Subgroups — C₂×C₇×#540 #11744404

Order 896.

1260 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#540	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#123	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×#121	C₂	1	4	yes	maximal, complemented
6	C₂×C₇×#121	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₇×D₄	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₄	C₂²	1	2	yes	—
10	C₂²×C₇×D₄	C₂²	1	4	yes	—
11	C₂²×C₇×D₄	C₂²	1	4	yes	—
12	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₇	C₂²	1	4	yes	—
14	C₇×#127	C₂²	1	8	yes	complemented
15	C₇×#125	C₂²	1	8	yes	complemented
16	C₇×#123	C₂²	1	32	yes	complemented
17	C₇×#121	C₂²	1	32	yes	complemented
18	C₇×#121	C₂²	1	16	yes	complemented
19	C₄×C₇×D₄	C₂²	1	16	yes	complemented
20	C₇×#117	C₂²	1	8	yes	complemented
21	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
24	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₂×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
27	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
29	C₂×#540	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂⁴×C₇	C₂³	1	1	yes	characteristic
31	C₂⁴×C₇	C₂³	1	4	yes	—
32	C₂×C₇×D₄	—	2	8	no	—
33	C₂×C₇×D₄	—	4	8	no	—
34	C₂×C₇×D₄	—	4	8	no	—
35	C₂×C₇×D₄	C₂³	1	8	yes	—
36	C₂×C₇×D₄	C₂³	1	16	yes	—
37	C₂×C₇×D₄	C₂³	1	16	yes	—
38	C₂²×C₄×C₇	—	2	4	no	—
39	C₂²×C₄×C₇	—	4	8	no	—
40	C₂²×C₄×C₇	C₂³	1	4	yes	—
41	C₂²×C₄×C₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂³	1	4	yes	—
45	C₂²×C₄×C₇	C₂³	1	16	yes	—
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₇×C₄⋊C₄	C₂³	1	8	yes	—
48	C₇×C₄⋊C₄	C₂³	1	8	yes	—
49	C₇×C₄⋊C₄	C₂³	1	16	yes	—
50	C₇×C₂²⋊C₄	C₂³	1	8	yes	—
51	C₇×C₂²⋊C₄	C₂³	1	16	yes	—
52	C₇×C₂²⋊C₄	C₂³	1	16	yes	—
53	C₄²×C₇	C₂³	1	4	yes	—
54	C₄²×C₇	C₂³	1	4	yes	—
55	#540	C₂×C₇	1	16	yes	direct product factor
56	C₂×#127	C₂×C₇	1	1	yes	characteristic, complemented
57	C₂×#125	C₂×C₇	1	1	yes	characteristic, complemented
58	C₂×#123	C₂×C₇	1	4	yes	complemented
59	C₂×#121	C₂×C₇	1	4	yes	complemented
60	C₂×#121	C₂×C₇	1	2	yes	complemented
61	C₂×C₄×D₄	C₂×C₇	1	2	yes	complemented
62	C₂×#117	C₂×C₇	1	1	yes	characteristic, complemented
63	C₂³×C₇	—	2	2	no	—
64	C₂³×C₇	—	2	8	no	—
65	C₂³×C₇	—	2	8	no	—
66	C₂³×C₇	—	2	4	no	—
67	C₂³×C₇	—	2	8	no	—
68	C₂³×C₇	C₂⁴	1	2	yes	—
69	C₂³×C₇	C₂⁴	1	8	yes	—
70	C₂³×C₇	C₂⁴	1	1	yes	characteristic
71	C₇×D₄	—	2	16	no	—
72	C₇×D₄	—	4	16	no	—
73	C₇×D₄	—	4	16	no	—
