Subgroups — C₃×#90315 #13060968

Order 1344.

468 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6493	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6461	C₂	1	2	yes	maximal, complemented
7	C₃×#6456	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#6455	C₂	1	2	yes	maximal, complemented
9	C₃×#6453	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#6449	C₂	1	2	yes	maximal, complemented
11	C₃×(C₂²⋊D₇)⋊C₄	C₂	1	2	yes	maximal, complemented
12	#90315	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
17	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
20	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
24	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
25	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
28	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
29	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
30	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
31	C₃×C₄×Dic₇	C₂²	1	2	yes	—
32	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
33	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
36	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
37	#6510	C₂×C₃	1	1	yes	characteristic, complemented
38	Q₈×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
39	#6493	C₂×C₃	1	1	yes	characteristic
40	C₂×C₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
41	#6461	C₂×C₃	1	2	yes	complemented
42	#6456	C₂×C₃	1	1	yes	characteristic, complemented
43	#6455	C₂×C₃	1	2	yes	complemented
44	#6453	C₂×C₃	1	1	yes	characteristic, complemented
45	#6449	C₂×C₃	1	2	yes	complemented
46	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	2	yes	complemented
47	C₃×#529	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
49	C₂²×C₃×D₇	C₂³	1	2	yes	—
50	C₃×C₇×Q₈	—	2	2	no	—
51	C₂×C₃×C₄×C₇	—	2	2	no	—
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	2	yes	—
55	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
56	C₃×C₂²⋊D₇	—	2	4	no	—
57	C₂×C₃×Dic₇	—	2	2	no	—
58	C₂×C₃×Dic₇	—	2	2	no	—
59	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
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	C₃×D₂₈	—	2	2	no	—
65	C₃×C₄×D₇	—	2	4	no	—
66	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
69	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
70	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
71	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
73	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
76	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
77	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
78	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
79	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
80	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
81	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
82	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
83	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
84	C₄×Dic₇	C₂²×C₃	1	2	yes	—
85	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
86	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
87	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
88	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
89	C₃×#126	—	7	2	no	—
90	C₃×#125	—	7	1	no	—
91	C₃×#124	—	7	1	no	—
92	C₃×#123	—	7	2	no	—
93	C₃×C₂²⋊Q₈	D₇	1	1	yes	characteristic, complemented
94	C₃×#121	—	7	1	no	—
95	C₃×C₄×Q₈	—	7	1	no	—
96	C₃×C₄×D₄	—	7	1	no	—
97	C₃×C₄×D₄	—	7	2	no	—
98	C₃×#117	—	7	2	no	—
99	C₂×C₃×C₄²	—	7	1	no	—
