Subgroups — C₂×C₇×#1426 #13190853

Order 1792.

190 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#368	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#367	C₂	1	1	yes	maximal, characteristic
4	C₇×#1426	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₇×#108	—	2	1	no	—
7	C₂×C₇×#107	—	2	1	no	—
8	C₇×M₆₄	C₄	1	2	yes	complemented
9	C₇×M₆₄	C₂²	1	2	yes	—
10	C₂×C₇×C₃₂	C₄	1	2	yes	complemented
11	C₇×#368	—	2	1	no	—
12	C₇×#368	C₂²	1	2	yes	—
13	C₇×#367	—	2	1	no	—
14	C₇×#367	C₂²	1	2	yes	—
15	C₂×#1426	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₇×M₁₆	—	4	1	no	—
17	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	—	4	1	no	—
19	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₇×C₁₆	Q₈	1	1	yes	characteristic
23	C₂×C₇×C₁₆	Q₈	1	1	yes	characteristic
24	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
25	C₇×#108	—	2	2	no	—
26	C₇×#108	—	4	1	no	—
27	C₇×#107	—	2	2	no	—
28	C₇×#107	—	4	1	no	—
29	C₇×C₃₂	C₂×C₄	1	4	yes	complemented
30	C₂×M₆₄	C₂×C₇	1	1	yes	characteristic
31	C₂×#368	C₂×C₇	1	1	yes	characteristic
32	C₂×#367	C₂×C₇	1	1	yes	characteristic
33	#1426	C₂×C₇	1	4	yes	direct product factor
34	C₂²×C₄×C₇	—	8	1	no	—
35	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
36	C₇×M₁₆	—	4	2	no	—
37	C₇×M₁₆	—	8	1	no	—
38	C₂×C₇×C₈	—	8	1	no	—
39	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₇×C₈	D₈	1	1	yes	characteristic
42	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
43	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
44	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
45	C₇×C₄⋊C₄	—	4	2	no	—
46	C₇×C₄⋊C₄	—	8	1	no	—
47	C₇×C₁₆	C₄⋊C₄	1	1	yes	characteristic
48	C₇×C₁₆	C₄⋊C₄	1	1	yes	characteristic
49	C₇×C₁₆	C₂²×C₄	1	1	yes	characteristic
50	C₇×C₁₆	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
52	C₂×#108	—	2	1	no	—
53	C₂×#107	—	2	1	no	—
54	M₆₄	C₄×C₇	1	2	yes	complemented
55	M₆₄	C₂²×C₇	1	2	yes	—
56	C₂×C₃₂	C₄×C₇	1	2	yes	complemented
57	#368	—	2	1	no	—
58	#368	C₂²×C₇	1	2	yes	—
59	#367	—	2	1	no	—
60	#367	C₂²×C₇	1	2	yes	—
61	C₂³×C₇	SD₃₂	1	1	yes	characteristic
62	C₂×C₄×C₇	—	8	2	no	—
63	C₂×C₄×C₇	—	16	2	no	—
64	C₂×C₄×C₇	#107	1	1	yes	characteristic
65	C₂×C₄×C₇	#107	1	1	yes	characteristic
66	C₂×C₄×C₇	SD₃₂	1	1	yes	characteristic
67	C₂×C₄×C₇	SD₃₂	1	1	yes	characteristic
68	C₂×C₄×C₇	SD₃₂	1	1	yes	characteristic
69	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
70	C₇×C₈	—	8	2	no	—
71	C₇×C₈	#107	1	1	yes	characteristic
72	C₇×C₈	#107	1	1	yes	characteristic
73	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×M₁₆	—	4	1	no	—
76	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	4	1	no	—
78	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
79	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
80	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
81	C₂×C₁₆	C₇×Q₈	1	1	yes	characteristic
82	C₂×C₁₆	C₇×Q₈	1	1	yes	characteristic
83	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
