Subgroups — C₃×C₇×#562 #13061982

Order 1344.

440 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	C₃×C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₇×#124	C₂	1	2	yes	maximal, complemented
3	C₃×C₇×#123	C₂	1	4	yes	maximal, complemented
4	C₃×C₇×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
5	C₃×C₇×#121	C₂	1	4	yes	maximal, complemented
6	C₇×#562	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₇×Q₈	C₂²	1	2	yes	—
8	C₂×C₃×C₇×D₄	C₂²	1	4	yes	—
9	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₇	C₂²	1	4	yes	—
11	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
12	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
16	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
17	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
18	C₇×#125	C₂×C₃	1	1	yes	characteristic, complemented
19	C₇×#124	C₂×C₃	1	2	yes	complemented
20	C₇×#123	C₂×C₃	1	4	yes	complemented
21	C₇×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
22	C₇×#121	C₂×C₃	1	4	yes	complemented
23	C₃×#562	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂³×C₃×C₇	C₂³	1	4	yes	—
25	C₃×C₇×Q₈	—	4	2	no	—
26	C₃×C₇×D₄	—	4	4	no	—
27	C₃×C₇×D₄	—	4	2	no	—
28	C₂×C₃×C₄×C₇	—	4	4	no	—
29	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
31	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
32	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
33	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
34	C₂×C₇×D₄	C₂²×C₃	1	4	yes	—
35	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
36	C₂²×C₄×C₇	C₂²×C₃	1	4	yes	—
37	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
38	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
39	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
40	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
41	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
42	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
43	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
44	C₃×#125	C₂×C₇	1	1	yes	characteristic, complemented
45	C₃×#124	C₂×C₇	1	2	yes	complemented
46	C₃×#123	C₂×C₇	1	4	yes	complemented
47	C₃×C₂²⋊Q₈	C₂×C₇	1	4	yes	complemented
48	C₃×#121	C₂×C₇	1	4	yes	complemented
49	C₂²×C₃×C₇	—	2	4	no	—
50	C₂²×C₃×C₇	—	2	4	no	—
51	C₂²×C₃×C₇	—	2	4	no	—
52	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
53	C₃×C₄×C₇	—	2	4	no	—
54	C₃×C₄×C₇	—	2	4	no	—
55	C₃×C₄×C₇	—	2	2	no	—
56	C₃×C₄×C₇	—	2	1	no	—
57	#562	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₇	C₂³×C₃	1	4	yes	—
59	C₇×Q₈	—	4	2	no	—
60	C₇×D₄	—	4	4	no	—
61	C₇×D₄	—	4	2	no	—
62	C₂×C₄×C₇	—	4	4	no	—
63	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
64	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
65	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
66	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
67	C₂×C₃×Q₈	C₂²×C₇	1	2	yes	—
68	C₂×C₃×D₄	C₂²×C₇	1	4	yes	—
69	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
70	C₂²×C₃×C₄	C₂²×C₇	1	4	yes	—
71	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
72	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
73	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
74	C₃×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
75	C₃×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
76	C₃×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
77	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
78	C₂×C₃×C₇	—	4	4	no	—
79	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	—
