Subgroups — C₂²×D₇×2₋¹⁺⁴ #14267684

Order 1792.

59806 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₇×2₋¹⁺⁴	C₂	1	96	yes	maximal, direct product factor
3	C₂²×D₇×#41	C₂	1	10	yes	maximal, complemented
4	C₂²×#6554	C₂	1	10	yes	maximal, complemented
5	C₂²×#6551	C₂	1	5	yes	maximal, complemented
6	C₂³×Q₈×D₇	C₂	1	5	yes	maximal, complemented
7	C₂×C₇×2₋¹⁺⁴	C₂²	1	48	yes	complemented
8	C₂²×C₇×#41	C₂²	1	10	yes	complemented
9	C₂³×C₇×Q₈	C₂²	1	5	yes	complemented
10	D₇×2₋¹⁺⁴	C₂²	1	1024	yes	direct product factor
11	C₂×#6554	C₂²	1	480	yes	complemented
12	C₂×D₇×#41	C₂²	1	480	yes	complemented
13	C₂×#6551	C₂²	1	240	yes	complemented
14	C₂²×#1161	C₂²	1	10	yes	complemented
15	C₂²×Q₈×D₇	C₂²	1	10	yes	—
16	C₂²×Q₈×D₇	C₂²	1	20	yes	complemented
17	C₂²×Q₈×D₇	C₂²	1	240	yes	complemented
18	C₂²×#1159	C₂²	1	30	yes	complemented
19	C₂²×D₄×D₇	C₂²	1	10	yes	—
20	C₂²×#1157	C₂²	1	30	yes	complemented
21	C₂³×C₄×D₇	C₂²	1	15	yes	—
22	C₂³×Dic₁₄	C₂²	1	15	yes	complemented
23	C₂³×2₋¹⁺⁴	—	7	1	no	maximal, Sylow
24	C₂⁴×D₇	C₂³	1	5	yes	—
25	C₇×2₋¹⁺⁴	C₂³	1	256	yes	complemented
26	C₂×C₇×#41	C₂³	1	240	yes	complemented
27	C₂²×C₇×Q₈	C₂³	1	10	yes	—
28	C₂²×C₇×Q₈	C₂³	1	120	yes	complemented
29	C₂²×C₇×D₄	C₂³	1	10	yes	—
30	C₂³×C₄×C₇	C₂³	1	15	yes	—
31	C₂²×C₂²⋊D₇	C₂³	1	20	yes	—
32	C₂³×Dic₇	C₂³	1	15	yes	—
33	#6554	C₂³	1	2560	yes	complemented
34	D₇×#41	C₂³	1	2560	yes	complemented
35	#6551	C₂³	1	1280	yes	complemented
36	C₂×#1161	C₂³	1	240	yes	complemented
37	C₂×Q₈×D₇	C₂³	1	240	yes	—
38	C₂×Q₈×D₇	C₂³	1	480	yes	complemented
39	C₂×Q₈×D₇	C₂³	1	1280	yes	complemented
40	C₂×#1159	C₂³	1	720	yes	complemented
41	C₂×D₄×D₇	C₂³	1	240	yes	—
42	C₂×#1157	C₂³	1	720	yes	complemented
43	C₂²×D₂₈	C₂³	1	10	yes	—
44	C₂²×C₄×D₇	C₂³	1	10	yes	—
45	C₂²×C₄×D₇	C₂³	1	30	yes	—
46	C₂²×C₄×D₇	C₂³	1	360	yes	—
47	C₂²×Dic₁₄	C₂³	1	30	yes	—
48	C₂²×Dic₁₄	C₂³	1	360	yes	complemented
49	C₂²×2₋¹⁺⁴	—	7	96	no	—
50	C₂²×2₋¹⁺⁴	—	7	10	no	—
51	C₂²×2₋¹⁺⁴	—	7	5	no	—
52	C₂²×2₋¹⁺⁴	D₇	1	1	yes	characteristic, direct product factor
53	C₂³×#41	—	7	10	no	—
54	C₂⁴×Q₈	—	7	5	no	—
55	C₂⁴×C₇	C₂⁴	1	5	yes	—
56	C₂³×D₇	—	2	40	no	—
57	C₂³×D₇	C₂⁴	1	5	yes	—
58	C₂³×D₇	C₂⁴	1	60	yes	—
59	C₂³×D₇	C₂⁴	1	1	yes	characteristic
