Subgroups — C₂×C₇×#1368 #13189214

Order 1792.

254 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#334	C₂	1	2	yes	maximal
3	C₇×#1368	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₇×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₇×C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₇×C₄⋊C₈	—	2	2	no	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×#346	C₄	1	2	yes	—
9	C₇×#346	C₂²	1	2	yes	—
10	C₇×#334	—	2	2	no	—
11	C₇×#334	C₂²	1	4	yes	—
12	C₂×#1368	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₇×C₈	—	4	2	no	—
14	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
17	C₂×C₇×C₁₆	—	2	2	no	—
18	C₂×C₇×C₁₆	C₈	1	4	yes	complemented
19	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
21	C₇×C₄⋊C₈	—	2	4	no	—
22	C₇×C₄⋊C₈	—	4	2	no	—
23	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₇×C₈	D₄	1	1	yes	characteristic
25	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
27	C₂×#346	C₂×C₇	1	1	yes	characteristic
28	C₂×#334	C₂×C₇	1	2	yes	—
29	#1368	C₂×C₇	1	4	yes	direct product factor
30	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
32	C₂×C₇×C₈	—	2	2	no	—
33	C₂×C₇×C₈	—	4	4	no	—
34	C₂×C₇×C₈	—	8	4	no	—
35	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
38	C₂×C₇×C₈	M₁₆	1	2	yes	—
39	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₇	SD₁₆	1	2	yes	—
43	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
44	C₇×C₁₆	—	2	2	no	—
45	C₇×C₁₆	C₂×C₈	1	4	yes	complemented
46	C₂²×C₁₆	C₄×C₇	1	1	yes	characteristic
47	C₂²×C₁₆	C₄×C₇	1	1	yes	characteristic
48	C₂×C₄⋊C₈	—	2	2	no	—
49	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
50	#346	C₄×C₇	1	2	yes	—
51	#346	C₂²×C₇	1	2	yes	—
52	#334	—	2	2	no	—
53	#334	C₂²×C₇	1	4	yes	—
54	C₂³×C₇	#106	1	1	yes	characteristic
55	C₂×C₄×C₇	—	2	4	no	—
56	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₇	#106	1	2	yes	—
58	C₂×C₄×C₇	#106	1	1	yes	characteristic
59	C₂×C₄×C₇	#108	1	2	yes	—
60	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
62	C₇×C₈	—	2	1	no	—
63	C₇×C₈	—	2	1	no	—
64	C₇×C₈	—	8	4	no	—
65	C₇×C₈	C₄⋊C₈	1	2	yes	—
66	C₇×C₈	C₂²×C₈	1	1	yes	characteristic
67	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
68	C₂²×C₈	—	4	2	no	—
69	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
70	C₂²×C₈	C₇×Q₈	1	1	yes	characteristic
71	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	C₇×C₈	1	4	yes	complemented
74	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
75	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
76	C₄⋊C₈	—	2	4	no	—
77	C₄⋊C₈	—	4	2	no	—
78	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
79	C₄×C₈	C₇×D₄	1	1	yes	characteristic
80	C₄×C₈	C₇×Q₈	1	1	yes	characteristic
81	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
82	C₂²×C₇	#334	1	2	yes	—
83	C₂²×C₇	#365	1	4	yes	—
84	C₂²×C₇	C₂×#106	1	1	yes	characteristic
85	C₄×C₇	—	2	2	no	—
86	C₄×C₇	—	2	2	no	—
87	C₄×C₇	#334	1	2	yes	—
88	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₄×C₇	C₂×#108	1	1	yes	characteristic
90	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄	C₇×SD₁₆	1	2	yes	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	4	4	no	—
94	C₂×C₈	—	8	4	no	—
95	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
98	C₂×C₈	C₇×M₁₆	1	2	yes	—
99	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
100	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
101	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
102	C₄²	C₇×SD₁₆	1	2	yes	—
103	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
104	C₁₆	—	2	2	no	—
105	C₁₆	C₂×C₇×C₈	1	4	yes	complemented
106	C₂×C₇	#1368	1	4	yes	direct product factor
107	C₂×C₇	C₂×#334	1	1	yes	characteristic
108	C₂×C₇	C₂×#365	1	2	yes	—
109	C₂³	C₇×#106	1	1	yes	characteristic
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	C₇×C₄⋊C₈	1	2	yes	—
112	C₂×C₄	C₇×#106	1	2	yes	—
113	C₂×C₄	C₇×#106	1	1	yes	characteristic
114	C₂×C₄	C₇×#108	1	2	yes	—
115	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
117	C₈	—	2	1	no	—
118	C₈	—	2	1	no	—
119	C₈	—	8	4	no	—
120	C₈	C₇×C₄⋊C₈	1	2	yes	—
121	C₈	C₂²×C₇×C₈	1	1	yes	characteristic
122	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
123	C₇	C₂×#1368	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²	C₇×#334	1	2	yes	—
125	C₂²	C₇×#365	1	4	yes	—
126	C₂²	C₂×C₇×#106	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	C₇×#334	1	2	yes	—
130	C₄	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
131	C₄	C₂×C₇×#108	1	1	yes	characteristic
132	C₂	C₇×#1368	1	4	yes	direct product factor
133	C₂	C₂×C₇×#334	1	1	yes	characteristic
134	C₂	C₂×C₇×#365	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#106
commutator	C₈	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1368	C₂	direct (N × H)
12. C₂×#1368	C₇	direct (N × H)
18. C₂×C₇×C₁₆	C₈	semidirect
29. #1368	C₂×C₇	direct (N × H)
45. C₇×C₁₆	C₂×C₈	semidirect
73. C₂×C₁₆	C₇×C₈	semidirect
105. C₁₆	C₂×C₇×C₈	semidirect
106. C₂×C₇	#1368	direct (N × H)
123. C₇	C₂×#1368	direct (N × H)
132. C₂	C₇×#1368	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×C₇×#346	2
C₂×C₇×#334	2
C₇×#1368	2
C₂×#1368	7

Subgroups — C2×C7×#1368 #13189214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1368 #13189214