Subgroups — C₃×#90365 #13061017

Order 1344.

668 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#1161	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6510	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6503	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6447	C₂	1	4	yes	maximal, complemented
7	C₃×#6445	C₂	1	4	yes	maximal, complemented
8	C₃×#6437	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#6435	C₂	1	1	yes	maximal, characteristic, complemented
10	#90365	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	C₂×C₃×C₂²⋊D₇	C₂²	1	4	yes	complemented
16	C₃×#1161	—	2	4	no	—
17	C₃×D₄×D₇	—	2	4	no	—
18	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
19	C₂×C₃×D₂₈	C₂²	1	2	yes	—
20	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×D₂₈	C₂²	1	4	yes	complemented
22	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄×D₇	C₂²	1	2	yes	complemented
24	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	complemented
25	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
27	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	complemented
28	C₃×Dic₇⋊C₄	C₂²	1	2	yes	complemented
29	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic, complemented
30	C₂×#1161	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×D₄×D₇	C₂×C₃	1	1	yes	characteristic, complemented
32	C₇×#124	C₂×C₃	1	1	yes	characteristic, complemented
33	#6510	C₂×C₃	1	1	yes	characteristic, complemented
34	#6503	C₂×C₃	1	1	yes	characteristic, complemented
35	#6447	C₂×C₃	1	4	yes	complemented
36	#6445	C₂×C₃	1	4	yes	complemented
37	#6437	C₂×C₃	1	1	yes	characteristic, complemented
38	#6435	C₂×C₃	1	1	yes	characteristic, complemented
39	C₃×#535	—	7	1	no	maximal, Hall
40	C₂³×C₃×C₇	C₂³	1	2	yes	—
41	C₂²×C₃×D₇	—	2	4	no	—
42	C₂²×C₃×D₇	—	4	4	no	—
43	C₂²×C₃×D₇	C₂³	1	2	yes	—
44	C₂²×C₃×D₇	C₂³	1	2	yes	—
45	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
46	C₃×C₇×Q₈	—	2	2	no	—
47	C₃×C₇×D₄	—	2	2	no	—
48	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₃×C₂²⋊D₇	—	2	4	no	—
52	C₃×C₂²⋊D₇	—	4	4	no	—
53	C₂×C₃×Dic₇	C₂³	1	2	yes	—
54	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
55	C₃×D₂₈	—	2	4	no	—
56	C₃×D₂₈	—	2	2	no	—
57	C₃×D₂₈	—	2	2	no	—
58	C₃×D₂₈	—	4	4	no	—
59	C₃×C₄×D₇	—	2	4	no	—
60	C₃×C₄×D₇	D₄	1	4	yes	complemented
61	C₂³×D₇	C₂²×C₃	1	2	yes	—
62	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
63	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
65	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	complemented
66	#1161	—	2	4	no	—
67	D₄×D₇	—	2	4	no	—
68	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
69	C₂×D₂₈	C₂²×C₃	1	2	yes	—
70	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic, complemented
71	C₂×D₂₈	C₂²×C₃	1	4	yes	complemented
