Subgroups — C₃×#90339 #13060991

Order 1344.

540 conjugacy classes of subgroups.

The subgroup lattice has 320 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	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#123	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6459	C₂	1	2	yes	maximal, complemented
7	C₃×#6451	C₂	1	2	yes	maximal, complemented
8	C₃×#6449	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#6448	C₂	1	2	yes	maximal, complemented
10	C₃×#6441	C₂	1	2	yes	maximal, complemented
11	C₃×#6440	C₂	1	1	yes	maximal, characteristic, complemented
12	#90339	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	1	yes	characteristic
15	C₂×C₃×C₂²⋊D₇	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₂²⋊D₇	C₂²	1	1	yes	characteristic
17	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
18	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
19	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
20	C₃×#1159	—	2	4	no	—
21	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
22	C₂×C₃×Dic₁₄	—	2	4	no	—
23	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
24	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
25	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
26	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
27	C₃×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
30	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
31	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
32	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
33	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
34	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
35	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
36	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
37	C₃×C₄×Dic₇	C₂²	1	2	yes	—
38	C₂×#1159	C₂×C₃	1	1	yes	characteristic
39	C₂²×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
40	C₇×#123	C₂×C₃	1	1	yes	characteristic, complemented
41	#6498	C₂×C₃	1	1	yes	characteristic, complemented
42	#6492	C₂×C₃	1	1	yes	characteristic, complemented
43	#6459	C₂×C₃	1	2	yes	complemented
44	#6451	C₂×C₃	1	2	yes	complemented
45	#6449	C₂×C₃	1	1	yes	characteristic, complemented
46	#6448	C₂×C₃	1	2	yes	complemented
47	#6441	C₂×C₃	1	2	yes	complemented
48	#6440	C₂×C₃	1	1	yes	characteristic, complemented
49	C₃×#536	—	7	1	no	maximal, Hall
50	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
51	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
52	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
53	C₃×C₇×D₄	—	2	2	no	—
54	C₂×C₃×C₄×C₇	—	2	2	no	—
55	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
56	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
58	C₃×C₂²⋊D₇	—	2	2	no	—
59	C₃×C₂²⋊D₇	—	2	2	no	—
60	C₂×C₃×Dic₇	—	2	2	no	—
61	C₂×C₃×Dic₇	—	2	2	no	—
62	C₂×C₃×Dic₇	D₄	1	4	yes	—
63	C₂×C₃×Dic₇	C₂³	1	2	yes	—
64	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
65	C₂×C₃×Dic₇	C₂³	1	2	yes	—
66	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
67	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
68	C₃×C₄×D₇	—	2	2	no	—
69	C₃×Dic₁₄	—	2	4	no	—
70	C₃×Dic₁₄	—	2	4	no	—
71	C₃×Dic₁₄	—	2	2	no	—
72	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
76	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
77	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
78	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
79	#1159	—	2	4	no	—
80	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
81	C₂×Dic₁₄	—	2	4	no	—
82	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
83	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
84	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
85	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
86	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
89	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
90	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
91	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
92	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
93	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
94	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
95	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
96	C₄×Dic₇	C₂²×C₃	1	2	yes	—
97	C₂×C₃×#41	—	7	1	no	—
98	C₂²×C₃×Q₈	—	7	1	no	—
99	C₃×C₄⋊Q₈	—	7	2	no	—
100	C₃×#124	—	7	2	no	—
101	C₃×#123	—	7	1	no	—
102	C₃×#123	—	7	1	no	—
103	C₃×#123	—	7	1	no	—
104	C₃×#123	D₇	1	1	yes	characteristic, complemented
105	C₃×C₂²⋊Q₈	—	7	2	no	—
106	C₃×C₂²⋊Q₈	—	7	2	no	—
107	C₃×#117	—	7	1	no	—
108	C₂²×C₃×C₇	—	2	2	no	—
109	C₂²×C₃×C₇	—	2	1	no	—
110	C₂²×C₃×C₇	—	2	2	no	—
111	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
112	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
113	C₂×C₃×D₇	—	2	1	no	—
114	C₂×C₃×D₇	—	2	2	no	—
115	C₃×C₄×C₇	—	2	2	no	—
116	C₃×C₄×C₇	—	2	1	no	—
117	C₃×C₄×C₇	—	2	2	no	—
118	C₃×Dic₇	—	2	2	no	—
119	C₃×Dic₇	—	2	1	no	—
120	C₃×Dic₇	—	2	2	no	—
121	C₃×Dic₇	C₂×D₄	1	2	yes	—
122	C₃×Dic₇	C₂×D₄	1	2	yes	—
123	#536	—	7	1	no	Sylow
124	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
125	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
126	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
127	C₇×D₄	—	2	2	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
130	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
132	C₂²⋊D₇	—	2	2	no	—
133	C₂²⋊D₇	—	2	2	no	—
134	C₂×Dic₇	—	2	2	no	—
135	C₂×Dic₇	—	2	2	no	—
136	C₂×Dic₇	C₃×D₄	1	4	yes	—
137	C₂×Dic₇	C₂³×C₃	1	2	yes	—
