Subgroups — C₃×#90294 #13060947

Order 1344.

432 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#121	C₂	1	1	yes	maximal, characteristic
2	C₃×#6499	C₂	1	2	yes	maximal, complemented
3	C₃×#6499	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6498	C₂	1	2	yes	maximal, complemented
5	C₃×D₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#6487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#6451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#6450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#6442	C₂	1	2	yes	maximal, complemented
11	C₃×#6441	C₂	1	2	yes	maximal, complemented
12	#90294	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
17	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
22	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
23	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
24	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
25	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
26	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
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	C₃×C₄×Dic₇	C₂²	1	2	yes	—
34	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
36	C₇×#121	C₂×C₃	1	1	yes	characteristic
37	#6499	C₂×C₃	1	2	yes	complemented
38	#6499	C₂×C₃	1	1	yes	characteristic, complemented
39	#6498	C₂×C₃	1	2	yes	complemented
40	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
41	#6489	C₂×C₃	1	1	yes	characteristic, complemented
42	#6487	C₂×C₃	1	1	yes	characteristic, complemented
43	#6451	C₂×C₃	1	1	yes	characteristic, complemented
44	#6450	C₂×C₃	1	1	yes	characteristic, complemented
45	#6442	C₂×C₃	1	2	yes	complemented
46	#6441	C₂×C₃	1	2	yes	complemented
47	C₃×#542	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
49	C₂³×C₃×C₇	C₂³	1	2	yes	—
50	C₃×C₇×D₄	—	2	2	no	—
51	C₃×C₇×D₄	—	4	2	no	—
52	C₂×C₃×C₄×C₇	—	2	2	no	—
53	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
56	C₂×C₃×Dic₇	—	4	2	no	—
57	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
58	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₇	C₂³	1	2	yes	—
61	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₇	C₂³	1	2	yes	—
63	C₃×Dic₁₄	—	2	2	no	—
64	C₃×Dic₁₄	—	4	2	no	—
65	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
67	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
68	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
69	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
71	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
74	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
75	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
76	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
77	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
78	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
79	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
80	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
81	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
82	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
83	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
84	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
85	C₄×Dic₇	C₂²×C₃	1	2	yes	—
86	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
87	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
88	C₃×C₄⋊Q₈	—	7	1	no	—
89	C₃×#126	—	7	2	no	—
90	C₃×#124	—	7	1	no	—
