Subgroups — C₂³×D₇×#41 #14267677

Order 1792.

65470 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₇×#41	C₂	1	112	yes	maximal, direct product factor
3	C₂³×Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#1161	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄×D₇	C₂	1	3	yes	maximal, complemented
6	C₂³×#1159	C₂	1	3	yes	maximal, complemented
7	C₂³×#1157	C₂	1	3	yes	maximal, complemented
8	C₂⁴×C₄×D₇	C₂	1	3	yes	maximal, complemented
9	C₂⁵×D₇	C₂²	1	3	yes	—
10	C₂²×C₇×#41	C₂²	1	56	yes	complemented
11	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₇×D₄	C₂²	1	3	yes	complemented
13	C₂⁴×C₄×C₇	C₂²	1	3	yes	complemented
14	C₂³×C₂²⋊D₇	C₂²	1	6	yes	complemented
15	C₂⁴×Dic₇	C₂²	1	3	yes	complemented
16	C₂×D₇×#41	C₂²	1	1792	yes	direct product factor
17	C₂²×#1161	C₂²	1	56	yes	complemented
18	C₂²×Q₈×D₇	C₂²	1	56	yes	complemented
19	C₂²×#1159	C₂²	1	168	yes	complemented
20	C₂²×D₄×D₇	C₂²	1	168	yes	complemented
21	C₂²×#1157	C₂²	1	168	yes	complemented
22	C₂³×D₂₈	C₂²	1	3	yes	complemented
23	C₂³×C₄×D₇	C₂²	1	1	yes	characteristic
24	C₂³×C₄×D₇	C₂²	1	3	yes	—
25	C₂³×C₄×D₇	C₂²	1	168	yes	complemented
26	C₂³×C₄×D₇	C₂²	1	3	yes	complemented
27	C₂³×C₄×D₇	C₂²	1	3	yes	complemented
28	C₂³×Dic₁₄	C₂²	1	3	yes	complemented
29	C₂⁴×#41	—	7	1	no	maximal, Sylow
30	C₂⁵×C₇	C₂³	1	3	yes	—
31	C₂⁴×D₇	—	2	48	no	—
32	C₂⁴×D₇	C₂³	1	3	yes	—
33	C₂⁴×D₇	C₂³	1	84	yes	—
34	C₂⁴×D₇	C₂³	1	1	yes	characteristic
35	C₂×C₇×#41	C₂³	1	448	yes	complemented
36	C₂²×C₇×Q₈	C₂³	1	28	yes	complemented
37	C₂²×C₇×D₄	C₂³	1	84	yes	complemented
38	C₂³×C₄×C₇	C₂³	1	3	yes	—
39	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₇	C₂³	1	84	yes	complemented
41	C₂²×C₂²⋊D₇	C₂³	1	168	yes	complemented
42	C₂³×Dic₇	C₂³	1	3	yes	—
43	C₂³×Dic₇	C₂³	1	1	yes	characteristic
44	C₂³×Dic₇	C₂³	1	84	yes	complemented
45	D₇×#41	C₂³	1	4096	yes	direct product factor
46	C₂×#1161	C₂³	1	448	yes	complemented
47	C₂×Q₈×D₇	C₂³	1	448	yes	complemented
48	C₂×#1159	C₂³	1	1344	yes	complemented
49	C₂×D₄×D₇	C₂³	1	1344	yes	complemented
50	C₂×#1157	C₂³	1	1344	yes	complemented
51	C₂²×D₂₈	C₂³	1	84	yes	complemented
52	C₂²×C₄×D₇	C₂³	1	28	yes	—
53	C₂²×C₄×D₇	C₂³	1	84	yes	—
54	C₂²×C₄×D₇	C₂³	1	1344	yes	complemented
55	C₂²×C₄×D₇	C₂³	1	84	yes	complemented
56	C₂²×C₄×D₇	C₂³	1	84	yes	complemented
57	C₂²×Dic₁₄	C₂³	1	84	yes	complemented
