Subgroups — C₃×#90302 #13060955

Order 1344.

584 conjugacy classes of subgroups.

The subgroup lattice has 328 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×#6501	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×C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×D₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#6451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#6446	C₂	1	2	yes	maximal, complemented
11	C₃×#6441	C₂	1	2	yes	maximal, complemented
12	#90302	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	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
17	C₂×C₃×C₂²⋊D₇	C₂²	1	1	yes	characteristic
18	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
19	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
20	C₃×#1159	—	2	8	no	—
21	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
22	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
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×C₂²⋊Dic₇	C₂²	1	2	yes	—
28	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
29	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
30	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
31	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
32	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
33	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
34	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
35	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
36	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
37	C₂×#1159	C₂×C₃	1	2	yes	complemented
38	C₇×#121	C₂×C₃	1	1	yes	characteristic, complemented
39	#6501	C₂×C₃	1	1	yes	characteristic, complemented
40	#6498	C₂×C₃	1	2	yes	complemented
41	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
42	C₄×C₂²⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂×C₄⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
44	D₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
45	#6451	C₂×C₃	1	1	yes	characteristic, complemented
46	#6446	C₂×C₃	1	2	yes	complemented
47	#6441	C₂×C₃	1	2	yes	complemented
48	C₃×#547	—	7	1	no	maximal, Hall
49	C₂³×C₃×C₇	C₂³	1	2	yes	—
50	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
51	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
52	C₃×C₇×D₄	—	2	4	no	—
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	1	yes	characteristic
58	C₃×C₂²⋊D₇	—	2	4	no	—
59	C₃×C₂²⋊D₇	D₄	1	4	yes	complemented
60	C₂×C₃×Dic₇	—	2	4	no	—
61	C₂×C₃×Dic₇	—	2	4	no	—
62	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
63	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
64	C₂×C₃×Dic₇	C₂³	1	2	yes	—
65	C₂×C₃×Dic₇	C₂³	1	2	yes	—
66	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
67	C₃×C₄×D₇	—	2	4	no	—
68	C₃×C₄×D₇	—	2	2	no	—
69	C₃×Dic₁₄	—	2	4	no	—
70	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
72	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
74	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
75	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
76	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
77	#1159	—	2	8	no	—
78	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
79	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
80	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
81	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
84	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
85	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
86	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
88	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
89	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
90	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
91	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
92	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
93	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
94	C₂×C₃×#41	—	7	2	no	—
95	C₃×C₄⋊Q₈	—	7	1	no	—
96	C₃×#123	—	7	2	no	—