91	C₃×#124	—	7	2	no	—
92	C₃×#123	—	7	2	no	—
93	C₃×C₂²⋊Q₈	—	7	1	no	—
94	C₃×C₂²⋊Q₈	—	7	2	no	—
95	C₃×#121	D₇	1	1	yes	characteristic
96	C₃×C₄×Q₈	—	7	1	no	—
97	C₃×C₄×D₄	—	7	1	no	—
98	C₃×#117	—	7	1	no	—
99	C₂²×C₃×C₇	—	2	1	no	—
100	C₂²×C₃×C₇	—	2	2	no	—
101	C₂²×C₃×C₇	—	2	2	no	—
102	C₂²×C₃×C₇	—	2	1	no	—
103	C₂²×C₃×C₇	—	2	1	no	—
104	C₂²×C₃×C₇	—	2	2	no	—
105	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
106	C₃×C₄×C₇	—	2	1	no	—
107	C₃×C₄×C₇	—	2	2	no	—
108	C₃×C₄×C₇	#41	1	2	yes	—
109	C₃×Dic₇	—	2	2	no	—
110	C₃×Dic₇	—	2	1	no	—
111	C₃×Dic₇	—	2	2	no	—
112	C₃×Dic₇	—	2	1	no	—
113	C₃×Dic₇	—	2	1	no	—
114	C₃×Dic₇	—	2	1	no	—
115	#542	—	7	1	no	Sylow
116	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
117	C₂³×C₇	C₂³×C₃	1	2	yes	—
118	C₇×D₄	—	2	2	no	—
119	C₇×D₄	—	4	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
122	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₇	—	4	2	no	—
125	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
126	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
127	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
128	C₂×Dic₇	C₂³×C₃	1	2	yes	—
129	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
130	C₂×Dic₇	C₂³×C₃	1	2	yes	—
131	Dic₁₄	—	2	2	no	—
132	Dic₁₄	—	4	2	no	—
133	C₂×C₃×Q₈	—	7	1	no	—
134	C₂×C₃×Q₈	—	7	2	no	—
135	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
136	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
137	C₂²×C₃×C₄	—	7	2	no	—
138	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	7	2	no	—
140	C₃×C₄⋊C₄	—	7	2	no	—
141	C₃×C₄⋊C₄	—	7	2	no	—
142	C₃×C₄⋊C₄	—	7	1	no	—
143	C₃×C₄⋊C₄	—	7	1	no	—
144	C₃×C₄⋊C₄	—	7	1	no	—
145	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
146	C₃×C₂²⋊C₄	—	7	1	no	—
147	C₃×C₂²⋊C₄	—	7	1	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	—	7	2	no	—
150	C₃×C₂²⋊C₄	—	7	2	no	—
151	C₃×C₂²⋊C₄	—	7	2	no	—
152	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
153	C₃×C₄²	—	7	1	no	—
154	C₃×C₄²	—	7	1	no	—
155	C₃×C₄²	—	7	2	no	—
156	C₂×C₃×C₇	—	4	1	no	—
157	C₂×C₃×C₇	—	4	2	no	—
158	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
159	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
161	C₄⋊Q₈	—	7	1	no	—
162	#126	—	7	2	no	—
163	#124	—	7	1	no	—
164	#124	—	7	2	no	—
165	#123	—	7	2	no	—
166	C₂²⋊Q₈	—	7	1	no	—
167	C₂²⋊Q₈	—	7	2	no	—
168	#121	C₃×D₇	1	1	yes	characteristic
169	C₄×Q₈	—	7	1	no	—
170	C₄×D₄	—	7	1	no	—
171	#117	—	7	1	no	—
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	—	2	2	no	—
174	C₂²×C₇	—	2	2	no	—
175	C₂²×C₇	—	2	1	no	—
176	C₂²×C₇	—	2	1	no	—
177	C₂²×C₇	—	2	2	no	—
178	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
179	C₄×C₇	—	2	1	no	—
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	C₃×#41	1	2	yes	—
182	Dic₇	—	2	2	no	—
183	Dic₇	—	2	1	no	—
184	Dic₇	—	2	2	no	—
185	Dic₇	—	2	1	no	—
186	Dic₇	—	2	1	no	—
187	Dic₇	—	2	1	no	—
188	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
189	C₂³×C₃	C₂²×D₇	1	2	yes	—
190	C₃×Q₈	—	14	2	no	—
191	C₃×Q₈	—	28	2	no	—
192	C₃×D₄	—	2	2	no	—
193	C₃×D₄	—	4	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	7	1	no	—
196	C₂×C₃×C₄	—	7	1	no	—
197	C₂×C₃×C₄	—	7	2	no	—
198	C₂×C₃×C₄	—	7	1	no	—
199	C₂×C₃×C₄	—	7	1	no	—
200	C₂×C₃×C₄	—	7	2	no	—
201	C₂×C₃×C₄	—	28	2	no	—
202	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
203	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