84	#108	—	2	2	no	—
85	#108	—	4	1	no	—
86	#107	—	2	2	no	—
87	#107	—	4	1	no	—
88	C₃₂	C₂×C₄×C₇	1	4	yes	complemented
89	C₂²×C₇	—	2	2	no	—
90	C₂²×C₇	#367	1	1	yes	characteristic
91	C₂²×C₇	#367	1	1	yes	characteristic
92	C₂²×C₇	C₂×SD₃₂	1	1	yes	characteristic
93	C₄×C₇	—	16	2	no	—
94	C₄×C₇	#367	1	1	yes	characteristic
95	C₄×C₇	#367	1	1	yes	characteristic
96	C₄×C₇	C₂×#107	1	1	yes	characteristic
97	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
98	C₂²×C₄	—	8	1	no	—
99	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
100	M₁₆	—	4	2	no	—
101	M₁₆	—	8	1	no	—
102	C₂×C₈	—	8	1	no	—
103	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₈	C₇×D₈	1	1	yes	characteristic
106	C₂×C₈	C₇×Q₁₆	1	1	yes	characteristic
107	C₂×C₈	C₇×Q₁₆	1	1	yes	characteristic
108	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	8	1	no	—
111	C₁₆	C₇×C₄⋊C₄	1	1	yes	characteristic
112	C₁₆	C₇×C₄⋊C₄	1	1	yes	characteristic
113	C₁₆	C₂²×C₄×C₇	1	1	yes	characteristic
114	C₁₆	C₂×C₇×Q₈	1	1	yes	characteristic
115	C₂×C₇	—	2	1	no	—
116	C₂×C₇	—	2	1	no	—
117	C₂×C₇	#1426	1	2	yes	direct product factor
118	C₂×C₇	C₂×#367	1	1	yes	characteristic
119	C₂³	C₇×SD₃₂	1	1	yes	characteristic
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	—	16	2	no	—
122	C₂×C₄	C₇×#107	1	1	yes	characteristic
123	C₂×C₄	C₇×#107	1	1	yes	characteristic
124	C₂×C₄	C₇×SD₃₂	1	1	yes	characteristic
125	C₂×C₄	C₇×SD₃₂	1	1	yes	characteristic
126	C₂×C₄	C₇×SD₃₂	1	1	yes	characteristic
127	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
128	C₈	—	8	2	no	—
129	C₈	C₇×#107	1	1	yes	characteristic
130	C₈	C₇×#107	1	1	yes	characteristic
131	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
132	C₈	C₂×C₇×Q₁₆	1	1	yes	characteristic
133	C₇	C₂×#1426	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²	—	2	2	no	—
135	C₂²	C₇×#367	1	1	yes	characteristic
136	C₂²	C₇×#367	1	1	yes	characteristic
137	C₂²	C₂×C₇×SD₃₂	1	1	yes	characteristic
138	C₄	—	16	2	no	—
139	C₄	C₇×#367	1	1	yes	characteristic
140	C₄	C₇×#367	1	1	yes	characteristic
141	C₄	C₂×C₇×#107	1	1	yes	characteristic
142	C₄	C₂×C₇×SD₃₂	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	C₇×#1426	1	2	yes	direct product factor
146	C₂	C₂×C₇×#367	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	#367
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₂³×C₇	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1426	C₂	direct (N × H)
8. C₇×M₆₄	C₄	semidirect
10. C₂×C₇×C₃₂	C₄	semidirect
15. C₂×#1426	C₇	direct (N × H)
29. C₇×C₃₂	C₂×C₄	semidirect
33. #1426	C₂×C₇	direct (N × H)
54. M₆₄	C₄×C₇	semidirect
56. C₂×C₃₂	C₄×C₇	semidirect
88. C₃₂	C₂×C₄×C₇	semidirect
117. C₂×C₇	#1426	direct (N × H)
133. C₇	C₂×#1426	direct (N × H)
145. C₂	C₇×#1426	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×M₆₄	2
C₂×C₇×#368	2
C₂×C₇×#367	2
C₇×#1426	2
C₂×#1426	7

Subgroups — C2×C7×#1426 #13190853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1426 #13190853