Subgroups — C₃×D₇×#124 #13061016

Order 1344.

660 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₄×D₇	C₂	1	1	yes	maximal, characteristic
3	C₃×C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6510	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6499	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6448	C₂	1	4	yes	maximal, complemented
7	C₃×D₇×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
8	C₃×#6438	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄²×D₇	C₂	1	1	yes	maximal, characteristic, complemented
10	D₇×#124	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂³×C₃×D₇	C₂²	1	2	yes	—
12	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
15	C₃×Q₈×D₇	—	2	4	no	—
16	C₃×D₄×D₇	—	2	4	no	—
17	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
19	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
20	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
21	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃×Dic₁₄	C₂²	1	2	yes	complemented
23	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	complemented
24	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
25	C₃×C₂²⋊Dic₇	C₂²	1	4	yes	complemented
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	complemented
27	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	complemented
28	C₃×C₄×Dic₇	C₂²	1	2	yes	complemented
29	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic, complemented
30	C₂×Q₈×D₇	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×D₄×D₇	C₂×C₃	1	1	yes	characteristic
32	C₇×#124	C₂×C₃	1	1	yes	characteristic, complemented
33	#6510	C₂×C₃	1	1	yes	characteristic, complemented
34	#6499	C₂×C₃	1	1	yes	characteristic, complemented
35	#6448	C₂×C₃	1	4	yes	complemented
36	D₇×C₂²⋊C₄	C₂×C₃	1	4	yes	complemented
37	#6438	C₂×C₃	1	1	yes	characteristic, complemented
38	C₄²×D₇	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂×C₃×#124	—	7	1	no	maximal, Hall
40	C₂³×C₃×C₇	C₂³	1	2	yes	—
41	C₂²×C₃×D₇	—	2	4	no	—
42	C₂²×C₃×D₇	—	2	8	no	—
43	C₂²×C₃×D₇	C₂³	1	2	yes	—
44	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
45	C₃×C₇×Q₈	—	2	2	no	—
46	C₃×C₇×D₄	—	2	2	no	—
47	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
48	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₃×C₂²⋊D₇	—	2	4	no	—
51	C₂×C₃×Dic₇	C₂³	1	2	yes	—
52	C₂×C₃×Dic₇	C₂³	1	2	yes	—
53	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
54	C₃×D₂₈	—	2	2	no	—
55	C₃×C₄×D₇	—	2	4	no	—
56	C₃×C₄×D₇	—	2	4	no	—
57	C₃×C₄×D₇	D₄	1	4	yes	—
58	C₃×Dic₁₄	—	2	2	no	—
59	C₃×Dic₁₄	—	2	4	no	—
60	C₂³×D₇	C₂²×C₃	1	2	yes	—
61	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
62	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
64	Q₈×D₇	—	2	4	no	—
65	D₄×D₇	—	2	4	no	—
66	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
68	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