60	C₇×#41	C₂⁴	1	640	yes	complemented
61	C₂×C₇×Q₈	C₂⁴	1	120	yes	—
62	C₂×C₇×Q₈	C₂⁴	1	320	yes	complemented
63	C₂×C₇×D₄	C₂⁴	1	120	yes	—
64	C₂²×C₄×C₇	C₂⁴	1	10	yes	—
65	C₂²×C₄×C₇	C₂⁴	1	180	yes	—
66	C₂×C₂²⋊D₇	C₂⁴	1	240	yes	—
67	C₂²×Dic₇	C₂⁴	1	10	yes	—
68	C₂²×Dic₇	C₂⁴	1	180	yes	—
69	#1161	C₂⁴	1	640	yes	complemented
70	Q₈×D₇	C₂⁴	1	640	yes	—
71	Q₈×D₇	C₂⁴	1	1280	yes	complemented
72	#1159	C₂⁴	1	1920	yes	complemented
73	D₄×D₇	C₂⁴	1	640	yes	—
74	#1157	C₂⁴	1	1920	yes	complemented
75	C₂×D₂₈	C₂⁴	1	120	yes	—
76	C₂×C₄×D₇	C₂⁴	1	120	yes	—
77	C₂×C₄×D₇	C₂⁴	1	360	yes	—
78	C₂×C₄×D₇	C₂⁴	1	960	yes	—
79	C₂×Dic₁₄	C₂⁴	1	360	yes	—
80	C₂×Dic₁₄	C₂⁴	1	960	yes	complemented
81	C₂×2₋¹⁺⁴	—	7	1024	no	—
82	C₂×2₋¹⁺⁴	—	7	480	no	—
83	C₂×2₋¹⁺⁴	—	7	240	no	—
84	C₂×2₋¹⁺⁴	C₂×D₇	1	48	yes	direct product factor
85	C₂²×#41	—	7	480	no	—
86	C₂²×#41	—	7	10	no	—
87	C₂²×#41	—	7	30	no	—
88	C₂²×#41	—	7	30	no	—
89	C₂²×#41	C₂×D₇	1	10	yes	complemented
90	C₂³×Q₈	—	7	10	no	—
91	C₂³×Q₈	—	7	20	no	—
92	C₂³×Q₈	—	7	240	no	—
93	C₂³×Q₈	—	7	15	no	—
94	C₂³×Q₈	C₂×D₇	1	5	yes	complemented
95	C₂³×D₄	—	7	10	no	—
96	C₂⁴×C₄	—	7	15	no	—
97	C₂³×C₇	—	2	20	no	—
98	C₂³×C₇	C₂⁵	1	30	yes	—
99	C₂³×C₇	C₂⁵	1	1	yes	characteristic
100	C₂²×D₇	—	2	20	no	—
101	C₂²×D₇	—	2	240	no	—
102	C₂²×D₇	2₋¹⁺⁴	1	8	yes	direct product factor
103	C₂²×D₇	C₂⁵	1	30	yes	—
104	C₂²×D₇	C₂⁵	1	80	yes	—
105	C₂²×D₇	C₂⁵	1	6	yes	—
106	C₇×Q₈	C₂⁵	1	160	yes	—
107	C₇×D₄	C₂⁵	1	160	yes	—
108	C₂×C₄×C₇	C₂⁵	1	60	yes	—
109	C₂×C₄×C₇	C₂⁵	1	240	yes	—
110	C₂²⋊D₇	C₂⁵	1	320	yes	—
111	C₂×Dic₇	C₂⁵	1	60	yes	—
112	C₂×Dic₇	C₂⁵	1	240	yes	—
113	D₂₈	C₂⁵	1	160	yes	—
114	C₄×D₇	C₂⁵	1	160	yes	—
115	C₄×D₇	C₂⁵	1	480	yes	—
116	Dic₁₄	C₂⁵	1	480	yes	—
117	C₂⁵	—	7	5	no	—
118	2₋¹⁺⁴	—	7	2560	no	—
119	2₋¹⁺⁴	—	7	1280	no	—
120	2₋¹⁺⁴	C₂²×D₇	1	256	yes	direct product factor
121	C₂×#41	—	7	2560	no	—
122	C₂×#41	—	7	240	no	—
123	C₂×#41	—	7	720	no	—
124	C₂×#41	—	7	720	no	—
125	C₂×#41	C₂²×D₇	1	240	yes	complemented
126	C₂²×Q₈	—	7	240	no	—
127	C₂²×Q₈	—	7	480	no	—
128	C₂²×Q₈	—	7	1280	no	—
129	C₂²×Q₈	—	7	30	no	—
130	C₂²×Q₈	—	7	360	no	—
131	C₂²×Q₈	C₂²×D₇	1	10	yes	—
132	C₂²×Q₈	C₂²×D₇	1	120	yes	complemented
133	C₂²×D₄	—	7	20	no	—
134	C₂²×D₄	—	7	240	no	—
135	C₂²×D₄	—	7	10	no	—
