Subgroups — C₇×#553 #90574

Order 448.

238 conjugacy classes of subgroups.

The subgroup lattice has 158 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×#123	C₂	1	2	yes	maximal, complemented
4	C₇×#121	C₂	1	2	yes	maximal, complemented
5	C₇×#121	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#121	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₇×D₄	C₂	1	2	yes	maximal, complemented
8	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×D₄	C₂²	1	2	yes	—
13	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×D₄	C₂²	1	2	yes	—
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	C₂²×C₄×C₇	C₂²	1	2	yes	—
20	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
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×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₇×C₄⋊C₄	C₂²	1	2	yes	—
27	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
32	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
33	C₄²×C₇	C₂²	1	2	yes	—
34	C₄²×C₇	C₂²	1	1	yes	characteristic
35	#553	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂³×C₇	C₂³	1	1	yes	characteristic
37	C₂³×C₇	C₂³	1	2	yes	—
38	C₂³×C₇	C₂³	1	1	yes	characteristic
39	C₇×D₄	—	2	4	no	—
40	C₇×D₄	—	2	2	no	—
41	C₇×D₄	—	2	2	no	—
42	C₇×D₄	—	4	2	no	—
43	C₂×C₄×C₇	—	2	2	no	—
44	C₂×C₄×C₇	—	2	2	no	—
45	C₂×C₄×C₇	—	4	2	no	—
46	C₂×C₄×C₇	—	4	2	no	—
47	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₇	C₂³	1	2	yes	—
49	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₇	C₂³	1	2	yes	—
51	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₇	C₂³	1	2	yes	—
54	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
55	#126	C₂×C₇	1	2	yes	complemented
56	#125	C₂×C₇	1	1	yes	characteristic, complemented
57	#123	C₂×C₇	1	2	yes	complemented
58	#121	C₂×C₇	1	2	yes	complemented
59	#121	C₂×C₇	1	1	yes	characteristic, complemented
60	#121	C₂×C₇	1	1	yes	characteristic, complemented
61	C₄×D₄	C₂×C₇	1	2	yes	complemented
62	C₄×D₄	C₂×C₇	1	1	yes	characteristic, complemented
63	C₄×D₄	C₂×C₇	1	1	yes	characteristic, complemented
64	#117	C₂×C₇	1	1	yes	characteristic, complemented
65	C₂×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
66	C₂²×C₇	—	2	1	no	—
67	C₂²×C₇	—	2	1	no	—
68	C₂²×C₇	—	2	1	no	—
69	C₂²×C₇	—	2	1	no	—
70	C₂²×C₇	—	2	2	no	—
71	C₂²×C₇	—	2	2	no	—
72	C₂²×C₇	—	2	2	no	—
73	C₂²×C₇	—	2	1	no	—
74	C₂²×C₇	#41	1	2	yes	—
75	C₂²×C₇	C₂⁴	1	1	yes	characteristic
76	C₄×C₇	—	2	1	no	—
77	C₄×C₇	—	2	1	no	—
78	C₄×C₇	—	2	2	no	—
79	C₄×C₇	—	2	1	no	—
80	C₄×C₇	—	2	1	no	—
81	C₄×C₇	—	2	2	no	—
82	C₄×C₇	—	2	2	no	—
83	C₄×C₇	#41	1	2	yes	—
84	C₂×D₄	C₂²×C₇	1	2	yes	—
85	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
86	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
87	C₂×D₄	C₂²×C₇	1	2	yes	—
88	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²×C₇	1	2	yes	—
91	C₂²×C₄	C₂²×C₇	1	2	yes	—
92	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
93	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
94	C₄⋊C₄	C₂²×C₇	1	2	yes	—
95	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
96	C₄⋊C₄	C₂²×C₇	1	2	yes	—
97	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
98	C₄⋊C₄	C₂²×C₇	1	2	yes	—
99	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
100	C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
101	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
102	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
103	C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
104	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
105	C₄²	C₂²×C₇	1	2	yes	—
106	C₄²	C₂²×C₇	1	1	yes	characteristic
107	C₂×C₇	—	2	2	no	—
108	C₂×C₇	—	4	1	no	—
109	C₂×C₇	—	4	2	no	—
110	C₂×C₇	C₂×#41	1	1	yes	characteristic
111	C₂×C₇	C₂×#41	1	1	yes	characteristic
112	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂³	C₂³×C₇	1	1	yes	characteristic
114	C₂³	C₂³×C₇	1	2	yes	—
115	C₂³	C₂³×C₇	1	1	yes	characteristic
116	D₄	—	2	4	no	—
117	D₄	—	2	2	no	—
118	D₄	—	2	2	no	—
119	D₄	—	4	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
125	C₂×C₄	C₂³×C₇	1	2	yes	—
126	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
127	C₂×C₄	C₂³×C₇	1	2	yes	—
128	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
129	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
130	C₂×C₄	C₂³×C₇	1	2	yes	—
131	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
132	C₇	#553	1	1	yes	characteristic, direct product factor, Sylow
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	2	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₇×#41	1	2	yes	—
142	C₂²	C₂⁴×C₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	2	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	C₇×#41	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	4	1	no	—
153	C₂	—	4	2	no	—
154	C₂	C₂×C₇×#41	1	1	yes	characteristic
155	C₂	C₂×C₇×#41	1	1	yes	characteristic
156	C₂	C₇×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×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×#123	C₂	semidirect
4. C₇×#121	C₂	semidirect
5. C₇×#121	C₂	semidirect
6. C₇×#121	C₂	semidirect
7. C₄×C₇×D₄	C₂	semidirect
8. C₄×C₇×D₄	C₂	semidirect
9. C₄×C₇×D₄	C₂	semidirect
10. C₇×#117	C₂	semidirect
11. C₂×C₇×C₄⋊C₄	C₂	semidirect
35. #553	C₇	direct (N × H)
55. #126	C₂×C₇	semidirect
56. #125	C₂×C₇	semidirect
57. #123	C₂×C₇	semidirect
58. #121	C₂×C₇	semidirect
59. #121	C₂×C₇	semidirect
60. #121	C₂×C₇	semidirect
61. C₄×D₄	C₂×C₇	semidirect
62. C₄×D₄	C₂×C₇	semidirect
63. C₄×D₄	C₂×C₇	semidirect
64. #117	C₂×C₇	semidirect
65. C₂×C₄⋊C₄	C₂×C₇	semidirect
132. C₇	#553	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×#123	2
C₇×#121	2
C₄×C₇×D₄	2
C₇×#117	2
C₂×C₇×C₄⋊C₄	2
#553	7

Subgroups — C7×#553 #90574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#553 #90574