Subgroups — C₃×C₇×C₈×D₄ #13056945

Order 1344.

268 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×C₈	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×C₈×D₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₇×D₄	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
9	C₂×C₃×C₇×C₈	—	2	4	no	—
10	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
14	C₃×C₇×C₂²⋊C₄	C₄	1	2	yes	—
15	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
16	C₂²×C₇×C₈	C₂×C₃	1	2	yes	complemented
17	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic
18	C₇×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
19	C₇×C₂²⋊C₈	C₂×C₃	1	2	yes	complemented
20	C₄×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃×C₈×D₄	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂³×C₃×C₇	C₂×C₄	1	2	yes	—
23	C₃×C₇×D₄	C₈	1	4	yes	direct product factor
24	C₂×C₃×C₄×C₇	—	2	2	no	—
25	C₂×C₃×C₄×C₇	—	2	2	no	—
26	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₃×C₇×C₈	—	2	2	no	—
31	C₃×C₇×C₈	—	2	1	no	—
32	C₃×C₇×C₈	D₄	1	2	yes	direct product factor
33	C₂×C₇×D₄	C₃×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
35	C₂×C₇×C₈	—	2	4	no	—
36	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
37	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
38	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
39	C₇×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
40	C₇×C₂²⋊C₄	C₃×C₄	1	2	yes	—
41	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂×C₇	1	2	yes	complemented
43	C₃×C₄×D₄	C₂×C₇	1	1	yes	characteristic
44	C₃×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
45	C₃×C₂²⋊C₈	C₂×C₇	1	2	yes	complemented
46	C₃×C₄×C₈	C₂×C₇	1	1	yes	characteristic, complemented
47	C₂²×C₃×C₇	—	2	2	no	—
48	C₂²×C₃×C₇	—	2	2	no	—
49	C₂²×C₃×C₇	C₂×C₈	1	4	yes	complemented
50	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₃×C₄×C₇	—	2	2	no	—
52	C₃×C₄×C₇	—	2	1	no	—
53	C₃×C₄×C₇	C₂×C₈	1	2	yes	complemented
54	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
55	C₃×C₄×C₇	#41	1	1	yes	characteristic
56	C₈×D₄	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₇	C₂×C₃×C₄	1	2	yes	—
58	C₇×D₄	C₃×C₈	1	4	yes	direct product factor
59	C₂×C₄×C₇	—	2	2	no	—
60	C₂×C₄×C₇	—	2	2	no	—
61	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
62	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
63	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
64	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
65	C₇×C₈	—	2	2	no	—
66	C₇×C₈	—	2	1	no	—
67	C₇×C₈	C₃×D₄	1	2	yes	direct product factor
68	C₂×C₃×D₄	C₄×C₇	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
70	C₂×C₃×C₈	—	2	4	no	—
71	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
73	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
74	C₃×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
75	C₃×C₂²⋊C₄	C₄×C₇	1	2	yes	—
76	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
77	C₂×C₃×C₇	—	2	4	no	—
78	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
79	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₃×C₇	#131	1	1	yes	characteristic
81	C₂²×C₈	C₂×C₃×C₇	1	2	yes	complemented
82	C₄×D₄	C₂×C₃×C₇	1	1	yes	characteristic
83	C₄⋊C₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
84	C₂²⋊C₈	C₂×C₃×C₇	1	2	yes	complemented
85	C₄×C₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
86	C₂²×C₇	—	2	2	no	—
87	C₂²×C₇	—	2	2	no	—
88	C₂²×C₇	C₂×C₃×C₈	1	4	yes	complemented
89	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
90	C₄×C₇	—	2	2	no	—
91	C₄×C₇	—	2	1	no	—
92	C₄×C₇	C₂×C₃×C₈	1	2	yes	complemented
93	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
94	C₄×C₇	C₃×#41	1	1	yes	characteristic
95	C₂³×C₃	C₂×C₄×C₇	1	2	yes	—
96	C₃×D₄	C₇×C₈	1	4	yes	direct product factor
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
100	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
