Subgroups — C₄×#1373 #129100

Order 512.

276 conjugacy classes of subgroups.

The subgroup lattice has 147 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#364	C₂	1	2	yes	maximal
2	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1373	C₂	1	2	yes	maximal
4	#7160	C₂	1	2	yes	maximal
5	C₂×#364	C₂²	1	4	yes	—
6	C₄×M₃₂	—	2	2	no	—
7	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	#1376	—	2	2	no	—
15	#1373	C₄	1	8	yes	direct product factor
16	C₂×M₃₂	—	2	4	no	—
17	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	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	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄³	Q₈	1	1	yes	characteristic
28	#364	—	2	4	no	—
29	#364	C₂×C₄	1	8	yes	—
30	#346	—	4	2	no	—
31	C₄×C₁₆	—	2	2	no	—
32	C₄×C₁₆	—	2	2	no	—
33	C₄×C₁₆	—	2	2	no	—
34	C₄×C₁₆	—	4	2	no	—
35	C₄×C₁₆	C₈	1	4	yes	—
36	C₄×C₁₆	C₈	1	4	yes	—
37	C₄×C₁₆	C₂×C₄	1	4	yes	—
38	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄²	—	2	2	no	—
43	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
46	M₃₂	—	2	8	no	—
47	M₃₂	—	4	4	no	—
48	C₂×C₁₆	—	2	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	—	4	4	no	—
55	C₂×C₁₆	C₄²	1	2	yes	—
56	C₂×C₁₆	C₄²	1	2	yes	—
57	C₂×C₁₆	C₂×C₈	1	2	yes	—
58	C₂×C₁₆	C₂×C₈	1	2	yes	—
59	C₂×C₁₆	C₂×C₈	1	2	yes	—
60	C₂×C₁₆	C₂×C₈	1	2	yes	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	C₂×C₈	1	2	yes	—
64	C₄×C₈	C₂×C₈	1	2	yes	—
65	C₄×C₈	M₁₆	1	2	yes	—
66	C₄×C₈	M₁₆	1	2	yes	—
67	C₄×C₈	D₈	1	1	yes	characteristic
68	C₄×C₈	D₈	1	1	yes	characteristic
69	C₄×C₈	Q₁₆	1	1	yes	characteristic
70	C₄×C₈	Q₁₆	1	1	yes	characteristic
71	C₄×C₈	C₂²×C₄	1	2	yes	—
72	C₄×C₈	#41	1	2	yes	—
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	—	2	1	no	—
75	C₂²×C₄	#108	1	2	yes	—
76	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	C₄⋊C₈	1	2	yes	—
80	C₂×C₈	C₄⋊C₈	1	2	yes	—
81	C₂×C₈	#107	1	1	yes	characteristic
82	C₂×C₈	#107	1	1	yes	characteristic
83	C₂×C₈	#108	1	2	yes	—
84	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₈	#117	1	1	yes	characteristic
86	C₂×C₈	C₄×D₄	1	2	yes	—
87	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₈	C₂×D₈	1	1	yes	characteristic
92	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×C₈	#135	1	2	yes	—
94	C₄²	—	2	8	no	—
95	C₄²	—	2	1	no	—
96	C₄²	—	2	1	no	—
97	C₄²	C₄⋊C₈	1	2	yes	—
98	C₄²	C₄⋊C₈	1	2	yes	—
99	C₄²	#107	1	1	yes	characteristic
100	C₄²	#107	1	1	yes	characteristic
101	C₄²	C₄×Q₈	1	2	yes	—
102	C₁₆	—	2	2	no	—
103	C₁₆	—	2	2	no	—
104	C₁₆	—	2	2	no	—
105	C₁₆	—	4	8	no	—
106	C₁₆	C₄×C₈	1	4	yes	—
107	C₂³	C₂×#108	1	1	yes	characteristic
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	#335	1	2	yes	—
114	C₂×C₄	#335	1	2	yes	—
115	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂×C₄	C₂×#107	1	1	yes	characteristic
119	C₂×C₄	#427	1	1	yes	characteristic
120	C₂×C₄	C₂×#108	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	#335	1	2	yes	—
123	C₈	#335	1	2	yes	—
124	C₈	C₂×C₄×C₈	1	1	yes	characteristic
125	C₈	C₄×M₁₆	1	1	yes	characteristic
126	C₈	#431	1	1	yes	characteristic
127	C₈	#431	1	1	yes	characteristic
128	C₈	C₄×D₈	1	2	yes	—
129	C₈	C₄×Q₁₆	1	2	yes	—
130	C₂²	—	2	2	no	—
131	C₂²	C₂×#335	1	1	yes	characteristic
132	C₂²	#1564	1	1	yes	characteristic
133	C₂²	C₄×#108	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	2	4	no	—
136	C₄	—	2	2	no	—
137	C₄	#1373	1	4	yes	direct product factor
138	C₄	C₂×#335	1	1	yes	characteristic
139	C₄	#1564	1	1	yes	characteristic
140	C₄	C₄×C₄⋊C₈	1	1	yes	characteristic
141	C₄	C₄×#107	1	1	yes	characteristic
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	C₄×#335	1	1	yes	characteristic
145	C₂	C₂×#1373	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	D₈
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#335
prime radical	C₂³	C₂×#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ 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. #1373	C₄	direct (N × H)
137. C₄	#1373	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×#364	2
C₄²×C₁₆	2
C₂×#1373	2
#7160	2

Subgroups — C4×#1373 #129100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#1373 #129100