Subgroups — C₃×#90288 #13060939

Order 1344.

588 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#1159	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6502	C₂	1	2	yes	maximal, complemented
4	C₃×#6502	C₂	1	2	yes	maximal, complemented
5	C₃×#6493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#6459	C₂	1	2	yes	maximal, complemented
8	C₃×#6447	C₂	1	2	yes	maximal, complemented
9	C₃×#6441	C₂	1	2	yes	maximal, complemented
10	#90288	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
12	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
15	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
17	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
18	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
19	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
20	C₃×#1159	—	2	8	no	—
21	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
23	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
24	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
27	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
28	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
29	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
30	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
31	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
32	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
33	C₂×#1159	C₂×C₃	1	2	yes	complemented
34	C₇×#121	C₂×C₃	1	1	yes	characteristic, complemented
35	#6502	C₂×C₃	1	2	yes	complemented
36	#6502	C₂×C₃	1	2	yes	complemented
37	#6493	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×Dic₇⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
39	#6459	C₂×C₃	1	2	yes	complemented
40	#6447	C₂×C₃	1	2	yes	complemented
41	#6441	C₂×C₃	1	2	yes	complemented
42	C₃×#537	—	7	1	no	maximal, Hall
43	C₂³×C₃×C₇	C₂³	1	2	yes	—
44	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
45	C₂²×C₃×D₇	C₂³	1	2	yes	—
46	C₃×C₇×D₄	—	2	4	no	—
47	C₃×C₇×D₄	—	4	1	no	—
48	C₂×C₃×C₄×C₇	—	4	1	no	—
49	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
52	C₃×C₂²⋊D₇	—	2	4	no	—
53	C₃×C₂²⋊D₇	—	2	4	no	—
54	C₃×C₂²⋊D₇	—	4	2	no	—
55	C₂×C₃×Dic₇	—	2	4	no	—
56	C₂×C₃×Dic₇	—	4	1	no	—
57	C₂×C₃×Dic₇	D₄	1	4	yes	complemented
58	C₂×C₃×Dic₇	C₂³	1	2	yes	—
59	C₂×C₃×Dic₇	C₂³	1	2	yes	—
60	C₂×C₃×Dic₇	C₂³	1	2	yes	—
61	C₃×D₂₈	—	4	1	no	—
62	C₃×C₄×D₇	—	2	4	no	—
63	C₃×Dic₁₄	—	2	4	no	—
64	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
65	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
68	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
69	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
70	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
71	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
72	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
73	#1159	—	2	8	no	—
74	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
76	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
77	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
80	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
81	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
82	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
83	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
84	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
85	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
86	C₂×C₃×#41	—	7	2	no	—
87	C₃×C₂²⋊Q₈	—	7	2	no	—
88	C₃×C₂²⋊Q₈	—	7	2	no	—
89	C₃×#121	—	7	2	no	—
90	C₃×#121	—	7	2	no	—
91	C₃×#121	—	7	1	no	—
