Subgroups — C₂×C₇×#949 #13061539

Order 1344.

596 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#949	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#215	C₂	1	2	yes	maximal, complemented
5	C₂×C₇×#213	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#208	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×AGL(1,ℤ/8)	—	3	1	no	maximal, Hall
8	C₂³×D₃×C₇	—	2	1	no	—
9	C₃×C₇×#41	—	2	2	no	—
10	C₃×C₇×#41	C₂²	1	4	yes	complemented
11	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₃×C₇×D₄	—	2	1	no	—
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₄×C₇	—	2	1	no	—
15	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₂×C₇×D₁₂	—	2	2	no	—
17	C₂×C₇×D₁₂	—	2	1	no	—
18	C₂×C₇×D₁₂	C₂²	1	2	yes	—
19	C₂×C₇×D₁₂	C₂²	1	4	yes	complemented
20	C₇×#215	C₂²	1	8	yes	complemented
21	C₇×#213	C₂²	1	8	yes	complemented
22	C₇×#208	C₂²	1	4	yes	complemented
23	C₂×C₇×C₃⋊C₈	C₂²	1	2	yes	complemented
24	C₂×C₇×#41	D₃	1	1	yes	characteristic, complemented
25	C₂²×C₇×D₄	—	3	1	no	—
26	C₇×AGL(1,ℤ/8)	—	3	8	no	—
27	C₂×C₇×SD₁₆	—	3	2	no	—
28	C₂×C₇×D₈	—	3	2	no	—
29	C₂×C₇×M₁₆	—	3	1	no	—
30	C₂×#949	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₂³×C₃×C₇	—	2	1	no	—
32	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
33	C₂²×D₃×C₇	—	2	2	no	—
34	C₂²×D₃×C₇	—	2	4	no	—
35	C₂²×D₃×C₇	—	4	4	no	—
36	C₃×C₇×Q₈	—	2	1	no	—
37	C₃×C₇×Q₈	C₂³	1	2	yes	complemented
38	C₃×C₇×D₄	—	2	4	no	—
39	C₃×C₇×D₄	—	2	1	no	—
40	C₃×C₇×D₄	C₂³	1	2	yes	complemented
41	C₂×C₃×C₄×C₇	—	2	1	no	—
42	C₂×C₃×C₄×C₇	—	2	4	no	—
43	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
45	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
47	C₇×D₁₂	—	2	2	no	—
48	C₇×D₁₂	—	2	4	no	—
49	C₇×D₁₂	C₂³	1	4	yes	—
50	C₇×C₃⋊C₈	C₂³	1	4	yes	complemented
51	C₂⁴×C₇	—	6	1	no	—
52	C₇×#41	—	2	2	no	—
53	C₇×#41	C₂×D₃	1	4	yes	complemented
54	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
55	C₂×C₇×D₄	—	2	1	no	—
56	C₂×C₇×D₄	—	3	2	no	—
57	C₂×C₇×D₄	—	3	4	no	—
58	C₂×C₇×D₄	—	6	2	no	—
59	C₂×C₇×D₄	—	6	1	no	—
60	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic, complemented
61	C₂²×C₄×C₇	—	2	1	no	—
62	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
63	C₇×SD₁₆	—	3	8	no	—
64	C₇×D₈	—	3	8	no	—
65	C₇×M₁₆	—	3	4	no	—
66	C₂×C₇×C₈	—	3	2	no	—
67	C₂×C₃×#41	C₂×C₇	1	1	yes	characteristic, complemented
68	C₂²×D₁₂	C₂×C₇	1	1	yes	characteristic, complemented
69	#949	C₂×C₇	1	8	yes	direct product factor
70	C₂×#215	C₂×C₇	1	2	yes	complemented
71	C₂×#213	C₂×C₇	1	2	yes	complemented
72	C₂×#208	C₂×C₇	1	1	yes	characteristic, complemented
73	C₂²×C₃×C₇	—	2	2	no	—
74	C₂²×C₃×C₇	—	2	2	no	—
75	C₂²×C₃×C₇	—	4	2	no	—
76	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
77	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂×D₃×C₇	—	2	4	no	—
79	C₂×D₃×C₇	—	4	8	no	—
80	C₂×D₃×C₇	—	4	4	no	—
81	C₃×C₄×C₇	—	2	2	no	—
82	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
83	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
85	C₂×AGL(1,ℤ/8)	—	3	1	no	Sylow
86	C₂³×C₇	—	2	1	no	—
87	C₂³×C₇	—	6	4	no	—
88	C₂³×C₇	—	6	2	no	—
89	C₂³×C₇	—	12	4	no	—
90	C₂³×C₇	#67	1	1	yes	characteristic
91	C₇×Q₈	—	2	1	no	—
92	C₇×Q₈	C₂²×D₃	1	2	yes	complemented
93	C₇×D₄	—	2	4	no	—
94	C₇×D₄	—	2	1	no	—
95	C₇×D₄	—	3	4	no	—
96	C₇×D₄	—	6	4	no	—
97	C₇×D₄	—	6	2	no	—
