Subgroups — C₃×C₇×#432 #13056943

Order 1344.

220 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₃×C₇×C₂²⋊C₈	C₂	1	2	yes	maximal
4	C₃×C₇×#97	C₂	1	2	yes	maximal, complemented
5	C₇×#432	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₄×C₇	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₇	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₇×C₈	C₂²	1	4	yes	—
10	C₃×C₄²×C₇	C₄	1	1	yes	characteristic
11	C₃×C₄²×C₇	C₄	1	1	yes	characteristic
12	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₇	C₂×C₃	1	1	yes	characteristic
15	C₇×C₄⋊C₈	C₂×C₃	1	2	yes	complemented
16	C₇×C₂²⋊C₈	C₂×C₃	1	2	yes	—
17	C₇×#97	C₂×C₃	1	2	yes	complemented
18	C₃×#432	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₇	—	2	2	no	—
21	C₂×C₃×C₄×C₇	—	2	2	no	—
22	C₂×C₃×C₄×C₇	—	2	2	no	—
23	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₃×C₇×C₈	—	2	4	no	—
30	C₂²×C₄×C₇	C₃×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇	C₃×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
33	C₂×C₇×C₈	C₂²×C₃	1	4	yes	—
34	C₄²×C₇	C₃×C₄	1	1	yes	characteristic
35	C₄²×C₇	C₃×C₄	1	1	yes	characteristic
36	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
37	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
38	C₂×C₃×C₄²	C₂×C₇	1	1	yes	characteristic
39	C₃×C₄⋊C₈	C₂×C₇	1	2	yes	complemented
40	C₃×C₂²⋊C₈	C₂×C₇	1	2	yes	—
41	C₃×#97	C₂×C₇	1	2	yes	complemented
42	C₂²×C₃×C₇	—	2	1	no	—
43	C₂²×C₃×C₇	—	2	1	no	—
44	C₂²×C₃×C₇	M₁₆	1	2	yes	—
45	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
46	C₃×C₄×C₇	—	2	1	no	—
47	C₃×C₄×C₇	—	2	1	no	—
48	C₃×C₄×C₇	—	2	1	no	—
49	C₃×C₄×C₇	—	2	1	no	—
50	C₃×C₄×C₇	M₁₆	1	2	yes	—
51	C₃×C₄×C₇	#41	1	2	yes	—
52	#432	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
54	C₂×C₄×C₇	—	2	2	no	—
55	C₂×C₄×C₇	—	2	2	no	—
56	C₂×C₄×C₇	—	2	2	no	—
57	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
58	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
59	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
60	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
61	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
62	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
63	C₇×C₈	—	2	4	no	—
64	C₂²×C₃×C₄	C₄×C₇	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₄×C₇	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂²×C₇	1	4	yes	—
68	C₃×C₄²	C₄×C₇	1	1	yes	characteristic
69	C₃×C₄²	C₄×C₇	1	1	yes	characteristic
70	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
71	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
72	C₂×C₃×C₇	—	2	2	no	—
73	C₂×C₃×C₇	#117	1	1	yes	characteristic
74	C₂×C₃×C₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₃×C₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄²	C₂×C₃×C₇	1	1	yes	characteristic
77	C₄⋊C₈	C₂×C₃×C₇	1	2	yes	complemented
78	C₂²⋊C₈	C₂×C₃×C₇	1	2	yes	—
79	#97	C₂×C₃×C₇	1	2	yes	complemented
80	C₂²×C₇	—	2	1	no	—
81	C₂²×C₇	—	2	1	no	—
82	C₂²×C₇	C₃×M₁₆	1	2	yes	—
83	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
84	C₄×C₇	—	2	1	no	—
85	C₄×C₇	—	2	1	no	—
86	C₄×C₇	—	2	1	no	—
87	C₄×C₇	—	2	1	no	—
88	C₄×C₇	C₃×M₁₆	1	2	yes	—
89	C₄×C₇	C₃×#41	1	2	yes	—
90	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
96	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
100	C₃×C₈	—	2	4	no	—
101	C₃×C₇	#432	1	1	yes	characteristic, direct product factor, Hall
102	C₂²×C₄	C₃×C₄×C₇	1	1	yes	characteristic
103	C₂²×C₄	C₃×C₄×C₇	1	1	yes	characteristic
104	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
105	C₂×C₈	C₂²×C₃×C₇	1	4	yes	—
106	C₄²	C₃×C₄×C₇	1	1	yes	characteristic
107	C₄²	C₃×C₄×C₇	1	1	yes	characteristic
108	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
109	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
110	C₂×C₇	—	2	2	no	—
111	C₂×C₇	C₃×#117	1	1	yes	characteristic
112	C₂×C₇	C₂×C₃×M₁₆	1	1	yes	characteristic
113	C₂×C₇	C₂×C₃×M₁₆	1	1	yes	characteristic
114	C₂²×C₃	—	2	1	no	—
115	C₂²×C₃	—	2	1	no	—
116	C₂²×C₃	C₇×M₁₆	1	2	yes	—
117	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
118	C₃×C₄	—	2	1	no	—
119	C₃×C₄	—	2	1	no	—
120	C₃×C₄	—	2	1	no	—
121	C₃×C₄	—	2	1	no	—
122	C₃×C₄	C₇×M₁₆	1	2	yes	—
123	C₃×C₄	C₇×#41	1	2	yes	—
124	C₂³	C₂×C₃×C₄×C₇	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
129	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
131	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
132	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
133	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
134	C₈	—	2	4	no	—
135	C₇	C₃×#432	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×C₃	—	2	2	no	—
137	C₂×C₃	C₇×#117	1	1	yes	characteristic
138	C₂×C₃	C₂×C₇×M₁₆	1	1	yes	characteristic
139	C₂×C₃	C₂×C₇×M₁₆	1	1	yes	characteristic
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	C₃×C₇×M₁₆	1	2	yes	—
143	C₂²	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	C₃×C₇×M₁₆	1	2	yes	—
149	C₄	C₃×C₇×#41	1	2	yes	—
150	C₃	C₇×#432	1	1	yes	characteristic, direct product factor, Sylow
151	C₂	—	2	2	no	—
152	C₂	C₃×C₇×#117	1	1	yes	characteristic
153	C₂	C₂×C₃×C₇×M₁₆	1	1	yes	characteristic
154	C₂	C₂×C₃×C₇×M₁₆	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×C₄⋊C₈	C₂	semidirect
4. C₃×C₇×#97	C₂	semidirect
5. C₇×#432	C₃	direct (N × H)
15. C₇×C₄⋊C₈	C₂×C₃	semidirect
17. C₇×#97	C₂×C₃	semidirect
18. C₃×#432	C₇	direct (N × H)
39. C₃×C₄⋊C₈	C₂×C₇	semidirect
41. C₃×#97	C₂×C₇	semidirect
52. #432	C₃×C₇	direct (N × H)
77. C₄⋊C₈	C₂×C₃×C₇	semidirect
79. #97	C₂×C₃×C₇	semidirect
101. C₃×C₇	#432	direct (N × H)
135. C₇	C₃×#432	direct (N × H)
150. C₃	C₇×#432	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×C₄²×C₇	2
C₃×C₇×C₄⋊C₈	2
C₃×C₇×C₂²⋊C₈	2
C₃×C₇×#97	2
C₇×#432	3
C₃×#432	7

Subgroups — C3×C7×#432 #13056943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#432 #13056943