Subgroups — C₂³×C₇×#124 #13239876

Order 1792.

14334 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	C₂⁴×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇×#124	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₇×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁶×C₇	C₂²	1	2	yes	—
7	C₂³×C₇×Q₈	—	2	16	no	—
8	C₂³×C₇×Q₈	C₂²	1	28	yes	complemented
9	C₂³×C₇×D₄	—	2	16	no	—
10	C₂³×C₇×D₄	C₂²	1	28	yes	—
11	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₇×#124	C₂²	1	448	yes	direct product factor
15	C₂²×C₇×C₂²⋊C₄	C₂²	1	112	yes	complemented
16	C₂²×C₄²×C₇	C₂²	1	28	yes	complemented
17	C₂³×#124	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂⁵×C₇	—	2	16	no	—
19	C₂⁵×C₇	—	2	32	no	—
20	C₂⁵×C₇	C₂³	1	28	yes	—
21	C₂⁵×C₇	C₂³	1	1	yes	characteristic
22	C₂²×C₇×Q₈	—	2	224	no	—
23	C₂²×C₇×Q₈	C₂³	1	112	yes	complemented
24	C₂²×C₇×D₄	—	2	224	no	—
25	C₂²×C₇×D₄	C₂³	1	112	yes	—
26	C₂³×C₄×C₇	—	2	16	no	—
27	C₂³×C₄×C₇	—	2	16	no	—
28	C₂³×C₄×C₇	D₄	1	16	yes	—
29	C₂³×C₄×C₇	C₂³	1	28	yes	—
30	C₂³×C₄×C₇	C₂³	1	28	yes	—
31	C₂³×C₄×C₇	C₂³	1	14	yes	—
32	C₇×#124	C₂³	1	512	yes	direct product factor
33	C₂×C₇×C₂²⋊C₄	C₂³	1	448	yes	complemented
34	C₂×C₄²×C₇	C₂³	1	112	yes	complemented
35	C₂⁴×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
36	C₂⁴×D₄	C₂×C₇	1	1	yes	characteristic
37	C₂²×#124	C₂×C₇	1	56	yes	direct product factor
38	C₂³×C₂²⋊C₄	C₂×C₇	1	4	yes	complemented
39	C₂³×C₄²	C₂×C₇	1	1	yes	characteristic, complemented
40	C₂⁴×C₇	—	2	112	no	—
41	C₂⁴×C₇	—	2	224	no	—
42	C₂⁴×C₇	—	4	128	no	—
43	C₂⁴×C₇	C₂×D₄	1	8	yes	—
44	C₂⁴×C₇	#41	1	16	yes	—
45	C₂⁴×C₇	C₂⁴	1	56	yes	—
46	C₂⁴×C₇	C₂⁴	1	7	yes	—
47	C₂×C₇×Q₈	—	2	448	no	—
48	C₂×C₇×Q₈	C₂⁴	1	64	yes	complemented
49	C₂×C₇×D₄	—	2	448	no	—
50	C₂×C₇×D₄	C₂⁴	1	64	yes	—
51	C₂²×C₄×C₇	—	2	112	no	—
52	C₂²×C₄×C₇	—	2	112	no	—
53	C₂²×C₄×C₇	C₂×D₄	1	112	yes	—
54	C₂²×C₄×C₇	C₂⁴	1	56	yes	—
55	C₂²×C₄×C₇	C₂⁴	1	56	yes	—
56	C₂²×C₄×C₇	C₂⁴	1	28	yes	—
57	C₇×C₂²⋊C₄	C₂⁴	1	256	yes	complemented
58	C₄²×C₇	C₂⁴	1	64	yes	complemented
59	C₂⁶	C₂²×C₇	1	2	yes	—
60	C₂³×Q₈	—	2	16	no	—
61	C₂³×Q₈	C₂²×C₇	1	28	yes	complemented
62	C₂³×D₄	—	2	16	no	—
63	C₂³×D₄	C₂²×C₇	1	28	yes	—
64	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
65	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
