Subgroups — C₂²×C₇×#493 #13237845

Order 1792.

4638 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×D₈	C₂	1	4	yes	maximal, complemented
2	C₂²×C₇×#127	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#493	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₇×D₄	—	2	4	no	—
6	C₂³×C₇×D₄	C₂²	1	4	yes	—
7	C₂²×C₇×D₈	—	2	32	no	—
8	C₂²×C₇×D₈	C₂²	1	48	yes	complemented
9	C₂³×C₇×C₈	C₂²	1	2	yes	—
10	C₂×C₇×#127	—	2	12	no	—
11	C₂×C₇×#127	C₂²	1	24	yes	complemented
12	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₇×#493	C₂²	1	64	yes	direct product factor
14	C₂×C₄×C₇×C₈	C₂²	1	12	yes	complemented
15	C₂²×#493	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂⁵×C₇	—	2	4	no	—
17	C₂²×C₇×D₄	—	2	32	no	—
18	C₂²×C₇×D₄	—	2	48	no	—
19	C₂²×C₇×D₄	—	2	12	no	—
20	C₂²×C₇×D₄	—	4	32	no	—
21	C₂²×C₇×D₄	C₂³	1	24	yes	—
22	C₂³×C₄×C₇	D₄	1	2	yes	—
23	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×C₇×D₈	—	2	192	no	—
25	C₂×C₇×D₈	C₂³	1	64	yes	complemented
26	C₂²×C₇×C₈	D₄	1	16	yes	complemented
27	C₂²×C₇×C₈	C₂³	1	12	yes	—
28	C₇×#127	—	2	48	no	—
29	C₇×#127	C₂³	1	32	yes	complemented
30	C₂×C₄²×C₇	D₄	1	6	yes	—
31	C₂×C₄²×C₇	C₂³	1	6	yes	—
32	C₄×C₇×C₈	C₂³	1	16	yes	complemented
33	C₂³×D₈	C₂×C₇	1	4	yes	complemented
34	C₂²×#127	C₂×C₇	1	2	yes	complemented
35	C₂×#493	C₂×C₇	1	24	yes	direct product factor
36	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic, complemented
37	C₂⁴×C₇	—	2	24	no	—
38	C₂⁴×C₇	—	4	16	no	—
39	C₂⁴×C₇	—	4	32	no	—
40	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₇×D₄	—	2	96	no	—
42	C₂×C₇×D₄	—	2	24	no	—
43	C₂×C₇×D₄	—	2	64	no	—
44	C₂×C₇×D₄	—	4	48	no	—
45	C₂×C₇×D₄	—	4	192	no	—
46	C₂×C₇×D₄	C₂⁴	1	16	yes	—
47	C₂²×C₄×C₇	D₈	1	16	yes	complemented
48	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
50	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
51	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
52	C₇×D₈	—	2	128	no	—
53	C₂×C₇×C₈	C₂×D₄	1	48	yes	complemented
54	C₂×C₇×C₈	C₂⁴	1	8	yes	—
55	C₄²×C₇	C₂×D₄	1	12	yes	—
56	C₄²×C₇	C₂⁴	1	4	yes	—
57	C₂³×D₄	—	2	4	no	—
58	C₂³×D₄	C₂²×C₇	1	4	yes	—
59	C₂²×D₈	—	2	32	no	—
60	C₂²×D₈	C₂²×C₇	1	48	yes	complemented
61	C₂³×C₈	C₂²×C₇	1	2	yes	—
62	C₂×#127	—	2	12	no	—
63	C₂×#127	C₂²×C₇	1	24	yes	complemented
64	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
65	#493	C₂²×C₇	1	64	yes	direct product factor
66	C₂×C₄×C₈	C₂²×C₇	1	12	yes	complemented
67	C₂³×C₇	—	2	16	no	—
68	C₂³×C₇	—	4	48	no	—
69	C₂³×C₇	—	4	96	no	—
70	C₂³×C₇	—	8	64	no	—
71	C₂³×C₇	#127	1	4	yes	—
72	C₂³×C₇	C₂×D₈	1	8	yes	—
73	C₂³×C₇	C₂²×D₄	1	3	yes	—
74	C₇×D₄	—	2	32	no	—
75	C₇×D₄	—	4	48	no	—
76	C₇×D₄	—	4	128	no	—
77	C₂×C₄×C₇	#127	1	12	yes	—
78	C₂×C₄×C₇	C₂×D₈	1	48	yes	complemented
79	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
80	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
81	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
82	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
83	C₇×C₈	C₂²×D₄	1	16	yes	complemented
84	C₂⁵	—	2	4	no	—
85	C₂²×D₄	—	2	32	no	—
86	C₂²×D₄	—	2	12	no	—
87	C₂²×D₄	—	2	48	no	—
88	C₂²×D₄	—	4	32	no	—
89	C₂²×D₄	C₂³×C₇	1	24	yes	—
90	C₂³×C₄	C₇×D₄	1	2	yes	—
91	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
92	C₂×D₈	—	2	192	no	—
93	C₂×D₈	C₂³×C₇	1	64	yes	complemented
94	C₂²×C₈	C₇×D₄	1	16	yes	complemented
95	C₂²×C₈	C₂³×C₇	1	12	yes	—
96	#127	—	2	48	no	—
97	#127	C₂³×C₇	1	32	yes	complemented
98	C₂×C₄²	C₇×D₄	1	6	yes	—
99	C₂×C₄²	C₂³×C₇	1	6	yes	—
100	C₄×C₈	C₂³×C₇	1	16	yes	complemented
101	C₂²×C₇	—	4	16	no	—
102	C₂²×C₇	—	4	32	no	—
103	C₂²×C₇	—	8	96	no	—
104	C₂²×C₇	#493	1	16	yes	direct product factor
