Subgroups — C₃×#90308 #13060959

Order 1344.

500 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×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6502	C₂	1	2	yes	maximal, complemented
3	C₃×#6500	C₂	1	2	yes	maximal, complemented
4	C₃×D₄×Dic₇	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#6461	C₂	1	2	yes	maximal, complemented
8	C₃×#6449	C₂	1	2	yes	maximal, complemented
9	C₃×#6448	C₂	1	2	yes	maximal, complemented
10	#90308	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
17	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
18	C₂²×C₃×Dic₇	C₂²	1	2	yes	complemented
19	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
20	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
21	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	complemented
22	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
23	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	complemented
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×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
28	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic, complemented
29	C₃×C₄⋊Dic₇	C₂²	1	2	yes	complemented
30	C₃×Dic₇⋊C₄	C₂²	1	2	yes	complemented
31	C₃×C₄×Dic₇	C₂²	1	2	yes	complemented
32	C₇×#121	C₂×C₃	1	1	yes	characteristic, complemented
33	#6502	C₂×C₃	1	2	yes	complemented
34	#6500	C₂×C₃	1	2	yes	complemented
35	D₄×Dic₇	C₂×C₃	1	2	yes	complemented
36	C₂×(C₂×D₇)⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
37	#6487	C₂×C₃	1	1	yes	characteristic, complemented
38	#6461	C₂×C₃	1	2	yes	complemented
39	#6449	C₂×C₃	1	2	yes	complemented
40	#6448	C₂×C₃	1	2	yes	complemented
41	C₃×#538	—	7	1	no	maximal, Hall
42	C₂³×C₃×C₇	C₂³	1	2	yes	—
43	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
44	C₂²×C₃×D₇	—	2	2	no	—
45	C₂²×C₃×D₇	—	4	1	no	—
46	C₂²×C₃×D₇	—	4	1	no	—
47	C₂²×C₃×D₇	C₂³	1	2	yes	—
48	C₃×C₇×D₄	—	2	4	no	—
49	C₃×C₇×D₄	—	4	1	no	—
50	C₂×C₃×C₄×C₇	—	4	1	no	—
51	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₃×C₂²⋊D₇	—	4	2	no	—
55	C₃×C₂²⋊D₇	—	4	2	no	—
56	C₂×C₃×Dic₇	—	4	1	no	—
57	C₂×C₃×Dic₇	—	4	2	no	—
58	C₂×C₃×Dic₇	C₂³	1	2	yes	—
59	C₂×C₃×Dic₇	C₂³	1	2	yes	—
60	C₂×C₃×Dic₇	C₂³	1	2	yes	—
61	C₃×Dic₁₄	—	4	1	no	—
62	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
64	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
65	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
67	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
68	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
69	C₂²×Dic₇	C₂²×C₃	1	2	yes	complemented
70	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
71	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
72	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