66	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
67	C₂×#124	C₂²×C₇	1	448	yes	direct product factor
68	C₂²×C₂²⋊C₄	C₂²×C₇	1	112	yes	complemented
69	C₂²×C₄²	C₂²×C₇	1	28	yes	complemented
70	C₂³×C₇	—	2	112	no	—
71	C₂³×C₇	—	2	224	no	—
72	C₂³×C₇	—	4	448	no	—
73	C₂³×C₇	#124	1	64	yes	direct product factor
74	C₂³×C₇	C₂²×D₄	1	28	yes	—
75	C₂³×C₇	C₂×#41	1	56	yes	—
76	C₂³×C₇	C₂⁵	1	16	yes	—
77	C₂³×C₇	C₂⁵	1	7	yes	—
78	C₇×Q₈	—	2	128	no	—
79	C₇×D₄	—	2	128	no	—
80	C₂×C₄×C₇	—	2	112	no	—
81	C₂×C₄×C₇	—	2	112	no	—
82	C₂×C₄×C₇	C₂²×D₄	1	112	yes	—
83	C₂×C₄×C₇	C₂⁵	1	16	yes	—
84	C₂×C₄×C₇	C₂⁵	1	16	yes	—
85	C₂×C₄×C₇	C₂⁵	1	8	yes	—
86	C₂⁵	—	2	16	no	—
87	C₂⁵	—	2	32	no	—
88	C₂⁵	C₂³×C₇	1	28	yes	—
89	C₂⁵	C₂³×C₇	1	1	yes	characteristic
90	C₂²×Q₈	—	2	224	no	—
91	C₂²×Q₈	C₂³×C₇	1	112	yes	complemented
92	C₂²×D₄	—	2	224	no	—
93	C₂²×D₄	C₂³×C₇	1	112	yes	—
94	C₂³×C₄	—	2	16	no	—
95	C₂³×C₄	—	2	16	no	—
96	C₂³×C₄	C₇×D₄	1	16	yes	—
97	C₂³×C₄	C₂³×C₇	1	28	yes	—
98	C₂³×C₄	C₂³×C₇	1	28	yes	—
99	C₂³×C₄	C₂³×C₇	1	14	yes	—
100	#124	C₂³×C₇	1	512	yes	direct product factor
101	C₂×C₂²⋊C₄	C₂³×C₇	1	448	yes	complemented
102	C₂×C₄²	C₂³×C₇	1	112	yes	complemented
103	C₂²×C₇	—	2	16	no	—
104	C₂²×C₇	—	2	32	no	—
105	C₂²×C₇	—	4	224	no	—
106	C₂²×C₇	C₂×#124	1	112	yes	direct product factor
107	C₂²×C₇	C₂³×D₄	1	14	yes	—
108	C₂²×C₇	C₂²×#41	1	28	yes	—
109	C₂²×C₇	C₂⁶	1	1	yes	characteristic
110	C₄×C₇	—	2	16	no	—
111	C₄×C₇	—	2	16	no	—
112	C₄×C₇	C₂³×D₄	1	16	yes	—
113	C₂⁴	—	2	112	no	—
114	C₂⁴	—	2	224	no	—
115	C₂⁴	—	4	128	no	—
116	C₂⁴	C₂×C₇×D₄	1	8	yes	—
117	C₂⁴	C₇×#41	1	16	yes	—
118	C₂⁴	C₂⁴×C₇	1	56	yes	—
119	C₂⁴	C₂⁴×C₇	1	7	yes	—
120	C₂×Q₈	—	2	448	no	—
121	C₂×Q₈	C₂⁴×C₇	1	64	yes	complemented
122	C₂×D₄	—	2	448	no	—
123	C₂×D₄	C₂⁴×C₇	1	64	yes	—
124	C₂²×C₄	—	2	112	no	—
125	C₂²×C₄	—	2	112	no	—
126	C₂²×C₄	C₂×C₇×D₄	1	112	yes	—
127	C₂²×C₄	C₂⁴×C₇	1	56	yes	—
128	C₂²×C₄	C₂⁴×C₇	1	56	yes	—
129	C₂²×C₄	C₂⁴×C₇	1	28	yes	—
130	C₂²⋊C₄	C₂⁴×C₇	1	256	yes	complemented
131	C₄²	C₂⁴×C₇	1	64	yes	complemented
132	C₂×C₇	—	4	16	no	—
133	C₂×C₇	C₂²×#124	1	28	yes	direct product factor
134	C₂×C₇	C₂⁴×D₄	1	1	yes	characteristic
135	C₂×C₇	C₂³×#41	1	2	yes	—
136	C₂³	—	2	112	no	—
137	C₂³	—	2	224	no	—
138	C₂³	—	4	448	no	—
139	C₂³	C₇×#124	1	64	yes	direct product factor
140	C₂³	C₂²×C₇×D₄	1	28	yes	—
141	C₂³	C₂×C₇×#41	1	56	yes	—
