Subgroups — C₇×#546 #90567

Order 448.

334 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×D₄	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, complemented
7	C₇×#121	C₂	1	2	yes	maximal, complemented
8	C₇×#120	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
12	C₂⁴×C₇	C₂²	1	2	yes	—
13	C₇×#41	—	2	4	no	—
14	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₇×D₄	—	2	4	no	—
16	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×D₄	C₂²	1	2	yes	—
18	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
19	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
20	C₂×C₇×D₄	C₂²	1	2	yes	complemented
21	C₂×C₇×D₄	C₂²	1	2	yes	complemented
22	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₇	C₂²	1	2	yes	complemented
24	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₄×C₇	C₂²	1	2	yes	complemented
26	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
27	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
28	C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
29	C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₇×C₂²⋊C₄	C₂²	1	2	yes	complemented
32	C₇×C₂²⋊C₄	C₂²	1	2	yes	complemented
33	C₇×C₂²⋊C₄	C₂²	1	2	yes	complemented
34	C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
35	C₇×C₂²⋊C₄	C₂²	1	2	yes	complemented
36	C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
37	#546	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂³×C₇	—	2	4	no	—
39	C₂³×C₇	—	2	4	no	—
40	C₂³×C₇	—	4	2	no	—
41	C₂³×C₇	C₂³	1	2	yes	—
42	C₂³×C₇	C₂³	1	1	yes	characteristic
43	C₂³×C₇	C₂³	1	2	yes	—
44	C₂³×C₇	C₂³	1	1	yes	characteristic
45	C₇×Q₈	—	2	2	no	—
46	C₇×D₄	—	2	4	no	—
47	C₇×D₄	—	2	4	no	—
48	C₇×D₄	—	2	2	no	—
49	C₇×D₄	—	2	2	no	—
50	C₇×D₄	—	4	2	no	—
51	C₇×D₄	D₄	1	4	yes	complemented
52	C₂×C₄×C₇	—	2	4	no	—
53	C₂×C₄×C₇	—	2	2	no	—
54	C₂×C₄×C₇	—	2	2	no	—
55	C₂×C₄×C₇	—	4	2	no	—
56	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₇	C₂³	1	2	yes	—
59	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₇	C₂³	1	2	yes	—
61	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
62	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
63	C₂×#41	C₂×C₇	1	1	yes	characteristic, complemented
64	C₂²×D₄	C₂×C₇	1	1	yes	characteristic, complemented
65	#124	C₂×C₇	1	1	yes	characteristic, complemented
66	#123	C₂×C₇	1	2	yes	complemented
67	C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
68	#121	C₂×C₇	1	1	yes	characteristic, complemented
69	#121	C₂×C₇	1	2	yes	complemented
70	#120	C₂×C₇	1	2	yes	complemented
71	C₄×D₄	C₂×C₇	1	1	yes	characteristic, complemented
72	C₄×D₄	C₂×C₇	1	1	yes	characteristic, complemented
73	C₂×C₂²⋊C₄	C₂×C₇	1	2	yes	complemented
74	C₂²×C₇	—	2	1	no	—
75	C₂²×C₇	—	2	2	no	—
76	C₂²×C₇	—	2	2	no	—
77	C₂²×C₇	—	2	1	no	—
78	C₂²×C₇	—	2	1	no	—
79	C₂²×C₇	—	2	2	no	—
80	C₂²×C₇	—	2	1	no	—
81	C₂²×C₇	—	2	2	no	—
82	C₂²×C₇	—	2	1	no	—
83	C₂²×C₇	—	2	2	no	—
84	C₂²×C₇	—	4	8	no	—
85	C₂²×C₇	C₂×D₄	1	4	yes	—
86	C₂²×C₇	#41	1	2	yes	complemented
87	C₂²×C₇	C₂⁴	1	1	yes	characteristic
88	C₄×C₇	—	2	1	no	—
89	C₄×C₇	—	2	1	no	—
90	C₄×C₇	—	2	2	no	—
91	C₄×C₇	—	2	1	no	—
92	C₄×C₇	—	2	1	no	—
93	C₄×C₇	—	2	2	no	—
94	C₄×C₇	C₂×D₄	1	2	yes	—
95	C₂⁴	C₂²×C₇	1	2	yes	—
96	#41	—	2	4	no	—
97	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
98	C₂×D₄	—	2	4	no	—
99	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
100	C₂×D₄	C₂²×C₇	1	2	yes	—
101	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
102	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
103	C₂×D₄	C₂²×C₇	1	2	yes	complemented
104	C₂×D₄	C₂²×C₇	1	2	yes	complemented
105	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²×C₇	1	2	yes	complemented
107	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic, complemented
108	C₂²×C₄	C₂²×C₇	1	2	yes	complemented
109	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic, complemented
