Subgroups — C₂×C₇×C₄⋊(C₃⋊C₈) #13055958

Order 1344.

276 conjugacy classes of subgroups.

The subgroup lattice has 140 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	C₂²×C₇×C₃⋊C₈	C₂	1	2	yes	maximal
3	C₇×C₄⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×C₄⋊C₈	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₄×C₇	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₄×C₇	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₃×C₄²×C₇	C₄	1	2	yes	—
9	C₃×C₄²×C₇	C₂²	1	2	yes	—
10	C₂×C₇×C₃⋊C₈	—	2	4	no	—
11	C₂×C₇×C₃⋊C₈	C₂²	1	4	yes	—
12	C₂²×C₇×C₈	—	3	2	no	—
13	C₂×C₄²×C₇	D₃	1	1	yes	characteristic
14	C₇×C₄⋊C₈	—	3	4	no	—
15	C₂×C₄⋊(C₃⋊C₈)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₄×C₇	—	2	2	no	—
18	C₂×C₃×C₄×C₇	C₈	1	4	yes	complemented
19	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
23	C₂×C₃×C₄×C₇	Q₈	1	2	yes	—
24	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₇×C₃⋊C₈	—	2	4	no	—
26	C₂²×C₄×C₇	Dic₃	1	1	yes	characteristic
27	C₂²×C₄×C₇	Dic₃	1	1	yes	characteristic
28	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
29	C₂×C₇×C₈	—	3	4	no	—
30	C₂×C₇×C₈	—	6	4	no	—
31	C₄²×C₇	Dic₃	1	2	yes	—
32	C₄²×C₇	C₂×D₃	1	2	yes	—
33	C₂×C₃×C₄²	C₂×C₇	1	1	yes	characteristic
34	C₂²×C₃⋊C₈	C₂×C₇	1	2	yes	—
35	C₄⋊(C₃⋊C₈)	C₂×C₇	1	4	yes	direct product factor
36	C₂²×C₃×C₇	C₄⋊C₄	1	2	yes	—
37	C₂²×C₃×C₇	C₂×C₈	1	2	yes	—
38	C₂²×C₃×C₇	M₁₆	1	2	yes	—
39	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
40	C₃×C₄×C₇	—	2	2	no	—
41	C₃×C₄×C₇	C₄⋊C₄	1	2	yes	—
42	C₃×C₄×C₇	C₂×C₈	1	4	yes	complemented
43	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₃×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄⋊C₈	—	3	1	no	Sylow
46	C₂³×C₇	C₂×Dic₃	1	1	yes	characteristic
47	C₂×C₄×C₇	—	2	2	no	—
48	C₂×C₄×C₇	C₃⋊C₈	1	4	yes	complemented
49	C₂×C₄×C₇	Dic₆	1	2	yes	—
50	C₂×C₄×C₇	D₁₂	1	2	yes	—
51	C₂×C₄×C₇	C₂×Dic₃	1	2	yes	—
52	C₂×C₄×C₇	C₂×Dic₃	1	2	yes	—
53	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
54	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
55	C₇×C₈	—	6	4	no	—
56	C₂²×C₃×C₄	C₄×C₇	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₄×C₇	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
59	C₃×C₄²	C₄×C₇	1	2	yes	—
60	C₃×C₄²	C₂²×C₇	1	2	yes	—
61	C₂×C₃⋊C₈	—	2	4	no	—
62	C₂×C₃⋊C₈	C₂²×C₇	1	4	yes	—
63	C₂×C₃×C₇	C₄⋊C₈	1	4	yes	complemented
64	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₃×C₇	C₂×M₁₆	1	1	yes	characteristic
67	C₂²×C₈	—	3	2	no	—
68	C₂×C₄²	D₃×C₇	1	1	yes	characteristic
69	C₄⋊C₈	—	3	4	no	—
70	C₂²×C₇	C₂×C₃⋊C₈	1	2	yes	—
71	C₂²×C₇	#208	1	2	yes	—
72	C₂²×C₇	C₄⋊Dic₃	1	2	yes	—
73	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
74	C₄×C₇	—	2	2	no	—
75	C₄×C₇	C₂×C₃⋊C₈	1	4	yes	complemented
76	C₄×C₇	C₄⋊Dic₃	1	2	yes	—
77	C₄×C₇	C₂×Dic₆	1	1	yes	characteristic
78	C₄×C₇	C₂×D₁₂	1	1	yes	characteristic
79	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	C₇×C₈	1	4	yes	complemented
82	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
83	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
85	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
86	C₂×C₃×C₄	C₇×Q₈	1	2	yes	—
87	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
88	C₃⋊C₈	—	2	4	no	—
89	C₃×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
90	C₂²×C₄	C₇×Dic₃	1	1	yes	characteristic
91	C₂²×C₄	C₇×Dic₃	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
93	C₂×C₈	—	3	4	no	—
94	C₂×C₈	—	6	4	no	—
95	C₄²	C₇×Dic₃	1	2	yes	—
96	C₄²	C₂×D₃×C₇	1	2	yes	—
97	C₂×C₇	C₄⋊(C₃⋊C₈)	1	4	yes	direct product factor
98	C₂×C₇	C₂²×C₃⋊C₈	1	1	yes	characteristic
99	C₂×C₇	C₂×#208	1	1	yes	characteristic
100	C₂×C₇	C₂×C₄⋊Dic₃	1	1	yes	characteristic
101	C₂²×C₃	C₇×C₄⋊C₄	1	2	yes	—
102	C₂²×C₃	C₂×C₇×C₈	1	2	yes	—
103	C₂²×C₃	C₇×M₁₆	1	2	yes	—
104	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
105	C₃×C₄	—	2	2	no	—
106	C₃×C₄	C₇×C₄⋊C₄	1	2	yes	—
107	C₃×C₄	C₂×C₇×C₈	1	4	yes	complemented
108	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
109	C₃×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
110	C₂³	C₂×C₇×Dic₃	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	C₇×C₃⋊C₈	1	4	yes	complemented
113	C₂×C₄	C₇×Dic₆	1	2	yes	—
114	C₂×C₄	C₇×D₁₂	1	2	yes	—
115	C₂×C₄	C₂×C₇×Dic₃	1	2	yes	—
116	C₂×C₄	C₂×C₇×Dic₃	1	2	yes	—
117	C₂×C₄	C₂×C₇×Dic₃	1	1	yes	characteristic
118	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
119	C₈	—	6	4	no	—
120	C₇	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×C₃	C₇×C₄⋊C₈	1	4	yes	complemented
122	C₂×C₃	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₃	C₂²×C₇×C₈	1	1	yes	characteristic
124	C₂×C₃	C₂×C₇×M₁₆	1	1	yes	characteristic
125	C₂²	C₂×C₇×C₃⋊C₈	1	2	yes	—
126	C₂²	C₇×#208	1	2	yes	—
127	C₂²	C₇×C₄⋊Dic₃	1	2	yes	—
128	C₂²	C₂²×C₇×Dic₃	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	C₂×C₇×C₃⋊C₈	1	4	yes	complemented
131	C₄	C₇×C₄⋊Dic₃	1	2	yes	—
132	C₄	C₂×C₇×Dic₆	1	1	yes	characteristic
133	C₄	C₂×C₇×D₁₂	1	1	yes	characteristic
134	C₃	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
135	C₂	C₇×C₄⋊(C₃⋊C₈)	1	4	yes	direct product factor
136	C₂	C₂²×C₇×C₃⋊C₈	1	1	yes	characteristic
137	C₂	C₂×C₇×#208	1	1	yes	characteristic
138	C₂	C₂×C₇×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₃
commutator	C₂×C₃	C₂²×C₇×C₈
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₂×C₃×C₄²×C₇	C₂
socle, prime radical	C₂³×C₃×C₇	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₂×C₄²×C₇	D₃
hypocenter	C₃	C₂×C₇×C₄⋊C₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂×C₄²×C₇

