Subgroups — C₄×#1370 #129092

Order 512.

354 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×#1370	C₂	1	2	yes	maximal
3	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
4	#7151	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
6	C₂×C₄×C₁₆	—	2	2	no	—
7	C₄²×C₈	C₄	1	1	yes	characteristic
8	C₄²×C₈	C₄	1	1	yes	characteristic
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₈²	C₄	1	1	yes	characteristic
11	C₂×C₈²	C₄	1	1	yes	characteristic
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1374	—	2	2	no	—
15	#1370	C₄	1	8	yes	direct product factor
16	C₂²×C₁₆	—	2	4	no	—
17	C₂×C₄×C₈	—	2	2	no	—
18	C₂×C₄×C₈	C₈	1	2	yes	—
19	C₂×C₄×C₈	C₈	1	2	yes	—
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
28	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
30	C₄³	C₂×C₄	1	1	yes	characteristic
31	C₄⋊C₁₆	—	2	4	no	—
32	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
33	C₄×C₁₆	—	4	4	no	—
34	C₈²	—	2	2	no	—
35	C₈²	—	2	2	no	—
36	C₈²	C₈	1	4	yes	—
37	C₈²	C₂×C₄	1	4	yes	—
38	C₂²×C₈	—	2	1	no	—
39	C₂²×C₈	—	2	1	no	—
40	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₈	C₂×C₈	1	2	yes	—
44	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄²	C₂×C₈	1	2	yes	—
50	C₂×C₄²	M₁₆	1	2	yes	—
51	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₁₆	—	2	8	no	—
53	C₂×C₁₆	—	4	4	no	—
54	C₂×C₁₆	—	4	4	no	—
55	C₄×C₈	—	2	8	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	—	2	4	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	C₁₆	1	8	yes	complemented
65	C₄×C₈	C₄²	1	2	yes	—
66	C₄×C₈	C₄²	1	2	yes	—
67	C₄×C₈	C₂×C₈	1	2	yes	—
68	C₄×C₈	C₂×C₈	1	2	yes	—
69	C₄×C₈	C₂×C₈	1	2	yes	—
70	C₄×C₈	M₁₆	1	2	yes	—
71	C₄×C₈	D₈	1	2	yes	—
72	C₄×C₈	Q₁₆	1	2	yes	—
73	C₄×C₈	C₂²×C₄	1	2	yes	—
74	C₄×C₈	#41	1	2	yes	—
75	C₂²×C₄	C₄⋊C₈	1	2	yes	—
76	C₂²×C₄	C₄⋊C₈	1	2	yes	—
77	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
79	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	C₄×C₈	1	2	yes	—
92	C₂×C₈	C₄×C₈	1	2	yes	—
93	C₂×C₈	C₄⋊C₈	1	2	yes	—
94	C₂×C₈	C₄⋊C₈	1	2	yes	—
95	C₂×C₈	#107	1	2	yes	—
96	C₂×C₈	#108	1	2	yes	—
97	C₂×C₈	C₂×C₁₆	1	4	yes	—
98	C₂×C₈	C₂×C₁₆	1	4	yes	—
99	C₂×C₈	C₄×D₄	1	2	yes	—
100	C₂×C₈	C₄×Q₈	1	2	yes	—
101	C₂×C₈	C₂×D₈	1	1	yes	characteristic
102	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₈	#135	1	2	yes	—
104	C₄²	—	2	4	no	—
105	C₄²	—	2	1	no	—
106	C₄²	—	2	1	no	—
107	C₄²	#107	1	2	yes	—
108	C₄²	#108	1	2	yes	—
109	C₄²	C₂×C₁₆	1	4	yes	—
110	C₄²	M₃₂	1	4	yes	—
111	C₄²	C₂×C₄²	1	1	yes	characteristic
112	C₄²	#117	1	1	yes	characteristic
113	C₄²	C₂²×C₈	1	1	yes	characteristic
114	C₄²	C₂×M₁₆	1	1	yes	characteristic
115	C₁₆	—	4	8	no	—
116	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	#335	1	4	yes	—
121	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
122	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
123	C₂×C₄	#364	1	4	yes	—
124	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
126	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₄	C₂×#107	1	1	yes	characteristic
129	C₂×C₄	#427	1	1	yes	characteristic
130	C₂×C₄	C₂×#108	1	1	yes	characteristic
131	C₂×C₄	C₂×#108	1	1	yes	characteristic
132	C₂×C₄	#431	1	1	yes	characteristic
133	C₂×C₄	#431	1	1	yes	characteristic
134	C₂×C₄	C₂²×C₁₆	1	2	yes	—
135	C₂×C₄	C₂×M₃₂	1	2	yes	—
136	C₈	—	2	4	no	—
137	C₈	—	2	4	no	—
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	2	4	no	—
141	C₈	—	2	2	no	—
142	C₈	#335	1	4	yes	—
143	C₈	C₄×C₁₆	1	4	yes	complemented
144	C₈	C₄×D₈	1	2	yes	—
145	C₈	C₄×Q₁₆	1	2	yes	—
146	C₂²	C₂×#335	1	1	yes	characteristic
147	C₂²	#1564	1	1	yes	characteristic
148	C₂²	C₄×C₄⋊C₈	1	1	yes	characteristic
149	C₂²	C₂×C₄⋊C₁₆	1	2	yes	—
150	C₂²	C₂×#364	1	2	yes	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	4	no	—
154	C₄	#1370	1	8	yes	direct product factor
155	C₄	C₂×#335	1	1	yes	characteristic
156	C₄	#1564	1	1	yes	characteristic
157	C₄	C₄×#107	1	1	yes	characteristic
158	C₄	C₄×#108	1	1	yes	characteristic
159	C₄	C₂×C₄×C₁₆	1	1	yes	characteristic
160	C₄	C₄×M₃₂	1	1	yes	characteristic
161	C₄	#2162	1	2	yes	—
162	C₂	C₄×#335	1	1	yes	characteristic
163	C₂	C₂×#1370	1	2	yes	—
164	C₂	#12012	1	2	yes	—
165	C₂	C₄×C₄⋊C₁₆	1	1	yes	characteristic
166	C₂	C₄×#364	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
15. #1370	C₄	direct (N × H)
64. C₄×C₈	C₁₆	semidirect
143. C₈	C₄×C₁₆	semidirect
154. C₄	#1370	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₁₆	2
C₂×#1370	2
C₄×C₈²	2
#7151	2

Subgroups — C4×#1370 #129092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#1370 #129092