98	C₇×D₄	C₂²×D₃	1	2	yes	complemented
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	4	no	—
101	C₂×C₄×C₇	#67	1	2	yes	—
102	C₂×C₄×C₇	#67	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
104	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
105	C₇×C₈	—	3	4	no	—
106	C₂³×D₃	—	2	1	no	—
107	C₃×#41	—	2	2	no	—
108	C₃×#41	C₂²×C₇	1	4	yes	complemented
109	C₂×C₃×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
110	C₂×C₃×D₄	—	2	1	no	—
111	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic, complemented
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
114	C₂×D₁₂	—	2	2	no	—
115	C₂×D₁₂	—	2	1	no	—
116	C₂×D₁₂	C₂²×C₇	1	2	yes	—
117	C₂×D₁₂	C₂²×C₇	1	4	yes	complemented
118	#215	C₂²×C₇	1	8	yes	complemented
119	#213	C₂²×C₇	1	8	yes	complemented
120	#208	C₂²×C₇	1	4	yes	complemented
121	C₂×C₃⋊C₈	C₂²×C₇	1	2	yes	complemented
122	C₂×C₃×C₇	—	2	2	no	—
123	C₂×C₃×C₇	—	4	2	no	—
124	C₂×C₃×C₇	AGL(1,ℤ/8)	1	2	yes	complemented
125	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
126	D₃×C₇	—	4	4	no	—
127	C₂×#41	D₃×C₇	1	1	yes	characteristic, complemented
128	C₂²×D₄	—	3	1	no	—
129	AGL(1,ℤ/8)	—	3	8	no	—
130	C₂×SD₁₆	—	3	2	no	—
131	C₂×D₈	—	3	2	no	—
132	C₂×M₁₆	—	3	1	no	—
133	C₂²×C₇	—	2	2	no	—
134	C₂²×C₇	—	2	2	no	—
135	C₂²×C₇	—	4	2	no	—
136	C₂²×C₇	—	6	4	no	—
137	C₂²×C₇	—	12	8	no	—
138	C₂²×C₇	—	12	4	no	—
139	C₂²×C₇	C₂×#67	1	2	yes	—
140	C₂²×C₇	C₂×#67	1	1	yes	characteristic
141	C₄×C₇	—	2	2	no	—
142	C₄×C₇	C₂×#67	1	2	yes	—
143	C₄×C₇	C₂×#67	1	1	yes	characteristic
144	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
145	C₂³×C₃	—	2	1	no	—
146	C₂³×C₃	C₇×D₄	1	1	yes	characteristic
147	C₂²×D₃	—	2	2	no	—
148	C₂²×D₃	—	2	4	no	—
149	C₂²×D₃	—	4	4	no	—
150	C₃×Q₈	—	2	1	no	—
151	C₃×Q₈	C₂³×C₇	1	2	yes	complemented
152	C₃×D₄	—	2	4	no	—
153	C₃×D₄	—	2	1	no	—
154	C₃×D₄	C₂³×C₇	1	2	yes	complemented
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
159	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
161	D₁₂	—	2	2	no	—
162	D₁₂	—	2	4	no	—
163	D₁₂	C₂³×C₇	1	4	yes	—
164	C₃⋊C₈	C₂³×C₇	1	4	yes	complemented
165	C₃×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Hall
166	C₂⁴	—	6	1	no	—
167	#41	—	2	2	no	—
168	#41	C₂×D₃×C₇	1	4	yes	complemented
169	C₂×Q₈	C₂×D₃×C₇	1	1	yes	characteristic, complemented
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	3	2	no	—
172	C₂×D₄	—	3	4	no	—
173	C₂×D₄	—	6	1	no	—
174	C₂×D₄	—	6	2	no	—
175	C₂×D₄	C₂×D₃×C₇	1	1	yes	characteristic, complemented
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
178	SD₁₆	—	3	8	no	—
179	D₈	—	3	8	no	—
180	M₁₆	—	3	4	no	—
181	C₂×C₈	—	3	2	no	—
182	C₂×C₇	—	2	2	no	—
183	C₂×C₇	—	4	2	no	—
184	C₂×C₇	—	12	4	no	—
185	C₂×C₇	#949	1	2	yes	direct product factor
186	C₂×C₇	C₂²×#67	1	1	yes	characteristic
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	4	2	no	—
190	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
191	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
192	C₂×D₃	—	2	4	no	—
193	C₂×D₃	—	4	8	no	—
194	C₂×D₃	—	4	4	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
197	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
198	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
199	C₂³	—	2	1	no	—
200	C₂³	—	6	4	no	—
201	C₂³	—	6	2	no	—
202	C₂³	—	12	4	no	—
203	C₂³	C₇×#67	1	1	yes	characteristic
204	Q₈	—	2	1	no	—