72	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄×D₇	C₂²×C₃	1	2	yes	complemented
74	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
75	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
76	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
77	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	complemented
78	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
79	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
80	C₂×C₃×#41	—	7	1	no	—
81	C₂²×C₃×D₄	—	7	1	no	—
82	C₃×#127	—	7	1	no	—
83	C₃×#127	—	7	1	no	—
84	C₃×#124	—	7	1	no	—
85	C₃×#124	D₇	1	1	yes	characteristic, complemented
86	C₃×#121	—	7	4	no	—
87	C₃×#120	—	7	4	no	—
88	C₃×#117	—	7	1	no	—
89	C₂²×C₃×C₇	—	2	2	no	—
90	C₂²×C₃×C₇	—	2	4	no	—
91	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
92	C₂×C₃×D₇	—	2	2	no	—
93	C₂×C₃×D₇	—	2	2	no	—
94	C₂×C₃×D₇	—	2	2	no	—
95	C₂×C₃×D₇	—	2	4	no	—
96	C₂×C₃×D₇	—	2	4	no	—
97	C₂×C₃×D₇	—	4	8	no	—
98	C₂×C₃×D₇	C₂×D₄	1	2	yes	—
99	C₃×C₄×C₇	—	2	2	no	—
100	C₃×C₄×C₇	—	2	2	no	—
101	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
102	C₃×Dic₇	—	2	2	no	—
103	C₃×Dic₇	C₂×D₄	1	2	yes	—
104	#535	—	7	1	no	Sylow
105	C₂³×C₇	C₂³×C₃	1	2	yes	—
106	C₂²×D₇	—	2	4	no	—
107	C₂²×D₇	—	4	4	no	—
108	C₂²×D₇	C₂³×C₃	1	2	yes	—
109	C₂²×D₇	C₂³×C₃	1	2	yes	—
110	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
111	C₇×Q₈	—	2	2	no	—
112	C₇×D₄	—	2	2	no	—
113	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
114	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
115	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
116	C₂²⋊D₇	—	2	4	no	—
117	C₂²⋊D₇	—	4	4	no	—
118	C₂×Dic₇	C₂³×C₃	1	2	yes	—
119	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
120	D₂₈	—	2	4	no	—
121	D₂₈	—	2	2	no	—
122	D₂₈	—	2	2	no	—
123	D₂₈	—	4	4	no	—
124	C₄×D₇	—	2	4	no	—
125	C₄×D₇	C₃×D₄	1	4	yes	complemented
126	C₂⁴×C₃	—	7	2	no	—
127	C₃×#41	—	14	4	no	—
128	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
129	C₂×C₃×D₄	—	7	2	no	—
130	C₂×C₃×D₄	—	7	4	no	—
131	C₂×C₃×D₄	—	7	2	no	—
132	C₂×C₃×D₄	—	7	1	no	—
133	C₂×C₃×D₄	—	7	1	no	—
134	C₂×C₃×D₄	—	7	4	no	—
135	C₂×C₃×D₄	—	14	4	no	—
136	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
137	C₂²×C₃×C₄	—	7	1	no	—
138	C₂²×C₃×C₄	—	7	2	no	—
139	C₃×C₄⋊C₄	—	7	2	no	—
140	C₃×C₂²⋊C₄	—	7	2	no	—
141	C₃×C₂²⋊C₄	—	7	4	no	—
142	C₃×C₂²⋊C₄	C₂×D₇	1	4	yes	complemented
143	C₃×C₄²	—	7	1	no	—
144	C₃×C₄²	C₂×D₇	1	1	yes	characteristic, complemented
145	C₂×C₃×C₇	—	4	2	no	—
146	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	—
148	C₃×D₇	—	2	2	no	—
149	C₃×D₇	—	4	2	no	—
150	C₃×D₇	—	4	2	no	—
151	C₂×#41	—	7	1	no	—
152	C₂²×D₄	—	7	1	no	—
153	#127	—	7	1	no	—
154	#127	—	7	1	no	—
155	#124	—	7	1	no	—
156	#124	C₃×D₇	1	1	yes	characteristic, complemented
157	#121	—	7	4	no	—
