Subgroups — C₃×#90239 #13060893

Order 1344.

620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#1157	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6503	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6501	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
6	C₃×#6458	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#6451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#6448	C₂	1	2	yes	maximal, complemented
9	C₃×#6447	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄²×D₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#90239	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	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
17	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
18	C₃×#1157	—	2	8	no	—
19	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
20	C₂×C₃×D₂₈	C₂²	1	2	yes	—
21	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
22	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
25	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
26	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
29	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
30	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
31	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
32	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
33	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
34	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
37	C₂×#1157	C₂×C₃	1	2	yes	complemented
38	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
39	#6503	C₂×C₃	1	1	yes	characteristic, complemented
40	#6501	C₂×C₃	1	1	yes	characteristic, complemented
41	C₄×C₂²⋊D₇	C₂×C₃	1	2	yes	complemented
42	#6458	C₂×C₃	1	1	yes	characteristic, complemented
43	#6451	C₂×C₃	1	1	yes	characteristic, complemented
44	#6448	C₂×C₃	1	2	yes	complemented
45	#6447	C₂×C₃	1	2	yes	complemented
46	C₄×D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄²×D₇	C₂×C₃	1	1	yes	characteristic, complemented
48	C₃×#532	—	7	1	no	maximal, Hall
49	C₂³×C₃×C₇	C₂³	1	2	yes	—
50	C₂²×C₃×D₇	C₂³	1	2	yes	—
51	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
52	C₃×C₇×D₄	—	2	2	no	—
53	C₂×C₃×C₄×C₇	—	2	4	no	—
54	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
58	C₃×C₂²⋊D₇	—	2	4	no	—
59	C₃×C₂²⋊D₇	—	2	4	no	—
60	C₃×C₂²⋊D₇	—	2	4	no	—
61	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₇	C₂³	1	2	yes	—
63	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
64	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
65	C₃×D₂₈	—	2	4	no	—
66	C₃×D₂₈	—	2	2	no	—
67	C₃×C₄×D₇	—	2	4	no	—
68	C₃×C₄×D₇	—	2	2	no	—
69	C₃×C₄×D₇	—	2	2	no	—
70	C₃×C₄×D₇	D₄	1	4	yes	complemented
71	C₃×Dic₁₄	—	2	4	no	—
72	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
74	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
75	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
76	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
77	#1157	—	2	8	no	—
78	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
79	C₂×D₂₈	C₂²×C₃	1	2	yes	—
80	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
83	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
84	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
85	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
88	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
89	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
90	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
91	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
92	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
93	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
94	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
95	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₃×#41	—	7	2	no	—
97	C₃×C₄⋊Q₈	—	7	1	no	—
98	C₃×#127	—	7	1	no	—
99	C₃×#124	—	7	2	no	—
100	C₃×#121	—	7	1	no	—
101	C₃×#121	—	7	1	no	—
102	C₃×#121	—	7	2	no	—
103	C₃×C₄×D₄	—	7	2	no	—
