Subgroups — C₃×C₄×C₇×D₈ #13056948

Order 1344.

268 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇×D₄	C₂	1	2	yes	maximal, complemented
3	C₃×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×#102	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₇×D₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₇	—	2	2	no	—
9	C₃×C₇×D₈	C₄	1	4	yes	direct product factor
10	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×C₇×C₂²⋊C₄	—	2	2	no	—
14	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₇×D₈	C₂×C₃	1	1	yes	characteristic
16	C₄×C₇×D₄	C₂×C₃	1	2	yes	complemented
17	C₇×#107	C₂×C₃	1	1	yes	characteristic, complemented
18	C₇×#102	C₂×C₃	1	2	yes	complemented
19	C₄×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
20	C₃×C₄×D₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂³×C₃×C₇	—	2	2	no	—
22	C₃×C₇×D₄	—	2	2	no	—
23	C₃×C₇×D₄	C₂×C₄	1	4	yes	complemented
24	C₂×C₃×C₄×C₇	—	2	2	no	—
25	C₂×C₃×C₄×C₇	—	4	4	no	—
26	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₃×C₇×C₈	—	2	1	no	—
30	C₃×C₇×C₈	C₂×C₄	1	2	yes	complemented
31	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
32	C₂²×C₄×C₇	—	2	2	no	—
33	C₇×D₈	C₃×C₄	1	4	yes	direct product factor
34	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
35	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
36	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
37	C₇×C₂²⋊C₄	—	2	2	no	—
38	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
39	C₂×C₃×D₈	C₂×C₇	1	1	yes	characteristic
40	C₃×C₄×D₄	C₂×C₇	1	2	yes	complemented
41	C₃×#107	C₂×C₇	1	1	yes	characteristic, complemented
42	C₃×#102	C₂×C₇	1	2	yes	complemented
43	C₃×C₄×C₈	C₂×C₇	1	1	yes	characteristic, complemented
44	C₂²×C₃×C₇	—	2	4	no	—
45	C₂²×C₃×C₇	—	4	2	no	—
46	C₂²×C₃×C₇	—	4	2	no	—
47	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
48	C₃×C₄×C₇	—	2	1	no	—
49	C₃×C₄×C₇	—	4	2	no	—
50	C₃×C₄×C₇	D₈	1	2	yes	direct product factor
51	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
52	C₃×C₄×C₇	#41	1	1	yes	characteristic
53	C₄×D₈	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₇	—	2	2	no	—
55	C₇×D₄	—	2	2	no	—
56	C₇×D₄	C₂×C₃×C₄	1	4	yes	complemented
57	C₂×C₄×C₇	—	2	2	no	—
58	C₂×C₄×C₇	—	4	4	no	—
59	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
62	C₇×C₈	—	2	1	no	—
63	C₇×C₈	C₂×C₃×C₄	1	2	yes	complemented
64	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₃×D₈	C₄×C₇	1	4	yes	direct product factor
67	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
69	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
70	C₃×C₂²⋊C₄	—	2	2	no	—
71	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
72	C₂×C₃×C₇	—	4	4	no	—
73	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₃×C₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₃×C₇	#135	1	1	yes	characteristic
76	C₂×D₈	C₂×C₃×C₇	1	1	yes	characteristic
77	C₄×D₄	C₂×C₃×C₇	1	2	yes	complemented
78	#107	C₂×C₃×C₇	1	1	yes	characteristic, complemented
79	#102	C₂×C₃×C₇	1	2	yes	complemented
80	C₄×C₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
81	C₂²×C₇	—	2	4	no	—
82	C₂²×C₇	—	4	2	no	—
83	C₂²×C₇	—	4	2	no	—
84	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
85	C₄×C₇	—	2	1	no	—
86	C₄×C₇	—	4	2	no	—
87	C₄×C₇	C₃×D₈	1	2	yes	direct product factor
88	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
89	C₄×C₇	C₃×#41	1	1	yes	characteristic
90	C₂³×C₃	—	2	2	no	—
91	C₃×D₄	—	2	2	no	—
92	C₃×D₄	C₂×C₄×C₇	1	4	yes	complemented
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	4	4	no	—
95	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
96	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
