Subgroups — C₂×#13647 #512747

Order 512.

426 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2258	C₂	1	2	yes	maximal, complemented
2	C₂×#2170	C₂	1	1	yes	maximal, characteristic
3	#13647	C₂	1	8	yes	maximal, direct product factor
4	C₂×#1422	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#1401	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#504	C₂²	1	2	yes	—
8	C₂×#443	C₄	1	1	yes	characteristic
9	#2258	—	2	4	no	—
10	#2258	C₂²	1	8	yes	complemented
11	C₂×M₆₄	—	2	2	no	—
12	C₂×M₆₄	C₂²	1	2	yes	—
13	C₂²×C₃₂	—	2	2	no	—
14	C₂²×C₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₃₂	C₂²	1	1	yes	characteristic
16	#2170	C₄	1	4	yes	—
17	#2170	C₂²	1	4	yes	—
18	C₂×#364	C₄	1	1	yes	characteristic
19	C₂×#350	C₄	1	2	yes	—
20	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
21	#1422	C₂²	1	4	yes	complemented
22	#1401	C₂²	1	8	yes	complemented
23	C₄×C₃₂	C₂²	1	4	yes	complemented
24	C₂×#135	C₈	1	1	yes	characteristic
25	C₂×#131	C₂×C₄	1	2	yes	—
26	#504	—	2	2	no	—
27	#504	—	2	2	no	—
28	#504	C₂×C₄	1	4	yes	—
29	#504	C₂³	1	4	yes	—
30	C₂×M₃₂	—	2	2	no	—
31	C₂×M₃₂	C₂×C₄	1	2	yes	—
32	C₂²×C₁₆	—	2	2	no	—
33	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₁₆	C₂³	1	1	yes	characteristic
35	#443	C₈	1	4	yes	—
36	#443	C₂×C₄	1	4	yes	—
37	C₂×#108	C₈	1	1	yes	characteristic
38	C₂×#104	C₈	1	2	yes	—
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	M₆₄	—	2	2	no	—
41	M₆₄	—	2	8	no	—
42	M₆₄	C₂³	1	4	yes	—
43	C₂×C₃₂	—	2	2	no	—
44	C₂×C₃₂	—	2	1	no	—
45	C₂×C₃₂	—	2	1	no	—
46	C₂×C₃₂	—	2	8	no	—
47	C₂×C₃₂	D₄	1	1	yes	characteristic
48	C₂×C₃₂	D₄	1	2	yes	—
49	C₂×C₃₂	D₄	1	1	yes	characteristic
50	C₂×C₃₂	C₂³	1	2	yes	—
51	C₂×C₃₂	C₂³	1	2	yes	—
52	#364	C₂×C₄	1	4	yes	—
53	#350	C₂×C₄	1	8	yes	—
54	C₄×C₁₆	C₂×C₄	1	2	yes	—
55	C₄×C₁₆	C₂³	1	2	yes	—
56	C₂×#41	C₂×C₈	1	2	yes	—
57	#135	C₁₆	1	4	yes	—
58	#135	C₂×C₈	1	4	yes	—
59	C₂×Q₁₆	C₁₆	1	1	yes	characteristic
60	C₂×SD₁₆	C₁₆	1	2	yes	—
61	C₂×D₈	C₁₆	1	1	yes	characteristic
62	#131	—	2	2	no	—
63	#131	—	2	2	no	—
64	#131	C₂×C₈	1	4	yes	—
65	#131	C₂²×C₄	1	4	yes	—
66	C₂×M₁₆	—	2	2	no	—
67	C₂×M₁₆	C₂×C₈	1	2	yes	—
68	C₂²×C₈	—	2	2	no	—
69	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
72	M₃₂	—	2	2	no	—
73	M₃₂	—	2	4	no	—
74	M₃₂	—	2	4	no	—
75	M₃₂	C₂²×C₄	1	4	yes	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	2	4	no	—
80	C₂×C₁₆	—	2	4	no	—
81	C₂×C₁₆	C₂²×C₄	1	2	yes	—
82	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
84	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
85	C₂×C₁₆	#41	1	1	yes	characteristic
86	C₂×C₁₆	#41	1	1	yes	characteristic
87	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
88	#108	C₂×C₈	1	4	yes	—
89	#104	C₂×C₈	1	8	yes	—
90	C₄×C₈	C₂×C₈	1	2	yes	—
91	C₄×C₈	C₂²×C₄	1	2	yes	—
92	C₃₂	—	2	4	no	—
93	C₃₂	—	2	2	no	—
94	C₃₂	C₂×D₄	1	2	yes	—
95	C₃₂	C₂×D₄	1	2	yes	—
96	#41	—	2	2	no	—
97	#41	—	2	2	no	—
98	#41	C₂×C₁₆	1	4	yes	—
99	#41	C₂²×C₈	1	4	yes	—