142	C₂³	C₂⁵×C₇	1	16	yes	—
143	C₂³	C₂⁵×C₇	1	7	yes	—
144	Q₈	—	2	128	no	—
145	D₄	—	2	128	no	—
146	C₂×C₄	—	2	112	no	—
147	C₂×C₄	—	2	112	no	—
148	C₂×C₄	C₂²×C₇×D₄	1	112	yes	—
149	C₂×C₄	C₂⁵×C₇	1	16	yes	—
150	C₂×C₄	C₂⁵×C₇	1	16	yes	—
151	C₂×C₄	C₂⁵×C₇	1	8	yes	—
152	C₇	C₂³×#124	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²	—	2	16	no	—
154	C₂²	—	2	32	no	—
155	C₂²	—	4	224	no	—
156	C₂²	C₂×C₇×#124	1	112	yes	direct product factor
157	C₂²	C₂³×C₇×D₄	1	14	yes	—
158	C₂²	C₂²×C₇×#41	1	28	yes	—
159	C₂²	C₂⁶×C₇	1	1	yes	characteristic
160	C₄	—	2	16	no	—
161	C₄	—	2	16	no	—
162	C₄	C₂³×C₇×D₄	1	16	yes	—
163	C₂	—	4	16	no	—
164	C₂	C₂²×C₇×#124	1	28	yes	direct product factor
165	C₂	C₂⁴×C₇×D₄	1	1	yes	characteristic
166	C₂	C₂³×C₇×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇
prime radical	C₂⁴×C₇×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₇×Q₈	C₂	semidirect
3. C₂²×C₇×#124	C₂	direct (N × H)
4. C₂³×C₇×C₂²⋊C₄	C₂	semidirect
5. C₂³×C₄²×C₇	C₂	semidirect
8. C₂³×C₇×Q₈	C₂²	semidirect
14. C₂×C₇×#124	C₂²	direct (N × H)
15. C₂²×C₇×C₂²⋊C₄	C₂²	semidirect
16. C₂²×C₄²×C₇	C₂²	semidirect
17. C₂³×#124	C₇	direct (N × H)
23. C₂²×C₇×Q₈	C₂³	semidirect
32. C₇×#124	C₂³	direct (N × H)
33. C₂×C₇×C₂²⋊C₄	C₂³	semidirect
34. C₂×C₄²×C₇	C₂³	semidirect
35. C₂⁴×Q₈	C₂×C₇	semidirect
37. C₂²×#124	C₂×C₇	direct (N × H)
38. C₂³×C₂²⋊C₄	C₂×C₇	semidirect
39. C₂³×C₄²	C₂×C₇	semidirect
48. C₂×C₇×Q₈	C₂⁴	semidirect
57. C₇×C₂²⋊C₄	C₂⁴	semidirect
58. C₄²×C₇	C₂⁴	semidirect
61. C₂³×Q₈	C₂²×C₇	semidirect
67. C₂×#124	C₂²×C₇	direct (N × H)
68. C₂²×C₂²⋊C₄	C₂²×C₇	semidirect
69. C₂²×C₄²	C₂²×C₇	semidirect
73. C₂³×C₇	#124	direct (N × H)
91. C₂²×Q₈	C₂³×C₇	semidirect
100. #124	C₂³×C₇	direct (N × H)
101. C₂×C₂²⋊C₄	C₂³×C₇	semidirect
102. C₂×C₄²	C₂³×C₇	semidirect
106. C₂²×C₇	C₂×#124	direct (N × H)
121. C₂×Q₈	C₂⁴×C₇	semidirect
130. C₂²⋊C₄	C₂⁴×C₇	semidirect
131. C₄²	C₂⁴×C₇	semidirect
133. C₂×C₇	C₂²×#124	direct (N × H)
139. C₂³	C₇×#124	direct (N × H)
152. C₇	C₂³×#124	direct (N × H)
156. C₂²	C₂×C₇×#124	direct (N × H)
164. C₂	C₂²×C₇×#124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×Q₈	2
C₂⁴×C₇×D₄	2
C₂²×C₇×#124	2
C₂³×C₇×C₂²⋊C₄	2
C₂³×C₄²×C₇	2
C₂³×#124	7

Subgroups — C23×C7×#124 #13239876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×#124 #13239876