98	C₃×C₈	—	2	1	no	—
99	C₃×C₈	C₂×C₄×C₇	1	2	yes	complemented
100	C₃×C₇	C₄×D₈	1	1	yes	characteristic, direct product factor, Hall
101	C₂×D₄	C₂²×C₃×C₇	1	2	yes	—
102	C₂²×C₄	—	2	2	no	—
103	D₈	C₃×C₄×C₇	1	4	yes	direct product factor
104	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
105	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
106	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
107	C₂²⋊C₄	—	2	2	no	—
108	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
109	C₂×C₇	—	4	4	no	—
110	C₂×C₇	C₃×C₄×D₄	1	1	yes	characteristic
111	C₂×C₇	C₂×C₃×D₈	1	1	yes	characteristic
112	C₂×C₇	C₃×#135	1	1	yes	characteristic
113	C₂²×C₃	—	2	4	no	—
114	C₂²×C₃	—	4	2	no	—
115	C₂²×C₃	—	4	2	no	—
116	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
117	C₃×C₄	—	2	1	no	—
118	C₃×C₄	—	4	2	no	—
119	C₃×C₄	C₇×D₈	1	2	yes	direct product factor
120	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
121	C₃×C₄	C₇×#41	1	1	yes	characteristic
122	C₂³	—	2	2	no	—
123	D₄	—	2	2	no	—
124	D₄	C₂×C₃×C₄×C₇	1	4	yes	complemented
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
128	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
129	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	C₂×C₃×C₄×C₇	1	2	yes	complemented
132	C₇	C₃×C₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
133	C₂×C₃	—	4	4	no	—
134	C₂×C₃	C₄×C₇×D₄	1	1	yes	characteristic
135	C₂×C₃	C₂×C₇×D₈	1	1	yes	characteristic
136	C₂×C₃	C₇×#135	1	1	yes	characteristic
137	C₂²	—	2	4	no	—
138	C₂²	—	4	2	no	—
139	C₂²	—	4	2	no	—
140	C₂²	C₂×C₃×C₇×D₄	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	2	no	—
143	C₄	C₃×C₇×D₈	1	2	yes	direct product factor
144	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
145	C₄	C₃×C₇×#41	1	1	yes	characteristic
146	C₃	C₄×C₇×D₈	1	1	yes	characteristic, direct product factor, Sylow
147	C₂	—	4	4	no	—
148	C₂	C₃×C₄×C₇×D₄	1	1	yes	characteristic
149	C₂	C₂×C₃×C₇×D₈	1	1	yes	characteristic
150	C₂	C₃×C₇×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₇	D₄
commutator	C₄	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₃×C₇×D₈	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₇ ◁ C₃×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₇×D₄	C₂	semidirect
3. C₃×C₇×#107	C₂	semidirect
4. C₃×C₇×#102	C₂	semidirect
5. C₃×C₄×C₇×C₈	C₂	semidirect
6. C₄×C₇×D₈	C₃	direct (N × H)
9. C₃×C₇×D₈	C₄	direct (N × H)
16. C₄×C₇×D₄	C₂×C₃	semidirect
17. C₇×#107	C₂×C₃	semidirect
18. C₇×#102	C₂×C₃	semidirect
19. C₄×C₇×C₈	C₂×C₃	semidirect
20. C₃×C₄×D₈	C₇	direct (N × H)
23. C₃×C₇×D₄	C₂×C₄	semidirect
30. C₃×C₇×C₈	C₂×C₄	semidirect
33. C₇×D₈	C₃×C₄	direct (N × H)
40. C₃×C₄×D₄	C₂×C₇	semidirect
41. C₃×#107	C₂×C₇	semidirect
42. C₃×#102	C₂×C₇	semidirect
43. C₃×C₄×C₈	C₂×C₇	semidirect
50. C₃×C₄×C₇	D₈	direct (N × H)
53. C₄×D₈	C₃×C₇	direct (N × H)
56. C₇×D₄	C₂×C₃×C₄	semidirect
63. C₇×C₈	C₂×C₃×C₄	semidirect
66. C₃×D₈	C₄×C₇	direct (N × H)
77. C₄×D₄	C₂×C₃×C₇	semidirect
78. #107	C₂×C₃×C₇	semidirect
79. #102	C₂×C₃×C₇	semidirect
80. C₄×C₈	C₂×C₃×C₇	semidirect
87. C₄×C₇	C₃×D₈	direct (N × H)
92. C₃×D₄	C₂×C₄×C₇	semidirect
99. C₃×C₈	C₂×C₄×C₇	semidirect
100. C₃×C₇	C₄×D₈	direct (N × H)
103. D₈	C₃×C₄×C₇	direct (N × H)
119. C₃×C₄	C₇×D₈	direct (N × H)
124. D₄	C₂×C₃×C₄×C₇	semidirect
131. C₈	C₂×C₃×C₄×C₇	semidirect
132. C₇	C₃×C₄×D₈	direct (N × H)
143. C₄	C₃×C₇×D₈	direct (N × H)
146. C₃	C₄×C₇×D₈	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×C₃×C₇×D₈	2
C₃×C₄×C₇×D₄	2
C₃×C₇×#107	2
C₃×C₇×#102	2
C₃×C₄×C₇×C₈	2
C₄×C₇×D₈	3
C₃×C₄×D₈	7

Subgroups — C3×C4×C7×D8 #13056948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₇×D₈ #13056948