Subgroups — C₇×D₄×C₃⋊C₈ #13056045

Order 1344.

268 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₇×#830	C₂	1	2	yes	maximal, complemented
4	C₇×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×C₈×D₄	—	3	1	no	maximal, Hall
7	C₂×C₃×C₇×D₄	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
9	C₃×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₃×C₇×C₂²⋊C₄	C₄	1	2	yes	—
11	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₃⋊C₈	—	2	4	no	—
13	C₂×C₇×C₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₇×C₈	—	3	2	no	—
17	C₄×C₇×D₄	D₃	1	1	yes	characteristic
18	C₇×C₄⋊C₈	—	3	1	no	—
19	C₇×C₂²⋊C₈	—	3	2	no	—
20	C₄×C₇×C₈	—	3	1	no	—
21	D₄×C₃⋊C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂³×C₃×C₇	C₂×C₄	1	2	yes	—
23	C₃×C₇×D₄	C₈	1	4	yes	complemented
24	C₂×C₃×C₄×C₇	—	2	2	no	—
25	C₂×C₃×C₄×C₇	—	2	2	no	—
26	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₇×C₃⋊C₈	—	2	2	no	—
31	C₇×C₃⋊C₈	—	2	1	no	—
32	C₇×C₃⋊C₈	D₄	1	2	yes	direct product factor
33	C₂×C₇×D₄	Dic₃	1	1	yes	characteristic
34	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
35	C₂×C₇×C₈	—	3	1	no	—
36	C₂×C₇×C₈	—	3	1	no	—
37	C₂×C₇×C₈	—	3	2	no	—
38	C₂×C₇×C₈	—	6	4	no	—
39	C₇×C₄⋊C₄	Dic₃	1	1	yes	characteristic
40	C₇×C₂²⋊C₄	Dic₃	1	2	yes	—
41	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
42	C₃×C₄×D₄	C₂×C₇	1	1	yes	characteristic
43	C₂²×C₃⋊C₈	C₂×C₇	1	2	yes	complemented
44	#830	C₂×C₇	1	2	yes	complemented
45	C₄⋊(C₃⋊C₈)	C₂×C₇	1	1	yes	characteristic, complemented
46	C₄×C₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
47	C₂²×C₃×C₇	—	2	2	no	—
48	C₂²×C₃×C₇	—	2	2	no	—
49	C₂²×C₃×C₇	C₂×C₈	1	4	yes	complemented
50	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₃×C₄×C₇	—	2	2	no	—
52	C₃×C₄×C₇	—	2	1	no	—
53	C₃×C₄×C₇	C₂×C₈	1	2	yes	complemented
54	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
55	C₃×C₄×C₇	#41	1	1	yes	characteristic
56	C₈×D₄	—	3	1	no	Sylow
57	C₂³×C₇	C₂×Dic₃	1	2	yes	—
58	C₇×D₄	C₃⋊C₈	1	4	yes	direct product factor
59	C₂×C₄×C₇	—	2	2	no	—
60	C₂×C₄×C₇	—	2	2	no	—
61	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
62	C₂×C₄×C₇	C₂×Dic₃	1	2	yes	—
63	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
64	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
65	C₇×C₈	—	3	2	no	—
66	C₇×C₈	—	6	2	no	—
67	C₇×C₈	—	6	1	no	—
68	C₂×C₃×D₄	C₄×C₇	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
70	C₃×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
71	C₃×C₂²⋊C₄	C₄×C₇	1	2	yes	—
72	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
73	C₂×C₃⋊C₈	—	2	4	no	—
74	C₂×C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic
75	C₂×C₃⋊C₈	C₂²×C₇	1	2	yes	—
76	C₂×C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic
77	C₂×C₃×C₇	—	2	4	no	—
78	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
79	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₃×C₇	#131	1	1	yes	characteristic
81	C₂²×C₈	—	3	2	no	—
82	C₄×D₄	D₃×C₇	1	1	yes	characteristic
83	C₄⋊C₈	—	3	1	no	—
84	C₂²⋊C₈	—	3	2	no	—
85	C₄×C₈	—	3	1	no	—
86	C₂²×C₇	—	2	2	no	—
87	C₂²×C₇	—	2	2	no	—
88	C₂²×C₇	C₂×C₃⋊C₈	1	4	yes	complemented
89	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
90	C₄×C₇	—	2	2	no	—
91	C₄×C₇	—	2	1	no	—
92	C₄×C₇	C₂×C₃⋊C₈	1	2	yes	complemented
93	C₄×C₇	D₃×D₄	1	1	yes	characteristic
94	C₄×C₇	#237	1	1	yes	characteristic
95	C₂³×C₃	C₂×C₄×C₇	1	2	yes	—
96	C₃×D₄	C₇×C₈	1	4	yes	complemented
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
100	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