100	C₂×Q₈	C₂×C₁₆	1	2	yes	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	C₂×C₁₆	1	2	yes	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
106	Q₁₆	C₂×C₁₆	1	4	yes	—
107	SD₁₆	C₂×C₁₆	1	8	yes	—
108	D₈	C₂×C₁₆	1	4	yes	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	4	no	—
111	M₁₆	—	2	4	no	—
112	M₁₆	C₂²×C₈	1	4	yes	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
117	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
118	C₂×C₈	C₂×C₁₆	1	2	yes	—
119	C₂×C₈	C₄×D₄	1	1	yes	characteristic
120	C₂×C₈	C₄×D₄	1	1	yes	characteristic
121	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₈	C₂²×C₈	1	2	yes	—
123	C₂×C₈	#131	1	1	yes	characteristic
124	C₂×C₈	#131	1	1	yes	characteristic
125	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
126	C₄²	—	2	1	no	—
127	C₄²	C₂²×C₈	1	2	yes	—
128	C₁₆	—	2	4	no	—
129	C₁₆	—	2	2	no	—
130	C₁₆	C₄×D₄	1	1	yes	characteristic
131	C₁₆	C₄×D₄	1	1	yes	characteristic
132	C₁₆	C₂²×D₄	1	1	yes	characteristic
133	C₁₆	C₂×#41	1	1	yes	characteristic
134	C₂³	—	2	2	no	—
135	C₂³	#504	1	1	yes	characteristic
136	Q₈	—	2	2	no	—
137	Q₈	C₂²×C₁₆	1	4	yes	—
138	D₄	—	2	4	no	—
139	D₄	—	2	4	no	—
140	D₄	—	2	2	no	—
141	D₄	C₂²×C₁₆	1	4	yes	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	C₈×D₄	1	1	yes	characteristic
148	C₂×C₄	C₈×D₄	1	1	yes	characteristic
149	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
150	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
151	C₂×C₄	#504	1	1	yes	characteristic
152	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
153	C₈	—	2	4	no	—
154	C₈	C₈×D₄	1	1	yes	characteristic
155	C₈	C₈×D₄	1	1	yes	characteristic
156	C₈	C₂²×C₁₆	1	2	yes	—
157	C₈	C₂²×C₁₆	1	2	yes	—
158	C₈	C₂×C₄×D₄	1	1	yes	characteristic
159	C₈	C₂×#131	1	1	yes	characteristic
160	C₂²	—	2	2	no	—
161	C₂²	—	2	4	no	—
162	C₂²	—	4	4	no	—
163	C₂²	D₄×C₁₆	1	1	yes	characteristic
164	C₂²	D₄×C₁₆	1	1	yes	characteristic
165	C₂²	C₂×#504	1	1	yes	characteristic
166	C₄	—	2	4	no	—
167	C₄	—	2	4	no	—
168	C₄	D₄×C₁₆	1	1	yes	characteristic
169	C₄	D₄×C₁₆	1	1	yes	characteristic
170	C₄	C₂×C₈×D₄	1	1	yes	characteristic
171	C₄	C₂³×C₁₆	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	—	4	4	no	—
175	C₂	#13647	1	2	yes	direct product factor
176	C₂	C₂×D₄×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂	D₄
commutator	C₄	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂²	D₄×C₁₆
prime radical	C₂×#135	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂ ◁ 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
1. C₂×#2258	C₂	semidirect
3. #13647	C₂	direct (N × H)
4. C₂×#1422	C₂	semidirect
5. C₂×#1401	C₂	semidirect
6. C₂×C₄×C₃₂	C₂	semidirect
10. #2258	C₂²	semidirect
21. #1422	C₂²	semidirect
22. #1401	C₂²	semidirect
23. C₄×C₃₂	C₂²	semidirect
175. C₂	#13647	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#2258	2
C₂×#2170	2
#13647	2
C₂×#1422	2
C₂×#1401	2
C₂×C₄×C₃₂	2

Subgroups — C2×#13647 #512747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13647 #512747