Subgroups — C₂×C₇×#819 #13056141

Order 1344.

212 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#208	C₂	1	2	yes	maximal
3	C₇×#819	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×#108	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₇×C₈	C₄	1	2	yes	—
7	C₂×C₃×C₇×C₈	C₄	1	2	yes	—
8	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
9	C₇×#208	—	2	2	no	—
10	C₇×#208	C₂²	1	4	yes	—
11	C₂×C₇×C₃⋊C₈	—	2	2	no	—
12	C₂×C₇×M₁₆	—	3	2	no	—
13	C₂²×C₇×C₈	D₃	1	1	yes	characteristic
14	C₇×#108	—	3	4	no	—
15	C₂×#819	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂³×C₃×C₇	Q₈	1	1	yes	characteristic
17	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×C₃×C₄×C₇	Q₈	1	1	yes	characteristic
22	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₃×C₇×C₈	C₂×C₄	1	4	yes	—
24	C₇×C₃⋊C₈	—	2	4	no	—
25	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
26	C₇×M₁₆	—	3	4	no	—
27	C₇×M₁₆	—	6	2	no	—
28	C₂×C₇×C₈	—	6	2	no	—
29	C₂×C₇×C₈	Dic₃	1	2	yes	—
30	C₂×C₇×C₈	Dic₃	1	2	yes	—
31	C₂×C₇×C₈	C₂×D₃	1	2	yes	—
32	C₂²×C₃×C₈	C₂×C₇	1	1	yes	characteristic
33	C₂×#208	C₂×C₇	1	2	yes	—
34	#819	C₂×C₇	1	4	yes	direct product factor
35	C₂²×C₃×C₇	—	2	2	no	—
36	C₂²×C₃×C₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₃×C₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₃×C₇	C₂×Q₈	1	1	yes	characteristic
39	C₃×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
40	C₃×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
41	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂×#108	—	3	1	no	Sylow
44	C₂³×C₇	Dic₆	1	1	yes	characteristic
45	C₂×C₄×C₇	Dic₆	1	1	yes	characteristic
46	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
47	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
48	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
49	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
50	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
51	C₇×C₈	—	6	4	no	—
52	C₇×C₈	C₂×Dic₃	1	4	yes	—
53	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
54	C₂×C₃×C₈	C₄×C₇	1	2	yes	—
55	C₂×C₃×C₈	C₄×C₇	1	2	yes	—
56	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
57	#208	—	2	2	no	—
58	#208	C₂²×C₇	1	4	yes	—
59	C₂×C₃⋊C₈	—	2	2	no	—
60	C₂×C₃×C₇	—	2	1	no	—
61	C₂×C₃×C₇	—	2	1	no	—
62	C₂×C₃×C₇	#108	1	2	yes	complemented
63	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×M₁₆	—	3	2	no	—
65	C₂²×C₈	D₃×C₇	1	1	yes	characteristic
66	#108	—	3	4	no	—
67	C₂²×C₇	—	2	2	no	—
68	C₂²×C₇	C₄⋊Dic₃	1	1	yes	characteristic
69	C₂²×C₇	C₄⋊Dic₃	1	1	yes	characteristic
70	C₂²×C₇	C₂×Dic₆	1	1	yes	characteristic
71	C₄×C₇	C₄⋊Dic₃	1	1	yes	characteristic
72	C₄×C₇	C₄⋊Dic₃	1	1	yes	characteristic
73	C₄×C₇	C₂×D₁₂	1	1	yes	characteristic
74	C₄×C₇	C₂²×Dic₃	1	1	yes	characteristic
75	C₂³×C₃	C₇×Q₈	1	1	yes	characteristic
76	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
78	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₇×Q₈	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
82	C₃×C₈	C₂×C₄×C₇	1	4	yes	—
83	C₃⋊C₈	—	2	4	no	—
84	C₃×C₇	C₂×#108	1	1	yes	characteristic, complemented, Hall
85	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
86	M₁₆	—	3	4	no	—
87	M₁₆	—	6	2	no	—
88	C₂×C₈	—	6	2	no	—
89	C₂×C₈	C₇×Dic₃	1	2	yes	—
90	C₂×C₈	C₇×Dic₃	1	2	yes	—
91	C₂×C₈	C₂×D₃×C₇	1	2	yes	—
92	C₂×C₇	—	2	1	no	—
93	C₂×C₇	—	2	1	no	—
94	C₂×C₇	#819	1	2	yes	direct product factor
95	C₂×C₇	C₂×C₄⋊Dic₃	1	1	yes	characteristic
96	C₂²×C₃	—	2	2	no	—
97	C₂²×C₃	C₇×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₃	C₇×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₃	C₂×C₇×Q₈	1	1	yes	characteristic
100	C₃×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
101	C₃×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
102	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
103	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
104	C₂³	C₇×Dic₆	1	1	yes	characteristic
105	C₂×C₄	C₇×Dic₆	1	1	yes	characteristic
106	C₂×C₄	C₇×D₁₂	1	1	yes	characteristic
107	C₂×C₄	C₇×D₁₂	1	1	yes	characteristic
108	C₂×C₄	C₂×C₇×Dic₃	1	1	yes	characteristic
109	C₂×C₄	C₂×C₇×Dic₃	1	1	yes	characteristic
110	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
111	C₈	—	6	4	no	—
112	C₈	C₂×C₇×Dic₃	1	4	yes	—
113	C₇	C₂×#819	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×C₃	—	2	1	no	—
115	C₂×C₃	—	2	1	no	—
116	C₂×C₃	C₇×#108	1	2	yes	complemented
117	C₂×C₃	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	C₇×C₄⋊Dic₃	1	1	yes	characteristic
120	C₂²	C₇×C₄⋊Dic₃	1	1	yes	characteristic
121	C₂²	C₂×C₇×Dic₆	1	1	yes	characteristic
122	C₄	C₇×C₄⋊Dic₃	1	1	yes	characteristic
123	C₄	C₇×C₄⋊Dic₃	1	1	yes	characteristic
124	C₄	C₂×C₇×D₁₂	1	1	yes	characteristic
125	C₄	C₂²×C₇×Dic₃	1	1	yes	characteristic
126	C₃	C₂×C₇×#108	1	1	yes	characteristic, complemented, Sylow
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	C₇×#819	1	2	yes	direct product factor
130	C₂	C₂×C₇×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	D₁₂
commutator	C₃×C₄	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₂²×C₃×C₇×C₈	C₂
socle	C₂²×C₃×C₇	C₄⋊C₄
prime radical	C₂³×C₃×C₇	Q₈
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₂²×C₇×C₈	D₃
hypocenter	C₃	C₂×C₇×#108
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈

Complements

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

N	H ≅ G/N	type
3. C₇×#819	C₂	direct (N × H)
15. C₂×#819	C₇	direct (N × H)
34. #819	C₂×C₇	direct (N × H)
62. C₂×C₃×C₇	#108	semidirect
84. C₃×C₇	C₂×#108	semidirect
94. C₂×C₇	#819	direct (N × H)
113. C₇	C₂×#819	direct (N × H)
116. C₂×C₃	C₇×#108	semidirect
126. C₃	C₂×C₇×#108	semidirect
129. C₂	C₇×#819	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₇×C₈	2
C₂×C₇×#208	2
C₇×#819	2
C₂×C₇×#108	3
C₂×#819	7

Subgroups — C2×C7×#819 #13056141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#819 #13056141