105	C₂²×C₇	C₂×#127	1	6	yes	—
106	C₂²×C₇	C₂²×D₈	1	12	yes	—
107	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
108	C₄×C₇	C₂×#127	1	6	yes	—
109	C₄×C₇	C₂²×D₈	1	16	yes	complemented
110	C₄×C₇	C₂³×D₄	1	2	yes	—
111	C₂⁴	—	2	24	no	—
112	C₂⁴	—	4	16	no	—
113	C₂⁴	—	4	32	no	—
114	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
115	C₂×D₄	—	2	96	no	—
116	C₂×D₄	—	2	24	no	—
117	C₂×D₄	—	2	64	no	—
118	C₂×D₄	—	4	48	no	—
119	C₂×D₄	—	4	192	no	—
120	C₂×D₄	C₂⁴×C₇	1	16	yes	—
121	C₂²×C₄	C₇×D₈	1	16	yes	complemented
122	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
123	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
124	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
125	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
126	D₈	—	2	128	no	—
127	C₂×C₈	C₂×C₇×D₄	1	48	yes	complemented
128	C₂×C₈	C₂⁴×C₇	1	8	yes	—
129	C₄²	C₂×C₇×D₄	1	12	yes	—
130	C₄²	C₂⁴×C₇	1	4	yes	—
131	C₂×C₇	—	8	16	no	—
132	C₂×C₇	C₂×#493	1	12	yes	direct product factor
133	C₂×C₇	C₂²×#127	1	1	yes	characteristic
134	C₂×C₇	C₂³×D₈	1	2	yes	—
135	C₂³	—	2	16	no	—
136	C₂³	—	4	48	no	—
137	C₂³	—	4	96	no	—
138	C₂³	—	8	64	no	—
139	C₂³	C₇×#127	1	4	yes	—
140	C₂³	C₂×C₇×D₈	1	8	yes	—
141	C₂³	C₂²×C₇×D₄	1	3	yes	—
142	D₄	—	2	32	no	—
143	D₄	—	4	48	no	—
144	D₄	—	4	128	no	—
145	C₂×C₄	C₇×#127	1	12	yes	—
146	C₂×C₄	C₂×C₇×D₈	1	48	yes	complemented
147	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
148	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
149	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
150	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
151	C₈	C₂²×C₇×D₄	1	16	yes	complemented
152	C₇	C₂²×#493	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²	—	4	32	no	—
154	C₂²	—	4	16	no	—
155	C₂²	—	8	96	no	—
156	C₂²	C₇×#493	1	16	yes	direct product factor
157	C₂²	C₂×C₇×#127	1	6	yes	—
158	C₂²	C₂²×C₇×D₈	1	12	yes	—
159	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
160	C₄	C₂×C₇×#127	1	6	yes	—
161	C₄	C₂²×C₇×D₈	1	16	yes	complemented
162	C₄	C₂³×C₇×D₄	1	2	yes	—
163	C₂	—	8	16	no	—
164	C₂	C₂×C₇×#493	1	12	yes	direct product factor
165	C₂	C₂²×C₇×#127	1	1	yes	characteristic
166	C₂	C₂³×C₇×D₈	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇×D₈	C₂	semidirect
2. C₂²×C₇×#127	C₂	semidirect
3. C₂×C₇×#493	C₂	direct (N × H)
4. C₂²×C₄×C₇×C₈	C₂	semidirect
8. C₂²×C₇×D₈	C₂²	semidirect
11. C₂×C₇×#127	C₂²	semidirect
13. C₇×#493	C₂²	direct (N × H)
14. C₂×C₄×C₇×C₈	C₂²	semidirect
15. C₂²×#493	C₇	direct (N × H)
25. C₂×C₇×D₈	C₂³	semidirect
26. C₂²×C₇×C₈	D₄	semidirect
29. C₇×#127	C₂³	semidirect
32. C₄×C₇×C₈	C₂³	semidirect
33. C₂³×D₈	C₂×C₇	semidirect
34. C₂²×#127	C₂×C₇	semidirect
35. C₂×#493	C₂×C₇	direct (N × H)
36. C₂²×C₄×C₈	C₂×C₇	semidirect
47. C₂²×C₄×C₇	D₈	semidirect
53. C₂×C₇×C₈	C₂×D₄	semidirect
60. C₂²×D₈	C₂²×C₇	semidirect
63. C₂×#127	C₂²×C₇	semidirect
65. #493	C₂²×C₇	direct (N × H)
66. C₂×C₄×C₈	C₂²×C₇	semidirect
78. C₂×C₄×C₇	C₂×D₈	semidirect
83. C₇×C₈	C₂²×D₄	semidirect
93. C₂×D₈	C₂³×C₇	semidirect
94. C₂²×C₈	C₇×D₄	semidirect
97. #127	C₂³×C₇	semidirect
100. C₄×C₈	C₂³×C₇	semidirect
104. C₂²×C₇	#493	direct (N × H)
109. C₄×C₇	C₂²×D₈	semidirect
121. C₂²×C₄	C₇×D₈	semidirect
127. C₂×C₈	C₂×C₇×D₄	semidirect
132. C₂×C₇	C₂×#493	direct (N × H)
146. C₂×C₄	C₂×C₇×D₈	semidirect
151. C₈	C₂²×C₇×D₄	semidirect
152. C₇	C₂²×#493	direct (N × H)
156. C₂²	C₇×#493	direct (N × H)
161. C₄	C₂²×C₇×D₈	semidirect
164. C₂	C₂×C₇×#493	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×D₈	2
C₂²×C₇×#127	2
C₂×C₇×#493	2
C₂²×C₄×C₇×C₈	2
C₂²×#493	7

Subgroups — C22×C7×#493 #13237845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#493 #13237845