Subgroups — C₃×#13096694 #72246633

Order 4704.

818 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#11716103	C₂	1	2	yes	maximal, complemented
2	C₃×#11716108	C₂	1	4	yes	maximal, complemented
3	C₃×C₇×#1159	C₂	1	2	yes	maximal, complemented
4	C₃×#11716101	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×D₇²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#87639	C₂	1	2	yes	maximal, complemented
7	C₃×D₄×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#11716115	C₂	1	2	yes	maximal, complemented
9	#13096694	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₃×C₇⋊D₇	C₂²	1	2	yes	—
11	C₂×C₃×D₇²	C₂²	1	1	yes	characteristic
12	C₃×C₇²×D₄	C₂²	1	1	yes	characteristic, complemented
13	C₃×C₂²⋊(C₇⋊D₇)	C₂²	1	2	yes	complemented
14	C₃×C₄⋊(C₇⋊D₇)	C₂²	1	1	yes	characteristic, complemented
15	C₃×C₄×C₇⋊D₇	C₂²	1	1	yes	characteristic
16	C₃×C₇×C₂²⋊D₇	C₂²	1	4	yes	complemented
17	C₂×C₃×C₇×Dic₇	C₂²	1	4	yes	complemented
18	C₃×C₄×C₇×D₇	C₂²	1	2	yes	complemented
19	C₃×C₇×Dic₁₄	C₂²	1	2	yes	complemented
20	C₃×#87642	C₂²	1	2	yes	complemented
21	C₃×#87641	C₂²	1	4	yes	complemented
22	C₃×#87639	C₂²	1	1	yes	characteristic
23	C₃×#87639	C₂²	1	2	yes	—
24	C₃×#87639	C₂²	1	4	yes	complemented
25	C₃×D₇×Dic₇	C₂²	1	2	yes	complemented
26	D₄×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
27	C₇×#1159	C₂×C₃	1	2	yes	complemented
28	#11716115	C₂×C₃	1	2	yes	complemented
29	C₂×#87639	C₂×C₃	1	2	yes	complemented
30	#11716108	C₂×C₃	1	4	yes	complemented
31	C₄×D₇²	C₂×C₃	1	1	yes	characteristic, complemented
32	#11716103	C₂×C₃	1	2	yes	complemented
33	#11716101	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×D₇×#41	—	7	2	no	maximal
35	C₂²×C₃×C₇²	C₂³	1	2	yes	—
36	C₂×C₃×C₇⋊D₇	—	2	4	no	—
37	C₂×C₃×C₇⋊D₇	C₂³	1	2	yes	—
38	C₂×C₃×C₇⋊D₇	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₇×D₇	C₂³	1	2	yes	—
40	C₃×D₇²	—	2	2	no	—
41	C₃×C₄×C₇²	C₂³	1	1	yes	characteristic
42	C₃×C₇⋊Dic₇	C₂³	1	1	yes	characteristic
43	C₃×C₇×Dic₇	C₂³	1	2	yes	—
44	C₃×C₇×Dic₇	C₂³	1	4	yes	—
45	C₂²×C₇⋊D₇	C₂²×C₃	1	2	yes	—
46	C₂×D₇²	C₂²×C₃	1	1	yes	characteristic
47	C₇²×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
48	C₂²⋊(C₇⋊D₇)	C₂²×C₃	1	2	yes	complemented
49	C₄⋊(C₇⋊D₇)	C₂²×C₃	1	1	yes	characteristic, complemented
50	C₄×C₇⋊D₇	C₂²×C₃	1	1	yes	characteristic