104	C₃×C₄×D₄	—	7	1	no	—
105	C₃×C₄×D₄	D₇	1	1	yes	characteristic, complemented
106	C₂×C₃×C₄²	—	7	1	no	—
107	C₂²×C₃×C₇	—	2	2	no	—
108	C₂²×C₃×C₇	—	2	2	no	—
109	C₂²×C₃×C₇	—	2	2	no	—
110	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
111	C₂×C₃×D₇	—	2	1	no	—
112	C₂×C₃×D₇	—	2	1	no	—
113	C₂×C₃×D₇	—	2	2	no	—
114	C₂×C₃×D₇	—	2	2	no	—
115	C₂×C₃×D₇	—	2	2	no	—
116	C₂×C₃×D₇	C₂×D₄	1	2	yes	—
117	C₃×C₄×C₇	—	2	2	no	—
118	C₃×C₄×C₇	—	2	1	no	—
119	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
120	C₃×C₄×C₇	#41	1	2	yes	complemented
121	C₃×Dic₇	—	2	2	no	—
122	C₃×Dic₇	—	2	1	no	—
123	C₃×Dic₇	C₂×D₄	1	2	yes	—
124	C₃×Dic₇	#41	1	2	yes	complemented
125	#532	—	7	1	no	Sylow
126	C₂³×C₇	C₂³×C₃	1	2	yes	—
127	C₂²×D₇	C₂³×C₃	1	2	yes	—
128	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
129	C₇×D₄	—	2	2	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
133	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
135	C₂²⋊D₇	—	2	4	no	—
136	C₂²⋊D₇	—	2	4	no	—
137	C₂²⋊D₇	—	2	4	no	—
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	D₂₈	—	2	4	no	—
143	D₂₈	—	2	2	no	—
144	C₄×D₇	—	2	4	no	—
145	C₄×D₇	—	2	2	no	—
146	C₄×D₇	—	2	2	no	—
147	C₄×D₇	C₃×D₄	1	4	yes	complemented
148	Dic₁₄	—	2	4	no	—
149	C₃×#41	—	14	8	no	—
150	C₂×C₃×Q₈	—	7	2	no	—
151	C₂×C₃×D₄	—	7	2	no	—
152	C₂×C₃×D₄	—	7	2	no	—
153	C₂×C₃×D₄	—	7	2	no	—
154	C₂×C₃×D₄	—	7	1	no	—
155	C₂×C₃×D₄	—	7	2	no	—
156	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
157	C₂²×C₃×C₄	—	7	1	no	—
158	C₂²×C₃×C₄	—	7	1	no	—
159	C₂²×C₃×C₄	—	7	1	no	—
160	C₂²×C₃×C₄	—	7	2	no	—
161	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
162	C₃×C₄⋊C₄	—	7	2	no	—
163	C₃×C₄⋊C₄	—	7	1	no	—
164	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
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	2	yes	—
169	C₃×C₄²	—	7	1	no	—
170	C₃×C₄²	—	7	1	no	—
171	C₃×C₄²	—	7	1	no	—
172	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
173	C₂×C₃×C₇	—	4	2	no	—
174	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
176	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
177	C₃×D₇	—	2	2	no	—
178	C₃×D₇	—	4	2	no	—
179	C₂×#41	—	7	2	no	—
180	C₄⋊Q₈	—	7	1	no	—
181	#127	—	7	1	no	—
182	#124	—	7	2	no	—
183	#121	—	7	1	no	—
184	#121	—	7	1	no	—
185	#121	—	7	2	no	—
186	C₄×D₄	—	7	2	no	—
187	C₄×D₄	—	7	1	no	—
188	C₄×D₄	C₃×D₇	1	1	yes	characteristic, complemented
189	C₂×C₄²	—	7	1	no	—
190	C₂²×C₇	—	2	2	no	—
191	C₂²×C₇	—	2	2	no	—
192	C₂²×C₇	—	2	2	no	—
193	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
194	C₂×D₇	—	2	1	no	—
195	C₂×D₇	—	2	1	no	—
196	C₂×D₇	—	2	2	no	—
197	C₂×D₇	—	2	2	no	—
198	C₂×D₇	—	2	2	no	—
199	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
200	C₄×C₇	—	2	2	no	—
201	C₄×C₇	—	2	1	no	—
202	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
203	C₄×C₇	C₃×#41	1	2	yes	complemented
204	Dic₇	—	2	2	no	—
205	Dic₇	—	2	1	no	—
206	Dic₇	C₂×C₃×D₄	1	2	yes	—
207	Dic₇	C₃×#41	1	2	yes	complemented
208	C₂³×C₃	—	7	1	no	—
209	C₂³×C₃	—	7	2	no	—
210	C₂³×C₃	C₂²×D₇	1	2	yes	—
211	C₃×Q₈	—	14	4	no	—
212	C₃×D₄	—	2	2	no	—
213	C₃×D₄	—	14	4	no	—
214	C₃×D₄	—	14	4	no	—
215	C₃×D₄	—	14	4	no	—
216	C₃×D₄	—	14	4	no	—
217	C₃×D₄	—	14	2	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	7	1	no	—
220	C₂×C₃×C₄	—	7	1	no	—
221	C₂×C₃×C₄	—	7	4	no	—
222	C₂×C₃×C₄	—	7	1	no	—
223	C₂×C₃×C₄	—	7	2	no	—
224	C₂×C₃×C₄	—	14	2	no	—
225	C₂×C₃×C₄	—	14	2	no	—
226	C₂×C₃×C₄	—	14	4	no	—
227	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
229	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
231	C₃×C₇	#532	1	1	yes	characteristic, complemented, Hall
232	#41	—	14	8	no	—
233	C₂×Q₈	—	7	2	no	—
234	C₂×D₄	—	7	2	no	—
235	C₂×D₄	—	7	2	no	—
236	C₂×D₄	—	7	2	no	—
237	C₂×D₄	—	7	1	no	—
238	C₂×D₄	—	7	2	no	—
239	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
240	C₂²×C₄	—	7	1	no	—
241	C₂²×C₄	—	7	1	no	—
242	C₂²×C₄	—	7	1	no	—
243	C₂²×C₄	—	7	2	no	—
244	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
245	C₄⋊C₄	—	7	2	no	—