73	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
74	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	complemented
75	(C₂×D₇)⋊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	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
80	C₄⋊Dic₇	C₂²×C₃	1	2	yes	complemented
81	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
82	C₄×Dic₇	C₂²×C₃	1	2	yes	complemented
83	C₃×#126	—	7	2	no	—
84	C₃×#124	—	7	2	no	—
85	C₃×#123	—	7	2	no	—
86	C₃×C₂²⋊Q₈	—	7	1	no	—
87	C₃×#121	—	7	2	no	—
88	C₃×#121	D₇	1	1	yes	characteristic, complemented
89	C₃×#120	—	7	2	no	—
90	C₃×C₄×D₄	—	7	2	no	—
91	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
92	C₂²×C₃×C₇	—	2	1	no	—
93	C₂²×C₃×C₇	—	2	1	no	—
94	C₂²×C₃×C₇	—	2	2	no	—
95	C₂²×C₃×C₇	—	2	2	no	—
96	C₂²×C₃×C₇	—	2	2	no	—
97	C₂²×C₃×C₇	#41	1	2	yes	—
98	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
99	C₂×C₃×D₇	—	2	2	no	—
100	C₂×C₃×D₇	—	2	2	no	—
101	C₂×C₃×D₇	—	2	2	no	—
102	C₂×C₃×D₇	—	4	4	no	—
103	C₃×C₄×C₇	—	2	2	no	—
104	C₃×C₄×C₇	—	2	1	no	—
105	C₃×C₄×C₇	—	2	1	no	—
106	C₃×Dic₇	—	2	2	no	—
107	C₃×Dic₇	—	2	2	no	—
108	C₃×Dic₇	—	2	2	no	—
109	#538	—	7	1	no	Sylow
110	C₂³×C₇	C₂³×C₃	1	2	yes	—
111	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
112	C₂²×D₇	—	2	2	no	—
113	C₂²×D₇	—	4	1	no	—
114	C₂²×D₇	—	4	1	no	—
115	C₂²×D₇	C₂³×C₃	1	2	yes	—
116	C₇×D₄	—	2	4	no	—
117	C₇×D₄	—	4	1	no	—
118	C₂×C₄×C₇	—	4	1	no	—
119	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
120	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
122	C₂²⋊D₇	—	4	2	no	—
123	C₂²⋊D₇	—	4	2	no	—
124	C₂×Dic₇	—	4	1	no	—
125	C₂×Dic₇	—	4	2	no	—
126	C₂×Dic₇	C₂³×C₃	1	2	yes	—
127	C₂×Dic₇	C₂³×C₃	1	2	yes	—
128	C₂×Dic₇	C₂³×C₃	1	2	yes	—
129	Dic₁₄	—	4	1	no	—
130	C₂⁴×C₃	—	7	1	no	—
131	C₂×C₃×Q₈	—	7	1	no	—
132	C₂×C₃×D₄	—	7	2	no	—
133	C₂×C₃×D₄	—	7	2	no	—
134	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
135	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic, complemented
136	C₂²×C₃×C₄	—	7	1	no	—
137	C₂²×C₃×C₄	—	7	2	no	—
138	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	7	2	no	—
140	C₃×C₄⋊C₄	—	7	2	no	—
141	C₃×C₄⋊C₄	—	7	1	no	—
142	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic, complemented
143	C₃×C₂²⋊C₄	—	7	2	no	—
144	C₃×C₂²⋊C₄	—	7	2	no	—
145	C₃×C₂²⋊C₄	—	7	2	no	—
146	C₃×C₂²⋊C₄	—	7	2	no	—
147	C₃×C₂²⋊C₄	—	7	2	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	complemented
150	C₃×C₄²	—	7	2	no	—
151	C₂×C₃×C₇	—	2	2	no	—
152	C₂×C₃×C₇	—	4	2	no	—
153	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₃×D₇	—	4	2	no	—
157	#126	—	7	2	no	—
158	#124	—	7	2	no	—
159	#123	—	7	2	no	—
160	C₂²⋊Q₈	—	7	1	no	—
161	#121	—	7	2	no	—
162	#121	C₃×D₇	1	1	yes	characteristic, complemented
163	#120	—	7	2	no	—
164	C₄×D₄	—	7	2	no	—
