Subgroups — C₃×C₇×#413 #13056924

Order 1344.

300 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×C₇×#101	C₂	1	2	yes	maximal, complemented
4	C₃×C₇×#100	C₂	1	2	yes	maximal, complemented
5	C₇×#413	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₇×#41	—	2	2	no	—
7	C₃×C₇×#41	—	2	2	no	—
8	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₇×D₄	C₄	1	2	yes	—
10	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₄	1	2	yes	—
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₃×C₇×M₁₆	—	2	2	no	—
14	C₃×C₇×M₁₆	C₂²	1	4	yes	—
15	C₂×C₃×C₇×C₈	—	2	2	no	—
16	C₂×C₇×#41	C₂×C₃	1	1	yes	characteristic
17	C₂×C₇×M₁₆	C₂×C₃	1	2	yes	complemented
18	C₇×#101	C₂×C₃	1	2	yes	complemented
19	C₇×#100	C₂×C₃	1	2	yes	complemented
20	C₃×#413	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂³×C₃×C₇	C₂×C₄	1	2	yes	—
22	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
23	C₃×C₇×Q₈	—	2	2	no	—
24	C₃×C₇×D₄	—	2	4	no	—
25	C₃×C₇×D₄	—	2	2	no	—
26	C₂×C₃×C₄×C₇	—	2	2	no	—
27	C₂×C₃×C₄×C₇	—	2	2	no	—
28	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
32	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₃×C₇×C₈	—	2	4	no	—
34	C₇×#41	—	2	2	no	—
35	C₇×#41	—	2	2	no	—
36	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
37	C₂×C₇×D₄	C₃×C₄	1	2	yes	—
38	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₃×C₄	1	2	yes	—
40	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
41	C₇×M₁₆	—	2	2	no	—
42	C₇×M₁₆	C₂²×C₃	1	4	yes	—
43	C₂×C₇×C₈	—	2	2	no	—
44	C₂×C₃×#41	C₂×C₇	1	1	yes	characteristic
45	C₂×C₃×M₁₆	C₂×C₇	1	2	yes	complemented
46	C₃×#101	C₂×C₇	1	2	yes	complemented
47	C₃×#100	C₂×C₇	1	2	yes	complemented
48	C₂²×C₃×C₇	—	2	2	no	—
49	C₂²×C₃×C₇	—	4	1	no	—
50	C₂²×C₃×C₇	—	4	2	no	—
51	C₂²×C₃×C₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₃×C₄×C₇	—	2	2	no	—
54	C₃×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₃×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
57	#413	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₇	C₂×C₃×C₄	1	2	yes	—
59	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
60	C₇×Q₈	—	2	2	no	—
61	C₇×D₄	—	2	4	no	—
62	C₇×D₄	—	2	2	no	—
63	C₂×C₄×C₇	—	2	2	no	—
64	C₂×C₄×C₇	—	2	2	no	—
65	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
66	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
67	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
68	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
69	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
70	C₇×C₈	—	2	4	no	—
71	C₃×#41	—	2	2	no	—
72	C₃×#41	—	2	2	no	—
73	C₂×C₃×Q₈	C₂²×C₇	1	1	yes	characteristic
74	C₂×C₃×D₄	C₄×C₇	1	2	yes	—
75	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₄×C₇	1	2	yes	—
77	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
78	C₃×M₁₆	—	2	2	no	—
79	C₃×M₁₆	C₂²×C₇	1	4	yes	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₇	—	2	1	no	—
82	C₂×C₃×C₇	—	2	2	no	—
83	C₂×C₃×C₇	—	4	2	no	—
84	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×#41	C₂×C₃×C₇	1	1	yes	characteristic
86	C₂×M₁₆	C₂×C₃×C₇	1	2	yes	complemented
87	#101	C₂×C₃×C₇	1	2	yes	complemented
88	#100	C₂×C₃×C₇	1	2	yes	complemented
89	C₂²×C₇	—	2	2	no	—
90	C₂²×C₇	—	4	1	no	—
91	C₂²×C₇	—	4	2	no	—
92	C₂²×C₇	C₃×C₂²⋊C₄	1	2	yes	—
93	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
94	C₄×C₇	—	2	2	no	—