51	C₇×C₂²⋊D₇	C₂²×C₃	1	4	yes	complemented
52	C₂×C₇×Dic₇	C₂²×C₃	1	4	yes	complemented
53	C₄×C₇×D₇	C₂²×C₃	1	2	yes	complemented
54	C₇×Dic₁₄	C₂²×C₃	1	2	yes	complemented
55	#87642	C₂²×C₃	1	2	yes	complemented
56	#87641	C₂²×C₃	1	4	yes	complemented
57	#87639	C₂²×C₃	1	2	yes	—
58	#87639	C₂²×C₃	1	1	yes	characteristic
59	#87639	C₂²×C₃	1	4	yes	complemented
60	D₇×Dic₇	C₂²×C₃	1	2	yes	complemented
61	C₃×C₇×#41	—	7	2	no	—
62	C₃×#1161	—	7	2	no	—
63	C₃×Q₈×D₇	—	7	2	no	—
64	C₃×#1159	—	7	4	no	—
65	C₃×#1159	D₇	1	2	yes	complemented
66	C₃×D₄×D₇	—	7	2	no	—
67	C₃×D₄×D₇	—	7	4	no	—
68	C₃×D₄×D₇	—	14	3	no	—
69	C₃×#1157	—	7	4	no	—
70	C₃×#1157	—	7	2	no	—
71	C₂×C₃×C₄×D₇	—	7	4	no	—
72	C₂×C₃×C₄×D₇	—	7	2	no	—
73	C₂×C₃×C₇²	—	2	2	no	—
74	C₂×C₃×C₇²	C₂⁴	1	1	yes	characteristic
75	C₃×C₇⋊D₇	—	2	2	no	—
76	C₃×C₇⋊D₇	#41	1	2	yes	complemented
77	C₃×C₇×D₇	—	2	2	no	—
78	D₇×#41	—	7	2	no	—
79	C₂²×C₇²	C₂³×C₃	1	2	yes	—
80	C₂×C₇⋊D₇	—	2	4	no	—
81	C₂×C₇⋊D₇	C₂³×C₃	1	2	yes	—
82	C₂×C₇⋊D₇	C₂³×C₃	1	1	yes	characteristic
83	C₂×C₇×D₇	C₂³×C₃	1	2	yes	—
84	D₇²	—	2	2	no	—
85	C₄×C₇²	C₂³×C₃	1	1	yes	characteristic
86	C₇⋊Dic₇	C₂³×C₃	1	1	yes	characteristic
87	C₇×Dic₇	C₂³×C₃	1	2	yes	—
88	C₇×Dic₇	C₂³×C₃	1	4	yes	—
89	C₂²×C₃×D₇	—	7	4	no	—
90	C₂²×C₃×D₇	—	7	2	no	—
91	C₂²×C₃×D₇	—	14	6	no	—
92	C₃×C₇×Q₈	—	7	2	no	—
93	C₃×C₇×D₄	—	2	3	no	—
94	C₃×C₇×D₄	—	7	4	no	—
95	C₃×C₇×D₄	C₂×D₇	1	2	yes	complemented
96	C₂×C₃×C₄×C₇	—	7	4	no	—
97	C₂×C₃×C₄×C₇	—	7	2	no	—
98	C₃×C₂²⋊D₇	—	7	4	no	—
99	C₃×C₂²⋊D₇	—	7	4	no	—
100	C₃×C₂²⋊D₇	—	14	6	no	—
101	C₃×C₂²⋊D₇	C₂×D₇	1	4	yes	complemented
102	C₂×C₃×Dic₇	—	7	2	no	—
103	C₂×C₃×Dic₇	C₂×D₇	1	4	yes	complemented
104	C₃×D₂₈	—	7	4	no	—
105	C₃×D₂₈	—	7	2	no	—
106	C₃×D₂₈	—	14	3	no	—
107	C₃×C₄×D₇	—	7	2	no	—
108	C₃×C₄×D₇	—	7	2	no	—
109	C₃×C₄×D₇	—	7	2	no	—
110	C₃×C₄×D₇	—	7	4	no	—
111	C₃×C₄×D₇	—	7	4	no	—
112	C₃×C₄×D₇	—	7	4	no	—
113	C₃×C₄×D₇	—	14	3	no	—
114	C₃×C₄×D₇	C₂×D₇	1	2	yes	complemented
115	C₃×Dic₁₄	—	7	4	no	—
116	C₃×Dic₁₄	C₂×D₇	1	2	yes	complemented
117	C₃×C₇²	C₂×#41	1	1	yes	characteristic, complemented, Hall
118	C₇×#41	—	7	2	no	—
119	#1161	—	7	2	no	—
120	Q₈×D₇	—	7	2	no	—
121	#1159	—	7	4	no	—
122	#1159	C₃×D₇	1	2	yes	complemented
123	D₄×D₇	—	7	2	no	—
124	D₄×D₇	—	7	4	no	—
125	D₄×D₇	—	14	3	no	—
126	#1157	—	7	4	no	—
127	#1157	—	7	2	no	—
128	C₂×C₄×D₇	—	7	4	no	—
129	C₂×C₄×D₇	—	7	2	no	—
130	C₂×C₇²	—	2	2	no	—