69	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
70	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
71	C₂×Dic₁₄	C₂²×C₃	1	2	yes	complemented
72	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
73	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
74	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	complemented
75	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	complemented
76	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	complemented
77	C₄×Dic₇	C₂²×C₃	1	2	yes	complemented
78	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
79	C₂²×C₃×Q₈	—	7	1	no	—
80	C₂²×C₃×D₄	—	7	1	no	—
81	C₃×#124	—	7	1	no	—
82	C₃×#124	—	7	1	no	—
83	C₃×#124	—	7	4	no	—
84	C₃×#124	—	7	1	no	—
85	C₃×#124	D₇	1	1	yes	characteristic, direct product factor
86	C₂×C₃×C₂²⋊C₄	—	7	4	no	—
87	C₂×C₃×C₄²	—	7	1	no	—
88	C₂²×C₃×C₇	—	2	2	no	—
89	C₂²×C₃×C₇	—	2	4	no	—
90	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₃×D₇	—	2	2	no	—
92	C₂×C₃×D₇	—	2	4	no	—
93	C₂×C₃×D₇	—	4	8	no	—
94	C₂×C₃×D₇	C₂×D₄	1	2	yes	—
95	C₂×C₃×D₇	#41	1	4	yes	—
96	C₃×C₄×C₇	—	2	2	no	—
97	C₃×C₄×C₇	—	2	2	no	—
98	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
99	C₃×Dic₇	—	2	2	no	—
100	C₃×Dic₇	—	2	2	no	—
101	C₃×Dic₇	C₂×D₄	1	2	yes	—
102	C₂×#124	—	7	1	no	Sylow
103	C₂³×C₇	C₂³×C₃	1	2	yes	—
104	C₂²×D₇	—	2	4	no	—
105	C₂²×D₇	—	2	8	no	—
106	C₂²×D₇	C₂³×C₃	1	2	yes	—
107	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
108	C₇×Q₈	—	2	2	no	—
109	C₇×D₄	—	2	2	no	—
110	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
111	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
112	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
113	C₂²⋊D₇	—	2	4	no	—
114	C₂×Dic₇	C₂³×C₃	1	2	yes	—
115	C₂×Dic₇	C₂³×C₃	1	2	yes	—
116	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
117	D₂₈	—	2	2	no	—
118	C₄×D₇	—	2	4	no	—
119	C₄×D₇	—	2	4	no	—
120	C₄×D₇	C₃×D₄	1	4	yes	—
121	Dic₁₄	—	2	2	no	—
122	Dic₁₄	—	2	4	no	—
123	C₂⁴×C₃	—	7	2	no	—
124	C₂×C₃×Q₈	—	7	2	no	—
125	C₂×C₃×Q₈	—	7	1	no	—
126	C₂×C₃×Q₈	—	14	4	no	—
127	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
128	C₂×C₃×D₄	—	7	2	no	—
129	C₂×C₃×D₄	—	7	1	no	—
130	C₂×C₃×D₄	—	14	4	no	—
131	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
132	C₂²×C₃×C₄	—	7	1	no	—
133	C₂²×C₃×C₄	—	7	2	no	—
134	C₂²×C₃×C₄	—	7	2	no	—
135	C₃×C₂²⋊C₄	—	7	4	no	—
136	C₃×C₂²⋊C₄	—	7	4	no	—
137	C₃×C₂²⋊C₄	—	7	4	no	—
138	C₃×C₂²⋊C₄	C₂×D₇	1	4	yes	complemented
139	C₃×C₄²	—	7	2	no	—
140	C₃×C₄²	—	7	1	no	—
141	C₃×C₄²	C₂×D₇	1	1	yes	characteristic, complemented
142	C₂×C₃×C₇	—	4	2	no	—
143	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₃×C₇	C₂×#41	1	2	yes	—
145	C₃×D₇	—	4	2	no	—
146	C₃×D₇	#124	1	4	yes	direct product factor
147	C₂²×Q₈	—	7	1	no	—
148	C₂²×D₄	—	7	1	no	—
149	#124	—	7	1	no	—
150	#124	—	7	1	no	—
151	#124	—	7	4	no	—
152	#124	—	7	1	no	—