80	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
81	#125	C₂×C₃×C₇	1	1	yes	characteristic, complemented
82	#124	C₂×C₃×C₇	1	2	yes	complemented
83	#123	C₂×C₃×C₇	1	4	yes	complemented
84	C₂²⋊Q₈	C₂×C₃×C₇	1	4	yes	complemented
85	#121	C₂×C₃×C₇	1	4	yes	complemented
86	C₂²×C₇	—	2	4	no	—
87	C₂²×C₇	—	2	4	no	—
88	C₂²×C₇	—	2	4	no	—
89	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
90	C₄×C₇	—	2	4	no	—
91	C₄×C₇	—	2	4	no	—
92	C₄×C₇	—	2	2	no	—
93	C₄×C₇	—	2	1	no	—
94	C₂³×C₃	C₂³×C₇	1	4	yes	—
95	C₃×Q₈	—	4	2	no	—
96	C₃×D₄	—	4	4	no	—
97	C₃×D₄	—	4	2	no	—
98	C₂×C₃×C₄	—	4	4	no	—
99	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
100	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
101	C₂×C₃×C₄	C₂³×C₇	1	4	yes	—
102	C₂×C₃×C₄	C₂³×C₇	1	4	yes	—
103	C₃×C₇	#562	1	1	yes	characteristic, direct product factor, Hall
104	C₂×Q₈	C₂²×C₃×C₇	1	2	yes	—
105	C₂×D₄	C₂²×C₃×C₇	1	4	yes	—
106	C₂×D₄	C₂²×C₃×C₇	1	2	yes	—
107	C₂²×C₄	C₂²×C₃×C₇	1	4	yes	—
108	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
109	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
110	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
111	C₂²⋊C₄	C₂²×C₃×C₇	1	4	yes	—
112	C₂²⋊C₄	C₂²×C₃×C₇	1	4	yes	—
113	C₂²⋊C₄	C₂²×C₃×C₇	1	4	yes	—
114	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
115	C₂×C₇	—	4	4	no	—
116	C₂×C₇	C₃×H(5,𝔽₂)	1	2	yes	—
117	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
118	C₂²×C₃	—	2	4	no	—
119	C₂²×C₃	—	2	4	no	—
120	C₂²×C₃	—	2	4	no	—
121	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
122	C₃×C₄	—	2	4	no	—
123	C₃×C₄	—	2	4	no	—
124	C₃×C₄	—	2	2	no	—
125	C₃×C₄	—	2	1	no	—
126	C₂³	C₂³×C₃×C₇	1	4	yes	—
127	Q₈	—	4	2	no	—
128	D₄	—	4	4	no	—
129	D₄	—	4	2	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
132	C₂×C₄	C₂³×C₃×C₇	1	2	yes	—
133	C₂×C₄	C₂³×C₃×C₇	1	4	yes	—
134	C₂×C₄	C₂³×C₃×C₇	1	4	yes	—
135	C₇	C₃×#562	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×C₃	—	4	4	no	—
137	C₂×C₃	C₇×H(5,𝔽₂)	1	2	yes	—
138	C₂×C₃	C₇×2₋¹⁺⁴	1	1	yes	characteristic
139	C₂²	—	2	4	no	—
140	C₂²	—	2	4	no	—
141	C₂²	—	2	4	no	—
142	C₂²	C₂⁴×C₃×C₇	1	1	yes	characteristic
143	C₄	—	2	4	no	—
144	C₄	—	2	4	no	—
145	C₄	—	2	2	no	—
146	C₄	—	2	1	no	—
147	C₃	C₇×#562	1	1	yes	characteristic, direct product factor, Sylow
148	C₂	—	4	4	no	—
149	C₂	C₃×C₇×H(5,𝔽₂)	1	2	yes	—
150	C₂	C₃×C₇×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×#125	C₂	semidirect
2. C₃×C₇×#124	C₂	semidirect
3. C₃×C₇×#123	C₂	semidirect
4. C₃×C₇×C₂²⋊Q₈	C₂	semidirect
5. C₃×C₇×#121	C₂	semidirect
6. C₇×#562	C₃	direct (N × H)
18. C₇×#125	C₂×C₃	semidirect
19. C₇×#124	C₂×C₃	semidirect
20. C₇×#123	C₂×C₃	semidirect
21. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
22. C₇×#121	C₂×C₃	semidirect
23. C₃×#562	C₇	direct (N × H)
44. C₃×#125	C₂×C₇	semidirect
45. C₃×#124	C₂×C₇	semidirect
46. C₃×#123	C₂×C₇	semidirect
47. C₃×C₂²⋊Q₈	C₂×C₇	semidirect
48. C₃×#121	C₂×C₇	semidirect
57. #562	C₃×C₇	direct (N × H)
81. #125	C₂×C₃×C₇	semidirect
82. #124	C₂×C₃×C₇	semidirect
83. #123	C₂×C₃×C₇	semidirect
84. C₂²⋊Q₈	C₂×C₃×C₇	semidirect
85. #121	C₂×C₃×C₇	semidirect
103. C₃×C₇	#562	direct (N × H)
135. C₇	C₃×#562	direct (N × H)
147. C₃	C₇×#562	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇×#125	2
C₃×C₇×#124	2
C₃×C₇×#123	2
C₃×C₇×C₂²⋊Q₈	2
C₃×C₇×#121	2
C₇×#562	3
C₃×#562	7

Subgroups — C3×C7×#562 #13061982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#562 #13061982