246	C₄⋊C₄	—	7	1	no	—
247	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
248	C₂²⋊C₄	—	7	2	no	—
249	C₂²⋊C₄	—	7	2	no	—
250	C₂²⋊C₄	—	7	2	no	—
251	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
252	C₄²	—	7	1	no	—
253	C₄²	—	7	1	no	—
254	C₄²	—	7	1	no	—
255	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
256	C₂×C₇	—	4	2	no	—
257	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
258	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
259	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
260	D₇	—	2	2	no	—
261	D₇	—	4	2	no	—
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	2	2	no	—
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	—	7	2	no	—
266	C₂²×C₃	—	14	1	no	—
267	C₂²×C₃	—	14	1	no	—
268	C₂²×C₃	—	14	2	no	—
269	C₂²×C₃	—	14	2	no	—
270	C₂²×C₃	—	14	2	no	—
271	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	7	2	no	—
275	C₃×C₄	—	7	2	no	—
276	C₃×C₄	—	14	1	no	—
277	C₃×C₄	—	14	2	no	—
278	C₃×C₄	#1157	1	2	yes	complemented
279	C₃×C₄	D₄×D₇	1	2	yes	—
280	C₂³	—	7	1	no	—
281	C₂³	—	7	2	no	—
282	C₂³	C₂²×C₃×D₇	1	2	yes	—
283	Q₈	—	14	4	no	—
284	D₄	—	2	2	no	—
285	D₄	—	14	4	no	—
286	D₄	—	14	4	no	—
287	D₄	—	14	4	no	—
288	D₄	—	14	4	no	—
289	D₄	—	14	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	7	1	no	—
292	C₂×C₄	—	7	1	no	—
293	C₂×C₄	—	7	4	no	—
294	C₂×C₄	—	7	1	no	—
295	C₂×C₄	—	7	2	no	—
296	C₂×C₄	—	14	2	no	—
297	C₂×C₄	—	14	2	no	—
298	C₂×C₄	—	14	4	no	—
299	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
301	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
303	C₇	C₃×#532	1	1	yes	characteristic, complemented, Sylow
304	C₂×C₃	—	4	2	no	—
305	C₂×C₃	—	14	2	no	—
306	C₂×C₃	—	28	2	no	—
307	C₂×C₃	C₂×#1157	1	1	yes	characteristic
308	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
309	C₂×C₃	D₇×#41	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	7	2	no	—
314	C₂²	—	14	1	no	—
315	C₂²	—	14	1	no	—
316	C₂²	—	14	2	no	—
317	C₂²	—	14	2	no	—
318	C₂²	—	14	2	no	—
319	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	7	2	no	—
323	C₄	—	7	2	no	—
324	C₄	—	14	1	no	—
325	C₄	—	14	2	no	—
326	C₄	C₃×#1157	1	2	yes	complemented
327	C₄	C₃×D₄×D₇	1	2	yes	—
328	C₃	#90239	1	1	yes	characteristic, direct product factor, Sylow
329	C₂	—	4	2	no	—
330	C₂	—	14	2	no	—
331	C₂	—	28	2	no	—
332	C₂	C₂×C₃×#1157	1	1	yes	characteristic
333	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
334	C₂	C₃×D₇×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₄×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×C₄×D₄	D₇
hypocenter	C₇	C₃×#532
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#1157	C₂	semidirect
2. C₃×C₄×C₇×D₄	C₂	semidirect
3. C₃×#6503	C₂	semidirect
4. C₃×#6501	C₂	semidirect
5. C₃×C₄×C₂²⋊D₇	C₂	semidirect
6. C₃×#6458	C₂	semidirect
7. C₃×#6451	C₂	semidirect
8. C₃×#6448	C₂	semidirect
9. C₃×#6447	C₂	semidirect
10. C₃×C₄×D₂₈	C₂	semidirect
11. C₃×C₄²×D₇	C₂	semidirect
12. #90239	C₃	direct (N × H)
37. C₂×#1157	C₂×C₃	semidirect
38. C₄×C₇×D₄	C₂×C₃	semidirect
39. #6503	C₂×C₃	semidirect
40. #6501	C₂×C₃	semidirect
41. C₄×C₂²⋊D₇	C₂×C₃	semidirect
42. #6458	C₂×C₃	semidirect
43. #6451	C₂×C₃	semidirect
44. #6448	C₂×C₃	semidirect
45. #6447	C₂×C₃	semidirect
46. C₄×D₂₈	C₂×C₃	semidirect
47. C₄²×D₇	C₂×C₃	semidirect
70. C₃×C₄×D₇	D₄	semidirect
105. C₃×C₄×D₄	D₇	semidirect
120. C₃×C₄×C₇	#41	semidirect
124. C₃×Dic₇	#41	semidirect
147. C₄×D₇	C₃×D₄	semidirect
188. C₄×D₄	C₃×D₇	semidirect
203. C₄×C₇	C₃×#41	semidirect
207. Dic₇	C₃×#41	semidirect
231. C₃×C₇	#532	semidirect
278. C₃×C₄	#1157	semidirect
303. C₇	C₃×#532	semidirect
326. C₄	C₃×#1157	semidirect
328. C₃	#90239	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×#1157	2
C₃×C₄×C₇×D₄	2
C₃×#6503	2
C₃×#6501	2
C₃×C₄×C₂²⋊D₇	2
C₃×#6458	2
C₃×#6451	2
C₃×#6448	2
C₃×#6447	2
C₃×C₄×D₂₈	2
C₃×C₄²×D₇	2
#90239	3
C₃×#532	7

Subgroups — C3×#90239 #13060893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90239 #13060893