153	#124	C₃×D₇	1	1	yes	characteristic, direct product factor
154	C₂×C₂²⋊C₄	—	7	4	no	—
155	C₂×C₄²	—	7	1	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	2	4	no	—
158	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
159	C₂×D₇	—	2	2	no	—
160	C₂×D₇	—	2	4	no	—
161	C₂×D₇	—	4	8	no	—
162	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
163	C₂×D₇	C₃×#41	1	4	yes	—
164	C₄×C₇	—	2	2	no	—
165	C₄×C₇	—	2	2	no	—
166	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
167	Dic₇	—	2	2	no	—
168	Dic₇	—	2	2	no	—
169	Dic₇	C₂×C₃×D₄	1	2	yes	—
170	C₂³×C₃	—	7	1	no	—
171	C₂³×C₃	—	7	2	no	—
172	C₂³×C₃	—	14	4	no	—
173	C₂³×C₃	—	14	8	no	—
174	C₂³×C₃	C₂²×D₇	1	2	yes	—
175	C₃×Q₈	—	2	2	no	—
176	C₃×Q₈	—	14	4	no	—
177	C₃×Q₈	—	14	2	no	—
178	C₃×D₄	—	2	2	no	—
179	C₃×D₄	—	14	4	no	—
180	C₃×D₄	—	14	2	no	—
181	C₂×C₃×C₄	—	7	1	no	—
182	C₂×C₃×C₄	—	7	4	no	—
183	C₂×C₃×C₄	—	7	2	no	—
184	C₂×C₃×C₄	—	7	2	no	—
185	C₂×C₃×C₄	—	14	4	no	—
186	C₂×C₃×C₄	—	14	4	no	—
187	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
188	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
189	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
190	C₃×C₇	C₂×#124	1	1	yes	characteristic, complemented, Hall
191	C₂⁴	—	7	2	no	—
192	C₂×Q₈	—	7	2	no	—
193	C₂×Q₈	—	7	1	no	—
194	C₂×Q₈	—	14	4	no	—
195	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic, complemented
196	C₂×D₄	—	7	2	no	—
197	C₂×D₄	—	7	1	no	—
198	C₂×D₄	—	14	4	no	—
199	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
200	C₂²×C₄	—	7	1	no	—
201	C₂²×C₄	—	7	2	no	—
202	C₂²×C₄	—	7	2	no	—
203	C₂²⋊C₄	—	7	4	no	—
204	C₂²⋊C₄	—	7	4	no	—
205	C₂²⋊C₄	—	7	4	no	—
206	C₂²⋊C₄	C₂×C₃×D₇	1	4	yes	complemented
207	C₄²	—	7	2	no	—
208	C₄²	—	7	1	no	—
209	C₄²	C₂×C₃×D₇	1	1	yes	characteristic, complemented
210	C₂×C₇	—	4	2	no	—
211	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
212	C₂×C₇	C₂×C₃×#41	1	2	yes	—
213	D₇	—	4	2	no	—
214	D₇	C₃×#124	1	4	yes	direct product factor
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	4	no	—
217	C₂²×C₃	—	7	2	no	—
218	C₂²×C₃	—	7	4	no	—
219	C₂²×C₃	—	14	2	no	—
220	C₂²×C₃	—	14	4	no	—
221	C₂²×C₃	—	28	8	no	—
222	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	7	2	no	—
226	C₃×C₄	—	14	2	no	—
227	C₃×C₄	—	14	2	no	—
228	C₃×C₄	D₄×D₇	1	2	yes	—
229	C₂³	—	7	1	no	—
230	C₂³	—	7	2	no	—
231	C₂³	—	14	4	no	—
232	C₂³	—	14	8	no	—
233	C₂³	C₂²×C₃×D₇	1	2	yes	—
234	Q₈	—	2	2	no	—
235	Q₈	—	14	4	no	—
236	Q₈	—	14	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	14	4	no	—
239	D₄	—	14	2	no	—
240	C₂×C₄	—	7	1	no	—
241	C₂×C₄	—	7	4	no	—
242	C₂×C₄	—	7	2	no	—
243	C₂×C₄	—	7	2	no	—
244	C₂×C₄	—	14	4	no	—
245	C₂×C₄	—	14	4	no	—
246	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
247	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
248	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
249	C₇	C₂×C₃×#124	1	1	yes	characteristic, complemented, Sylow