131	C₂×C₇²	C₂⁴×C₃	1	1	yes	characteristic
132	C₇⋊D₇	—	2	2	no	—
133	C₇⋊D₇	C₃×#41	1	2	yes	complemented
134	C₇×D₇	—	2	2	no	—
135	C₂×C₃×#41	—	49	1	no	Hall
136	C₂²×C₃×C₇	—	2	6	no	—
137	C₂²×C₃×C₇	—	7	2	no	—
138	C₂²×C₃×C₇	C₂²×D₇	1	4	yes	—
139	C₂×C₃×D₇	—	7	2	no	—
140	C₂×C₃×D₇	—	7	4	no	—
141	C₂×C₃×D₇	—	14	3	no	—
142	C₂×C₃×D₇	—	14	6	no	—
143	C₂×C₃×D₇	—	14	4	no	—
144	C₂×C₃×D₇	—	14	8	no	—
145	C₂×C₃×D₇	—	28	12	no	—
146	C₂×C₃×D₇	C₂²×D₇	1	2	yes	—
147	C₃×C₄×C₇	—	2	3	no	—
148	C₃×C₄×C₇	—	7	4	no	—
149	C₃×C₄×C₇	—	7	2	no	—
150	C₃×C₄×C₇	C₂²×D₇	1	2	yes	—
151	C₃×Dic₇	—	7	2	no	—
152	C₃×Dic₇	—	14	3	no	—
153	C₃×Dic₇	C₂²×D₇	1	4	yes	—
154	C₃×Dic₇	C₂²×D₇	1	2	yes	—
155	C₂²×D₇	—	7	4	no	—
156	C₂²×D₇	—	7	2	no	—
157	C₂²×D₇	—	14	6	no	—
158	C₇×Q₈	—	7	2	no	—
159	C₇×D₄	—	2	3	no	—
160	C₇×D₄	—	7	4	no	—
161	C₇×D₄	C₂×C₃×D₇	1	2	yes	complemented
162	C₂×C₄×C₇	—	7	4	no	—
163	C₂×C₄×C₇	—	7	2	no	—
164	C₂²⋊D₇	—	7	4	no	—
165	C₂²⋊D₇	—	7	4	no	—
166	C₂²⋊D₇	—	14	6	no	—
167	C₂²⋊D₇	C₂×C₃×D₇	1	4	yes	complemented
168	C₂×Dic₇	—	7	2	no	—
169	C₂×Dic₇	C₂×C₃×D₇	1	4	yes	complemented
170	D₂₈	—	7	4	no	—
171	D₂₈	—	7	2	no	—
172	D₂₈	—	14	3	no	—
173	C₄×D₇	—	7	2	no	—
174	C₄×D₇	—	7	2	no	—
175	C₄×D₇	—	7	2	no	—
176	C₄×D₇	—	7	4	no	—
177	C₄×D₇	—	7	4	no	—
178	C₄×D₇	—	7	4	no	—
179	C₄×D₇	—	14	3	no	—
180	C₄×D₇	C₂×C₃×D₇	1	2	yes	complemented
181	Dic₁₄	—	7	4	no	—
182	Dic₁₄	C₂×C₃×D₇	1	2	yes	complemented
183	C₇²	C₂×C₃×#41	1	1	yes	characteristic, complemented, Sylow
184	C₃×#41	—	7	2	no	—
185	C₃×#41	—	49	4	no	—
186	C₃×#41	—	49	2	no	—
187	C₂×C₃×Q₈	—	49	1	no	—
188	C₂×C₃×D₄	—	49	2	no	—
189	C₂×C₃×D₄	—	49	1	no	—
190	C₂²×C₃×C₄	—	49	2	no	—
191	C₂²×C₃×C₄	—	49	1	no	—
192	C₂×C₃×C₇	—	2	4	no	—
193	C₂×C₃×C₇	—	2	3	no	—
194	C₂×C₃×C₇	—	4	6	no	—
195	C₂×C₃×C₇	—	14	2	no	—
196	C₂×C₃×C₇	C₂³×D₇	1	2	yes	—
197	C₃×D₇	—	2	2	no	—
198	C₃×D₇	—	7	4	no	—
199	C₃×D₇	—	14	6	no	—
200	C₃×D₇	—	14	4	no	—
201	C₃×D₇	—	28	6	no	—
202	C₂×#41	—	49	1	no	Sylow
203	C₂²×C₇	—	2	6	no	—
204	C₂²×C₇	—	7	2	no	—
205	C₂²×C₇	C₂²×C₃×D₇	1	4	yes	—
206	C₂×D₇	—	7	2	no	—
207	C₂×D₇	—	7	4	no	—
208	C₂×D₇	—	14	3	no	—
209	C₂×D₇	—	14	6	no	—
210	C₂×D₇	—	14	4	no	—
211	C₂×D₇	—	14	8	no	—
212	C₂×D₇	—	28	12	no	—
213	C₂×D₇	C₂²×C₃×D₇	1	2	yes	—
214	C₄×C₇	—	2	3	no	—
215	C₄×C₇	—	7	4	no	—
216	C₄×C₇	—	7	2	no	—
217	C₄×C₇	C₂²×C₃×D₇	1	2	yes	—
218	Dic₇	—	7	2	no	—