205	Q₈	C₂²×D₃×C₇	1	2	yes	complemented
206	D₄	—	2	4	no	—
207	D₄	—	2	1	no	—
208	D₄	—	3	4	no	—
209	D₄	—	6	4	no	—
210	D₄	—	6	2	no	—
211	D₄	C₂²×D₃×C₇	1	2	yes	complemented
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	C₇×#67	1	1	yes	characteristic
215	C₂×C₄	C₇×#67	1	2	yes	—
216	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
218	C₈	—	3	4	no	—
219	C₇	C₂×#949	1	1	yes	characteristic, direct product factor, Sylow
220	C₂×C₃	—	2	2	no	—
221	C₂×C₃	—	4	2	no	—
222	C₂×C₃	C₇×AGL(1,ℤ/8)	1	2	yes	complemented
223	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
224	D₃	—	4	4	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	6	4	no	—
229	C₂²	—	12	8	no	—
230	C₂²	—	12	4	no	—
231	C₂²	C₂×C₇×#67	1	2	yes	—
232	C₂²	C₂×C₇×#67	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	C₂×C₇×#67	1	2	yes	—
235	C₄	C₂×C₇×#67	1	1	yes	characteristic
236	C₄	C₂³×D₃×C₇	1	1	yes	characteristic
237	C₃	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
238	C₂	—	2	2	no	—
239	C₂	—	4	2	no	—
240	C₂	—	12	4	no	—
241	C₂	C₇×#949	1	2	yes	direct product factor
242	C₂	C₂²×C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂×#67
commutator, cosocle	C₃×C₄	C₂⁴×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₂×C₃×C₇×#41	C₂
socle	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₂×C₇×#41	D₃
hypocenter	C₃	C₂×C₇×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×#41	C₂	semidirect
2. C₂²×C₇×D₁₂	C₂	semidirect
3. C₇×#949	C₂	direct (N × H)
4. C₂×C₇×#215	C₂	semidirect
5. C₂×C₇×#213	C₂	semidirect
6. C₂×C₇×#208	C₂	semidirect
10. C₃×C₇×#41	C₂²	semidirect
11. C₂×C₃×C₇×Q₈	C₂²	semidirect
13. C₂×C₃×C₇×D₄	C₂²	semidirect
19. C₂×C₇×D₁₂	C₂²	semidirect
20. C₇×#215	C₂²	semidirect
21. C₇×#213	C₂²	semidirect
22. C₇×#208	C₂²	semidirect
23. C₂×C₇×C₃⋊C₈	C₂²	semidirect
24. C₂×C₇×#41	D₃	semidirect
30. C₂×#949	C₇	direct (N × H)
37. C₃×C₇×Q₈	C₂³	semidirect
40. C₃×C₇×D₄	C₂³	semidirect
50. C₇×C₃⋊C₈	C₂³	semidirect
53. C₇×#41	C₂×D₃	semidirect
54. C₂×C₇×Q₈	C₂×D₃	semidirect
60. C₂×C₇×D₄	C₂×D₃	semidirect
67. C₂×C₃×#41	C₂×C₇	semidirect
68. C₂²×D₁₂	C₂×C₇	semidirect
69. #949	C₂×C₇	direct (N × H)
70. C₂×#215	C₂×C₇	semidirect
71. C₂×#213	C₂×C₇	semidirect
72. C₂×#208	C₂×C₇	semidirect
92. C₇×Q₈	C₂²×D₃	semidirect
98. C₇×D₄	C₂²×D₃	semidirect
108. C₃×#41	C₂²×C₇	semidirect
109. C₂×C₃×Q₈	C₂²×C₇	semidirect
111. C₂×C₃×D₄	C₂²×C₇	semidirect
117. C₂×D₁₂	C₂²×C₇	semidirect
118. #215	C₂²×C₇	semidirect
119. #213	C₂²×C₇	semidirect
120. #208	C₂²×C₇	semidirect
121. C₂×C₃⋊C₈	C₂²×C₇	semidirect
124. C₂×C₃×C₇	AGL(1,ℤ/8)	semidirect
127. C₂×#41	D₃×C₇	semidirect
151. C₃×Q₈	C₂³×C₇	semidirect
154. C₃×D₄	C₂³×C₇	semidirect
164. C₃⋊C₈	C₂³×C₇	semidirect
165. C₃×C₇	C₂×AGL(1,ℤ/8)	semidirect
168. #41	C₂×D₃×C₇	semidirect
169. C₂×Q₈	C₂×D₃×C₇	semidirect
175. C₂×D₄	C₂×D₃×C₇	semidirect
185. C₂×C₇	#949	direct (N × H)
205. Q₈	C₂²×D₃×C₇	semidirect
211. D₄	C₂²×D₃×C₇	semidirect
219. C₇	C₂×#949	direct (N × H)
222. C₂×C₃	C₇×AGL(1,ℤ/8)	semidirect
237. C₃	C₂×C₇×AGL(1,ℤ/8)	semidirect
241. C₂	C₇×#949	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×#41	2
C₂²×C₇×D₁₂	2
C₇×#949	2
C₂×C₇×#215	2
C₂×C₇×#213	2
C₂×C₇×#208	2
C₂×C₇×AGL(1,ℤ/8)	3
C₂×#949	7

Subgroups — C2×C7×#949 #13061539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#949 #13061539