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	—	2	4	no	—
76	C₂×C₄×C₇	—	2	8	no	—
77	C₂×C₄×C₇	—	2	4	no	—
78	C₂×C₄×C₇	—	2	8	no	—
79	C₂×C₄×C₇	—	4	16	no	—
80	C₂×C₄×C₇	#41	1	4	yes	—
81	C₂×C₄×C₇	C₂⁴	1	4	yes	—
82	C₂×C₄×C₇	C₂⁴	1	4	yes	—
83	C₂×C₄×C₇	C₂⁴	1	2	yes	—
84	C₂×C₄×C₇	C₂⁴	1	8	yes	—
85	C₂×C₄×C₇	C₂⁴	1	2	yes	—
86	C₂²×D₄	C₂²×C₇	1	2	yes	—
87	C₂²×D₄	C₂²×C₇	1	4	yes	—
88	C₂²×D₄	C₂²×C₇	1	4	yes	—
89	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
90	C₂³×C₄	C₂²×C₇	1	4	yes	—
91	#127	C₂²×C₇	1	8	yes	complemented
92	#125	C₂²×C₇	1	8	yes	complemented
93	#123	C₂²×C₇	1	32	yes	complemented
94	#121	C₂²×C₇	1	32	yes	complemented
95	#121	C₂²×C₇	1	16	yes	complemented
96	C₄×D₄	C₂²×C₇	1	16	yes	complemented
97	#117	C₂²×C₇	1	8	yes	complemented
98	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
99	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
101	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
102	C₂×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
103	C₂×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
104	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
105	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
106	C₂²×C₇	—	2	2	no	—
107	C₂²×C₇	—	2	8	no	—
108	C₂²×C₇	—	2	8	no	—
109	C₂²×C₇	—	2	4	no	—
110	C₂²×C₇	—	2	8	no	—
111	C₂²×C₇	—	4	4	no	—
112	C₂²×C₇	—	4	16	no	—
113	C₂²×C₇	C₂×#41	1	2	yes	—
114	C₂²×C₇	H(5,𝔽₂)	1	4	yes	—
115	C₂²×C₇	C₂⁵	1	1	yes	characteristic
116	C₄×C₇	—	2	2	no	—
117	C₄×C₇	—	2	4	no	—
118	C₄×C₇	—	2	8	no	—
119	C₄×C₇	—	2	4	no	—
120	C₄×C₇	C₂×#41	1	4	yes	—
121	C₂⁴	C₂³×C₇	1	1	yes	characteristic
122	C₂⁴	C₂³×C₇	1	4	yes	—
123	C₂×D₄	—	2	8	no	—
124	C₂×D₄	—	4	8	no	—
125	C₂×D₄	—	4	8	no	—
126	C₂×D₄	C₂³×C₇	1	8	yes	—
127	C₂×D₄	C₂³×C₇	1	16	yes	—
128	C₂×D₄	C₂³×C₇	1	16	yes	—
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	4	8	no	—
131	C₂²×C₄	C₂³×C₇	1	2	yes	—
132	C₂²×C₄	C₂³×C₇	1	2	yes	—
133	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₇	1	4	yes	—
135	C₂²×C₄	C₂³×C₇	1	4	yes	—
136	C₂²×C₄	C₂³×C₇	1	16	yes	—
137	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
138	C₄⋊C₄	C₂³×C₇	1	8	yes	—
139	C₄⋊C₄	C₂³×C₇	1	8	yes	—
140	C₄⋊C₄	C₂³×C₇	1	16	yes	—
141	C₂²⋊C₄	C₂³×C₇	1	8	yes	—
142	C₂²⋊C₄	C₂³×C₇	1	16	yes	—
143	C₂²⋊C₄	C₂³×C₇	1	16	yes	—
144	C₄²	C₂³×C₇	1	4	yes	—
145	C₄²	C₂³×C₇	1	4	yes	—
146	C₂×C₇	—	4	2	no	—
147	C₂×C₇	—	4	8	no	—
148	C₂×C₇	#540	1	4	yes	direct product factor
149	C₂×C₇	C₂²×#41	1	1	yes	characteristic
