Subgroups — C₇×#552 #90573

Order 448.

214 conjugacy classes of subgroups.

The subgroup lattice has 146 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	2	yes	maximal, complemented
3	C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#123	C₂	1	2	yes	maximal
5	C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
9	C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#117	C₂	1	2	yes	maximal, complemented
11	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₇	C₂²	1	2	yes	—
15	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
17	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
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×C₄⋊C₄	C₂²	1	2	yes	—
26	C₇×C₄⋊C₄	C₂²	1	2	yes	—
27	C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₄²×C₇	C₂²	1	2	yes	—
33	C₄²×C₇	C₂²	1	2	yes	—
34	C₄²×C₇	C₂²	1	1	yes	characteristic
35	#552	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂³×C₇	C₂³	1	1	yes	characteristic
37	C₂³×C₇	C₂³	1	1	yes	characteristic
38	C₇×Q₈	—	2	2	no	—
39	C₇×D₄	—	2	2	no	—
40	C₂×C₄×C₇	—	2	4	no	—
41	C₂×C₄×C₇	—	2	2	no	—
42	C₂×C₄×C₇	—	2	2	no	—
43	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₇	C₂³	1	2	yes	—
46	C₂×C₄×C₇	C₂³	1	2	yes	—
47	C₂×C₄×C₇	C₂³	1	2	yes	—
48	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
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	#126	C₂×C₇	1	2	yes	complemented
53	#125	C₂×C₇	1	2	yes	complemented
54	#125	C₂×C₇	1	1	yes	characteristic, complemented
55	#123	C₂×C₇	1	2	yes	—
56	C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
57	C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
58	C₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
59	C₄×D₄	C₂×C₇	1	1	yes	characteristic
60	#117	C₂×C₇	1	1	yes	characteristic, complemented
61	#117	C₂×C₇	1	2	yes	complemented
62	C₂×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
63	C₂²×C₇	—	2	1	no	—
64	C₂²×C₇	—	2	1	no	—
65	C₂²×C₇	—	2	1	no	—
66	C₂²×C₇	—	2	1	no	—
67	C₂²×C₇	—	2	1	no	—
68	C₂²×C₇	#41	1	2	yes	—
69	C₂²×C₇	C₂⁴	1	1	yes	characteristic
70	C₄×C₇	—	2	2	no	—
71	C₄×C₇	—	2	2	no	—
72	C₄×C₇	—	2	1	no	—
73	C₄×C₇	—	2	1	no	—
74	C₄×C₇	—	2	1	no	—
75	C₄×C₇	—	2	2	no	—
76	C₄×C₇	—	2	1	no	—
77	C₄×C₇	—	2	2	no	—
78	C₄×C₇	#41	1	2	yes	—
79	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
80	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
81	C₂²×C₄	C₂²×C₇	1	2	yes	—
82	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
83	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
84	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
85	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
86	C₄⋊C₄	C₂²×C₇	1	2	yes	—
87	C₄⋊C₄	C₂²×C₇	1	2	yes	—
88	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
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	2	yes	—
93	C₄⋊C₄	C₂²×C₇	1	2	yes	—
94	C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
95	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
96	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
97	C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
98	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
99	C₄²	C₂²×C₇	1	2	yes	—
100	C₄²	C₂²×C₇	1	2	yes	—
101	C₄²	C₂²×C₇	1	1	yes	characteristic
102	C₂×C₇	—	2	2	no	—
103	C₂×C₇	—	4	1	no	—
104	C₂×C₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₇	C₂×#41	1	1	yes	characteristic
106	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
107	C₂³	C₂³×C₇	1	1	yes	characteristic
108	C₂³	C₂³×C₇	1	1	yes	characteristic
109	Q₈	—	2	2	no	—
110	D₄	—	2	2	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
115	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
116	C₂×C₄	C₂³×C₇	1	2	yes	—
117	C₂×C₄	C₂³×C₇	1	2	yes	—
118	C₂×C₄	C₂³×C₇	1	2	yes	—
119	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
120	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
121	C₂×C₄	C₂³×C₇	1	2	yes	—
122	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
123	C₇	#552	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₇×#41	1	2	yes	—
130	C₂²	C₂⁴×C₇	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	2	no	—
139	C₄	C₇×#41	1	2	yes	—
140	C₂	—	2	2	no	—
141	C₂	—	4	1	no	—
142	C₂	C₂×C₇×#41	1	1	yes	characteristic
143	C₂	C₂×C₇×#41	1	1	yes	characteristic
144	C₂	C₇×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂×C₇×D₄	C₂²
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×#125	C₂	semidirect
5. C₇×C₂²⋊Q₈	C₂	semidirect
6. C₇×C₂²⋊Q₈	C₂	semidirect
7. C₄×C₇×Q₈	C₂	semidirect
9. C₇×#117	C₂	semidirect
10. C₇×#117	C₂	semidirect
11. C₂×C₇×C₄⋊C₄	C₂	semidirect
35. #552	C₇	direct (N × H)
52. #126	C₂×C₇	semidirect
53. #125	C₂×C₇	semidirect
54. #125	C₂×C₇	semidirect
56. C₂²⋊Q₈	C₂×C₇	semidirect
57. C₂²⋊Q₈	C₂×C₇	semidirect
58. C₄×Q₈	C₂×C₇	semidirect
60. #117	C₂×C₇	semidirect
61. #117	C₂×C₇	semidirect
62. C₂×C₄⋊C₄	C₂×C₇	semidirect
123. C₇	#552	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×C₂²⋊Q₈	2
C₄×C₇×Q₈	2
C₄×C₇×D₄	2
C₇×#117	2
C₂×C₇×C₄⋊C₄	2
#552	7

Subgroups — C7×#552 #90573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#552 #90573