Subgroups — #14260419

Order 1792.

10930 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	C₂²×C₇×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	Dic₇×2₋¹⁺⁴	C₂	1	32	yes	maximal, complemented
3	#11744328	C₂	1	20	yes	maximal, complemented
4	C₂×#90503	C₂	1	10	yes	maximal, complemented
5	C₂×C₇×2₋¹⁺⁴	C₄	1	32	yes	complemented
6	C₂×C₇×2₋¹⁺⁴	C₂²	1	16	yes	—
7	C₂²×C₇×#41	C₂²	1	10	yes	—
8	C₂³×C₇×Q₈	C₂²	1	5	yes	—
9	#90518	C₂²	1	160	yes	complemented
10	#41×Dic₇	C₂²	1	160	yes	complemented
11	#90503	C₂²	1	40	yes	—
12	#90503	C₂²	1	80	yes	complemented
13	C₂×Q₈×Dic₇	C₂²	1	80	yes	complemented
14	#90489	C₂²	1	40	yes	—
15	C₂×#6489	C₂²	1	60	yes	—
16	#62653	—	7	1	no	maximal, Sylow
17	C₇×2₋¹⁺⁴	C₂×C₄	1	256	yes	complemented
18	C₂×C₇×#41	C₂×C₄	1	160	yes	complemented
19	C₂×C₇×#41	C₂³	1	80	yes	—
20	C₂²×C₇×Q₈	C₂×C₄	1	80	yes	complemented
21	C₂²×C₇×Q₈	C₂³	1	10	yes	—
22	C₂²×C₇×Q₈	C₂³	1	40	yes	—
23	C₂²×C₇×D₄	C₂³	1	10	yes	—
24	C₂³×C₄×C₇	C₂³	1	15	yes	—
25	C₂×C₂²⋊Dic₇	C₂³	1	40	yes	—
26	Q₈×Dic₇	C₂³	1	320	yes	—
27	D₄×Dic₇	C₂³	1	320	yes	—
28	#6489	C₂³	1	80	yes	—
29	#6489	C₂³	1	240	yes	—
30	C₂×C₄⋊Dic₇	C₂³	1	120	yes	—
31	C₂×C₄×Dic₇	C₂³	1	120	yes	—
32	C₂²×2₋¹⁺⁴	D₇	1	1	yes	characteristic
33	C₄×2₋¹⁺⁴	—	7	32	no	—
34	#3428	—	7	20	no	—
35	C₂×#519	—	7	10	no	—
36	C₂⁴×C₇	C₂⁴	1	5	yes	—
37	C₇×#41	C₂²×C₄	1	640	yes	—
38	C₂×C₇×Q₈	C₂²×C₄	1	80	yes	—
39	C₂×C₇×Q₈	C₂²×C₄	1	320	yes	—
40	C₂×C₇×Q₈	C₂⁴	1	40	yes	—
41	C₂×C₇×D₄	C₂²×C₄	1	80	yes	—
42	C₂×C₇×D₄	C₂⁴	1	40	yes	—
43	C₂²×C₄×C₇	C₂²×C₄	1	120	yes	—
44	C₂²×C₄×C₇	C₂⁴	1	60	yes	—
45	C₂²×C₄×C₇	C₂⁴	1	10	yes	—
46	C₂²×Dic₇	C₂⁴	1	80	yes	—
47	C₂²⋊Dic₇	C₂⁴	1	16	yes	—
48	C₂²⋊Dic₇	C₂⁴	1	80	yes	—
49	C₄⋊Dic₇	C₂⁴	1	160	yes	—
50	C₄×Dic₇	C₂⁴	1	160	yes	—
51	C₂×2₋¹⁺⁴	Dic₇	1	32	yes	complemented
52	C₂×2₋¹⁺⁴	C₂×D₇	1	16	yes	—
53	C₂²×#41	C₂×D₇	1	10	yes	—
54	C₂³×Q₈	C₂×D₇	1	5	yes	—
55	#520	—	7	160	no	—
56	#519	—	7	40	no	—
57	#519	—	7	80	no	—
58	#518	—	7	40	no	—
59	C₄×#41	—	7	160	no	—
60	C₂×C₄×Q₈	—	7	80	no	—
61	C₂×#117	—	7	60	no	—
62	C₂³×C₇	—	4	10	no	—
63	C₂³×C₇	C₂³×C₄	1	20	yes	—
64	C₂³×C₇	C₂⁵	1	1	yes	characteristic
65	C₂³×C₇	C₂⁵	1	10	yes	—
66	C₇×Q₈	C₂³×C₄	1	160	yes	—
67	C₇×D₄	C₂³×C₄	1	160	yes	—
68	C₂×C₄×C₇	C₂³×C₄	1	40	yes	—
69	C₂×C₄×C₇	C₂³×C₄	1	240	yes	—
70	C₂×C₄×C₇	C₂⁵	1	20	yes	—
71	C₂×Dic₇	—	4	80	no	—
72	C₂×Dic₇	C₂⁵	1	32	yes	—
73	2₋¹⁺⁴	C₂×Dic₇	1	256	yes	complemented
74	C₂×#41	C₂×Dic₇	1	160	yes	complemented
75	C₂×#41	C₂²×D₇	1	80	yes	—
76	C₂²×Q₈	C₂×Dic₇	1	80	yes	complemented
77	C₂²×Q₈	C₂²×D₇	1	40	yes	—