95	C₄×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
98	C₂³×C₃	C₂×C₄×C₇	1	2	yes	—
99	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
100	C₃×Q₈	—	2	2	no	—
101	C₃×D₄	—	2	4	no	—
102	C₃×D₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
106	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
107	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
108	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
109	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
110	C₃×C₈	—	2	4	no	—
111	C₃×C₇	#413	1	1	yes	characteristic, direct product factor, Hall
112	#41	—	2	2	no	—
113	#41	—	2	2	no	—
114	C₂×Q₈	C₂²×C₃×C₇	1	1	yes	characteristic
115	C₂×D₄	C₃×C₄×C₇	1	2	yes	—
116	C₂×D₄	C₂²×C₃×C₇	1	1	yes	characteristic
117	C₂²×C₄	C₃×C₄×C₇	1	2	yes	—
118	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
119	M₁₆	—	2	2	no	—
120	M₁₆	C₂²×C₃×C₇	1	4	yes	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₇	—	2	1	no	—
123	C₂×C₇	—	2	2	no	—
124	C₂×C₇	—	4	2	no	—
125	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₃	—	2	2	no	—
127	C₂²×C₃	—	4	1	no	—
128	C₂²×C₃	—	4	2	no	—
129	C₂²×C₃	C₇×C₂²⋊C₄	1	2	yes	—
130	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
131	C₃×C₄	—	2	2	no	—
132	C₃×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
133	C₃×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
134	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
135	C₂³	C₂×C₃×C₄×C₇	1	2	yes	—
136	C₂³	C₂×C₃×C₄×C₇	1	1	yes	characteristic
137	Q₈	—	2	2	no	—
138	D₄	—	2	4	no	—
139	D₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
143	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
144	C₂×C₄	C₃×C₇×D₄	1	2	yes	—
145	C₂×C₄	C₃×C₇×D₄	1	2	yes	—
146	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
147	C₈	—	2	4	no	—
148	C₇	C₃×#413	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×C₃	—	2	1	no	—
150	C₂×C₃	—	2	2	no	—
151	C₂×C₃	—	4	2	no	—
152	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²	—	2	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	C₃×C₇×C₂²⋊C₄	1	2	yes	—
157	C₂²	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
160	C₄	C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
161	C₄	C₂×C₃×C₇×D₄	1	2	yes	—
162	C₃	C₇×#413	1	1	yes	characteristic, direct product factor, Sylow
163	C₂	—	2	1	no	—
164	C₂	—	2	2	no	—
165	C₂	—	4	2	no	—
166	C₂	C₂×C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle	C₂×C₃×C₇	C₂×C₂²⋊C₄
prime radical	C₂×C₃×C₇×#41	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₇ ◁ C₂²×C₃×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇×M₁₆	C₂	semidirect
3. C₃×C₇×#101	C₂	semidirect
4. C₃×C₇×#100	C₂	semidirect
5. C₇×#413	C₃	direct (N × H)
17. C₂×C₇×M₁₆	C₂×C₃	semidirect
18. C₇×#101	C₂×C₃	semidirect
19. C₇×#100	C₂×C₃	semidirect
20. C₃×#413	C₇	direct (N × H)
45. C₂×C₃×M₁₆	C₂×C₇	semidirect
46. C₃×#101	C₂×C₇	semidirect
47. C₃×#100	C₂×C₇	semidirect
57. #413	C₃×C₇	direct (N × H)
86. C₂×M₁₆	C₂×C₃×C₇	semidirect
87. #101	C₂×C₃×C₇	semidirect
88. #100	C₂×C₃×C₇	semidirect
111. C₃×C₇	#413	direct (N × H)
148. C₇	C₃×#413	direct (N × H)
162. C₃	C₇×#413	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×#41	2
C₂×C₃×C₇×M₁₆	2
C₃×C₇×#101	2
C₃×C₇×#100	2
C₇×#413	3
C₃×#413	7

Subgroups — C3×C7×#413 #13056924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#413 #13056924