92	C₃×#121	—	7	2	no	—
93	C₃×#121	D₇	1	1	yes	characteristic, complemented
94	C₂×C₃×C₄⋊C₄	—	7	1	no	—
95	C₂²×C₃×C₇	—	2	1	no	—
96	C₂²×C₃×C₇	—	2	1	no	—
97	C₂²×C₃×C₇	—	2	2	no	—
98	C₂²×C₃×C₇	—	2	2	no	—
99	C₂²×C₃×C₇	—	2	2	no	—
100	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
101	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
102	C₂×C₃×D₇	—	2	2	no	—
103	C₂×C₃×D₇	—	2	2	no	—
104	C₂×C₃×D₇	—	2	2	no	—
105	C₃×C₄×C₇	—	2	2	no	—
106	C₃×C₄×C₇	—	2	1	no	—
107	C₃×C₄×C₇	—	2	1	no	—
108	C₃×Dic₇	—	2	2	no	—
109	C₃×Dic₇	—	2	2	no	—
110	C₃×Dic₇	C₂×D₄	1	4	yes	—
111	#537	—	7	1	no	Sylow
112	C₂³×C₇	C₂³×C₃	1	2	yes	—
113	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
114	C₂²×D₇	C₂³×C₃	1	2	yes	—
115	C₇×D₄	—	2	4	no	—
116	C₇×D₄	—	4	1	no	—
117	C₂×C₄×C₇	—	4	1	no	—
118	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
121	C₂²⋊D₇	—	2	4	no	—
122	C₂²⋊D₇	—	2	4	no	—
123	C₂²⋊D₇	—	4	2	no	—
124	C₂×Dic₇	—	2	4	no	—
125	C₂×Dic₇	—	4	1	no	—
126	C₂×Dic₇	C₃×D₄	1	4	yes	complemented
127	C₂×Dic₇	C₂³×C₃	1	2	yes	—
128	C₂×Dic₇	C₂³×C₃	1	2	yes	—
129	C₂×Dic₇	C₂³×C₃	1	2	yes	—
130	D₂₈	—	4	1	no	—
131	C₄×D₇	—	2	4	no	—
132	Dic₁₄	—	2	4	no	—
133	C₃×#41	—	14	8	no	—
134	C₂×C₃×Q₈	—	7	2	no	—
135	C₂×C₃×D₄	—	7	2	no	—
136	C₂×C₃×D₄	—	7	2	no	—
137	C₂×C₃×D₄	—	7	2	no	—
138	C₂×C₃×D₄	—	7	1	no	—
139	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
140	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
141	C₂²×C₃×C₄	—	7	1	no	—
142	C₂²×C₃×C₄	—	7	2	no	—
143	C₂²×C₃×C₄	—	7	1	no	—
144	C₂²×C₃×C₄	—	7	2	no	—
145	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
146	C₃×C₄⋊C₄	—	7	2	no	—
147	C₃×C₄⋊C₄	—	7	2	no	—
148	C₃×C₄⋊C₄	—	7	2	no	—
149	C₃×C₄⋊C₄	—	7	1	no	—
150	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
151	C₃×C₂²⋊C₄	—	7	2	no	—
152	C₃×C₂²⋊C₄	—	7	2	no	—
153	C₃×C₂²⋊C₄	—	7	2	no	—
154	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
155	C₂×C₃×C₇	—	2	2	no	—
156	C₂×C₃×C₇	—	4	2	no	—
157	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
160	C₃×D₇	—	4	2	no	—
161	C₂×#41	—	7	2	no	—
162	C₂²⋊Q₈	—	7	2	no	—
163	C₂²⋊Q₈	—	7	2	no	—
164	#121	—	7	2	no	—
165	#121	—	7	2	no	—
166	#121	—	7	1	no	—
167	#121	—	7	2	no	—
168	#121	C₃×D₇	1	1	yes	characteristic, complemented
169	C₂×C₄⋊C₄	—	7	1	no	—
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	—	2	1	no	—
172	C₂²×C₇	—	2	2	no	—
173	C₂²×C₇	—	2	2	no	—
174	C₂²×C₇	—	2	2	no	—
175	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
176	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
177	C₂×D₇	—	2	2	no	—
178	C₂×D₇	—	2	2	no	—
179	C₂×D₇	—	2	2	no	—
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	Dic₇	—	2	2	no	—
184	Dic₇	—	2	2	no	—
185	Dic₇	C₂×C₃×D₄	1	4	yes	—
186	C₂³×C₃	—	7	2	no	—
187	C₂³×C₃	C₂²×D₇	1	2	yes	—
188	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
189	C₃×Q₈	—	14	4	no	—
190	C₃×D₄	—	2	4	no	—
191	C₃×D₄	—	4	1	no	—
192	C₃×D₄	—	14	4	no	—
193	C₃×D₄	—	14	4	no	—
194	C₃×D₄	—	28	1	no	—
195	C₃×D₄	—	28	2	no	—
196	C₂×C₃×C₄	—	4	1	no	—
197	C₂×C₃×C₄	—	7	4	no	—
198	C₂×C₃×C₄	—	7	2	no	—
199	C₂×C₃×C₄	—	7	2	no	—
200	C₂×C₃×C₄	—	7	2	no	—
201	C₂×C₃×C₄	—	14	4	no	—
202	C₂×C₃×C₄	—	14	4	no	—
203	C₂×C₃×C₄	—	28	1	no	—
204	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
206	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
207	C₃×C₇	#537	1	1	yes	characteristic, complemented, Hall
208	#41	—	14	8	no	—
209	C₂×Q₈	—	7	2	no	—
210	C₂×D₄	—	7	2	no	—
211	C₂×D₄	—	7	2	no	—
212	C₂×D₄	—	7	2	no	—
213	C₂×D₄	—	7	1	no	—
214	C₂×D₄	C₂×C₃×D₇	1	2	yes	—