150	C₂×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
151	C₂³	—	2	2	no	—
152	C₂³	—	2	8	no	—
153	C₂³	—	2	8	no	—
154	C₂³	—	2	4	no	—
155	C₂³	—	2	8	no	—
156	C₂³	C₂⁴×C₇	1	2	yes	—
157	C₂³	C₂⁴×C₇	1	8	yes	—
158	C₂³	C₂⁴×C₇	1	1	yes	characteristic
159	D₄	—	2	16	no	—
160	D₄	—	4	16	no	—
161	D₄	—	4	16	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	8	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	4	16	no	—
168	C₂×C₄	C₇×#41	1	4	yes	—
169	C₂×C₄	C₂⁴×C₇	1	4	yes	—
170	C₂×C₄	C₂⁴×C₇	1	4	yes	—
171	C₂×C₄	C₂⁴×C₇	1	2	yes	—
172	C₂×C₄	C₂⁴×C₇	1	8	yes	—
173	C₂×C₄	C₂⁴×C₇	1	2	yes	—
174	C₇	C₂×#540	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	—	2	2	no	—
176	C₂²	—	2	8	no	—
177	C₂²	—	2	8	no	—
178	C₂²	—	2	4	no	—
179	C₂²	—	2	8	no	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	16	no	—
182	C₂²	C₂×C₇×#41	1	2	yes	—
183	C₂²	C₇×H(5,𝔽₂)	1	4	yes	—
184	C₂²	C₂⁵×C₇	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	2	8	no	—
188	C₄	—	2	4	no	—
189	C₄	C₂×C₇×#41	1	4	yes	—
190	C₂	—	4	2	no	—
191	C₂	—	4	8	no	—
192	C₂	C₇×#540	1	4	yes	direct product factor
193	C₂	C₂²×C₇×#41	1	1	yes	characteristic
194	C₂	C₂×C₇×H(5,𝔽₂)	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₇×#540	C₂	direct (N × H)
2. C₂×C₇×#127	C₂	semidirect
3. C₂×C₇×#125	C₂	semidirect
4. C₂×C₇×#123	C₂	semidirect
5. C₂×C₇×#121	C₂	semidirect
6. C₂×C₇×#121	C₂	semidirect
7. C₂×C₄×C₇×D₄	C₂	semidirect
8. C₂×C₇×#117	C₂	semidirect
14. C₇×#127	C₂²	semidirect
15. C₇×#125	C₂²	semidirect
16. C₇×#123	C₂²	semidirect
17. C₇×#121	C₂²	semidirect
18. C₇×#121	C₂²	semidirect
19. C₄×C₇×D₄	C₂²	semidirect
20. C₇×#117	C₂²	semidirect
29. C₂×#540	C₇	direct (N × H)
55. #540	C₂×C₇	direct (N × H)
56. C₂×#127	C₂×C₇	semidirect
57. C₂×#125	C₂×C₇	semidirect
58. C₂×#123	C₂×C₇	semidirect
59. C₂×#121	C₂×C₇	semidirect
60. C₂×#121	C₂×C₇	semidirect
61. C₂×C₄×D₄	C₂×C₇	semidirect
62. C₂×#117	C₂×C₇	semidirect
91. #127	C₂²×C₇	semidirect
92. #125	C₂²×C₇	semidirect
93. #123	C₂²×C₇	semidirect
94. #121	C₂²×C₇	semidirect
95. #121	C₂²×C₇	semidirect
96. C₄×D₄	C₂²×C₇	semidirect
97. #117	C₂²×C₇	semidirect
148. C₂×C₇	#540	direct (N × H)
174. C₇	C₂×#540	direct (N × H)
192. C₂	C₇×#540	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×#540	2
C₂×C₇×#127	2
C₂×C₇×#125	2
C₂×C₇×#123	2
C₂×C₇×#121	2
C₂×C₄×C₇×D₄	2
C₂×C₇×#117	2
C₂×#540	7

Subgroups — C2×C7×#540 #11744404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#540 #11744404