136	C₂²×D₄	C₂²×D₇	1	10	yes	—
137	C₂³×C₄	—	7	15	no	—
138	C₂³×C₄	—	7	10	no	—
139	C₂³×C₄	—	7	360	no	—
140	C₂³×C₄	—	7	30	no	—
141	C₂³×C₄	C₂²×D₇	1	15	yes	—
142	C₂²×C₇	—	2	60	no	—
143	C₂²×C₇	C₂×2₋¹⁺⁴	1	4	yes	complemented
144	C₂²×C₇	C₂⁶	1	20	yes	—
145	C₂²×C₇	C₂⁶	1	3	yes	—
146	C₂×D₇	—	2	60	no	—
147	C₂×D₇	—	2	160	no	—
148	C₂×D₇	C₂×2₋¹⁺⁴	1	24	yes	direct product factor
149	C₂×D₇	C₂⁶	1	20	yes	—
150	C₂×D₇	C₂⁶	1	4	yes	—
151	C₄×C₇	C₂⁶	1	40	yes	—
152	Dic₇	C₂⁶	1	40	yes	—
153	C₂⁴	—	7	1	no	—
154	C₂⁴	—	7	60	no	—
155	C₂⁴	—	7	5	no	—
156	C₂⁴	—	14	40	no	—
157	C₂⁴	C₂³×D₇	1	5	yes	—
158	#41	—	7	640	no	—
159	#41	—	7	1920	no	—
160	#41	—	7	1920	no	—
161	#41	C₂³×D₇	1	640	yes	complemented
162	C₂×Q₈	—	7	640	no	—
163	C₂×Q₈	—	7	1280	no	—
164	C₂×Q₈	—	7	360	no	—
165	C₂×Q₈	—	7	960	no	—
166	C₂×Q₈	C₂³×D₇	1	120	yes	—
167	C₂×Q₈	C₂³×D₇	1	320	yes	complemented
168	C₂×D₄	—	7	240	no	—
169	C₂×D₄	—	7	640	no	—
170	C₂×D₄	—	7	120	no	—
171	C₂×D₄	C₂³×D₇	1	120	yes	—
172	C₂²×C₄	—	7	180	no	—
173	C₂²×C₄	—	7	10	no	—
174	C₂²×C₄	—	7	120	no	—
175	C₂²×C₄	—	7	360	no	—
176	C₂²×C₄	—	7	960	no	—
177	C₂²×C₄	C₂³×D₇	1	180	yes	—
178	C₂²×C₄	C₂³×D₇	1	10	yes	—
179	C₂×C₇	—	2	20	no	—
180	C₂×C₇	C₂²×2₋¹⁺⁴	1	6	yes	complemented
181	C₂×C₇	C₂⁷	1	1	yes	characteristic
182	D₇	—	2	20	no	—
183	D₇	C₂²×2₋¹⁺⁴	1	8	yes	direct product factor
184	C₂³	—	2	20	no	—
185	C₂³	—	7	6	no	—
186	C₂³	—	7	8	no	—
187	C₂³	—	7	80	no	—
188	C₂³	—	7	30	no	—
189	C₂³	—	14	240	no	—
190	C₂³	—	14	20	no	—
191	C₂³	C₂⁴×D₇	1	30	yes	—
192	C₂³	C₂⁴×D₇	1	1	yes	characteristic
193	Q₈	—	7	480	no	—
194	Q₈	C₂⁴×D₇	1	160	yes	—
195	D₄	—	7	320	no	—
196	D₄	—	7	160	no	—
197	D₄	C₂⁴×D₇	1	160	yes	—
198	C₂×C₄	—	7	60	no	—
199	C₂×C₄	—	7	240	no	—
200	C₂×C₄	—	7	160	no	—
201	C₂×C₄	—	7	480	no	—
202	C₂×C₄	C₂⁴×D₇	1	60	yes	—
203	C₂×C₄	C₂⁴×D₇	1	240	yes	—
204	C₇	C₂³×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
205	C₂²	—	2	60	no	—
206	C₂²	—	7	24	no	—
207	C₂²	—	7	4	no	—
208	C₂²	—	7	20	no	—
209	C₂²	—	14	160	no	—
210	C₂²	—	14	60	no	—
211	C₂²	D₇×2₋¹⁺⁴	1	4	yes	direct product factor
212	C₂²	C₂⁵×D₇	1	20	yes	—
213	C₂²	C₂⁵×D₇	1	3	yes	—
214	C₄	—	7	40	no	—
215	C₄	C₂⁵×D₇	1	40	yes	—
216	C₂	—	2	20	no	—
217	C₂	—	7	8	no	—
218	C₂	—	14	20	no	—