215	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
216	C₂²×C₄	—	7	1	no	—
217	C₂²×C₄	—	7	2	no	—
218	C₂²×C₄	—	7	1	no	—
219	C₂²×C₄	—	7	2	no	—
220	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
221	C₄⋊C₄	—	7	2	no	—
222	C₄⋊C₄	—	7	2	no	—
223	C₄⋊C₄	—	7	2	no	—
224	C₄⋊C₄	—	7	1	no	—
225	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
226	C₂²⋊C₄	—	7	2	no	—
227	C₂²⋊C₄	—	7	2	no	—
228	C₂²⋊C₄	—	7	2	no	—
229	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
230	C₂×C₇	—	2	2	no	—
231	C₂×C₇	—	4	2	no	—
232	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
233	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
235	D₇	—	4	2	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	14	2	no	—
242	C₂²×C₃	—	14	2	no	—
243	C₂²×C₃	—	14	2	no	—
244	C₂²×C₃	D₄×D₇	1	2	yes	—
245	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	7	4	no	—
250	C₃×C₄	—	14	2	no	—
251	C₃×C₄	—	14	2	no	—
252	C₂³	—	7	2	no	—
253	C₂³	C₂²×C₃×D₇	1	2	yes	—
254	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
255	Q₈	—	14	4	no	—
256	D₄	—	2	4	no	—
257	D₄	—	4	1	no	—
258	D₄	—	14	4	no	—
259	D₄	—	14	4	no	—
260	D₄	—	28	1	no	—
261	D₄	—	28	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	7	4	no	—
264	C₂×C₄	—	7	2	no	—
265	C₂×C₄	—	7	2	no	—
266	C₂×C₄	—	7	2	no	—
267	C₂×C₄	—	14	4	no	—
268	C₂×C₄	—	14	4	no	—
269	C₂×C₄	—	28	1	no	—
270	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
272	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
273	C₇	C₃×#537	1	1	yes	characteristic, complemented, Sylow
274	C₂×C₃	—	2	2	no	—
275	C₂×C₃	—	4	2	no	—
276	C₂×C₃	—	28	2	no	—
277	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
278	C₂×C₃	#6548	1	1	yes	characteristic
279	C₂×C₃	#6554	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	14	2	no	—
286	C₂²	—	14	2	no	—
287	C₂²	—	14	2	no	—
288	C₂²	C₃×D₄×D₇	1	2	yes	—
289	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	7	4	no	—
294	C₄	—	14	2	no	—
295	C₄	—	14	2	no	—
296	C₃	#90288	1	1	yes	characteristic, direct product factor, Sylow
297	C₂	—	2	2	no	—
298	C₂	—	4	2	no	—
299	C₂	—	28	2	no	—
300	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
301	C₂	C₃×#6548	1	1	yes	characteristic
302	C₂	C₃×#6554	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₇×#121	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×#121	D₇
hypocenter	C₇	C₃×#537
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#121

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#1159	C₂	semidirect
2. C₃×C₇×#121	C₂	semidirect
3. C₃×#6502	C₂	semidirect
4. C₃×#6502	C₂	semidirect
5. C₃×#6493	C₂	semidirect
6. C₂×C₃×Dic₇⋊C₄	C₂	semidirect
7. C₃×#6459	C₂	semidirect
8. C₃×#6447	C₂	semidirect
9. C₃×#6441	C₂	semidirect
10. #90288	C₃	direct (N × H)
33. C₂×#1159	C₂×C₃	semidirect
34. C₇×#121	C₂×C₃	semidirect
35. #6502	C₂×C₃	semidirect
36. #6502	C₂×C₃	semidirect
37. #6493	C₂×C₃	semidirect
38. C₂×Dic₇⋊C₄	C₂×C₃	semidirect
39. #6459	C₂×C₃	semidirect
40. #6447	C₂×C₃	semidirect
41. #6441	C₂×C₃	semidirect
57. C₂×C₃×Dic₇	D₄	semidirect
93. C₃×#121	D₇	semidirect
126. C₂×Dic₇	C₃×D₄	semidirect
168. #121	C₃×D₇	semidirect
207. C₃×C₇	#537	semidirect
273. C₇	C₃×#537	semidirect
296. C₃	#90288	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×#1159	2
C₃×C₇×#121	2
C₃×#6502	2
C₃×#6493	2
C₂×C₃×Dic₇⋊C₄	2
C₃×#6459	2
C₃×#6447	2
C₃×#6441	2
#90288	3
C₃×#537	7

Subgroups — C3×#90288 #13060939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90288 #13060939