Subgroups — C₃×C₇×#563 #13061983

Order 1344.

392 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×#126	C₂	1	4	yes	maximal, complemented
3	C₃×C₇×#125	C₂	1	2	yes	maximal, complemented
4	C₃×C₇×#124	C₂	1	1	yes	maximal, characteristic
5	C₃×C₇×#123	C₂	1	2	yes	maximal
6	C₃×C₇×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
7	C₇×#563	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₇×Q₈	C₂²	1	2	yes	—
9	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
12	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
14	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
15	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
16	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
18	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₇	C₂²	1	2	yes	—
22	C₇×C₄⋊Q₈	C₂×C₃	1	2	yes	complemented
23	C₇×#126	C₂×C₃	1	4	yes	complemented
24	C₇×#125	C₂×C₃	1	2	yes	complemented
25	C₇×#124	C₂×C₃	1	1	yes	characteristic
26	C₇×#123	C₂×C₃	1	2	yes	—
27	C₇×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
28	C₃×#563	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂³×C₃×C₇	C₂³	1	2	yes	—
30	C₃×C₇×Q₈	—	4	2	no	—
31	C₃×C₇×Q₈	—	4	1	no	—
32	C₃×C₇×D₄	—	4	1	no	—
33	C₂×C₃×C₄×C₇	—	4	2	no	—
34	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
35	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
36	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
37	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
39	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
40	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
43	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
44	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
45	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
46	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
47	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
49	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
50	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
51	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
52	C₄²×C₇	C₂²×C₃	1	2	yes	—
53	C₃×C₄⋊Q₈	C₂×C₇	1	2	yes	complemented
54	C₃×#126	C₂×C₇	1	4	yes	complemented
55	C₃×#125	C₂×C₇	1	2	yes	complemented
56	C₃×#124	C₂×C₇	1	1	yes	characteristic
57	C₃×#123	C₂×C₇	1	2	yes	—
58	C₃×C₂²⋊Q₈	C₂×C₇	1	4	yes	complemented
59	C₂²×C₃×C₇	—	2	2	no	—
60	C₂²×C₃×C₇	—	2	4	no	—
61	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
62	C₃×C₄×C₇	—	2	4	no	—
63	C₃×C₄×C₇	—	2	1	no	—
64	C₃×C₄×C₇	—	2	2	no	—
65	C₃×C₄×C₇	—	2	2	no	—
66	C₃×C₄×C₇	—	2	4	no	—
67	#563	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
68	C₂³×C₇	C₂³×C₃	1	2	yes	—
69	C₇×Q₈	—	4	2	no	—
70	C₇×Q₈	—	4	1	no	—
71	C₇×D₄	—	4	1	no	—
72	C₂×C₄×C₇	—	4	2	no	—
73	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
74	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
75	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
76	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
77	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
78	C₂×C₃×Q₈	C₂²×C₇	1	2	yes	—
79	C₂×C₃×Q₈	C₂²×C₇	1	1	yes	characteristic
80	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
82	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
83	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
84	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
85	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
86	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
87	C₃×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
88	C₃×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
89	C₃×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
90	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
91	C₃×C₄²	C₂²×C₇	1	2	yes	—
92	C₂×C₃×C₇	—	4	2	no	—