58	C₂³×#41	—	7	112	no	—
59	C₂³×#41	—	7	1	no	—
60	C₂³×#41	—	7	3	no	—
61	C₂³×#41	—	7	3	no	—
62	C₂³×#41	D₇	1	1	yes	characteristic, direct product factor
63	C₂⁴×Q₈	—	7	1	no	—
64	C₂⁴×D₄	—	7	3	no	—
65	C₂⁵×C₄	—	7	3	no	—
66	C₂⁴×C₇	—	2	24	no	—
67	C₂⁴×C₇	C₂⁴	1	42	yes	—
68	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
69	C₂³×D₇	—	2	24	no	—
70	C₂³×D₇	—	2	672	no	—
71	C₂³×D₇	#41	1	16	yes	direct product factor
72	C₂³×D₇	C₂⁴	1	42	yes	—
73	C₂³×D₇	C₂⁴	1	336	yes	—
74	C₂³×D₇	C₂⁴	1	14	yes	—
75	C₇×#41	C₂⁴	1	512	yes	complemented
76	C₂×C₇×Q₈	C₂⁴	1	112	yes	complemented
77	C₂×C₇×D₄	C₂⁴	1	336	yes	complemented
78	C₂²×C₄×C₇	C₂⁴	1	42	yes	—
79	C₂²×C₄×C₇	C₂⁴	1	14	yes	—
80	C₂²×C₄×C₇	C₂⁴	1	336	yes	complemented
81	C₂×C₂²⋊D₇	C₂⁴	1	672	yes	complemented
82	C₂²×Dic₇	C₂⁴	1	42	yes	—
83	C₂²×Dic₇	C₂⁴	1	14	yes	—
84	C₂²×Dic₇	C₂⁴	1	336	yes	complemented
85	#1161	C₂⁴	1	512	yes	complemented
86	Q₈×D₇	C₂⁴	1	512	yes	complemented
87	#1159	C₂⁴	1	1536	yes	complemented
88	D₄×D₇	C₂⁴	1	1536	yes	complemented
89	#1157	C₂⁴	1	1536	yes	complemented
90	C₂×D₂₈	C₂⁴	1	336	yes	complemented
91	C₂×C₄×D₇	C₂⁴	1	112	yes	—
92	C₂×C₄×D₇	C₂⁴	1	336	yes	—
93	C₂×C₄×D₇	C₂⁴	1	1536	yes	complemented
94	C₂×C₄×D₇	C₂⁴	1	336	yes	complemented
95	C₂×C₄×D₇	C₂⁴	1	336	yes	complemented
96	C₂×Dic₁₄	C₂⁴	1	336	yes	complemented
97	C₂⁶	—	7	3	no	—
98	C₂²×#41	—	7	1792	no	—
99	C₂²×#41	—	7	56	no	—
100	C₂²×#41	—	7	168	no	—
101	C₂²×#41	—	7	168	no	—
102	C₂²×#41	C₂×D₇	1	56	yes	direct product factor
103	C₂³×Q₈	—	7	56	no	—
104	C₂³×Q₈	—	7	3	no	—
105	C₂³×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
106	C₂³×D₄	—	7	6	no	—
107	C₂³×D₄	—	7	168	no	—
108	C₂³×D₄	—	7	3	no	—
109	C₂³×D₄	C₂×D₇	1	3	yes	complemented
110	C₂⁴×C₄	—	7	3	no	—
111	C₂⁴×C₄	—	7	3	no	—
112	C₂⁴×C₄	—	7	3	no	—
113	C₂⁴×C₄	—	7	1	no	—
114	C₂⁴×C₄	—	7	3	no	—
115	C₂⁴×C₄	—	7	168	no	—
116	C₂⁴×C₄	C₂×D₇	1	3	yes	complemented
117	C₂³×C₇	—	2	168	no	—
118	C₂³×C₇	C₂×#41	1	8	yes	complemented
119	C₂³×C₇	C₂⁵	1	84	yes	—
120	C₂³×C₇	C₂⁵	1	7	yes	—
121	C₂²×D₇	—	2	168	no	—
122	C₂²×D₇	—	2	1344	no	—
123	C₂²×D₇	C₂×#41	1	112	yes	direct product factor
124	C₂²×D₇	C₂⁵	1	84	yes	—
125	C₂²×D₇	C₂⁵	1	192	yes	—
126	C₂²×D₇	C₂⁵	1	28	yes	—
127	C₇×Q₈	C₂⁵	1	64	yes	complemented
128	C₇×D₄	C₂⁵	1	192	yes	complemented
129	C₂×C₄×C₇	C₂⁵	1	84	yes	—
130	C₂×C₄×C₇	C₂⁵	1	28	yes	—