250	C₂×C₃	—	4	2	no	—
251	C₂×C₃	—	7	4	no	—
252	C₂×C₃	—	28	2	no	—
253	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
254	C₂×C₃	D₇×#41	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	7	2	no	—
258	C₂²	—	7	4	no	—
259	C₂²	—	14	2	no	—
260	C₂²	—	14	4	no	—
261	C₂²	—	28	8	no	—
262	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	7	2	no	—
266	C₄	—	14	2	no	—
267	C₄	—	14	2	no	—
268	C₄	C₃×D₄×D₇	1	2	yes	—
269	C₃	D₇×#124	1	1	yes	characteristic, direct product factor, Sylow
270	C₂	—	4	2	no	—
271	C₂	—	7	4	no	—
272	C₂	—	28	2	no	—
273	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
274	C₂	C₃×D₇×#41	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₇×#124	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×D₄×D₇	C₂
hypercenter	C₃×#124	D₇
hypocenter	C₇	C₂×C₃×#124
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#124

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×Q₈×D₇	C₂	semidirect
3. C₃×C₇×#124	C₂	semidirect
4. C₃×#6510	C₂	semidirect
5. C₃×#6499	C₂	semidirect
6. C₃×#6448	C₂	semidirect
7. C₃×D₇×C₂²⋊C₄	C₂	semidirect
8. C₃×#6438	C₂	semidirect
9. C₃×C₄²×D₇	C₂	semidirect
10. D₇×#124	C₃	direct (N × H)
12. C₂×C₃×C₇×Q₈	C₂²	semidirect
21. C₂×C₃×Dic₁₄	C₂²	semidirect
22. C₂×C₃×Dic₁₄	C₂²	semidirect
23. C₃×C₇×C₂²⋊C₄	C₂²	semidirect
24. C₃×C₄²×C₇	C₂²	semidirect
25. C₃×C₂²⋊Dic₇	C₂²	semidirect
26. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
27. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
28. C₃×C₄×Dic₇	C₂²	semidirect
29. C₃×C₄×Dic₇	C₂²	semidirect
30. C₂×Q₈×D₇	C₂×C₃	semidirect
32. C₇×#124	C₂×C₃	semidirect
33. #6510	C₂×C₃	semidirect
34. #6499	C₂×C₃	semidirect
35. #6448	C₂×C₃	semidirect
36. D₇×C₂²⋊C₄	C₂×C₃	semidirect
37. #6438	C₂×C₃	semidirect
38. C₄²×D₇	C₂×C₃	semidirect
61. C₂×C₇×Q₈	C₂²×C₃	semidirect
70. C₂×Dic₁₄	C₂²×C₃	semidirect
71. C₂×Dic₁₄	C₂²×C₃	semidirect
72. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
73. C₄²×C₇	C₂²×C₃	semidirect
74. C₂²⋊Dic₇	C₂²×C₃	semidirect
75. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
76. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
77. C₄×Dic₇	C₂²×C₃	semidirect
78. C₄×Dic₇	C₂²×C₃	semidirect
85. C₃×#124	D₇	direct (N × H)
127. C₂×C₃×Q₈	C₂×D₇	semidirect
138. C₃×C₂²⋊C₄	C₂×D₇	semidirect
141. C₃×C₄²	C₂×D₇	semidirect
146. C₃×D₇	#124	direct (N × H)
153. #124	C₃×D₇	direct (N × H)
190. C₃×C₇	C₂×#124	semidirect
195. C₂×Q₈	C₂×C₃×D₇	semidirect
206. C₂²⋊C₄	C₂×C₃×D₇	semidirect
209. C₄²	C₂×C₃×D₇	semidirect
214. D₇	C₃×#124	direct (N × H)
249. C₇	C₂×C₃×#124	semidirect
269. C₃	D₇×#124	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×Q₈×D₇	2
C₂×C₃×D₄×D₇	2
C₃×C₇×#124	2
C₃×#6510	2
C₃×#6499	2
C₃×#6448	2
C₃×D₇×C₂²⋊C₄	2
C₃×#6438	2
C₃×C₄²×D₇	2
D₇×#124	3
C₂×C₃×#124	7

Subgroups — C3×D7×#124 #13061016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₇×#124 #13061016