219	Dic₇	—	14	3	no	—
220	Dic₇	C₂²×C₃×D₇	1	4	yes	—
221	Dic₇	C₂²×C₃×D₇	1	2	yes	—
222	C₂³×C₃	—	49	1	no	—
223	C₂³×C₃	—	49	2	no	—
224	C₃×Q₈	—	7	2	no	—
225	C₃×Q₈	—	49	2	no	—
226	C₃×D₄	—	7	4	no	—
227	C₃×D₄	—	49	4	no	—
228	C₃×D₄	—	49	2	no	—
229	C₃×D₄	—	49	1	no	—
230	C₃×D₄	D₇²	1	1	yes	characteristic, complemented
231	C₂×C₃×C₄	—	7	4	no	—
232	C₂×C₃×C₄	—	7	2	no	—
233	C₂×C₃×C₄	—	49	1	no	—
234	C₂×C₃×C₄	—	49	2	no	—
235	C₂×C₃×C₄	—	49	4	no	—
236	C₂×C₃×C₄	—	49	2	no	—
237	C₂×C₃×C₄	—	49	1	no	—
238	C₃×C₇	—	2	3	no	—
239	C₃×C₇	D₇×#41	1	2	yes	complemented
240	#41	—	7	2	no	—
241	#41	—	49	4	no	—
242	#41	—	49	2	no	—
243	C₂×Q₈	—	49	1	no	—
244	C₂×D₄	—	49	2	no	—
245	C₂×D₄	—	49	1	no	—
246	C₂²×C₄	—	49	2	no	—
247	C₂²×C₄	—	49	1	no	—
248	C₂×C₇	—	2	4	no	—
249	C₂×C₇	—	2	3	no	—
250	C₂×C₇	—	4	6	no	—
251	C₂×C₇	—	14	2	no	—
252	C₂×C₇	C₂³×C₃×D₇	1	2	yes	—
253	D₇	—	2	2	no	—
254	D₇	—	7	4	no	—
255	D₇	—	14	6	no	—
256	D₇	—	14	4	no	—
257	D₇	—	28	6	no	—
258	C₂²×C₃	—	7	2	no	—
259	C₂²×C₃	—	49	1	no	—
260	C₂²×C₃	—	49	2	no	—
261	C₂²×C₃	—	98	4	no	—
262	C₂²×C₃	—	98	2	no	—
263	C₂²×C₃	C₂×D₇²	1	2	yes	—
264	C₃×C₄	—	7	2	no	—
265	C₃×C₄	—	7	4	no	—
266	C₃×C₄	—	49	1	no	—
267	C₃×C₄	C₂×D₇²	1	1	yes	characteristic
268	C₂³	—	49	1	no	—
269	C₂³	—	49	2	no	—
270	Q₈	—	7	2	no	—
271	Q₈	—	49	2	no	—
272	D₄	—	7	4	no	—
273	D₄	—	49	4	no	—
274	D₄	—	49	2	no	—
275	D₄	—	49	1	no	—
276	D₄	C₃×D₇²	1	1	yes	characteristic, complemented
277	C₂×C₄	—	7	4	no	—
278	C₂×C₄	—	7	2	no	—
279	C₂×C₄	—	49	1	no	—
280	C₂×C₄	—	49	2	no	—
281	C₂×C₄	—	49	4	no	—
282	C₂×C₄	—	49	2	no	—
283	C₂×C₄	—	49	1	no	—
284	C₇	—	2	3	no	—
285	C₇	C₃×D₇×#41	1	2	yes	complemented
286	C₂×C₃	—	2	2	no	—
287	C₂×C₃	—	14	2	no	—
288	C₂×C₃	—	49	2	no	—
289	C₂×C₃	—	98	2	no	—
290	C₂×C₃	C₂²×D₇²	1	1	yes	characteristic
291	C₂²	—	7	2	no	—
292	C₂²	—	49	1	no	—
293	C₂²	—	49	2	no	—
294	C₂²	—	98	4	no	—
295	C₂²	—	98	2	no	—
296	C₂²	C₂×C₃×D₇²	1	2	yes	—
297	C₄	—	7	2	no	—
298	C₄	—	7	4	no	—
299	C₄	—	49	1	no	—
300	C₄	C₂×C₃×D₇²	1	1	yes	characteristic
301	C₃	#13096694	1	1	yes	characteristic, direct product factor, Sylow
302	C₂	—	2	2	no	—
303	C₂	—	14	2	no	—
304	C₂	—	49	2	no	—
305	C₂	—	98	2	no	—
306	C₂	C₂²×C₃×D₇²	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×D₇²
commutator, cosocle	C₂×C₇²	C₂⁴×C₃
Frattini	C₂	C₂²×C₃×D₇²
Fitting	C₃×C₇²×D₄	C₂²
socle	C₂×C₃×C₇²	C₂⁴
hypercenter	C₃×D₄	D₇²