101	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
102	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
103	C₃×C₈	—	2	2	no	—
104	C₃×C₈	—	2	1	no	—
105	C₃×C₈	C₇×D₄	1	2	yes	direct product factor
106	C₃×C₇	C₈×D₄	1	1	yes	characteristic, direct product factor, Hall
107	C₂×D₄	C₃×C₄×C₇	1	1	yes	characteristic
108	C₂²×C₄	C₂²×C₃×C₇	1	2	yes	—
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
111	C₂×C₈	C₂²×C₃×C₇	1	2	yes	—
112	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
113	C₄⋊C₄	C₃×C₄×C₇	1	1	yes	characteristic
114	C₂²⋊C₄	C₃×C₄×C₇	1	2	yes	—
115	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
116	C₂×C₇	—	2	4	no	—
117	C₂×C₇	C₃×C₄×D₄	1	1	yes	characteristic
118	C₂×C₇	C₂²×C₃×C₈	1	1	yes	characteristic
119	C₂×C₇	C₃×#131	1	1	yes	characteristic
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	2	2	no	—
122	C₂²×C₃	C₂×C₇×C₈	1	4	yes	complemented
123	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
124	C₃×C₄	—	2	2	no	—
125	C₃×C₄	—	2	1	no	—
126	C₃×C₄	C₂×C₇×C₈	1	2	yes	complemented
127	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
128	C₃×C₄	C₇×#41	1	1	yes	characteristic
129	C₂³	C₂×C₃×C₄×C₇	1	2	yes	—
130	D₄	C₃×C₇×C₈	1	4	yes	direct product factor
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
134	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
135	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
136	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	1	no	—
139	C₈	C₃×C₇×D₄	1	2	yes	direct product factor
140	C₇	C₃×C₈×D₄	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×C₃	—	2	4	no	—
142	C₂×C₃	C₄×C₇×D₄	1	1	yes	characteristic
143	C₂×C₃	C₂²×C₇×C₈	1	1	yes	characteristic
144	C₂×C₃	C₇×#131	1	1	yes	characteristic
145	C₂²	—	2	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	C₂×C₃×C₇×C₈	1	4	yes	complemented
148	C₂²	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	C₂×C₃×C₇×C₈	1	2	yes	complemented
152	C₄	C₂×C₃×C₇×D₄	1	1	yes	characteristic
153	C₄	C₃×C₇×#41	1	1	yes	characteristic
154	C₃	C₇×C₈×D₄	1	1	yes	characteristic, direct product factor, Sylow
155	C₂	—	2	4	no	—
156	C₂	C₃×C₄×C₇×D₄	1	1	yes	characteristic
157	C₂	C₂²×C₃×C₇×C₈	1	1	yes	characteristic
158	C₂	C₃×C₇×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₇×C₈	C₂²
commutator	C₂	C₂²×C₃×C₇×C₈
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂×C₃×C₇×D₄	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇×C₈	C₂	semidirect
3. C₃×C₇×C₄⋊C₈	C₂	semidirect
4. C₃×C₇×C₂²⋊C₈	C₂	semidirect
5. C₃×C₄×C₇×C₈	C₂	semidirect
6. C₇×C₈×D₄	C₃	direct (N × H)
16. C₂²×C₇×C₈	C₂×C₃	semidirect
18. C₇×C₄⋊C₈	C₂×C₃	semidirect
19. C₇×C₂²⋊C₈	C₂×C₃	semidirect
20. C₄×C₇×C₈	C₂×C₃	semidirect
21. C₃×C₈×D₄	C₇	direct (N × H)
23. C₃×C₇×D₄	C₈	direct (N × H)
32. C₃×C₇×C₈	D₄	direct (N × H)
42. C₂²×C₃×C₈	C₂×C₇	semidirect
44. C₃×C₄⋊C₈	C₂×C₇	semidirect
45. C₃×C₂²⋊C₈	C₂×C₇	semidirect
46. C₃×C₄×C₈	C₂×C₇	semidirect
49. C₂²×C₃×C₇	C₂×C₈	semidirect
53. C₃×C₄×C₇	C₂×C₈	semidirect
56. C₈×D₄	C₃×C₇	direct (N × H)
58. C₇×D₄	C₃×C₈	direct (N × H)
67. C₇×C₈	C₃×D₄	direct (N × H)
81. C₂²×C₈	C₂×C₃×C₇	semidirect
83. C₄⋊C₈	C₂×C₃×C₇	semidirect
84. C₂²⋊C₈	C₂×C₃×C₇	semidirect
85. C₄×C₈	C₂×C₃×C₇	semidirect
88. C₂²×C₇	C₂×C₃×C₈	semidirect
92. C₄×C₇	C₂×C₃×C₈	semidirect
96. C₃×D₄	C₇×C₈	direct (N × H)
105. C₃×C₈	C₇×D₄	direct (N × H)
106. C₃×C₇	C₈×D₄	direct (N × H)
122. C₂²×C₃	C₂×C₇×C₈	semidirect
126. C₃×C₄	C₂×C₇×C₈	semidirect
130. D₄	C₃×C₇×C₈	direct (N × H)
139. C₈	C₃×C₇×D₄	direct (N × H)
140. C₇	C₃×C₈×D₄	direct (N × H)
147. C₂²	C₂×C₃×C₇×C₈	semidirect
151. C₄	C₂×C₃×C₇×C₈	semidirect
154. C₃	C₇×C₈×D₄	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×C₃×C₇×C₈	2
C₃×C₄×C₇×D₄	2
C₃×C₇×C₄⋊C₈	2
C₃×C₇×C₂²⋊C₈	2
C₃×C₄×C₇×C₈	2
C₇×C₈×D₄	3
C₃×C₈×D₄	7

Subgroups — C3×C7×C8×D4 #13056945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×C₈×D₄ #13056945