Subgroups — C₂×#62843 #10585997

Order 512.

34554 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#62843	C₂	1	64	yes	maximal, direct product factor
2	C₂×#3501	C₂	1	48	yes	maximal, complemented
3	C₂×#3458	C₂	1	4	yes	maximal, complemented
4	C₂²×#543	C₂	1	8	yes	maximal, complemented
5	C₂²×#518	C₂	1	3	yes	maximal
6	C₂⁴×D₄	C₂²	1	1	yes	characteristic
7	#3501	C₂²	1	1536	yes	complemented
8	C₂×#554	C₂²	1	192	yes	complemented
9	C₂×D₄×Q₈	C₂²	1	64	yes	complemented
10	#3458	C₂²	1	128	yes	complemented
11	C₂×#544	C₂²	1	48	yes	complemented
12	C₂×#543	C₂²	1	16	yes	—
13	C₂×#543	C₂²	1	96	yes	complemented
14	C₂×#543	C₂²	1	256	yes	complemented
15	C₂×#533	C₂²	1	48	yes	complemented
16	C₂²×C₂²⋊Q₈	C₂²	1	96	yes	complemented
17	C₂×#518	C₂²	1	48	yes	—
18	C₂×#518	C₂²	1	96	yes	—
19	C₂²×C₄×D₄	C₂²	1	24	yes	—
20	C₂²×#117	C₂²	1	6	yes	—
21	C₂³×C₂²⋊C₄	C₂²	1	12	yes	—
22	C₂⁶	C₂³	1	2	yes	—
23	C₂³×Q₈	C₂³	1	32	yes	complemented
24	C₂³×D₄	Q₈	1	32	yes	complemented
25	C₂³×D₄	C₂³	1	12	yes	—
26	C₂³×D₄	C₂³	1	16	yes	—
27	C₂⁴×C₄	C₂³	1	24	yes	—
28	C₂⁴×C₄	C₂³	1	1	yes	characteristic
29	#554	C₂³	1	3072	yes	complemented
30	D₄×Q₈	C₂³	1	1024	yes	complemented
31	#544	C₂³	1	768	yes	complemented
32	#543	C₂³	1	256	yes	—
33	#543	C₂³	1	1536	yes	complemented
34	#533	C₂³	1	768	yes	complemented
35	C₂×C₄⋊Q₈	C₂³	1	96	yes	complemented
36	C₂×#125	C₂³	1	96	yes	complemented
37	C₂×C₂²⋊Q₈	C₂³	1	192	yes	—
38	C₂×C₂²⋊Q₈	C₂³	1	384	yes	complemented
39	C₂×C₂²⋊Q₈	C₂³	1	1536	yes	complemented
40	#518	C₂³	1	768	yes	—
41	C₂×C₄×Q₈	C₂³	1	64	yes	complemented
42	C₂×C₄×D₄	C₂³	1	192	yes	—
43	C₂×C₄×D₄	C₂³	1	384	yes	—
44	C₂×#117	C₂³	1	48	yes	—
45	C₂×#117	C₂³	1	96	yes	—
46	C₂²×C₄⋊C₄	C₂³	1	12	yes	—
47	C₂²×C₄⋊C₄	C₂³	1	96	yes	complemented
48	C₂²×C₂²⋊C₄	C₂³	1	12	yes	—
49	C₂²×C₂²⋊C₄	C₂³	1	24	yes	—
50	C₂²×C₂²⋊C₄	C₂³	1	96	yes	—
51	C₂²×C₂²⋊C₄	C₂³	1	192	yes	—
52	C₂²×C₄²	C₂³	1	12	yes	—
53	C₂⁵	—	8	8	no	—
54	C₂⁵	C₂×Q₈	1	32	yes	complemented
55	C₂⁵	C₂⁴	1	1	yes	characteristic
56	C₂⁵	C₂⁴	1	12	yes	—
57	C₂⁵	C₂⁴	1	16	yes	—
58	C₂²×Q₈	—	8	64	no	—
59	C₂²×Q₈	C₂⁴	1	64	yes	—
60	C₂²×Q₈	C₂⁴	1	256	yes	complemented