138	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
139	C₂×Dic₇	C₂³×C₃	1	2	yes	—
140	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
141	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
142	C₄×D₇	—	2	2	no	—
143	Dic₁₄	—	2	4	no	—
144	Dic₁₄	—	2	4	no	—
145	Dic₁₄	—	2	2	no	—
146	C₃×#41	—	14	4	no	—
147	C₂×C₃×Q₈	—	7	2	no	—
148	C₂×C₃×Q₈	—	7	2	no	—
149	C₂×C₃×Q₈	—	7	1	no	—
150	C₂×C₃×Q₈	—	14	4	no	—
151	C₂×C₃×D₄	—	7	1	no	—
152	C₂×C₃×D₄	—	7	1	no	—
153	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
154	C₂²×C₃×C₄	—	7	1	no	—
155	C₂²×C₃×C₄	—	7	1	no	—
156	C₂²×C₃×C₄	—	7	1	no	—
157	C₂²×C₃×C₄	—	7	1	no	—
158	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
159	C₃×C₄⋊C₄	—	7	2	no	—
160	C₃×C₄⋊C₄	—	7	2	no	—
161	C₃×C₄⋊C₄	—	7	2	no	—
162	C₃×C₄⋊C₄	—	7	2	no	—
163	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
164	C₃×C₂²⋊C₄	—	7	1	no	—
165	C₃×C₂²⋊C₄	—	7	2	no	—
166	C₃×C₂²⋊C₄	—	7	2	no	—
167	C₃×C₂²⋊C₄	—	7	2	no	—
168	C₃×C₂²⋊C₄	C₂×D₇	1	1	yes	characteristic
169	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
170	C₃×C₄²	—	7	2	no	—
171	C₂×C₃×C₇	—	2	2	no	—
172	C₂×C₃×C₇	—	4	1	no	—
173	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₃×C₇	2₋¹⁺⁴	1	2	yes	—
175	C₃×D₇	—	4	1	no	—
176	C₂×#41	—	7	1	no	—
177	C₂²×Q₈	—	7	1	no	—
178	C₄⋊Q₈	—	7	2	no	—
179	#124	—	7	2	no	—
180	#123	—	7	1	no	—
181	#123	—	7	1	no	—
182	#123	—	7	1	no	—
183	#123	C₃×D₇	1	1	yes	characteristic, complemented
184	C₂²⋊Q₈	—	7	2	no	—
185	C₂²⋊Q₈	—	7	2	no	—
186	#117	—	7	1	no	—
187	C₂²×C₇	—	2	2	no	—
188	C₂²×C₇	—	2	1	no	—
189	C₂²×C₇	—	2	2	no	—
190	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
191	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
192	C₂×D₇	—	2	1	no	—
193	C₂×D₇	—	2	2	no	—
194	C₄×C₇	—	2	2	no	—
195	C₄×C₇	—	2	1	no	—
196	C₄×C₇	—	2	2	no	—
197	Dic₇	—	2	2	no	—
198	Dic₇	—	2	1	no	—
199	Dic₇	—	2	2	no	—
200	Dic₇	C₂×C₃×D₄	1	2	yes	—
201	Dic₇	C₂×C₃×D₄	1	2	yes	—
202	C₂³×C₃	—	7	1	no	—
203	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
204	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
205	C₃×Q₈	—	14	4	no	—
206	C₃×Q₈	—	14	4	no	—
207	C₃×Q₈	—	14	2	no	—
208	C₃×D₄	—	2	2	no	—
209	C₃×D₄	—	14	2	no	—
210	C₃×D₄	—	14	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	7	4	no	—
213	C₂×C₃×C₄	—	7	1	no	—
214	C₂×C₃×C₄	—	7	1	no	—
215	C₂×C₃×C₄	—	7	2	no	—
216	C₂×C₃×C₄	—	7	2	no	—
217	C₂×C₃×C₄	—	7	1	no	—
218	C₂×C₃×C₄	—	14	2	no	—
219	C₂×C₃×C₄	—	14	2	no	—
220	C₂×C₃×C₄	—	14	2	no	—
221	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
222	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
223	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
224	C₃×C₇	#536	1	1	yes	characteristic, complemented, Hall
225	#41	—	14	4	no	—
226	C₂×Q₈	—	7	2	no	—
227	C₂×Q₈	—	7	2	no	—
228	C₂×Q₈	—	7	1	no	—
229	C₂×Q₈	—	14	4	no	—
230	C₂×D₄	—	7	1	no	—
231	C₂×D₄	—	7	1	no	—
232	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
233	C₂²×C₄	—	7	1	no	—
234	C₂²×C₄	—	7	1	no	—
235	C₂²×C₄	—	7	1	no	—
236	C₂²×C₄	—	7	1	no	—
237	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
238	C₄⋊C₄	—	7	2	no	—
239	C₄⋊C₄	—	7	2	no	—
240	C₄⋊C₄	—	7	2	no	—
241	C₄⋊C₄	—	7	2	no	—
242	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	—
243	C₂²⋊C₄	—	7	1	no	—
244	C₂²⋊C₄	—	7	2	no	—
245	C₂²⋊C₄	—	7	2	no	—
246	C₂²⋊C₄	—	7	2	no	—
247	C₂²⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
248	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
249	C₄²	—	7	2	no	—
250	C₂×C₇	—	2	2	no	—
251	C₂×C₇	—	4	1	no	—
252	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
253	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	—
254	D₇	—	4	1	no	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	14	1	no	—
259	C₂²×C₃	—	14	2	no	—
260	C₂²×C₃	D₄×D₇	1	2	yes	—
261	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	7	2	no	—
266	C₃×C₄	—	7	2	no	—
267	C₃×C₄	—	14	1	no	—
268	C₃×C₄	—	14	2	no	—
269	C₃×C₄	—	14	2	no	—
270	C₂³	—	7	1	no	—
271	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
272	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
273	Q₈	—	14	4	no	—
274	Q₈	—	14	4	no	—
275	Q₈	—	14	2	no	—
276	D₄	—	2	2	no	—
277	D₄	—	14	2	no	—
278	D₄	—	14	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	7	4	no	—
281	C₂×C₄	—	7	1	no	—
282	C₂×C₄	—	7	1	no	—
283	C₂×C₄	—	7	2	no	—
284	C₂×C₄	—	7	2	no	—
285	C₂×C₄	—	7	1	no	—
286	C₂×C₄	—	14	2	no	—
287	C₂×C₄	—	14	2	no	—
288	C₂×C₄	—	14	2	no	—
289	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
290	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
291	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
292	C₇	C₃×#536	1	1	yes	characteristic, complemented, Sylow
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	—	4	1	no	—
295	C₂×C₃	—	28	1	no	—
296	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
297	C₂×C₃	#6554	1	2	yes	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	14	1	no	—
302	C₂²	—	14	2	no	—
303	C₂²	C₃×D₄×D₇	1	2	yes	—
304	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	7	2	no	—
309	C₄	—	7	2	no	—
310	C₄	—	14	1	no	—
311	C₄	—	14	2	no	—
312	C₄	—	14	2	no	—
313	C₃	#90339	1	1	yes	characteristic, direct product factor, Sylow
314	C₂	—	2	2	no	—
315	C₂	—	4	1	no	—
316	C₂	—	28	1	no	—
317	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
318	C₂	C₃×#6554	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×#123	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×#1159	C₂
hypercenter	C₃×#123	D₇
hypocenter	C₇	C₃×#536
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#123