219	C₂	C₂×D₇×2₋¹⁺⁴	1	6	yes	direct product factor
220	C₂	C₂⁶×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₇
commutator, cosocle	C₂×C₇	C₂⁷
Frattini	C₂	C₂⁶×D₇
Fitting	C₂²×C₇×2₋¹⁺⁴	C₂
socle	C₂³×C₇	C₂⁵
hypercenter	C₂²×2₋¹⁺⁴	D₇
hypocenter	C₇	C₂³×2₋¹⁺⁴
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×2₋¹⁺⁴	C₂	semidirect
2. C₂×D₇×2₋¹⁺⁴	C₂	direct (N × H)
3. C₂²×D₇×#41	C₂	semidirect
4. C₂²×#6554	C₂	semidirect
5. C₂²×#6551	C₂	semidirect
6. C₂³×Q₈×D₇	C₂	semidirect
7. C₂×C₇×2₋¹⁺⁴	C₂²	semidirect
8. C₂²×C₇×#41	C₂²	semidirect
9. C₂³×C₇×Q₈	C₂²	semidirect
10. D₇×2₋¹⁺⁴	C₂²	direct (N × H)
11. C₂×#6554	C₂²	semidirect
12. C₂×D₇×#41	C₂²	semidirect
13. C₂×#6551	C₂²	semidirect
14. C₂²×#1161	C₂²	semidirect
16. C₂²×Q₈×D₇	C₂²	semidirect
17. C₂²×Q₈×D₇	C₂²	semidirect
18. C₂²×#1159	C₂²	semidirect
20. C₂²×#1157	C₂²	semidirect
22. C₂³×Dic₁₄	C₂²	semidirect
25. C₇×2₋¹⁺⁴	C₂³	semidirect
26. C₂×C₇×#41	C₂³	semidirect
28. C₂²×C₇×Q₈	C₂³	semidirect
33. #6554	C₂³	semidirect
34. D₇×#41	C₂³	semidirect
35. #6551	C₂³	semidirect
36. C₂×#1161	C₂³	semidirect
38. C₂×Q₈×D₇	C₂³	semidirect
39. C₂×Q₈×D₇	C₂³	semidirect
40. C₂×#1159	C₂³	semidirect
42. C₂×#1157	C₂³	semidirect
48. C₂²×Dic₁₄	C₂³	semidirect
52. C₂²×2₋¹⁺⁴	D₇	direct (N × H)
60. C₇×#41	C₂⁴	semidirect
62. C₂×C₇×Q₈	C₂⁴	semidirect
69. #1161	C₂⁴	semidirect
71. Q₈×D₇	C₂⁴	semidirect
72. #1159	C₂⁴	semidirect
74. #1157	C₂⁴	semidirect
80. C₂×Dic₁₄	C₂⁴	semidirect
84. C₂×2₋¹⁺⁴	C₂×D₇	direct (N × H)
89. C₂²×#41	C₂×D₇	semidirect
94. C₂³×Q₈	C₂×D₇	semidirect
102. C₂²×D₇	2₋¹⁺⁴	direct (N × H)
120. 2₋¹⁺⁴	C₂²×D₇	direct (N × H)
125. C₂×#41	C₂²×D₇	semidirect
132. C₂²×Q₈	C₂²×D₇	semidirect
143. C₂²×C₇	C₂×2₋¹⁺⁴	semidirect
148. C₂×D₇	C₂×2₋¹⁺⁴	direct (N × H)
161. #41	C₂³×D₇	semidirect
167. C₂×Q₈	C₂³×D₇	semidirect
180. C₂×C₇	C₂²×2₋¹⁺⁴	semidirect
183. D₇	C₂²×2₋¹⁺⁴	direct (N × H)
204. C₇	C₂³×2₋¹⁺⁴	semidirect
211. C₂²	D₇×2₋¹⁺⁴	direct (N × H)
219. C₂	C₂×D₇×2₋¹⁺⁴	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×2₋¹⁺⁴	2
C₂×D₇×2₋¹⁺⁴	2
C₂²×D₇×#41	2
C₂²×#6554	2
C₂²×#6551	2
C₂³×Q₈×D₇	2
C₂³×2₋¹⁺⁴	7

Subgroups — C22×D7×2−1+4 #14267684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₇×2₋¹⁺⁴ #14267684