158	#120	—	7	4	no	—
159	#117	—	7	1	no	—
160	C₂²×C₇	—	2	2	no	—
161	C₂²×C₇	—	2	4	no	—
162	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
163	C₂×D₇	—	2	2	no	—
164	C₂×D₇	—	2	2	no	—
165	C₂×D₇	—	2	2	no	—
166	C₂×D₇	—	2	4	no	—
167	C₂×D₇	—	2	4	no	—
168	C₂×D₇	—	4	8	no	—
169	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
170	C₄×C₇	—	2	2	no	—
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
173	Dic₇	—	2	2	no	—
174	Dic₇	C₂×C₃×D₄	1	2	yes	—
175	C₂³×C₃	—	7	1	no	—
176	C₂³×C₃	—	7	2	no	—
177	C₂³×C₃	—	7	2	no	—
178	C₂³×C₃	—	14	4	no	—
179	C₂³×C₃	—	28	4	no	—
180	C₂³×C₃	C₂²×D₇	1	2	yes	—
181	C₃×Q₈	—	2	2	no	—
182	C₃×D₄	—	2	2	no	—
183	C₃×D₄	—	14	4	no	—
184	C₃×D₄	—	14	4	no	—
185	C₃×D₄	—	14	2	no	—
186	C₃×D₄	—	14	2	no	—
187	C₃×D₄	—	28	4	no	—
188	C₃×D₄	—	28	4	no	—
189	C₂×C₃×C₄	—	7	1	no	—
190	C₂×C₃×C₄	—	7	4	no	—
191	C₂×C₃×C₄	—	7	2	no	—
192	C₂×C₃×C₄	—	14	4	no	—
193	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
194	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
195	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
196	C₃×C₇	#535	1	1	yes	characteristic, complemented, Hall
197	C₂⁴	—	7	2	no	—
198	#41	—	14	4	no	—
199	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic, complemented
200	C₂×D₄	—	7	2	no	—
201	C₂×D₄	—	7	4	no	—
202	C₂×D₄	—	7	2	no	—
203	C₂×D₄	—	7	1	no	—
204	C₂×D₄	—	7	1	no	—
205	C₂×D₄	—	7	4	no	—
206	C₂×D₄	—	14	4	no	—
207	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
208	C₂²×C₄	—	7	1	no	—
209	C₂²×C₄	—	7	2	no	—
210	C₄⋊C₄	—	7	2	no	—
211	C₂²⋊C₄	—	7	2	no	—
212	C₂²⋊C₄	—	7	4	no	—
213	C₂²⋊C₄	C₂×C₃×D₇	1	4	yes	complemented
214	C₄²	—	7	1	no	—
215	C₄²	C₂×C₃×D₇	1	1	yes	characteristic, complemented
216	C₂×C₇	—	4	2	no	—
217	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
218	C₂×C₇	C₃×H(5,𝔽₂)	1	2	yes	—
219	D₇	—	2	2	no	—
220	D₇	—	4	2	no	—
221	D₇	—	4	2	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	4	no	—
224	C₂²×C₃	—	7	2	no	—
225	C₂²×C₃	—	14	2	no	—
226	C₂²×C₃	—	14	2	no	—
227	C₂²×C₃	—	14	2	no	—
228	C₂²×C₃	—	14	4	no	—
229	C₂²×C₃	—	14	4	no	—
230	C₂²×C₃	—	28	8	no	—
231	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	7	2	no	—
235	C₃×C₄	—	14	2	no	—
236	C₃×C₄	D₄×D₇	1	2	yes	—
237	C₂³	—	7	1	no	—
238	C₂³	—	7	2	no	—
239	C₂³	—	7	2	no	—
240	C₂³	—	14	4	no	—
241	C₂³	—	28	4	no	—
242	C₂³	C₂²×C₃×D₇	1	2	yes	—
243	Q₈	—	2	2	no	—
244	D₄	—	2	2	no	—
245	D₄	—	14	4	no	—
246	D₄	—	14	4	no	—
247	D₄	—	14	2	no	—
248	D₄	—	14	2	no	—
249	D₄	—	28	4	no	—
250	D₄	—	28	4	no	—
251	C₂×C₄	—	7	1	no	—
252	C₂×C₄	—	7	4	no	—
253	C₂×C₄	—	7	2	no	—
254	C₂×C₄	—	14	4	no	—
255	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
256	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
257	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
258	C₇	C₃×#535	1	1	yes	characteristic, complemented, Sylow
