Subgroups — C₃×#90230 #13060882

Order 1344.

456 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₃×D₄×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₃×D₄×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₃×#6487	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×Dic₇⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×#6453	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#6451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#6441	C₂	1	2	yes	maximal, complemented
9	C₃×#6441	C₂	1	2	yes	maximal, complemented
10	C₃×#6433	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#90230	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
15	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
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×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
23	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
25	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
28	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
29	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
30	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
31	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
32	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
33	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
36	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic
37	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic
38	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic
39	#6487	C₂×C₃	1	2	yes	complemented
40	C₂×Dic₇⋊C₄	C₂×C₃	1	2	yes	complemented
41	#6453	C₂×C₃	1	1	yes	characteristic, complemented
42	#6451	C₂×C₃	1	1	yes	characteristic, complemented
43	#6441	C₂×C₃	1	2	yes	complemented
44	#6441	C₂×C₃	1	2	yes	complemented
45	#6433	C₂×C₃	1	1	yes	characteristic, complemented
46	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×#554	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	2	yes	—
49	C₃×C₇×D₄	Q₈	1	4	yes	complemented
50	C₂×C₃×C₄×C₇	—	4	2	no	—
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
55	C₂×C₃×Dic₇	—	2	4	no	—
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	2	yes	—
62	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
63	C₃×Dic₁₄	—	2	2	no	—
64	C₃×Dic₁₄	—	4	2	no	—
65	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
67	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
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	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
74	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
75	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
76	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
77	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
78	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
79	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
80	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
81	Dic₇⋊C₄	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	Dic₇⋊C₄	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×#125	—	7	1	no	—
90	C₃×#125	—	7	1	no	—
91	C₃×C₂²⋊Q₈	—	7	2	no	—
92	C₃×C₂²⋊Q₈	—	7	2	no	—
93	C₃×C₂²⋊Q₈	—	7	2	no	—
94	C₃×C₄×Q₈	—	7	1	no	—
95	C₃×C₄×D₄	—	7	1	no	—
96	C₃×C₄×D₄	—	7	1	no	—