100	C₂²×C₃×C₇	—	2	1	no	—
101	C₂²×C₃×C₇	—	2	1	no	—
102	C₂²×C₃×C₇	#41	1	2	yes	—
103	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
104	C₂×C₃×D₇	—	2	2	no	—
105	C₂×C₃×D₇	—	2	2	no	—
106	C₂×C₃×D₇	—	2	2	no	—
107	C₃×C₄×C₇	—	2	2	no	—
108	C₃×C₄×C₇	—	2	1	no	—
109	C₃×C₄×C₇	—	2	2	no	—
110	C₃×C₄×C₇	#41	1	2	yes	—
111	C₃×Dic₇	—	2	1	no	—
112	C₃×Dic₇	—	2	1	no	—
113	C₃×Dic₇	—	2	1	no	—
114	C₃×Dic₇	—	2	2	no	—
115	C₃×Dic₇	#41	1	2	yes	—
116	#529	—	7	1	no	Sylow
117	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
118	C₂²×D₇	C₂³×C₃	1	2	yes	—
119	C₇×Q₈	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
123	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
124	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
125	C₂²⋊D₇	—	2	4	no	—
126	C₂×Dic₇	—	2	2	no	—
127	C₂×Dic₇	—	2	2	no	—
128	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
129	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
130	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
131	C₂×Dic₇	C₂³×C₃	1	2	yes	—
132	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
133	D₂₈	—	2	2	no	—
134	C₄×D₇	—	2	4	no	—
135	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
136	C₂×C₃×D₄	—	7	2	no	—
137	C₂×C₃×D₄	—	7	1	no	—
138	C₂²×C₃×C₄	—	7	1	no	—
139	C₂²×C₃×C₄	—	7	1	no	—
140	C₂²×C₃×C₄	—	7	2	no	—
141	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
142	C₃×C₄⋊C₄	—	7	2	no	—
143	C₃×C₄⋊C₄	—	7	2	no	—
144	C₃×C₄⋊C₄	—	7	1	no	—
145	C₃×C₄⋊C₄	—	7	2	no	—
146	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
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	—	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	1	no	—
156	C₃×C₄²	—	7	1	no	—
157	C₃×C₄²	—	7	2	no	—
158	C₂×C₃×C₇	—	2	2	no	—
159	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
160	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
162	C₃×D₇	—	4	2	no	—
163	#126	—	7	2	no	—
164	#125	—	7	1	no	—
165	#124	—	7	1	no	—
166	#123	—	7	2	no	—
167	C₂²⋊Q₈	C₃×D₇	1	1	yes	characteristic, complemented
168	#121	—	7	1	no	—
169	C₄×Q₈	—	7	1	no	—
170	C₄×D₄	—	7	1	no	—
171	C₄×D₄	—	7	2	no	—
172	#117	—	7	2	no	—
173	C₂×C₄²	—	7	1	no	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	—	2	1	no	—
176	C₂²×C₇	C₃×#41	1	2	yes	—
177	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
178	C₂×D₇	—	2	2	no	—
179	C₂×D₇	—	2	2	no	—
180	C₂×D₇	—	2	2	no	—
181	C₄×C₇	—	2	2	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	C₃×#41	1	2	yes	—
185	Dic₇	—	2	1	no	—
186	Dic₇	—	2	1	no	—
187	Dic₇	—	2	1	no	—
188	Dic₇	—	2	2	no	—
189	Dic₇	C₃×#41	1	2	yes	—
190	C₂³×C₃	—	7	2	no	—
191	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
192	C₃×Q₈	—	2	2	no	—
193	C₃×D₄	—	14	4	no	—
194	C₃×D₄	—	14	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	7	1	no	—
197	C₂×C₃×C₄	—	7	1	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	—	14	4	no	—
202	C₂×C₃×C₄	—	14	2	no	—
203	C₂×C₃×C₄	—	14	2	no	—
204	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
206	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
207	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
208	C₃×C₇	#529	1	1	yes	characteristic, complemented, Hall
209	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
210	C₂×D₄	—	7	2	no	—
211	C₂×D₄	—	7	1	no	—
212	C₂²×C₄	—	7	1	no	—
213	C₂²×C₄	—	7	1	no	—
214	C₂²×C₄	—	7	2	no	—
215	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
216	C₄⋊C₄	—	7	2	no	—
217	C₄⋊C₄	—	7	2	no	—
218	C₄⋊C₄	—	7	1	no	—
219	C₄⋊C₄	—	7	2	no	—