259	C₂×C₃	—	4	2	no	—
260	C₂×C₃	—	14	2	no	—
261	C₂×C₃	—	28	2	no	—
262	C₂×C₃	—	28	2	no	—
263	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
264	C₂×C₃	#6553	1	2	yes	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	7	2	no	—
268	C₂²	—	14	2	no	—
269	C₂²	—	14	2	no	—
270	C₂²	—	14	2	no	—
271	C₂²	—	14	4	no	—
272	C₂²	—	14	4	no	—
273	C₂²	—	28	8	no	—
274	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	7	2	no	—
278	C₄	—	14	2	no	—
279	C₄	C₃×D₄×D₇	1	2	yes	—
280	C₃	#90365	1	1	yes	characteristic, direct product factor, Sylow
281	C₂	—	4	2	no	—
282	C₂	—	14	2	no	—
283	C₂	—	28	2	no	—
284	C₂	—	28	2	no	—
285	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
286	C₂	C₃×#6553	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
hypercenter	C₃×#124	D₇
hypocenter	C₇	C₃×#535
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×#1161	C₂	semidirect
2. C₂×C₃×D₄×D₇	C₂	semidirect
3. C₃×C₇×#124	C₂	semidirect
4. C₃×#6510	C₂	semidirect
5. C₃×#6503	C₂	semidirect
6. C₃×#6447	C₂	semidirect
7. C₃×#6445	C₂	semidirect
8. C₃×#6437	C₂	semidirect
9. C₃×#6435	C₂	semidirect
10. #90365	C₃	direct (N × H)
12. C₂×C₃×C₇×Q₈	C₂²	semidirect
15. C₂×C₃×C₂²⋊D₇	C₂²	semidirect
20. C₂×C₃×D₂₈	C₂²	semidirect
21. C₂×C₃×D₂₈	C₂²	semidirect
23. C₂×C₃×C₄×D₇	C₂²	semidirect
24. C₃×C₇×C₂²⋊C₄	C₂²	semidirect
25. C₃×C₄²×C₇	C₂²	semidirect
27. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
28. C₃×Dic₇⋊C₄	C₂²	semidirect
29. C₃×C₄×Dic₇	C₂²	semidirect
30. C₂×#1161	C₂×C₃	semidirect
31. C₂×D₄×D₇	C₂×C₃	semidirect
32. C₇×#124	C₂×C₃	semidirect
33. #6510	C₂×C₃	semidirect
34. #6503	C₂×C₃	semidirect
35. #6447	C₂×C₃	semidirect
36. #6445	C₂×C₃	semidirect
37. #6437	C₂×C₃	semidirect
38. #6435	C₂×C₃	semidirect
60. C₃×C₄×D₇	D₄	semidirect
62. C₂×C₇×Q₈	C₂²×C₃	semidirect
65. C₂×C₂²⋊D₇	C₂²×C₃	semidirect
70. C₂×D₂₈	C₂²×C₃	semidirect
71. C₂×D₂₈	C₂²×C₃	semidirect
73. C₂×C₄×D₇	C₂²×C₃	semidirect
74. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
75. C₄²×C₇	C₂²×C₃	semidirect
77. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
78. Dic₇⋊C₄	C₂²×C₃	semidirect
79. C₄×Dic₇	C₂²×C₃	semidirect
85. C₃×#124	D₇	semidirect
125. C₄×D₇	C₃×D₄	semidirect
128. C₂×C₃×Q₈	C₂×D₇	semidirect
142. C₃×C₂²⋊C₄	C₂×D₇	semidirect
144. C₃×C₄²	C₂×D₇	semidirect
156. #124	C₃×D₇	semidirect
196. C₃×C₇	#535	semidirect
199. C₂×Q₈	C₂×C₃×D₇	semidirect
213. C₂²⋊C₄	C₂×C₃×D₇	semidirect
215. C₄²	C₂×C₃×D₇	semidirect
258. C₇	C₃×#535	semidirect
280. C₃	#90365	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×#1161	2
C₂×C₃×D₄×D₇	2
C₃×C₇×#124	2
C₃×#6510	2
C₃×#6503	2
C₃×#6447	2
C₃×#6445	2
C₃×#6437	2
C₃×#6435	2
#90365	3
C₃×#535	7

Subgroups — C3×#90365 #13061017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90365 #13061017