
111	Q₁₆	C₂⁴	1	64	yes	complemented
112	SD₁₆	C₂⁴	1	128	yes	complemented
113	D₈	C₂⁴	1	64	yes	complemented
114	M₁₆	C₂⁴	1	64	yes	complemented
115	C₂×C₈	C₂⁴	1	24	yes	—
116	C₂×C₈	C₂⁴	1	32	yes	complemented
117	C₂³	—	2	8	no	—
118	C₂³	—	2	4	no	—
119	C₂³	—	2	24	no	—
120	C₂³	—	2	32	no	—
121	C₂³	—	4	24	no	—
122	C₂³	—	4	96	no	—
123	C₂³	—	4	16	no	—
124	C₂³	C₂²×D₄	1	12	yes	—
125	C₂³	C₂²×D₄	1	6	yes	—
126	C₂³	C₂²×D₄	1	16	yes	—
127	C₂³	C₂²×D₄	1	1	yes	characteristic
128	Q₈	—	2	24	no	—
129	Q₈	—	2	32	no	—
130	Q₈	C₂⁵	1	16	yes	—
131	D₄	—	2	64	no	—
132	D₄	—	2	64	no	—
133	D₄	—	2	24	no	—
134	D₄	—	2	32	no	—
135	D₄	C₂⁵	1	16	yes	—
136	C₂×C₄	—	2	24	no	—
137	C₂×C₄	—	2	32	no	—
138	C₂×C₄	—	2	64	no	—
139	C₂×C₄	—	2	64	no	—
140	C₂×C₄	C₂²×D₄	1	6	yes	—
141	C₂×C₄	C₂²×D₄	1	12	yes	—
142	C₂×C₄	C₂²×D₄	1	32	yes	—
143	C₂×C₄	C₂²×D₄	1	3	yes	—
144	C₂×C₄	C₂²×D₄	1	24	yes	—
145	C₂×C₄	C₂²×D₄	1	12	yes	—
146	C₂×C₄	C₂²×D₄	1	16	yes	—
147	C₂×C₄	C₂⁵	1	3	yes	—
148	C₂×C₄	C₂⁵	1	4	yes	—
149	C₂×C₄	C₂⁵	1	8	yes	—
150	C₈	C₂⁵	1	16	yes	—
151	C₂²	—	2	24	no	—
152	C₂²	—	2	12	no	—
153	C₂²	—	2	16	no	—
154	C₂²	—	4	16	no	—
155	C₂²	—	4	64	no	—
156	C₂²	—	4	48	no	—
157	C₂²	#575	1	4	yes	direct product factor
158	C₂²	C₂³×D₄	1	8	yes	—
159	C₂²	C₂³×D₄	1	4	yes	—
160	C₂²	C₂³×D₄	1	3	yes	—
161	C₄	—	2	16	no	—
162	C₄	C₂³×D₄	1	4	yes	—
163	C₄	C₂³×D₄	1	8	yes	—
164	C₄	C₂³×D₄	1	3	yes	—
165	C₄	C₂⁶	1	1	yes	characteristic
166	C₂	—	2	8	no	—
167	C₂	—	2	4	no	—
168	C₂	—	4	16	no	—
169	C₂	C₂×#575	1	6	yes	direct product factor
170	C₂	C₂⁴×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁶
socle	C₂³	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂³×#41	C₂	semidirect
2. C₂×#575	C₂	direct (N × H)
3. C₂²×#137	C₂	semidirect
4. C₂²×AGL(1,ℤ/8)	C₂	semidirect
5. C₂²×#135	C₂	semidirect
6. C₂³×M₁₆	C₂	semidirect
11. C₂²×#41	C₂²	semidirect
12. C₂²×#41	C₂²	semidirect
13. C₂³×Q₈	C₂²	semidirect
15. C₂³×D₄	C₂²	semidirect
18. #575	C₂²	direct (N × H)
19. C₂×#137	C₂²	semidirect
20. C₂×AGL(1,ℤ/8)	C₂²	semidirect
21. C₂×#135	C₂²	semidirect
22. C₂²×Q₁₆	C₂²	semidirect
23. C₂²×SD₁₆	C₂²	semidirect
24. C₂²×D₈	C₂²	semidirect
25. C₂²×M₁₆	C₂²	semidirect
26. C₂²×M₁₆	C₂²	semidirect
27. C₂³×C₈	C₂²	semidirect
36. C₂×#41	C₂³	semidirect
37. C₂×#41	C₂³	semidirect
41. C₂²×Q₈	C₂³	semidirect
48. C₂²×D₄	C₂³	semidirect
59. #137	C₂³	semidirect
60. AGL(1,ℤ/8)	C₂³	semidirect
61. #135	C₂³	semidirect
62. C₂×Q₁₆	C₂³	semidirect
63. C₂×SD₁₆	C₂³	semidirect
64. C₂×D₈	C₂³	semidirect
65. C₂×M₁₆	C₂³	semidirect
66. C₂×M₁₆	C₂³	semidirect
68. C₂²×C₈	C₂³	semidirect
81. #41	C₂⁴	semidirect
86. C₂×Q₈	C₂⁴	semidirect
94. C₂×D₄	C₂⁴	semidirect
111. Q₁₆	C₂⁴	semidirect
112. SD₁₆	C₂⁴	semidirect
113. D₈	C₂⁴	semidirect
114. M₁₆	C₂⁴	semidirect
116. C₂×C₈	C₂⁴	semidirect
157. C₂²	#575	direct (N × H)
169. C₂	C₂×#575	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41	2
C₂×#575	2
C₂²×#137	2
C₂²×AGL(1,ℤ/8)	2
C₂²×#135	2
C₂³×M₁₆	2

Subgroups — C22×#575 #63083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#575 #63083