131	C₂×C₄×C₇	C₂⁵	1	192	yes	complemented
132	C₂²⋊D₇	C₂⁵	1	384	yes	complemented
133	C₂×Dic₇	C₂⁵	1	84	yes	—
134	C₂×Dic₇	C₂⁵	1	28	yes	—
135	C₂×Dic₇	C₂⁵	1	192	yes	complemented
136	D₂₈	C₂⁵	1	192	yes	complemented
137	C₄×D₇	C₂⁵	1	64	yes	—
138	C₄×D₇	C₂⁵	1	192	yes	—
139	C₄×D₇	C₂⁵	1	192	yes	complemented
140	C₄×D₇	C₂⁵	1	192	yes	complemented
141	Dic₁₄	C₂⁵	1	192	yes	complemented
142	C₂⁵	—	7	1	no	—
143	C₂⁵	—	7	84	no	—
144	C₂⁵	—	7	3	no	—
145	C₂⁵	—	14	48	no	—
146	C₂⁵	C₂²×D₇	1	3	yes	—
147	C₂×#41	—	7	4096	no	—
148	C₂×#41	—	7	448	no	—
149	C₂×#41	—	7	1344	no	—
150	C₂×#41	—	7	1344	no	—
151	C₂×#41	C₂²×D₇	1	448	yes	direct product factor
152	C₂²×Q₈	—	7	448	no	—
153	C₂²×Q₈	—	7	84	no	—
154	C₂²×Q₈	C₂²×D₇	1	28	yes	complemented
155	C₂²×D₄	—	7	168	no	—
156	C₂²×D₄	—	7	1344	no	—
157	C₂²×D₄	—	7	84	no	—
158	C₂²×D₄	C₂²×D₇	1	84	yes	complemented
159	C₂³×C₄	—	7	1	no	—
160	C₂³×C₄	—	7	3	no	—
161	C₂³×C₄	—	7	84	no	—
162	C₂³×C₄	—	7	28	no	—
163	C₂³×C₄	—	7	84	no	—
164	C₂³×C₄	—	7	84	no	—
165	C₂³×C₄	—	7	84	no	—
166	C₂³×C₄	—	7	1344	no	—
167	C₂³×C₄	C₂²×D₇	1	3	yes	—
168	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
169	C₂³×C₄	C₂²×D₇	1	84	yes	complemented
170	C₂²×C₇	—	2	168	no	—
171	C₂²×C₇	C₂²×#41	1	28	yes	complemented
172	C₂²×C₇	C₂⁶	1	24	yes	—
173	C₂²×C₇	C₂⁶	1	7	yes	—
174	C₂×D₇	—	2	168	no	—
175	C₂×D₇	—	2	384	no	—
176	C₂×D₇	C₂²×#41	1	112	yes	direct product factor
177	C₂×D₇	C₂⁶	1	24	yes	—
178	C₂×D₇	C₂⁶	1	8	yes	—
179	C₄×C₇	C₂⁶	1	24	yes	—
180	C₄×C₇	C₂⁶	1	8	yes	—
181	Dic₇	C₂⁶	1	24	yes	—
182	Dic₇	C₂⁶	1	8	yes	—
183	C₂⁴	—	2	24	no	—
184	C₂⁴	—	7	16	no	—
185	C₂⁴	—	7	14	no	—
186	C₂⁴	—	7	336	no	—
187	C₂⁴	—	7	42	no	—
188	C₂⁴	—	14	672	no	—
189	C₂⁴	—	14	24	no	—
190	C₂⁴	C₂³×D₇	1	42	yes	—
191	C₂⁴	C₂³×D₇	1	1	yes	characteristic
192	#41	—	7	512	no	—
193	#41	—	7	1536	no	—
194	#41	—	7	1536	no	—
195	#41	C₂³×D₇	1	512	yes	direct product factor
196	C₂×Q₈	—	7	512	no	—
197	C₂×Q₈	—	7	336	no	—
198	C₂×Q₈	C₂³×D₇	1	112	yes	complemented
199	C₂×D₄	—	7	672	no	—
200	C₂×D₄	—	7	1536	no	—
201	C₂×D₄	—	7	336	no	—
202	C₂×D₄	C₂³×D₇	1	336	yes	complemented
203	C₂²×C₄	—	7	14	no	—
204	C₂²×C₄	—	7	42	no	—
205	C₂²×C₄	—	7	336	no	—
206	C₂²×C₄	—	7	112	no	—
207	C₂²×C₄	—	7	336	no	—
208	C₂²×C₄	—	7	336	no	—
209	C₂²×C₄	—	7	336	no	—
210	C₂²×C₄	—	7	1536	no	—
211	C₂²×C₄	C₂³×D₇	1	14	yes	—
212	C₂²×C₄	C₂³×D₇	1	42	yes	—