78	C₂²×Q₈	C₂²×D₇	1	10	yes	—
79	C₂²×D₄	C₂²×D₇	1	10	yes	—
80	C₂³×C₄	C₂²×D₇	1	15	yes	—
81	C₄×Q₈	—	7	320	no	—
82	C₄×D₄	—	7	320	no	—
83	#117	—	7	80	no	—
84	#117	—	7	240	no	—
85	C₂×C₄⋊C₄	—	7	120	no	—
86	C₂×C₂²⋊C₄	—	7	40	no	—
87	C₂×C₄²	—	7	120	no	—
88	C₂²×C₇	—	2	20	no	—
89	C₂²×C₇	—	2	2	no	—
90	C₂²×C₇	—	4	20	no	—
91	C₂²×C₇	C₂⁴×C₄	1	2	yes	—
92	C₂²×C₇	C₂⁴×C₄	1	20	yes	—
93	C₂²×C₇	C₂⁶	1	1	yes	characteristic
94	C₄×C₇	C₂⁴×C₄	1	40	yes	—
95	Dic₇	—	2	32	no	—
96	C₂⁴	C₂³×D₇	1	5	yes	—
97	#41	C₂²×Dic₇	1	640	yes	—
98	C₂×Q₈	C₂²×Dic₇	1	80	yes	—
99	C₂×Q₈	C₂²×Dic₇	1	320	yes	—
100	C₂×Q₈	C₂³×D₇	1	40	yes	—
101	C₂×D₄	C₂²×Dic₇	1	80	yes	—
102	C₂×D₄	C₂³×D₇	1	40	yes	—
103	C₂²×C₄	—	7	80	no	—
104	C₂²×C₄	C₂²×Dic₇	1	120	yes	—
105	C₂²×C₄	C₂³×D₇	1	60	yes	—
106	C₂²×C₄	C₂³×D₇	1	10	yes	—
107	C₄⋊C₄	—	7	160	no	—
108	C₂²⋊C₄	—	7	16	no	—
109	C₂²⋊C₄	—	7	80	no	—
110	C₄²	—	7	160	no	—
111	C₂×C₇	—	2	2	no	—
112	C₂×C₇	—	2	20	no	—
113	C₂×C₇	C₂⁵×C₄	1	1	yes	characteristic
114	C₂×C₇	2₋¹⁺⁶	1	2	yes	—
115	C₂³	—	4	10	no	—
116	C₂³	C₂³×Dic₇	1	20	yes	—
117	C₂³	C₂⁴×D₇	1	1	yes	characteristic
118	C₂³	C₂⁴×D₇	1	10	yes	—
119	Q₈	C₂³×Dic₇	1	160	yes	—
120	D₄	C₂³×Dic₇	1	160	yes	—
121	C₂×C₄	—	7	32	no	—
122	C₂×C₄	—	28	80	no	—
123	C₂×C₄	C₂³×Dic₇	1	40	yes	—
124	C₂×C₄	C₂³×Dic₇	1	240	yes	—
125	C₂×C₄	C₂⁴×D₇	1	20	yes	—
126	C₇	#62653	1	1	yes	characteristic, complemented, Sylow
127	C₂²	—	2	20	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	20	no	—
130	C₂²	C₂⁴×Dic₇	1	2	yes	—
131	C₂²	C₂⁴×Dic₇	1	20	yes	—
132	C₂²	C₂⁵×D₇	1	1	yes	characteristic
133	C₄	—	14	32	no	—
134	C₄	C₂⁴×Dic₇	1	40	yes	—
135	C₂	—	2	2	no	—
136	C₂	—	2	20	no	—
137	C₂	#11744563	1	2	yes	—
138	C₂	C₂⁵×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₇
commutator	C₂×C₇	C₂⁵×C₄
Fitting, prime radical	C₂²×C₇×2₋¹⁺⁴	C₂
socle, cosocle	C₂²×C₇	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₇
hypocenter	C₇	#62653
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
2. Dic₇×2₋¹⁺⁴	C₂	semidirect
3. #11744328	C₂	semidirect
4. C₂×#90503	C₂	semidirect
5. C₂×C₇×2₋¹⁺⁴	C₄	semidirect
9. #90518	C₂²	semidirect
10. #41×Dic₇	C₂²	semidirect
12. #90503	C₂²	semidirect
13. C₂×Q₈×Dic₇	C₂²	semidirect
17. C₇×2₋¹⁺⁴	C₂×C₄	semidirect
18. C₂×C₇×#41	C₂×C₄	semidirect
20. C₂²×C₇×Q₈	C₂×C₄	semidirect
51. C₂×2₋¹⁺⁴	Dic₇	semidirect
73. 2₋¹⁺⁴	C₂×Dic₇	semidirect
74. C₂×#41	C₂×Dic₇	semidirect
76. C₂²×Q₈	C₂×Dic₇	semidirect
126. C₇	#62653	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×2₋¹⁺⁴	2
Dic₇×2₋¹⁺⁴	2
#11744328	2
C₂×#90503	2
#62653	7