97	C₃×#123	—	7	2	no	—
98	C₃×C₂²⋊Q₈	—	7	2	no	—
99	C₃×#121	—	7	1	no	—
100	C₃×#121	D₇	1	1	yes	characteristic, complemented
101	C₃×C₄×D₄	—	7	1	no	—
102	C₃×C₄×D₄	—	7	1	no	—
103	C₂×C₃×C₄⋊C₄	—	7	1	no	—
104	C₂×C₃×C₄⋊C₄	—	7	1	no	—
105	C₂²×C₃×C₇	—	2	1	no	—
106	C₂²×C₃×C₇	—	2	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	C₂×D₄	1	2	yes	—
111	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
112	C₂×C₃×D₇	—	2	1	no	—
113	C₂×C₃×D₇	—	2	1	no	—
114	C₂×C₃×D₇	C₂×D₄	1	2	yes	—
115	C₃×C₄×C₇	—	2	2	no	—
116	C₃×C₄×C₇	—	2	1	no	—
117	C₃×C₄×C₇	#41	1	2	yes	complemented
118	C₃×Dic₇	—	2	2	no	—
119	C₃×Dic₇	—	2	2	no	—
120	C₃×Dic₇	—	2	1	no	—
121	C₃×Dic₇	—	2	1	no	—
122	C₃×Dic₇	C₂×D₄	1	2	yes	—
123	#547	—	7	1	no	Sylow
124	C₂³×C₇	C₂³×C₃	1	2	yes	—
125	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
126	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
127	C₇×D₄	—	2	4	no	—
128	C₇×D₄	—	2	2	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
131	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
133	C₂²⋊D₇	—	2	4	no	—
134	C₂²⋊D₇	C₃×D₄	1	4	yes	complemented
135	C₂×Dic₇	—	2	4	no	—
136	C₂×Dic₇	—	2	4	no	—
137	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
138	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
139	C₂×Dic₇	C₂³×C₃	1	2	yes	—
140	C₂×Dic₇	C₂³×C₃	1	2	yes	—
141	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
142	C₄×D₇	—	2	4	no	—
143	C₄×D₇	—	2	2	no	—
144	Dic₁₄	—	2	4	no	—
145	C₃×#41	—	14	8	no	—
146	C₂×C₃×Q₈	—	7	2	no	—
147	C₂×C₃×D₄	—	7	1	no	—
148	C₂×C₃×D₄	—	7	2	no	—
149	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
150	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
151	C₂²×C₃×C₄	—	7	2	no	—
152	C₂²×C₃×C₄	—	7	2	no	—
153	C₂²×C₃×C₄	—	7	1	no	—
154	C₂²×C₃×C₄	—	7	2	no	—
155	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
156	C₃×C₄⋊C₄	—	7	2	no	—
157	C₃×C₄⋊C₄	—	7	1	no	—
158	C₃×C₄⋊C₄	—	7	2	no	—
159	C₃×C₄⋊C₄	—	7	1	no	—
160	C₃×C₄⋊C₄	—	7	1	no	—
161	C₃×C₄⋊C₄	—	7	2	no	—
162	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
163	C₃×C₂²⋊C₄	—	7	1	no	—
164	C₃×C₂²⋊C₄	—	7	2	no	—
165	C₃×C₂²⋊C₄	—	7	2	no	—
166	C₃×C₂²⋊C₄	—	7	1	no	—
167	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
168	C₃×C₄²	—	7	1	no	—
169	C₂×C₃×C₇	—	2	2	no	—
170	C₂×C₃×C₇	—	4	2	no	—
171	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
173	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
174	C₃×D₇	—	2	2	no	—
175	C₂×#41	—	7	2	no	—
176	C₄⋊Q₈	—	7	1	no	—
177	#123	—	7	2	no	—
178	#123	—	7	2	no	—
179	C₂²⋊Q₈	—	7	2	no	—
180	#121	—	7	1	no	—
181	#121	C₃×D₇	1	1	yes	characteristic, complemented
182	C₄×D₄	—	7	1	no	—
183	C₄×D₄	—	7	1	no	—
184	C₂×C₄⋊C₄	—	7	1	no	—
185	C₂×C₄⋊C₄	—	7	1	no	—
186	C₂²×C₇	—	2	1	no	—
187	C₂²×C₇	—	2	1	no	—
188	C₂²×C₇	—	2	2	no	—
189	C₂²×C₇	—	2	2	no	—
190	C₂²×C₇	—	2	2	no	—
191	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
192	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
193	C₂×D₇	—	2	1	no	—
194	C₂×D₇	—	2	1	no	—
195	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
196	C₄×C₇	—	2	2	no	—
197	C₄×C₇	—	2	1	no	—
198	C₄×C₇	C₃×#41	1	2	yes	complemented
199	Dic₇	—	2	2	no	—
200	Dic₇	—	2	2	no	—
201	Dic₇	—	2	1	no	—
202	Dic₇	—	2	1	no	—
203	Dic₇	C₂×C₃×D₄	1	2	yes	—
204	C₂³×C₃	—	7	1	no	—
205	C₂³×C₃	C₂²×D₇	1	2	yes	—
206	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
207	C₃×Q₈	—	14	4	no	—
208	C₃×D₄	—	2	4	no	—
209	C₃×D₄	—	2	2	no	—
210	C₃×D₄	—	7	4	no	—
211	C₃×D₄	—	14	4	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	7	1	no	—
214	C₂×C₃×C₄	—	7	1	no	—
215	C₂×C₃×C₄	—	7	1	no	—
216	C₂×C₃×C₄	—	7	2	no	—
217	C₂×C₃×C₄	—	7	2	no	—
218	C₂×C₃×C₄	—	14	4	no	—
219	C₂×C₃×C₄	—	14	4	no	—
220	C₂×C₃×C₄	—	14	4	no	—
221	C₂×C₃×C₄	—	14	2	no	—
222	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
224	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
225	C₃×C₇	#547	1	1	yes	characteristic, complemented, Hall
226	#41	—	14	8	no	—
227	C₂×Q₈	—	7	2	no	—
228	C₂×D₄	—	7	1	no	—
229	C₂×D₄	—	7	2	no	—