213	C₂²×C₄	C₂³×D₇	1	336	yes	complemented
214	C₂×C₇	—	2	24	no	—
215	C₂×C₇	C₂³×#41	1	14	yes	complemented
216	C₂×C₇	C₂⁷	1	1	yes	characteristic
217	D₇	—	2	24	no	—
218	D₇	C₂³×#41	1	16	yes	direct product factor
219	C₂³	—	2	168	no	—
220	C₂³	—	7	28	no	—
221	C₂³	—	7	112	no	—
222	C₂³	—	7	192	no	—
223	C₂³	—	7	84	no	—
224	C₂³	—	14	1344	no	—
225	C₂³	—	14	168	no	—
226	C₂³	D₇×#41	1	8	yes	direct product factor
227	C₂³	C₂⁴×D₇	1	84	yes	—
228	C₂³	C₂⁴×D₇	1	7	yes	—
229	Q₈	—	7	192	no	—
230	Q₈	C₂⁴×D₇	1	64	yes	complemented
231	D₄	—	7	384	no	—
232	D₄	—	7	192	no	—
233	D₄	C₂⁴×D₇	1	192	yes	complemented
234	C₂×C₄	—	7	28	no	—
235	C₂×C₄	—	7	84	no	—
236	C₂×C₄	—	7	192	no	—
237	C₂×C₄	—	7	64	no	—
238	C₂×C₄	—	7	192	no	—
239	C₂×C₄	—	7	192	no	—
240	C₂×C₄	—	7	192	no	—
241	C₂×C₄	C₂⁴×D₇	1	28	yes	—
242	C₂×C₄	C₂⁴×D₇	1	84	yes	—
243	C₂×C₄	C₂⁴×D₇	1	192	yes	complemented
244	C₇	C₂⁴×#41	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	2	168	no	—
246	C₂²	—	7	112	no	—
247	C₂²	—	7	8	no	—
248	C₂²	—	7	24	no	—
249	C₂²	—	14	384	no	—
250	C₂²	—	14	168	no	—
251	C₂²	C₂×D₇×#41	1	28	yes	direct product factor
252	C₂²	C₂⁵×D₇	1	24	yes	—
253	C₂²	C₂⁵×D₇	1	7	yes	—
254	C₄	—	7	8	no	—
255	C₄	—	7	24	no	—
256	C₄	C₂⁵×D₇	1	8	yes	—
257	C₄	C₂⁵×D₇	1	24	yes	—
258	C₂	—	2	24	no	—
259	C₂	—	7	16	no	—
260	C₂	—	14	24	no	—
261	C₂	C₂²×D₇×#41	1	14	yes	direct product factor
262	C₂	C₂⁶×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇
commutator, cosocle	C₂×C₇	C₂⁷
Frattini	C₂	C₂⁶×D₇
Fitting	C₂³×C₇×#41	C₂
socle	C₂⁴×C₇	C₂⁴
hypercenter	C₂³×#41	D₇
hypocenter	C₇	C₂⁴×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇×#41	C₂	semidirect
2. C₂²×D₇×#41	C₂	direct (N × H)
3. C₂³×Q₈×D₇	C₂	semidirect
4. C₂³×#1161	C₂	semidirect
5. C₂³×D₄×D₇	C₂	semidirect
6. C₂³×#1159	C₂	semidirect
7. C₂³×#1157	C₂	semidirect
8. C₂⁴×C₄×D₇	C₂	semidirect
10. C₂²×C₇×#41	C₂²	semidirect
11. C₂³×C₇×Q₈	C₂²	semidirect
12. C₂³×C₇×D₄	C₂²	semidirect
13. C₂⁴×C₄×C₇	C₂²	semidirect
14. C₂³×C₂²⋊D₇	C₂²	semidirect
15. C₂⁴×Dic₇	C₂²	semidirect
16. C₂×D₇×#41	C₂²	direct (N × H)
17. C₂²×#1161	C₂²	semidirect
18. C₂²×Q₈×D₇	C₂²	semidirect
19. C₂²×#1159	C₂²	semidirect
20. C₂²×D₄×D₇	C₂²	semidirect
21. C₂²×#1157	C₂²	semidirect
22. C₂³×D₂₈	C₂²	semidirect
25. C₂³×C₄×D₇	C₂²	semidirect
26. C₂³×C₄×D₇	C₂²	semidirect
27. C₂³×C₄×D₇	C₂²	semidirect