97	C₃×C₄×D₄	D₇	1	1	yes	characteristic
98	C₂×C₃×C₄⋊C₄	—	7	2	no	—
99	C₂²×C₃×C₇	—	2	2	no	—
100	C₂²×C₃×C₇	—	2	2	no	—
101	C₂²×C₃×C₇	C₂×Q₈	1	4	yes	—
102	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
103	C₃×C₄×C₇	—	2	1	no	—
104	C₃×C₄×C₇	—	2	2	no	—
105	C₃×C₄×C₇	—	2	1	no	—
106	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
107	C₃×Dic₇	—	2	1	no	—
108	C₃×Dic₇	—	2	2	no	—
109	C₃×Dic₇	—	2	1	no	—
110	C₃×Dic₇	—	2	2	no	—
111	C₃×Dic₇	—	2	1	no	—
112	C₃×Dic₇	#41	1	2	yes	—
113	#554	—	7	1	no	Sylow
114	C₂³×C₇	C₂³×C₃	1	2	yes	—
115	C₇×D₄	C₃×Q₈	1	4	yes	complemented
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
120	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
121	C₂×Dic₇	—	2	4	no	—
122	C₂×Dic₇	—	4	2	no	—
123	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
125	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
126	C₂×Dic₇	C₂³×C₃	1	2	yes	—
127	C₂×Dic₇	C₂³×C₃	1	2	yes	—
128	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
129	Dic₁₄	—	2	2	no	—
130	Dic₁₄	—	4	2	no	—
131	C₂×C₃×Q₈	—	7	1	no	—
132	C₂×C₃×Q₈	—	7	2	no	—
133	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
134	C₂²×C₃×C₄	—	7	2	no	—
135	C₂²×C₃×C₄	—	7	2	no	—
136	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
137	C₃×C₄⋊C₄	—	7	1	no	—
138	C₃×C₄⋊C₄	—	7	2	no	—
139	C₃×C₄⋊C₄	—	7	2	no	—
140	C₃×C₄⋊C₄	—	7	2	no	—
141	C₃×C₄⋊C₄	—	7	1	no	—
142	C₃×C₄⋊C₄	—	7	2	no	—
143	C₃×C₄⋊C₄	—	7	1	no	—
144	C₃×C₄⋊C₄	—	7	1	no	—
145	C₃×C₄⋊C₄	—	7	1	no	—
146	C₃×C₄⋊C₄	—	7	2	no	—
147	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	—	7	2	no	—
150	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
151	C₃×C₄²	—	7	1	no	—
152	C₃×C₄²	—	7	1	no	—
153	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
154	C₂×C₃×C₇	—	2	4	no	—
155	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
157	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
158	C₄⋊Q₈	—	7	1	no	—
159	#125	—	7	1	no	—
160	#125	—	7	1	no	—
161	C₂²⋊Q₈	—	7	2	no	—
162	C₂²⋊Q₈	—	7	2	no	—
163	C₂²⋊Q₈	—	7	2	no	—
164	C₄×Q₈	—	7	1	no	—
165	C₄×D₄	—	7	1	no	—
166	C₄×D₄	—	7	1	no	—
167	C₄×D₄	C₃×D₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	7	2	no	—
169	C₂²×C₇	—	2	2	no	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	C₂×C₃×Q₈	1	4	yes	—
172	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	—	2	2	no	—
175	C₄×C₇	—	2	1	no	—
176	C₄×C₇	C₂×C₃×Q₈	1	2	yes	—
177	Dic₇	—	2	1	no	—
178	Dic₇	—	2	2	no	—
179	Dic₇	—	2	1	no	—
180	Dic₇	—	2	2	no	—
181	Dic₇	—	2	1	no	—
182	Dic₇	C₃×#41	1	2	yes	—
183	C₂³×C₃	C₂²×D₇	1	2	yes	—
184	C₃×Q₈	—	14	2	no	—
185	C₃×Q₈	—	28	2	no	—
186	C₃×D₄	Dic₁₄	1	4	yes	complemented
187	C₂×C₃×C₄	—	4	2	no	—
188	C₂×C₃×C₄	—	7	1	no	—
189	C₂×C₃×C₄	—	7	1	no	—
190	C₂×C₃×C₄	—	7	1	no	—
191	C₂×C₃×C₄	—	7	1	no	—
192	C₂×C₃×C₄	—	7	2	no	—
193	C₂×C₃×C₄	—	7	2	no	—
194	C₂×C₃×C₄	—	14	4	no	—
195	C₂×C₃×C₄	—	28	2	no	—
196	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
199	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
200	C₃×C₇	#554	1	1	yes	characteristic, complemented, Hall
201	C₂×Q₈	—	7	1	no	—
202	C₂×Q₈	—	7	2	no	—
203	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
204	C₂²×C₄	—	7	2	no	—
205	C₂²×C₄	—	7	2	no	—
206	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