230	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
231	C₂×D₄	C₂×C₃×D₇	1	2	yes	—
232	C₂²×C₄	—	7	2	no	—
233	C₂²×C₄	—	7	2	no	—
234	C₂²×C₄	—	7	1	no	—
235	C₂²×C₄	—	7	2	no	—
236	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
237	C₄⋊C₄	—	7	2	no	—
238	C₄⋊C₄	—	7	1	no	—
239	C₄⋊C₄	—	7	2	no	—
240	C₄⋊C₄	—	7	1	no	—
241	C₄⋊C₄	—	7	1	no	—
242	C₄⋊C₄	—	7	2	no	—
243	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
244	C₂²⋊C₄	—	7	1	no	—
245	C₂²⋊C₄	—	7	2	no	—
246	C₂²⋊C₄	—	7	2	no	—
247	C₂²⋊C₄	—	7	1	no	—
248	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
249	C₄²	—	7	1	no	—
250	C₂×C₇	—	2	2	no	—
251	C₂×C₇	—	4	2	no	—
252	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
253	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
254	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
255	D₇	—	2	2	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	2	no	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	7	2	no	—
262	C₂²×C₃	—	14	1	no	—
263	C₂²×C₃	—	14	1	no	—
264	C₂²×C₃	D₄×D₇	1	2	yes	—
265	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	7	2	no	—
269	C₃×C₄	—	14	2	no	—
270	C₃×C₄	—	14	2	no	—
271	C₃×C₄	—	14	1	no	—
272	C₃×C₄	—	14	1	no	—
273	C₃×C₄	#1159	1	2	yes	complemented
274	C₂³	—	7	1	no	—
275	C₂³	C₂²×C₃×D₇	1	2	yes	—
276	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
277	Q₈	—	14	4	no	—
278	D₄	—	2	4	no	—
279	D₄	—	2	2	no	—
280	D₄	—	7	4	no	—
281	D₄	—	14	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	7	1	no	—
284	C₂×C₄	—	7	1	no	—
285	C₂×C₄	—	7	1	no	—
286	C₂×C₄	—	7	2	no	—
287	C₂×C₄	—	7	2	no	—
288	C₂×C₄	—	14	4	no	—
289	C₂×C₄	—	14	4	no	—
290	C₂×C₄	—	14	4	no	—
291	C₂×C₄	—	14	2	no	—
292	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
294	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
295	C₇	C₃×#547	1	1	yes	characteristic, complemented, Sylow
296	C₂×C₃	—	2	2	no	—
297	C₂×C₃	—	4	2	no	—
298	C₂×C₃	—	14	2	no	—
299	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
300	C₂×C₃	C₂×#1159	1	1	yes	characteristic
301	C₂×C₃	#6554	1	1	yes	characteristic
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	7	2	no	—
308	C₂²	—	14	1	no	—
309	C₂²	—	14	1	no	—
310	C₂²	C₃×D₄×D₇	1	2	yes	—
311	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	7	2	no	—
315	C₄	—	14	2	no	—
316	C₄	—	14	2	no	—
317	C₄	—	14	1	no	—
318	C₄	—	14	1	no	—
319	C₄	C₃×#1159	1	2	yes	complemented
320	C₃	#90302	1	1	yes	characteristic, direct product factor, Sylow
321	C₂	—	2	2	no	—
322	C₂	—	4	2	no	—
323	C₂	—	14	2	no	—
324	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
325	C₂	C₂×C₃×#1159	1	1	yes	characteristic
326	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×#547
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×#6501	C₂	semidirect
4. C₃×#6498	C₂	semidirect
5. C₃×D₄×Dic₇	C₂	semidirect
6. C₃×C₄×C₂²⋊D₇	C₂	semidirect
7. C₂×C₃×C₄⋊Dic₇	C₂	semidirect
8. C₃×D₇×C₄⋊C₄	C₂	semidirect
9. C₃×#6451	C₂	semidirect
10. C₃×#6446	C₂	semidirect
11. C₃×#6441	C₂	semidirect
12. #90302	C₃	direct (N × H)
37. C₂×#1159	C₂×C₃	semidirect
38. C₇×#121	C₂×C₃	semidirect
39. #6501	C₂×C₃	semidirect
40. #6498	C₂×C₃	semidirect
41. D₄×Dic₇	C₂×C₃	semidirect
42. C₄×C₂²⋊D₇	C₂×C₃	semidirect
43. C₂×C₄⋊Dic₇	C₂×C₃	semidirect
44. D₇×C₄⋊C₄	C₂×C₃	semidirect
45. #6451	C₂×C₃	semidirect
46. #6446	C₂×C₃	semidirect
47. #6441	C₂×C₃	semidirect
59. C₃×C₂²⋊D₇	D₄	semidirect
100. C₃×#121	D₇	semidirect
117. C₃×C₄×C₇	#41	semidirect
134. C₂²⋊D₇	C₃×D₄	semidirect
181. #121	C₃×D₇	semidirect
198. C₄×C₇	C₃×#41	semidirect
225. C₃×C₇	#547	semidirect
273. C₃×C₄	#1159	semidirect
295. C₇	C₃×#547	semidirect
319. C₄	C₃×#1159	semidirect
320. C₃	#90302	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×#6501	2
C₃×#6498	2
C₃×D₄×Dic₇	2
C₃×C₄×C₂²⋊D₇	2
C₂×C₃×C₄⋊Dic₇	2
C₃×D₇×C₄⋊C₄	2
C₃×#6451	2
C₃×#6446	2
C₃×#6441	2
#90302	3
C₃×#547	7

Subgroups — C3×#90302 #13060955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90302 #13060955