61	C₂²×D₄	C₂×Q₈	1	192	yes	complemented
62	C₂²×D₄	C₂×Q₈	1	256	yes	complemented
63	C₂²×D₄	C₂⁴	1	16	yes	—
64	C₂²×D₄	C₂⁴	1	96	yes	—
65	C₂³×C₄	—	8	48	no	—
66	C₂³×C₄	C₂×Q₈	1	16	yes	complemented
67	C₂³×C₄	C₂⁴	1	24	yes	—
68	C₂³×C₄	C₂⁴	1	96	yes	—
69	C₂³×C₄	C₂⁴	1	192	yes	—
70	C₂³×C₄	C₂⁴	1	6	yes	—
71	C₂³×C₄	C₂⁴	1	8	yes	—
72	C₄⋊Q₈	C₂⁴	1	768	yes	complemented
73	#125	C₂⁴	1	768	yes	complemented
74	C₂²⋊Q₈	C₂⁴	1	1536	yes	—
75	C₂²⋊Q₈	C₂⁴	1	3072	yes	complemented
76	C₄×Q₈	C₂⁴	1	512	yes	complemented
77	C₄×D₄	C₂⁴	1	1536	yes	—
78	#117	C₂⁴	1	384	yes	—
79	C₂×C₄⋊C₄	C₂⁴	1	48	yes	—
80	C₂×C₄⋊C₄	C₂⁴	1	96	yes	—
81	C₂×C₄⋊C₄	C₂⁴	1	192	yes	—
82	C₂×C₄⋊C₄	C₂⁴	1	768	yes	complemented
83	C₂×C₂²⋊C₄	C₂⁴	1	48	yes	—
84	C₂×C₂²⋊C₄	C₂⁴	1	96	yes	—
85	C₂×C₂²⋊C₄	C₂⁴	1	96	yes	—
86	C₂×C₂²⋊C₄	C₂⁴	1	192	yes	—
87	C₂×C₂²⋊C₄	C₂⁴	1	768	yes	—
88	C₂×C₄²	C₂⁴	1	48	yes	—
89	C₂×C₄²	C₂⁴	1	96	yes	—
90	C₂⁴	—	4	48	no	—
91	C₂⁴	—	4	4	no	—
92	C₂⁴	—	8	64	no	—
93	C₂⁴	—	8	32	no	—
94	C₂⁴	C₂²×Q₈	1	8	yes	—
95	C₂⁴	C₂²×Q₈	1	96	yes	—
96	C₂⁴	C₂²×Q₈	1	128	yes	complemented
97	C₂⁴	C₂⁵	1	3	yes	—
98	C₂⁴	C₂⁵	1	8	yes	—
99	C₂⁴	C₂⁵	1	4	yes	—
100	C₂⁴	C₂⁵	1	48	yes	—
101	C₂×Q₈	—	4	128	no	—
102	C₂×Q₈	—	8	256	no	—
103	C₂×Q₈	C₂⁵	1	256	yes	—
104	C₂×D₄	C₂²×Q₈	1	128	yes	—
105	C₂×D₄	C₂²×Q₈	1	768	yes	complemented
106	C₂×D₄	C₂⁵	1	64	yes	—
107	C₂²×C₄	—	4	48	no	—
108	C₂²×C₄	—	4	192	no	—
109	C₂²×C₄	—	8	192	no	—
110	C₂²×C₄	C₂²×Q₈	1	48	yes	—
111	C₂²×C₄	C₂²×Q₈	1	64	yes	complemented
112	C₂²×C₄	C₂⁵	1	48	yes	—
113	C₂²×C₄	C₂⁵	1	96	yes	—
114	C₂²×C₄	C₂⁵	1	384	yes	—
115	C₂²×C₄	C₂⁵	1	24	yes	—
116	C₂²×C₄	C₂⁵	1	4	yes	—
117	C₄⋊C₄	C₂⁵	1	192	yes	—
118	C₄⋊C₄	C₂⁵	1	768	yes	—
119	C₂²⋊C₄	C₂⁵	1	192	yes	—
120	C₂²⋊C₄	C₂⁵	1	384	yes	—
121	C₄²	C₂⁵	1	192	yes	—
122	C₂³	—	2	32	no	—
123	C₂³	—	2	12	no	—
124	C₂³	—	4	96	no	—
125	C₂³	—	4	16	no	—
126	C₂³	—	4	8	no	—
127	C₂³	—	4	192	no	—
128	C₂³	—	8	128	no	—
129	C₂³	C₂³×Q₈	1	12	yes	—
130	C₂³	C₂³×Q₈	1	32	yes	—
131	C₂³	C₂³×Q₈	1	16	yes	—
132	C₂³	C₂³×Q₈	1	192	yes	—
133	C₂³	C₂⁶	1	6	yes	—
134	C₂³	C₂⁶	1	16	yes	—
135	C₂³	C₂⁶	1	1	yes	characteristic
136	Q₈	—	4	256	no	—
137	D₄	C₂³×Q₈	1	256	yes	—