28. C₂³×Dic₁₄	C₂²	semidirect
35. C₂×C₇×#41	C₂³	semidirect
36. C₂²×C₇×Q₈	C₂³	semidirect
37. C₂²×C₇×D₄	C₂³	semidirect
40. C₂³×C₄×C₇	C₂³	semidirect
41. C₂²×C₂²⋊D₇	C₂³	semidirect
44. C₂³×Dic₇	C₂³	semidirect
45. D₇×#41	C₂³	direct (N × H)
46. C₂×#1161	C₂³	semidirect
47. C₂×Q₈×D₇	C₂³	semidirect
48. C₂×#1159	C₂³	semidirect
49. C₂×D₄×D₇	C₂³	semidirect
50. C₂×#1157	C₂³	semidirect
51. C₂²×D₂₈	C₂³	semidirect
54. C₂²×C₄×D₇	C₂³	semidirect
55. C₂²×C₄×D₇	C₂³	semidirect
56. C₂²×C₄×D₇	C₂³	semidirect
57. C₂²×Dic₁₄	C₂³	semidirect
62. C₂³×#41	D₇	direct (N × H)
71. C₂³×D₇	#41	direct (N × H)
75. C₇×#41	C₂⁴	semidirect
76. C₂×C₇×Q₈	C₂⁴	semidirect
77. C₂×C₇×D₄	C₂⁴	semidirect
80. C₂²×C₄×C₇	C₂⁴	semidirect
81. C₂×C₂²⋊D₇	C₂⁴	semidirect
84. C₂²×Dic₇	C₂⁴	semidirect
85. #1161	C₂⁴	semidirect
86. Q₈×D₇	C₂⁴	semidirect
87. #1159	C₂⁴	semidirect
88. D₄×D₇	C₂⁴	semidirect
89. #1157	C₂⁴	semidirect
90. C₂×D₂₈	C₂⁴	semidirect
93. C₂×C₄×D₇	C₂⁴	semidirect
94. C₂×C₄×D₇	C₂⁴	semidirect
95. C₂×C₄×D₇	C₂⁴	semidirect
96. C₂×Dic₁₄	C₂⁴	semidirect
102. C₂²×#41	C₂×D₇	direct (N × H)
105. C₂³×Q₈	C₂×D₇	semidirect
109. C₂³×D₄	C₂×D₇	semidirect
116. C₂⁴×C₄	C₂×D₇	semidirect
118. C₂³×C₇	C₂×#41	semidirect
123. C₂²×D₇	C₂×#41	direct (N × H)
127. C₇×Q₈	C₂⁵	semidirect
128. C₇×D₄	C₂⁵	semidirect
131. C₂×C₄×C₇	C₂⁵	semidirect
132. C₂²⋊D₇	C₂⁵	semidirect
135. C₂×Dic₇	C₂⁵	semidirect
136. D₂₈	C₂⁵	semidirect
139. C₄×D₇	C₂⁵	semidirect
140. C₄×D₇	C₂⁵	semidirect
141. Dic₁₄	C₂⁵	semidirect
151. C₂×#41	C₂²×D₇	direct (N × H)
154. C₂²×Q₈	C₂²×D₇	semidirect
158. C₂²×D₄	C₂²×D₇	semidirect
169. C₂³×C₄	C₂²×D₇	semidirect
171. C₂²×C₇	C₂²×#41	semidirect
176. C₂×D₇	C₂²×#41	direct (N × H)
195. #41	C₂³×D₇	direct (N × H)
198. C₂×Q₈	C₂³×D₇	semidirect
202. C₂×D₄	C₂³×D₇	semidirect
213. C₂²×C₄	C₂³×D₇	semidirect
215. C₂×C₇	C₂³×#41	semidirect
218. D₇	C₂³×#41	direct (N × H)
226. C₂³	D₇×#41	direct (N × H)
230. Q₈	C₂⁴×D₇	semidirect
233. D₄	C₂⁴×D₇	semidirect
243. C₂×C₄	C₂⁴×D₇	semidirect
244. C₇	C₂⁴×#41	semidirect
251. C₂²	C₂×D₇×#41	direct (N × H)
261. C₂	C₂²×D₇×#41	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×#41	2
C₂²×D₇×#41	2
C₂³×Q₈×D₇	2
C₂³×#1161	2
C₂³×D₄×D₇	2
C₂³×#1159	2
C₂³×#1157	2
C₂⁴×C₄×D₇	2
C₂⁴×#41	7

Subgroups — C23×D7×#41 #14267677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₇×#41 #14267677