93	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂×C₃×C₇	2₋¹⁺⁴	1	2	yes	—
95	C₄⋊Q₈	C₂×C₃×C₇	1	2	yes	complemented
96	#126	C₂×C₃×C₇	1	4	yes	complemented
97	#125	C₂×C₃×C₇	1	2	yes	complemented
98	#124	C₂×C₃×C₇	1	1	yes	characteristic
99	#123	C₂×C₃×C₇	1	2	yes	—
100	C₂²⋊Q₈	C₂×C₃×C₇	1	4	yes	complemented
101	C₂²×C₇	—	2	2	no	—
102	C₂²×C₇	—	2	4	no	—
103	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
104	C₄×C₇	—	2	4	no	—
105	C₄×C₇	—	2	1	no	—
106	C₄×C₇	—	2	2	no	—
107	C₄×C₇	—	2	2	no	—
108	C₄×C₇	—	2	4	no	—
109	C₂³×C₃	C₂³×C₇	1	2	yes	—
110	C₃×Q₈	—	4	2	no	—
111	C₃×Q₈	—	4	1	no	—
112	C₃×D₄	—	4	1	no	—
113	C₂×C₃×C₄	—	4	2	no	—
114	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
115	C₂×C₃×C₄	C₂³×C₇	1	4	yes	—
116	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
117	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂³×C₇	1	4	yes	—
119	C₃×C₇	#563	1	1	yes	characteristic, direct product factor, Hall
120	C₂×Q₈	C₂²×C₃×C₇	1	2	yes	—
121	C₂×Q₈	C₂²×C₃×C₇	1	1	yes	characteristic
122	C₂×D₄	C₂²×C₃×C₇	1	1	yes	characteristic
123	C₂²×C₄	C₂²×C₃×C₇	1	2	yes	—
124	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
125	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
126	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
127	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
128	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
129	C₂²⋊C₄	C₂²×C₃×C₇	1	4	yes	—
130	C₂²⋊C₄	C₂²×C₃×C₇	1	4	yes	—
131	C₂²⋊C₄	C₂²×C₃×C₇	1	2	yes	—
132	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
133	C₄²	C₂²×C₃×C₇	1	2	yes	—
134	C₂×C₇	—	4	2	no	—
135	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	—
137	C₂²×C₃	—	2	2	no	—
138	C₂²×C₃	—	2	4	no	—
139	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
140	C₃×C₄	—	2	4	no	—
141	C₃×C₄	—	2	1	no	—
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	2	2	no	—
144	C₃×C₄	—	2	4	no	—
145	C₂³	C₂³×C₃×C₇	1	2	yes	—
146	Q₈	—	4	2	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	4	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	C₂³×C₃×C₇	1	2	yes	—
151	C₂×C₄	C₂³×C₃×C₇	1	4	yes	—
152	C₂×C₄	C₂³×C₃×C₇	1	2	yes	—
153	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
154	C₂×C₄	C₂³×C₃×C₇	1	4	yes	—
155	C₇	C₃×#563	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×C₃	—	4	2	no	—
157	C₂×C₃	C₇×H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₃	C₇×2₋¹⁺⁴	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	—	2	4	no	—
161	C₂²	C₂⁴×C₃×C₇	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	2	no	—
165	C₄	—	2	2	no	—
166	C₄	—	2	4	no	—
167	C₃	C₇×#563	1	1	yes	characteristic, direct product factor, Sylow
168	C₂	—	4	2	no	—
169	C₂	C₃×C₇×H(5,𝔽₂)	1	1	yes	characteristic
170	C₂	C₃×C₇×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₇
prime radical	C₂×C₃×C₇×D₄	C₂²
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×C₄⋊Q₈	C₂	semidirect
2. C₃×C₇×#126	C₂	semidirect
3. C₃×C₇×#125	C₂	semidirect
6. C₃×C₇×C₂²⋊Q₈	C₂	semidirect
7. C₇×#563	C₃	direct (N × H)
22. C₇×C₄⋊Q₈	C₂×C₃	semidirect
23. C₇×#126	C₂×C₃	semidirect
24. C₇×#125	C₂×C₃	semidirect
27. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
28. C₃×#563	C₇	direct (N × H)
53. C₃×C₄⋊Q₈	C₂×C₇	semidirect
54. C₃×#126	C₂×C₇	semidirect
55. C₃×#125	C₂×C₇	semidirect
58. C₃×C₂²⋊Q₈	C₂×C₇	semidirect
67. #563	C₃×C₇	direct (N × H)
95. C₄⋊Q₈	C₂×C₃×C₇	semidirect
96. #126	C₂×C₃×C₇	semidirect
97. #125	C₂×C₃×C₇	semidirect
100. C₂²⋊Q₈	C₂×C₃×C₇	semidirect
119. C₃×C₇	#563	direct (N × H)
155. C₇	C₃×#563	direct (N × H)
167. C₃	C₇×#563	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×C₄⋊Q₈	2
C₃×C₇×#126	2
C₃×C₇×#125	2
C₃×C₇×#124	2
C₃×C₇×#123	2
C₃×C₇×C₂²⋊Q₈	2
C₇×#563	3
C₃×#563	7

Subgroups — C3×C7×#563 #13061983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#563 #13061983