205	C₃×C₇	#542	1	1	yes	characteristic, complemented, Hall
206	C₂×Q₈	—	7	1	no	—
207	C₂×Q₈	—	7	2	no	—
208	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
209	C₂×D₄	C₂×C₃×D₇	1	2	yes	—
210	C₂²×C₄	—	7	2	no	—
211	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
212	C₄⋊C₄	—	7	2	no	—
213	C₄⋊C₄	—	7	2	no	—
214	C₄⋊C₄	—	7	2	no	—
215	C₄⋊C₄	—	7	1	no	—
216	C₄⋊C₄	—	7	1	no	—
217	C₄⋊C₄	—	7	1	no	—
218	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
219	C₂²⋊C₄	—	7	1	no	—
220	C₂²⋊C₄	—	7	1	no	—
221	C₂²⋊C₄	—	7	2	no	—
222	C₂²⋊C₄	—	7	2	no	—
223	C₂²⋊C₄	—	7	2	no	—
224	C₂²⋊C₄	—	7	2	no	—
225	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
226	C₄²	—	7	1	no	—
227	C₄²	—	7	1	no	—
228	C₄²	—	7	2	no	—
229	C₂×C₇	—	4	1	no	—
230	C₂×C₇	—	4	2	no	—
231	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
232	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
233	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	14	2	no	—
244	C₃×C₄	—	14	1	no	—
245	C₃×C₄	—	14	2	no	—
246	C₃×C₄	—	14	1	no	—
247	C₃×C₄	—	14	1	no	—
248	C₃×C₄	—	14	1	no	—
249	C₃×C₄	#1159	1	2	yes	—
250	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
251	C₂³	C₂²×C₃×D₇	1	2	yes	—
252	Q₈	—	14	2	no	—
253	Q₈	—	28	2	no	—
254	D₄	—	2	2	no	—
255	D₄	—	4	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	7	1	no	—
258	C₂×C₄	—	7	1	no	—
259	C₂×C₄	—	7	2	no	—
260	C₂×C₄	—	7	1	no	—
261	C₂×C₄	—	7	1	no	—
262	C₂×C₄	—	7	2	no	—
263	C₂×C₄	—	28	2	no	—
264	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
265	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
267	C₇	C₃×#542	1	1	yes	characteristic, complemented, Sylow
268	C₂×C₃	—	4	1	no	—
269	C₂×C₃	—	4	2	no	—
270	C₂×C₃	C₂×#1159	1	1	yes	characteristic
271	C₂×C₃	#6548	1	1	yes	characteristic
272	C₂×C₃	#6554	1	1	yes	characteristic
273	C₂²	—	2	1	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	14	2	no	—
283	C₄	—	14	1	no	—
284	C₄	—	14	2	no	—
285	C₄	—	14	1	no	—
286	C₄	—	14	1	no	—
287	C₄	—	14	1	no	—
288	C₄	C₃×#1159	1	2	yes	—
289	C₃	#90294	1	1	yes	characteristic, direct product factor, Sylow
290	C₂	—	4	1	no	—
291	C₂	—	4	2	no	—
292	C₂	C₂×C₃×#1159	1	1	yes	characteristic
293	C₂	C₃×#6548	1	1	yes	characteristic
294	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, prime radical	C₃×C₇×#121	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×#121	D₇
hypocenter	C₇	C₃×#542
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
2. C₃×#6499	C₂	semidirect
3. C₃×#6499	C₂	semidirect
4. C₃×#6498	C₂	semidirect
5. C₃×D₄×Dic₇	C₂	semidirect
6. C₃×#6489	C₂	semidirect
7. C₃×#6487	C₂	semidirect
8. C₃×#6451	C₂	semidirect
9. C₃×#6450	C₂	semidirect
10. C₃×#6442	C₂	semidirect
11. C₃×#6441	C₂	semidirect
12. #90294	C₃	direct (N × H)
37. #6499	C₂×C₃	semidirect
38. #6499	C₂×C₃	semidirect
39. #6498	C₂×C₃	semidirect
40. D₄×Dic₇	C₂×C₃	semidirect
41. #6489	C₂×C₃	semidirect
42. #6487	C₂×C₃	semidirect
43. #6451	C₂×C₃	semidirect
44. #6450	C₂×C₃	semidirect
45. #6442	C₂×C₃	semidirect
46. #6441	C₂×C₃	semidirect
205. C₃×C₇	#542	semidirect
267. C₇	C₃×#542	semidirect
289. C₃	#90294	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇×#121	2
C₃×#6499	2
C₃×#6498	2
C₃×D₄×Dic₇	2
C₃×#6489	2
C₃×#6487	2
C₃×#6451	2
C₃×#6450	2
C₃×#6442	2
C₃×#6441	2
#90294	3
C₃×#542	7

Subgroups — C3×#90294 #13060947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90294 #13060947