110	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic, complemented
111	C₄⋊C₄	C₂²×C₇	1	2	yes	complemented
112	C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
113	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
114	C₂²⋊C₄	C₂²×C₇	1	2	yes	complemented
115	C₂²⋊C₄	C₂²×C₇	1	2	yes	complemented
116	C₂²⋊C₄	C₂²×C₇	1	2	yes	complemented
117	C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic, complemented
118	C₂²⋊C₄	C₂²×C₇	1	2	yes	complemented
119	C₄²	C₂²×C₇	1	1	yes	characteristic, complemented
120	C₂×C₇	—	2	2	no	—
121	C₂×C₇	—	2	4	no	—
122	C₂×C₇	—	4	1	no	—
123	C₂×C₇	—	4	2	no	—
124	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂³	—	2	4	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	4	2	no	—
130	C₂³	C₂³×C₇	1	2	yes	—
131	C₂³	C₂³×C₇	1	1	yes	characteristic
132	C₂³	C₂³×C₇	1	2	yes	—
133	C₂³	C₂³×C₇	1	1	yes	characteristic
134	Q₈	—	2	2	no	—
135	D₄	—	2	4	no	—
136	D₄	—	2	4	no	—
137	D₄	—	2	2	no	—
138	D₄	—	2	2	no	—
139	D₄	—	4	2	no	—
140	D₄	C₇×D₄	1	4	yes	complemented
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
146	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
147	C₂×C₄	C₂³×C₇	1	2	yes	—
148	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
149	C₂×C₄	C₂³×C₇	1	2	yes	—
150	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
151	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
152	C₇	#546	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²	—	2	1	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	8	no	—
164	C₂²	C₂×C₇×D₄	1	4	yes	—
165	C₂²	C₇×#41	1	2	yes	complemented
166	C₂²	C₂⁴×C₇	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	2	no	—
173	C₄	C₂×C₇×D₄	1	2	yes	—
174	C₂	—	2	2	no	—
175	C₂	—	2	4	no	—
176	C₂	—	4	2	no	—
177	C₂	—	4	1	no	—
178	C₂	C₂²×C₇×D₄	1	1	yes	characteristic
179	C₂	C₂×C₇×#41	1	1	yes	characteristic
180	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. C₂×C₇×#41	C₂	semidirect
2. C₂²×C₇×D₄	C₂	semidirect
3. C₇×#124	C₂	semidirect
4. C₇×#123	C₂	semidirect
5. C₇×C₂²⋊Q₈	C₂	semidirect
6. C₇×#121	C₂	semidirect
7. C₇×#121	C₂	semidirect
8. C₇×#120	C₂	semidirect
9. C₄×C₇×D₄	C₂	semidirect
10. C₄×C₇×D₄	C₂	semidirect
11. C₂×C₇×C₂²⋊C₄	C₂	semidirect
14. C₂×C₇×Q₈	C₂²	semidirect
20. C₂×C₇×D₄	C₂²	semidirect
21. C₂×C₇×D₄	C₂²	semidirect
23. C₂²×C₄×C₇	C₂²	semidirect
24. C₂²×C₄×C₇	C₂²	semidirect
25. C₂²×C₄×C₇	C₂²	semidirect
26. C₇×C₄⋊C₄	C₂²	semidirect
27. C₇×C₄⋊C₄	C₂²	semidirect
28. C₇×C₄⋊C₄	C₂²	semidirect
31. C₇×C₂²⋊C₄	C₂²	semidirect
32. C₇×C₂²⋊C₄	C₂²	semidirect
33. C₇×C₂²⋊C₄	C₂²	semidirect
34. C₇×C₂²⋊C₄	C₂²	semidirect
35. C₇×C₂²⋊C₄	C₂²	semidirect
36. C₄²×C₇	C₂²	semidirect
37. #546	C₇	direct (N × H)
51. C₇×D₄	D₄	semidirect
63. C₂×#41	C₂×C₇	semidirect
64. C₂²×D₄	C₂×C₇	semidirect
65. #124	C₂×C₇	semidirect
66. #123	C₂×C₇	semidirect
67. C₂²⋊Q₈	C₂×C₇	semidirect
68. #121	C₂×C₇	semidirect
69. #121	C₂×C₇	semidirect
70. #120	C₂×C₇	semidirect
71. C₄×D₄	C₂×C₇	semidirect
72. C₄×D₄	C₂×C₇	semidirect
73. C₂×C₂²⋊C₄	C₂×C₇	semidirect
86. C₂²×C₇	#41	semidirect
97. C₂×Q₈	C₂²×C₇	semidirect
103. C₂×D₄	C₂²×C₇	semidirect
104. C₂×D₄	C₂²×C₇	semidirect
106. C₂²×C₄	C₂²×C₇	semidirect
107. C₂²×C₄	C₂²×C₇	semidirect
108. C₂²×C₄	C₂²×C₇	semidirect
109. C₄⋊C₄	C₂²×C₇	semidirect
110. C₄⋊C₄	C₂²×C₇	semidirect
111. C₄⋊C₄	C₂²×C₇	semidirect
114. C₂²⋊C₄	C₂²×C₇	semidirect
115. C₂²⋊C₄	C₂²×C₇	semidirect
116. C₂²⋊C₄	C₂²×C₇	semidirect
117. C₂²⋊C₄	C₂²×C₇	semidirect
118. C₂²⋊C₄	C₂²×C₇	semidirect
119. C₄²	C₂²×C₇	semidirect
140. D₄	C₇×D₄	semidirect
152. C₇	#546	direct (N × H)
165. C₂²	C₇×#41	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#41	2
C₂²×C₇×D₄	2
C₇×#124	2
C₇×#123	2
C₇×C₂²⋊Q₈	2
C₇×#121	2
C₇×#120	2
C₄×C₇×D₄	2
C₂×C₇×C₂²⋊C₄	2
#546	7

Subgroups — C7×#546 #90567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#546 #90567