138	C₂×C₄	—	2	96	no	—
139	C₂×C₄	—	4	96	no	—
140	C₂×C₄	—	4	384	no	—
141	C₂×C₄	C₂³×Q₈	1	16	yes	—
142	C₂×C₄	C₂³×Q₈	1	96	yes	—
143	C₂×C₄	C₂⁶	1	96	yes	—
144	C₂×C₄	C₂⁶	1	8	yes	—
145	C₂²	—	2	24	no	—
146	C₂²	—	2	64	no	—
147	C₂²	—	2	32	no	—
148	C₂²	—	2	12	no	—
149	C₂²	—	4	16	no	—
150	C₂²	—	4	192	no	—
151	C₂²	C₂⁴×Q₈	1	12	yes	—
152	C₂²	C₂⁴×Q₈	1	32	yes	—
153	C₂²	C₂⁴×Q₈	1	2	yes	—
154	C₂²	H(7,𝔽₂)	1	4	yes	—
155	C₂²	C₂⁷	1	1	yes	characteristic
156	C₄	—	2	96	no	—
157	C₄	C₂⁴×Q₈	1	16	yes	—
158	C₂	—	2	12	no	—
159	C₂	—	2	32	no	—
160	C₂	#62843	1	4	yes	direct product factor
161	C₂	C₂⁵×Q₈	1	1	yes	characteristic
162	C₂	C₂×H(7,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁷
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #62843	C₂	direct (N × H)
2. C₂×#3501	C₂	semidirect
3. C₂×#3458	C₂	semidirect
4. C₂²×#543	C₂	semidirect
7. #3501	C₂²	semidirect
8. C₂×#554	C₂²	semidirect
9. C₂×D₄×Q₈	C₂²	semidirect
10. #3458	C₂²	semidirect
11. C₂×#544	C₂²	semidirect
13. C₂×#543	C₂²	semidirect
14. C₂×#543	C₂²	semidirect
15. C₂×#533	C₂²	semidirect
16. C₂²×C₂²⋊Q₈	C₂²	semidirect
23. C₂³×Q₈	C₂³	semidirect
24. C₂³×D₄	Q₈	semidirect
29. #554	C₂³	semidirect
30. D₄×Q₈	C₂³	semidirect
31. #544	C₂³	semidirect
33. #543	C₂³	semidirect
34. #533	C₂³	semidirect
35. C₂×C₄⋊Q₈	C₂³	semidirect
36. C₂×#125	C₂³	semidirect
38. C₂×C₂²⋊Q₈	C₂³	semidirect
39. C₂×C₂²⋊Q₈	C₂³	semidirect
41. C₂×C₄×Q₈	C₂³	semidirect
47. C₂²×C₄⋊C₄	C₂³	semidirect
54. C₂⁵	C₂×Q₈	semidirect
60. C₂²×Q₈	C₂⁴	semidirect
61. C₂²×D₄	C₂×Q₈	semidirect
62. C₂²×D₄	C₂×Q₈	semidirect
66. C₂³×C₄	C₂×Q₈	semidirect
72. C₄⋊Q₈	C₂⁴	semidirect
73. #125	C₂⁴	semidirect
75. C₂²⋊Q₈	C₂⁴	semidirect
76. C₄×Q₈	C₂⁴	semidirect
82. C₂×C₄⋊C₄	C₂⁴	semidirect
96. C₂⁴	C₂²×Q₈	semidirect
105. C₂×D₄	C₂²×Q₈	semidirect
111. C₂²×C₄	C₂²×Q₈	semidirect
160. C₂	#62843	direct (N × H)

Maximal subgroups

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

subgroup	index
#62843	2
C₂×#3501	2
C₂×#3458	2
C₂²×#543	2
C₂²×#518	2

Subgroups — C2×#62843 #10585997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#62843 #10585997