Subgroups — C₂×#13630 #512709

Order 512.

460 conjugacy classes of subgroups.

The subgroup lattice has 167 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₆₄	C₂	1	2	yes	maximal
2	C₂×#2108	C₂	1	1	yes	maximal, characteristic
3	#13630	C₂	1	8	yes	maximal, direct product factor
4	C₂×#1422	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂	C₄	1	1	yes	characteristic
6	C₂³×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#405	C₄	1	1	yes	characteristic
8	C₂×M₆₄	—	2	4	no	—
9	C₂×M₆₄	—	2	2	no	—
10	C₂×M₆₄	C₂²	1	4	yes	—
11	C₂×M₆₄	C₂²	1	8	yes	—
12	C₂²×C₃₂	—	2	2	no	—
13	#2108	C₄	1	4	yes	—
14	#2108	C₂²	1	4	yes	—
15	C₂×#346	C₄	1	2	yes	—
16	C₂×C₄×C₁₆	C₂²	1	2	yes	—
17	#1422	C₂²	1	16	yes	complemented
18	C₂²×M₁₆	C₈	1	1	yes	characteristic
19	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×#117	C₈	1	1	yes	characteristic
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	—	2	2	no	—
23	C₂×M₃₂	C₂×C₄	1	4	yes	—
24	C₂²×C₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₁₆	D₄	1	2	yes	—
28	C₂²×C₁₆	D₄	1	2	yes	—
29	C₂²×C₁₆	Q₈	1	2	yes	—
30	C₂²×C₁₆	Q₈	1	2	yes	—
31	C₂²×C₁₆	C₂³	1	2	yes	—
32	C₂²×C₁₆	C₂³	1	1	yes	characteristic
33	#405	C₈	1	4	yes	—
34	#405	C₂×C₄	1	4	yes	—
35	C₂×#97	C₈	1	2	yes	—
36	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
37	M₆₄	—	2	8	no	—
38	M₆₄	—	2	4	no	—
39	M₆₄	C₂³	1	8	yes	—
40	C₂×C₃₂	—	2	8	no	—
41	C₂×C₃₂	—	2	4	no	—
42	#346	C₂×C₄	1	8	yes	—
43	C₄×C₁₆	C₂×C₄	1	4	yes	—
44	C₄×C₁₆	C₂³	1	4	yes	—
45	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
46	C₂×M₁₆	C₁₆	1	4	yes	—
47	C₂×M₁₆	C₁₆	1	4	yes	—
48	C₂×M₁₆	C₂×C₈	1	4	yes	—
49	C₂²×C₈	C₄⋊C₄	1	2	yes	—
50	C₂²×C₈	C₄⋊C₄	1	2	yes	—
51	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₈	C₂×C₈	1	2	yes	—
53	C₂²×C₈	C₂×C₈	1	2	yes	—
54	C₂²×C₈	M₁₆	1	2	yes	—
55	C₂²×C₈	M₁₆	1	2	yes	—
56	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₄	1	2	yes	—
58	#117	—	2	2	no	—
59	#117	C₂×C₈	1	4	yes	—
60	C₂×C₄⋊C₄	—	2	1	no	—
61	C₂×C₂²⋊C₄	—	2	1	no	—
62	C₂×C₄²	C₂×C₈	1	2	yes	—
63	M₃₂	—	2	8	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
67	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
68	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
69	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
70	C₂×C₁₆	C₂²×C₄	1	4	yes	—
71	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₁₆	C₂²×C₄	1	2	yes	—
73	C₂×C₁₆	C₂×D₄	1	4	yes	—
74	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
75	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
76	C₂×C₁₆	C₂×Q₈	1	4	yes	—
77	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
80	#97	C₂×C₈	1	8	yes	—
81	C₄×C₈	C₂×C₈	1	4	yes	—
82	C₄×C₈	C₂²×C₄	1	4	yes	—
83	C₃₂	—	2	8	no	—
84	C₂⁴	M₃₂	1	1	yes	characteristic
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₄⋊C₈	1	2	yes	—
87	C₂²×C₄	C₄⋊C₈	1	2	yes	—
88	C₂²×C₄	C₂×C₁₆	1	2	yes	—
89	C₂²×C₄	C₂×C₁₆	1	2	yes	—
90	C₂²×C₄	M₃₂	1	1	yes	characteristic
91	C₂²×C₄	M₃₂	1	2	yes	—
92	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
93	C₂²×C₄	C₂²×C₈	1	2	yes	—
94	M₁₆	C₂×C₁₆	1	16	yes	—
95	C₂×C₈	C₄⋊C₈	1	2	yes	—
96	C₂×C₈	C₄⋊C₈	1	2	yes	—
97	C₂×C₈	C₄⋊C₈	1	4	yes	—
98	C₂×C₈	C₄⋊C₈	1	4	yes	—
99	C₂×C₈	C₂×C₁₆	1	4	yes	—
100	C₂×C₈	C₂×C₁₆	1	4	yes	—
101	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₈	C₂²×C₈	1	2	yes	—
106	C₂×C₈	C₂²×C₈	1	4	yes	—
107	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₈	C₂×M₁₆	1	4	yes	—
110	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
111	C₄⋊C₄	—	2	4	no	—
112	C₂²⋊C₄	—	2	4	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²×C₈	1	4	yes	—
115	C₁₆	—	2	4	no	—
116	C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₁₆	C₂×C₄⋊C₄	1	4	yes	—
118	C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₁₆	C₂²×D₄	1	1	yes	characteristic
120	C₁₆	C₂²×Q₈	1	1	yes	characteristic
121	C₂³	—	4	2	no	—
122	C₂³	C₄⋊C₁₆	1	2	yes	—
123	C₂³	C₄⋊C₁₆	1	2	yes	—
124	C₂³	C₂×M₃₂	1	1	yes	characteristic
125	C₂³	C₂×M₃₂	1	2	yes	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	4	4	no	—
128	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
129	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
130	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
131	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
132	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄⋊C₈	1	4	yes	—
134	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
136	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
137	C₂×C₄	C₂²×C₁₆	1	4	yes	—
138	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
139	C₂×C₄	C₂×M₃₂	1	2	yes	—
140	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
141	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
142	C₈	C₂×C₄⋊C₈	1	4	yes	—
143	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₈	C₂²×C₁₆	1	8	yes	—
145	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₈	C₂²×M₁₆	1	1	yes	characteristic
147	C₂²	—	2	2	no	—
148	C₂²	—	2	4	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
152	C₂²	C₂×C₄⋊C₁₆	1	4	yes	—
153	C₂²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
154	C₂²	C₂²×M₃₂	1	1	yes	characteristic
155	C₄	—	4	4	no	—
156	C₄	C₂×C₄⋊C₁₆	1	1	yes	characteristic
157	C₄	C₂×C₄⋊C₁₆	1	1	yes	characteristic
158	C₄	C₂×C₄⋊C₁₆	1	4	yes	—
159	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
160	C₄	C₂³×C₁₆	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	4	no	—
164	C₂	#13630	1	2	yes	direct product factor
165	C₂	C₂²×C₄⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₄	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂²	C₂×C₄⋊C₁₆
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #13630	C₂	direct (N × H)
4. C₂×#1422	C₂	semidirect
17. #1422	C₂²	semidirect
164. C₂	#13630	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×M₆₄	2
C₂×#2108	2
#13630	2
C₂×#1422	2

Subgroups — C2×#13630 #512709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13630 #512709