165	C₂×C₂²⋊C₄	—	7	1	no	—
166	C₂²×C₇	—	2	1	no	—
167	C₂²×C₇	—	2	1	no	—
168	C₂²×C₇	—	2	2	no	—
169	C₂²×C₇	—	2	2	no	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	C₃×#41	1	2	yes	—
172	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
173	C₂×D₇	—	2	2	no	—
174	C₂×D₇	—	2	2	no	—
175	C₂×D₇	—	2	2	no	—
176	C₂×D₇	—	4	4	no	—
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	2	1	no	—
179	C₄×C₇	—	2	1	no	—
180	Dic₇	—	2	2	no	—
181	Dic₇	—	2	2	no	—
182	Dic₇	—	2	2	no	—
183	C₂³×C₃	—	7	2	no	—
184	C₂³×C₃	—	14	2	no	—
185	C₂³×C₃	—	28	1	no	—
186	C₂³×C₃	—	28	1	no	—
187	C₂³×C₃	C₂²×D₇	1	2	yes	—
188	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
189	C₃×Q₈	—	28	1	no	—
190	C₃×D₄	—	2	4	no	—
191	C₃×D₄	—	4	1	no	—
192	C₃×D₄	—	28	2	no	—
193	C₃×D₄	—	28	2	no	—
194	C₂×C₃×C₄	—	4	1	no	—
195	C₂×C₃×C₄	—	7	2	no	—
196	C₂×C₃×C₄	—	7	2	no	—
197	C₂×C₃×C₄	—	7	2	no	—
198	C₂×C₃×C₄	—	28	1	no	—
199	C₂×C₃×C₄	—	28	2	no	—
200	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
201	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
203	C₃×C₇	#538	1	1	yes	characteristic, complemented, Hall
204	C₂⁴	—	7	1	no	—
205	C₂×Q₈	—	7	1	no	—
206	C₂×D₄	—	7	2	no	—
207	C₂×D₄	—	7	2	no	—
208	C₂×D₄	C₂×C₃×D₇	1	2	yes	—
209	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic, complemented
210	C₂²×C₄	—	7	1	no	—
211	C₂²×C₄	—	7	2	no	—
212	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
213	C₄⋊C₄	—	7	2	no	—
214	C₄⋊C₄	—	7	2	no	—
215	C₄⋊C₄	—	7	1	no	—
216	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic, complemented
217	C₂²⋊C₄	—	7	2	no	—
218	C₂²⋊C₄	—	7	2	no	—
219	C₂²⋊C₄	—	7	2	no	—
220	C₂²⋊C₄	—	7	2	no	—
221	C₂²⋊C₄	—	7	2	no	—
222	C₂²⋊C₄	—	7	2	no	—
223	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	complemented
224	C₄²	—	7	2	no	—
225	C₂×C₇	—	2	2	no	—
226	C₂×C₇	—	4	2	no	—
227	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
228	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
230	D₇	—	4	2	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	14	2	no	—
237	C₂²×C₃	—	14	2	no	—
238	C₂²×C₃	—	14	2	no	—
239	C₂²×C₃	—	28	4	no	—
240	C₂²×C₃	#1159	1	2	yes	—
241	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	14	2	no	—
246	C₃×C₄	—	14	2	no	—
247	C₃×C₄	—	14	2	no	—
248	C₂³	—	7	2	no	—
249	C₂³	—	14	2	no	—
250	C₂³	—	28	1	no	—
251	C₂³	—	28	1	no	—
252	C₂³	C₂²×C₃×D₇	1	2	yes	—
253	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
254	Q₈	—	28	1	no	—
255	D₄	—	2	4	no	—
256	D₄	—	4	1	no	—
257	D₄	—	28	2	no	—
258	D₄	—	28	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	7	2	no	—
261	C₂×C₄	—	7	2	no	—
262	C₂×C₄	—	7	2	no	—
263	C₂×C₄	—	28	1	no	—
264	C₂×C₄	—	28	2	no	—
265	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
266	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