220	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	—
221	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
222	C₂²⋊C₄	—	7	2	no	—
223	C₂²⋊C₄	—	7	2	no	—
224	C₂²⋊C₄	—	7	2	no	—
225	C₂²⋊C₄	—	7	2	no	—
226	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
227	C₄²	—	7	1	no	—
228	C₄²	—	7	1	no	—
229	C₄²	—	7	1	no	—
230	C₄²	—	7	1	no	—
231	C₄²	—	7	2	no	—
232	C₂×C₇	—	2	2	no	—
233	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
234	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
235	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
236	D₇	—	4	2	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	14	2	no	—
240	C₂²×C₃	—	14	2	no	—
241	C₂²×C₃	—	14	2	no	—
242	C₂²×C₃	#1161	1	2	yes	—
243	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	7	2	no	—
248	C₃×C₄	—	14	1	no	—
249	C₃×C₄	—	14	1	no	—
250	C₃×C₄	—	14	2	no	—
251	C₃×C₄	—	14	1	no	—
252	C₃×C₄	#1159	1	2	yes	—
253	C₂³	—	7	2	no	—
254	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	D₄	—	14	4	no	—
257	D₄	—	14	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	7	1	no	—
260	C₂×C₄	—	7	1	no	—
261	C₂×C₄	—	7	1	no	—
262	C₂×C₄	—	7	1	no	—
263	C₂×C₄	—	7	2	no	—
264	C₂×C₄	—	14	4	no	—
265	C₂×C₄	—	14	2	no	—
266	C₂×C₄	—	14	2	no	—
267	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
269	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
270	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
271	C₇	C₃×#529	1	1	yes	characteristic, complemented, Sylow
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	—	28	2	no	—
274	C₂×C₃	C₂×#1159	1	1	yes	characteristic
275	C₂×C₃	C₂×#1161	1	1	yes	characteristic
276	C₂×C₃	D₇×#41	1	1	yes	characteristic
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	14	2	no	—
280	C₂²	—	14	2	no	—
281	C₂²	—	14	2	no	—
282	C₂²	C₃×#1161	1	2	yes	—
283	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	7	2	no	—
288	C₄	—	14	1	no	—
289	C₄	—	14	1	no	—
290	C₄	—	14	2	no	—
291	C₄	—	14	1	no	—
292	C₄	C₃×#1159	1	2	yes	—
293	C₃	#90315	1	1	yes	characteristic, direct product factor, Sylow
294	C₂	—	2	2	no	—
295	C₂	—	28	2	no	—
296	C₂	C₂×C₃×#1159	1	1	yes	characteristic
297	C₂	C₂×C₃×#1161	1	1	yes	characteristic
298	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×C₇×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₃×#6493	C₂
hypercenter	C₃×C₂²⋊Q₈	D₇
hypocenter	C₇	C₃×#529
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₂²⋊Q₈	C₂	semidirect
2. C₃×#6510	C₂	semidirect
3. C₃×Q₈×Dic₇	C₂	semidirect
5. C₂×C₃×C₄×Dic₇	C₂	semidirect
6. C₃×#6461	C₂	semidirect
7. C₃×#6456	C₂	semidirect
8. C₃×#6455	C₂	semidirect
9. C₃×#6453	C₂	semidirect
10. C₃×#6449	C₂	semidirect
11. C₃×(C₂²⋊D₇)⋊C₄	C₂	semidirect
12. #90315	C₃	direct (N × H)
36. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
37. #6510	C₂×C₃	semidirect
38. Q₈×Dic₇	C₂×C₃	semidirect
40. C₂×C₄×Dic₇	C₂×C₃	semidirect
41. #6461	C₂×C₃	semidirect
42. #6456	C₂×C₃	semidirect
43. #6455	C₂×C₃	semidirect
44. #6453	C₂×C₃	semidirect
45. #6449	C₂×C₃	semidirect
46. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
93. C₃×C₂²⋊Q₈	D₇	semidirect
167. C₂²⋊Q₈	C₃×D₇	semidirect
208. C₃×C₇	#529	semidirect
271. C₇	C₃×#529	semidirect
293. C₃	#90315	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×C₂²⋊Q₈	2
C₃×#6510	2
C₃×Q₈×Dic₇	2
C₃×#6493	2
C₂×C₃×C₄×Dic₇	2
C₃×#6461	2
C₃×#6456	2
C₃×#6455	2
C₃×#6453	2
C₃×#6449	2
C₃×(C₂²⋊D₇)⋊C₄	2
#90315	3
C₃×#529	7

Subgroups — C3×#90315 #13060968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90315 #13060968