hypocenter	C₇²	C₂×C₃×#41
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂×C₃ ◁ C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×#11716103	C₂	semidirect
2. C₃×#11716108	C₂	semidirect
3. C₃×C₇×#1159	C₂	semidirect
4. C₃×#11716101	C₂	semidirect
5. C₃×C₄×D₇²	C₂	semidirect
6. C₂×C₃×#87639	C₂	semidirect
7. C₃×D₄×C₇⋊D₇	C₂	semidirect
8. C₃×#11716115	C₂	semidirect
9. #13096694	C₃	direct (N × H)
12. C₃×C₇²×D₄	C₂²	semidirect
13. C₃×C₂²⋊(C₇⋊D₇)	C₂²	semidirect
14. C₃×C₄⋊(C₇⋊D₇)	C₂²	semidirect
16. C₃×C₇×C₂²⋊D₇	C₂²	semidirect
17. C₂×C₃×C₇×Dic₇	C₂²	semidirect
18. C₃×C₄×C₇×D₇	C₂²	semidirect
19. C₃×C₇×Dic₁₄	C₂²	semidirect
20. C₃×#87642	C₂²	semidirect
21. C₃×#87641	C₂²	semidirect
24. C₃×#87639	C₂²	semidirect
25. C₃×D₇×Dic₇	C₂²	semidirect
26. D₄×C₇⋊D₇	C₂×C₃	semidirect
27. C₇×#1159	C₂×C₃	semidirect
28. #11716115	C₂×C₃	semidirect
29. C₂×#87639	C₂×C₃	semidirect
30. #11716108	C₂×C₃	semidirect
31. C₄×D₇²	C₂×C₃	semidirect
32. #11716103	C₂×C₃	semidirect
33. #11716101	C₂×C₃	semidirect
47. C₇²×D₄	C₂²×C₃	semidirect
48. C₂²⋊(C₇⋊D₇)	C₂²×C₃	semidirect
49. C₄⋊(C₇⋊D₇)	C₂²×C₃	semidirect
51. C₇×C₂²⋊D₇	C₂²×C₃	semidirect
52. C₂×C₇×Dic₇	C₂²×C₃	semidirect
53. C₄×C₇×D₇	C₂²×C₃	semidirect
54. C₇×Dic₁₄	C₂²×C₃	semidirect
55. #87642	C₂²×C₃	semidirect
56. #87641	C₂²×C₃	semidirect
59. #87639	C₂²×C₃	semidirect
60. D₇×Dic₇	C₂²×C₃	semidirect
65. C₃×#1159	D₇	semidirect
76. C₃×C₇⋊D₇	#41	semidirect
95. C₃×C₇×D₄	C₂×D₇	semidirect
101. C₃×C₂²⋊D₇	C₂×D₇	semidirect
103. C₂×C₃×Dic₇	C₂×D₇	semidirect
114. C₃×C₄×D₇	C₂×D₇	semidirect
116. C₃×Dic₁₄	C₂×D₇	semidirect
117. C₃×C₇²	C₂×#41	semidirect
122. #1159	C₃×D₇	semidirect
133. C₇⋊D₇	C₃×#41	semidirect
161. C₇×D₄	C₂×C₃×D₇	semidirect
167. C₂²⋊D₇	C₂×C₃×D₇	semidirect
169. C₂×Dic₇	C₂×C₃×D₇	semidirect
180. C₄×D₇	C₂×C₃×D₇	semidirect
182. Dic₁₄	C₂×C₃×D₇	semidirect
183. C₇²	C₂×C₃×#41	semidirect
230. C₃×D₄	D₇²	semidirect
239. C₃×C₇	D₇×#41	semidirect
276. D₄	C₃×D₇²	semidirect
285. C₇	C₃×D₇×#41	semidirect
301. C₃	#13096694	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×#11716103	2
C₃×#11716108	2
C₃×C₇×#1159	2
C₃×#11716101	2
C₃×C₄×D₇²	2
C₂×C₃×#87639	2
C₃×D₄×C₇⋊D₇	2
C₃×#11716115	2
#13096694	3
C₃×D₇×#41	7

Subgroups — C3×#13096694 #72246633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13096694 #72246633