268	C₇	C₃×#538	1	1	yes	characteristic, complemented, Sylow
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	—	4	2	no	—
271	C₂×C₃	—	28	2	no	—
272	C₂×C₃	C₂×#1159	1	1	yes	characteristic
273	C₂×C₃	#6548	1	1	yes	characteristic
274	C₂×C₃	#6553	1	1	yes	characteristic
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	14	2	no	—
281	C₂²	—	14	2	no	—
282	C₂²	—	14	2	no	—
283	C₂²	—	28	4	no	—
284	C₂²	C₃×#1159	1	2	yes	—
285	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	14	2	no	—
290	C₄	—	14	2	no	—
291	C₄	—	14	2	no	—
292	C₃	#90308	1	1	yes	characteristic, direct product factor, Sylow
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	—	28	2	no	—
296	C₂	C₂×C₃×#1159	1	1	yes	characteristic
297	C₂	C₃×#6548	1	1	yes	characteristic
298	C₂	C₃×#6553	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×#538
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. C₃×C₇×#121	C₂	semidirect
2. C₃×#6502	C₂	semidirect
3. C₃×#6500	C₂	semidirect
4. C₃×D₄×Dic₇	C₂	semidirect
5. C₂×C₃×(C₂×D₇)⋊C₄	C₂	semidirect
6. C₃×#6487	C₂	semidirect
7. C₃×#6461	C₂	semidirect
8. C₃×#6449	C₂	semidirect
9. C₃×#6448	C₂	semidirect
10. #90308	C₃	direct (N × H)
13. C₂×C₃×C₇×D₄	C₂²	semidirect
18. C₂²×C₃×Dic₇	C₂²	semidirect
19. C₂×C₃×Dic₁₄	C₂²	semidirect
20. C₃×C₇×C₄⋊C₄	C₂²	semidirect
21. C₃×C₇×C₂²⋊C₄	C₂²	semidirect
23. C₃×C₂²⋊Dic₇	C₂²	semidirect
28. C₃×C₄⋊Dic₇	C₂²	semidirect
29. C₃×C₄⋊Dic₇	C₂²	semidirect
30. C₃×Dic₇⋊C₄	C₂²	semidirect
31. C₃×C₄×Dic₇	C₂²	semidirect
32. C₇×#121	C₂×C₃	semidirect
33. #6502	C₂×C₃	semidirect
34. #6500	C₂×C₃	semidirect
35. D₄×Dic₇	C₂×C₃	semidirect
36. C₂×(C₂×D₇)⋊C₄	C₂×C₃	semidirect
37. #6487	C₂×C₃	semidirect
38. #6461	C₂×C₃	semidirect
39. #6449	C₂×C₃	semidirect
40. #6448	C₂×C₃	semidirect
64. C₂×C₇×D₄	C₂²×C₃	semidirect
69. C₂²×Dic₇	C₂²×C₃	semidirect
70. C₂×Dic₁₄	C₂²×C₃	semidirect
71. C₇×C₄⋊C₄	C₂²×C₃	semidirect
72. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
74. C₂²⋊Dic₇	C₂²×C₃	semidirect
79. C₄⋊Dic₇	C₂²×C₃	semidirect
80. C₄⋊Dic₇	C₂²×C₃	semidirect
81. Dic₇⋊C₄	C₂²×C₃	semidirect
82. C₄×Dic₇	C₂²×C₃	semidirect
88. C₃×#121	D₇	semidirect
135. C₂×C₃×D₄	C₂×D₇	semidirect
142. C₃×C₄⋊C₄	C₂×D₇	semidirect
149. C₃×C₂²⋊C₄	C₂×D₇	semidirect
162. #121	C₃×D₇	semidirect
203. C₃×C₇	#538	semidirect
209. C₂×D₄	C₂×C₃×D₇	semidirect
216. C₄⋊C₄	C₂×C₃×D₇	semidirect
223. C₂²⋊C₄	C₂×C₃×D₇	semidirect
268. C₇	C₃×#538	semidirect
292. C₃	#90308	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×#121	2
C₃×#6502	2
C₃×#6500	2
C₃×D₄×Dic₇	2
C₂×C₃×(C₂×D₇)⋊C₄	2
C₃×#6487	2
C₃×#6461	2
C₃×#6449	2
C₃×#6448	2
#90308	3
C₃×#538	7

Subgroups — C3×#90308 #13060959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90308 #13060959