207	C₄⋊C₄	—	7	1	no	—
208	C₄⋊C₄	—	7	2	no	—
209	C₄⋊C₄	—	7	2	no	—
210	C₄⋊C₄	—	7	2	no	—
211	C₄⋊C₄	—	7	1	no	—
212	C₄⋊C₄	—	7	2	no	—
213	C₄⋊C₄	—	7	1	no	—
214	C₄⋊C₄	—	7	1	no	—
215	C₄⋊C₄	—	7	1	no	—
216	C₄⋊C₄	—	7	2	no	—
217	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
218	C₂²⋊C₄	—	7	2	no	—
219	C₂²⋊C₄	—	7	2	no	—
220	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
221	C₄²	—	7	1	no	—
222	C₄²	—	7	1	no	—
223	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
224	C₂×C₇	—	2	4	no	—
225	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
226	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
227	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	C₂×Dic₁₄	1	4	yes	—
231	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	7	2	no	—
236	C₃×C₄	—	14	1	no	—
237	C₃×C₄	—	14	2	no	—
238	C₃×C₄	—	14	1	no	—
239	C₃×C₄	—	14	2	no	—
240	C₃×C₄	—	14	1	no	—
241	C₃×C₄	C₂×Dic₁₄	1	2	yes	—
242	C₂³	C₂²×C₃×D₇	1	2	yes	—
243	Q₈	—	14	2	no	—
244	Q₈	—	28	2	no	—
245	D₄	C₃×Dic₁₄	1	4	yes	complemented
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	7	1	no	—
248	C₂×C₄	—	7	1	no	—
249	C₂×C₄	—	7	1	no	—
250	C₂×C₄	—	7	1	no	—
251	C₂×C₄	—	7	2	no	—
252	C₂×C₄	—	7	2	no	—
253	C₂×C₄	—	14	4	no	—
254	C₂×C₄	—	28	2	no	—
255	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
258	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
259	C₇	C₃×#554	1	1	yes	characteristic, complemented, Sylow
260	C₂×C₃	—	2	4	no	—
261	C₂×C₃	C₂²×Dic₁₄	1	1	yes	characteristic
262	C₂×C₃	#6548	1	1	yes	characteristic
263	C₂×C₃	D₇×#41	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	C₂×C₃×Dic₁₄	1	4	yes	—
267	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	1	no	—
271	C₄	—	7	2	no	—
272	C₄	—	14	1	no	—
273	C₄	—	14	2	no	—
274	C₄	—	14	1	no	—
275	C₄	—	14	2	no	—
276	C₄	—	14	1	no	—
277	C₄	C₂×C₃×Dic₁₄	1	2	yes	—
278	C₃	#90230	1	1	yes	characteristic, direct product factor, Sylow
279	C₂	—	2	4	no	—
280	C₂	C₂²×C₃×Dic₁₄	1	1	yes	characteristic
281	C₂	C₃×#6548	1	1	yes	characteristic
282	C₂	C₃×D₇×#41	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×C₄×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×C₇×D₄	C₂²
hypercenter	C₃×C₄×D₄	D₇
hypocenter	C₇	C₃×#554
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
4. C₃×#6487	C₂	semidirect
5. C₂×C₃×Dic₇⋊C₄	C₂	semidirect
6. C₃×#6453	C₂	semidirect
7. C₃×#6451	C₂	semidirect
8. C₃×#6441	C₂	semidirect
9. C₃×#6441	C₂	semidirect
10. C₃×#6433	C₂	semidirect
11. C₃×C₄×Dic₁₄	C₂	semidirect
12. #90230	C₃	direct (N × H)
39. #6487	C₂×C₃	semidirect
40. C₂×Dic₇⋊C₄	C₂×C₃	semidirect
41. #6453	C₂×C₃	semidirect
42. #6451	C₂×C₃	semidirect
43. #6441	C₂×C₃	semidirect
44. #6441	C₂×C₃	semidirect
45. #6433	C₂×C₃	semidirect
46. C₄×Dic₁₄	C₂×C₃	semidirect
49. C₃×C₇×D₄	Q₈	semidirect
115. C₇×D₄	C₃×Q₈	semidirect
186. C₃×D₄	Dic₁₄	semidirect
200. C₃×C₇	#554	semidirect
245. D₄	C₃×Dic₁₄	semidirect
259. C₇	C₃×#554	semidirect
278. C₃	#90230	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×C₄×C₇×D₄	2
C₃×D₄×Dic₇	2
C₃×#6487	2
C₂×C₃×Dic₇⋊C₄	2
C₃×#6453	2
C₃×#6451	2
C₃×#6441	2
C₃×#6433	2
C₃×C₄×Dic₁₄	2
#90230	3
C₃×#554	7

Subgroups — C3×#90230 #13060882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90230 #13060882