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×Dic₁₄	C₂	semidirect
3. C₃×C₇×#123	C₂	semidirect
4. C₃×#6498	C₂	semidirect
5. C₃×#6492	C₂	semidirect
6. C₃×#6459	C₂	semidirect
7. C₃×#6451	C₂	semidirect
8. C₃×#6449	C₂	semidirect
9. C₃×#6448	C₂	semidirect
10. C₃×#6441	C₂	semidirect
11. C₃×#6440	C₂	semidirect
12. #90339	C₃	direct (N × H)
39. C₂²×Dic₁₄	C₂×C₃	semidirect
40. C₇×#123	C₂×C₃	semidirect
41. #6498	C₂×C₃	semidirect
42. #6492	C₂×C₃	semidirect
43. #6459	C₂×C₃	semidirect
44. #6451	C₂×C₃	semidirect
45. #6449	C₂×C₃	semidirect
46. #6448	C₂×C₃	semidirect
47. #6441	C₂×C₃	semidirect
48. #6440	C₂×C₃	semidirect
104. C₃×#123	D₇	semidirect
183. #123	C₃×D₇	semidirect
224. C₃×C₇	#536	semidirect
292. C₇	C₃×#536	semidirect
313. C₃	#90339	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
C₂²×C₃×Dic₁₄	2
C₃×C₇×#123	2
C₃×#6498	2
C₃×#6492	2
C₃×#6459	2
C₃×#6451	2
C₃×#6449	2
C₃×#6448	2
C₃×#6441	2
C₃×#6440	2
#90339	3
C₃×#536	7

Subgroups — C3×#90339 #13060991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90339 #13060991