Subgroups — C₇×#529 #90550

Order 448.

234 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#126	C₂	1	2	yes	maximal, complemented
2	C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#123	C₂	1	2	yes	maximal, complemented
5	C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#121	C₂	1	1	yes	maximal, characteristic
7	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₇×D₄	C₂	1	2	yes	maximal, complemented
10	C₇×#117	C₂	1	2	yes	maximal, complemented
11	C₂×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×D₄	C₂²	1	2	yes	—
14	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₇	C₂²	1	2	yes	—
19	C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₇×C₄⋊C₄	C₂²	1	2	yes	—
23	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₄²×C₇	C₂²	1	1	yes	characteristic
31	C₄²×C₇	C₂²	1	1	yes	characteristic
32	C₄²×C₇	C₂²	1	1	yes	characteristic
33	C₄²×C₇	C₂²	1	1	yes	characteristic
34	C₄²×C₇	C₂²	1	2	yes	—
35	#529	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂³×C₇	C₂³	1	2	yes	—
37	C₂³×C₇	C₂³	1	1	yes	characteristic
38	C₇×Q₈	—	2	2	no	—
39	C₇×D₄	—	2	4	no	—
40	C₇×D₄	—	2	2	no	—
41	C₂×C₄×C₇	—	2	2	no	—
42	C₂×C₄×C₇	—	2	2	no	—
43	C₂×C₄×C₇	—	2	4	no	—
44	C₂×C₄×C₇	—	2	2	no	—
45	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₇	C₂³	1	2	yes	—
50	C₂×C₄×C₇	C₂³	1	2	yes	—
51	C₂×C₄×C₇	C₂³	1	2	yes	—
52	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
54	#126	C₂×C₇	1	2	yes	complemented
55	#125	C₂×C₇	1	1	yes	characteristic, complemented
56	#124	C₂×C₇	1	1	yes	characteristic, complemented
57	#123	C₂×C₇	1	2	yes	complemented
58	C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
59	#121	C₂×C₇	1	1	yes	characteristic
60	C₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
61	C₄×D₄	C₂×C₇	1	1	yes	characteristic, complemented
62	C₄×D₄	C₂×C₇	1	2	yes	complemented
63	#117	C₂×C₇	1	2	yes	complemented
64	C₂×C₄²	C₂×C₇	1	1	yes	characteristic, complemented
65	C₂²×C₇	—	2	1	no	—
66	C₂²×C₇	—	2	1	no	—
67	C₂²×C₇	—	2	2	no	—
68	C₂²×C₇	—	2	2	no	—
69	C₂²×C₇	—	2	2	no	—
70	C₂²×C₇	#41	1	2	yes	—
71	C₂²×C₇	C₂⁴	1	1	yes	characteristic
72	C₄×C₇	—	2	1	no	—
73	C₄×C₇	—	2	1	no	—
74	C₄×C₇	—	2	1	no	—
75	C₄×C₇	—	2	1	no	—
76	C₄×C₇	—	2	2	no	—
77	C₄×C₇	—	2	2	no	—
78	C₄×C₇	—	2	2	no	—
79	C₄×C₇	#41	1	2	yes	—
80	C₄×C₇	#41	1	2	yes	—
81	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
82	C₂×D₄	C₂²×C₇	1	2	yes	—
83	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
84	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
86	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²×C₇	1	2	yes	—
88	C₄⋊C₄	C₂²×C₇	1	2	yes	—
89	C₄⋊C₄	C₂²×C₇	1	2	yes	—
90	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
91	C₄⋊C₄	C₂²×C₇	1	2	yes	—
92	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
93	C₄⋊C₄	C₂²×C₇	1	2	yes	—
94	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
95	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
96	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
97	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
98	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
99	C₄²	C₂²×C₇	1	1	yes	characteristic
100	C₄²	C₂²×C₇	1	1	yes	characteristic
101	C₄²	C₂²×C₇	1	1	yes	characteristic
102	C₄²	C₂²×C₇	1	1	yes	characteristic
103	C₄²	C₂²×C₇	1	2	yes	—
104	C₂×C₇	—	2	2	no	—
105	C₂×C₇	—	4	2	no	—
106	C₂×C₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₇	C₂×#41	1	1	yes	characteristic
109	C₂³	C₂³×C₇	1	2	yes	—
110	C₂³	C₂³×C₇	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	D₄	—	2	4	no	—
113	D₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
119	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
120	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
121	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
122	C₂×C₄	C₂³×C₇	1	2	yes	—
123	C₂×C₄	C₂³×C₇	1	2	yes	—
124	C₂×C₄	C₂³×C₇	1	2	yes	—
125	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
126	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
127	C₇	#529	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	C₇×#41	1	2	yes	—
134	C₂²	C₂⁴×C₇	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	C₇×#41	1	2	yes	—
143	C₄	C₇×#41	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	4	2	no	—
146	C₂	C₂×C₇×#41	1	1	yes	characteristic
147	C₂	C₂×C₇×#41	1	1	yes	characteristic
148	C₂	C₂×C₇×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
socle	C₂²×C₇	C₂⁴
prime radical	C₇×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#126	C₂	semidirect
2. C₇×#125	C₂	semidirect
3. C₇×#124	C₂	semidirect
4. C₇×#123	C₂	semidirect
5. C₇×C₂²⋊Q₈	C₂	semidirect
7. C₄×C₇×Q₈	C₂	semidirect
8. C₄×C₇×D₄	C₂	semidirect
9. C₄×C₇×D₄	C₂	semidirect
10. C₇×#117	C₂	semidirect
11. C₂×C₄²×C₇	C₂	semidirect
35. #529	C₇	direct (N × H)
54. #126	C₂×C₇	semidirect
55. #125	C₂×C₇	semidirect
56. #124	C₂×C₇	semidirect
57. #123	C₂×C₇	semidirect
58. C₂²⋊Q₈	C₂×C₇	semidirect
60. C₄×Q₈	C₂×C₇	semidirect
61. C₄×D₄	C₂×C₇	semidirect
62. C₄×D₄	C₂×C₇	semidirect
63. #117	C₂×C₇	semidirect
64. C₂×C₄²	C₂×C₇	semidirect
127. C₇	#529	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#126	2
C₇×#125	2
C₇×#124	2
C₇×#123	2
C₇×C₂²⋊Q₈	2
C₇×#121	2
C₄×C₇×Q₈	2
C₄×C₇×D₄	2
C₇×#117	2
C₂×C₄²×C₇	2
#529	7

Subgroups — C7×#529 #90550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#529 #90550