Complements

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

N	H ≅ G/N	type
3. C₇×C₄⋊(C₃⋊C₈)	C₂	direct (N × H)
15. C₂×C₄⋊(C₃⋊C₈)	C₇	direct (N × H)
18. C₂×C₃×C₄×C₇	C₈	semidirect
35. C₄⋊(C₃⋊C₈)	C₂×C₇	direct (N × H)
42. C₃×C₄×C₇	C₂×C₈	semidirect
48. C₂×C₄×C₇	C₃⋊C₈	semidirect
63. C₂×C₃×C₇	C₄⋊C₈	semidirect
75. C₄×C₇	C₂×C₃⋊C₈	semidirect
81. C₂×C₃×C₄	C₇×C₈	semidirect
89. C₃×C₇	C₂×C₄⋊C₈	semidirect
97. C₂×C₇	C₄⋊(C₃⋊C₈)	direct (N × H)
107. C₃×C₄	C₂×C₇×C₈	semidirect
112. C₂×C₄	C₇×C₃⋊C₈	semidirect
120. C₇	C₂×C₄⋊(C₃⋊C₈)	direct (N × H)
121. C₂×C₃	C₇×C₄⋊C₈	semidirect
130. C₄	C₂×C₇×C₃⋊C₈	semidirect
134. C₃	C₂×C₇×C₄⋊C₈	semidirect
135. C₂	C₇×C₄⋊(C₃⋊C₈)	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
C₂²×C₇×C₃⋊C₈	2
C₇×C₄⋊(C₃⋊C₈)	2
C₂×C₇×C₄⋊C₈	3
C₂×C₄⋊(C₃⋊C₈)	7

Subgroups — C2×C7×C4⋊(C3⋊C8) #13055958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₄⋊(C₃⋊C₈) #13055958