101	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
102	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
103	C₃⋊C₈	—	2	2	no	—
104	C₃⋊C₈	—	2	1	no	—
105	C₃⋊C₈	C₇×D₄	1	2	yes	direct product factor
106	C₃×C₇	C₈×D₄	1	1	yes	characteristic, complemented, Hall
107	C₂×D₄	C₇×Dic₃	1	1	yes	characteristic
108	C₂²×C₄	C₂×D₃×C₇	1	2	yes	—
109	C₂×C₈	—	3	1	no	—
110	C₂×C₈	—	3	1	no	—
111	C₂×C₈	—	3	2	no	—
112	C₂×C₈	—	6	4	no	—
113	C₄⋊C₄	C₇×Dic₃	1	1	yes	characteristic
114	C₂²⋊C₄	C₇×Dic₃	1	2	yes	—
115	C₄²	C₂×D₃×C₇	1	1	yes	characteristic
116	C₂×C₇	—	2	4	no	—
117	C₂×C₇	C₂²×C₃⋊C₈	1	1	yes	characteristic
118	C₂×C₇	D₄×Dic₃	1	1	yes	characteristic
119	C₂×C₇	#948	1	1	yes	characteristic
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	2	2	no	—
122	C₂²×C₃	C₂×C₇×C₈	1	4	yes	complemented
123	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
124	C₃×C₄	—	2	2	no	—
125	C₃×C₄	—	2	1	no	—
126	C₃×C₄	C₂×C₇×C₈	1	2	yes	complemented
127	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
128	C₃×C₄	C₇×#41	1	1	yes	characteristic
129	C₂³	C₂×C₇×Dic₃	1	2	yes	—
130	D₄	C₇×C₃⋊C₈	1	4	yes	direct product factor
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	C₂×C₇×Dic₃	1	1	yes	characteristic
134	C₂×C₄	C₂×C₇×Dic₃	1	2	yes	—
135	C₂×C₄	C₂×C₇×Dic₃	1	1	yes	characteristic
136	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
137	C₈	—	3	2	no	—
138	C₈	—	6	2	no	—
139	C₈	—	6	1	no	—
140	C₇	D₄×C₃⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×C₃	—	2	4	no	—
142	C₂×C₃	C₄×C₇×D₄	1	1	yes	characteristic
143	C₂×C₃	C₂²×C₇×C₈	1	1	yes	characteristic
144	C₂×C₃	C₇×#131	1	1	yes	characteristic
145	C₂²	—	2	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	C₂×C₇×C₃⋊C₈	1	4	yes	complemented
148	C₂²	C₂²×C₇×Dic₃	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	C₂×C₇×C₃⋊C₈	1	2	yes	complemented
152	C₄	D₃×C₇×D₄	1	1	yes	characteristic
153	C₄	C₇×#237	1	1	yes	characteristic
154	C₃	C₇×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
155	C₂	—	2	4	no	—
156	C₂	C₂²×C₇×C₃⋊C₈	1	1	yes	characteristic
157	C₂	C₇×D₄×Dic₃	1	1	yes	characteristic
158	C₂	C₇×#948	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator	C₂×C₃	C₂²×C₇×C₈
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₄×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂×C₃×C₇×D₄	C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₄×C₇×D₄	D₃
hypocenter	C₃	C₇×C₈×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₄×C₇×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×C₃⋊C₈	C₂	semidirect
3. C₇×#830	C₂	semidirect
4. C₇×C₄⋊(C₃⋊C₈)	C₂	semidirect
5. C₄×C₇×C₃⋊C₈	C₂	semidirect
21. D₄×C₃⋊C₈	C₇	direct (N × H)
23. C₃×C₇×D₄	C₈	semidirect
32. C₇×C₃⋊C₈	D₄	direct (N × H)
43. C₂²×C₃⋊C₈	C₂×C₇	semidirect
44. #830	C₂×C₇	semidirect
45. C₄⋊(C₃⋊C₈)	C₂×C₇	semidirect
46. C₄×C₃⋊C₈	C₂×C₇	semidirect
49. C₂²×C₃×C₇	C₂×C₈	semidirect
53. C₃×C₄×C₇	C₂×C₈	semidirect
58. C₇×D₄	C₃⋊C₈	direct (N × H)
88. C₂²×C₇	C₂×C₃⋊C₈	semidirect
92. C₄×C₇	C₂×C₃⋊C₈	semidirect
96. C₃×D₄	C₇×C₈	semidirect
105. C₃⋊C₈	C₇×D₄	direct (N × H)
106. C₃×C₇	C₈×D₄	semidirect
122. C₂²×C₃	C₂×C₇×C₈	semidirect
126. C₃×C₄	C₂×C₇×C₈	semidirect
130. D₄	C₇×C₃⋊C₈	direct (N × H)
140. C₇	D₄×C₃⋊C₈	direct (N × H)
147. C₂²	C₂×C₇×C₃⋊C₈	semidirect
151. C₄	C₂×C₇×C₃⋊C₈	semidirect
154. C₃	C₇×C₈×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₇×D₄	2
C₂²×C₇×C₃⋊C₈	2
C₇×#830	2
C₇×C₄⋊(C₃⋊C₈)	2
C₄×C₇×C₃⋊C₈	2
C₇×C₈×D₄	3
D₄×C₃⋊C₈	7

Subgroups — C7×D4×C3⋊C8 #13056045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×D₄×C₃⋊C₈ #13056045