Subgroups — C₃×#90495 #13061146

Order 1344.

692 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6516	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×C₂²⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6502	C₂	1	4	yes	maximal, complemented
6	C₃×#6500	C₂	1	2	yes	maximal, complemented
7	C₃×#6498	C₂	1	2	yes	maximal, complemented
8	C₃×#6492	C₂	1	2	yes	maximal, complemented
9	#90495	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃×C₇×D₄	—	2	4	no	—
14	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
15	C₂×C₃×C₇×D₄	C₂²	1	2	yes	complemented
16	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×C₂²⋊D₇	—	2	4	no	—
18	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
19	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	complemented
20	C₂×C₃×C₂²⋊D₇	C₂²	1	4	yes	complemented
21	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic, complemented
22	C₂²×C₃×Dic₇	C₂²	1	2	yes	complemented
23	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	complemented
24	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	complemented
25	C₃×C₂²⋊Dic₇	C₂²	1	4	yes	complemented
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	complemented
27	C₃×Dic₇⋊C₄	C₂²	1	4	yes	complemented
28	C₂²×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₂²⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
30	#6516	C₂×C₃	1	2	yes	complemented
31	C₂×C₂²⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
32	#6502	C₂×C₃	1	4	yes	complemented
33	#6500	C₂×C₃	1	2	yes	complemented
34	#6498	C₂×C₃	1	2	yes	complemented
35	#6492	C₂×C₃	1	2	yes	complemented
36	C₃×#534	—	7	1	no	maximal, Hall
37	C₂³×C₃×C₇	—	2	2	no	—
38	C₂³×C₃×C₇	—	4	2	no	—
39	C₂³×C₃×C₇	—	4	2	no	—
40	C₂³×C₃×C₇	D₄	1	4	yes	complemented
41	C₂³×C₃×C₇	C₂³	1	2	yes	—
42	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
43	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
44	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
45	C₂²×C₃×D₇	—	2	2	no	—
46	C₂²×C₃×D₇	—	4	2	no	—
47	C₂²×C₃×D₇	C₂³	1	2	yes	—
48	C₃×C₇×D₄	—	2	4	no	—
49	C₃×C₇×D₄	—	2	4	no	—
50	C₂×C₃×C₄×C₇	—	2	2	no	—
51	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
52	C₃×C₂²⋊D₇	—	2	4	no	—
53	C₃×C₂²⋊D₇	—	2	4	no	—
54	C₃×C₂²⋊D₇	—	4	4	no	—
55	C₂×C₃×Dic₇	—	2	2	no	—
56	C₂×C₃×Dic₇	—	4	2	no	—
57	C₂×C₃×Dic₇	C₂³	1	2	yes	—
58	C₂×C₃×Dic₇	C₂³	1	4	yes	—
59	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
60	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
61	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic, complemented
62	C₂×C₇×D₄	—	2	4	no	—
63	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
64	C₂×C₇×D₄	C₂²×C₃	1	2	yes	complemented
65	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
66	C₂×C₂²⋊D₇	—	2	4	no	—
67	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
68	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	complemented
69	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	complemented
70	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
71	C₂²×Dic₇	C₂²×C₃	1	2	yes	complemented
72	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	complemented
73	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	complemented
74	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	complemented
75	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	complemented
76	Dic₇⋊C₄	C₂²×C₃	1	4	yes	complemented
77	C₂²×C₃×D₄	—	7	1	no	—
78	C₂²×C₃×D₄	D₇	1	1	yes	characteristic, complemented
79	C₃×#123	—	7	2	no	—
80	C₃×#123	—	7	2	no	—
81	C₃×#121	—	7	4	no	—
82	C₃×#120	—	7	2	no	—
83	C₃×#120	—	7	2	no	—
84	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
85	C₂²×C₃×C₇	—	2	2	no	—
86	C₂²×C₃×C₇	—	2	2	no	—
87	C₂²×C₃×C₇	—	2	2	no	—
88	C₂²×C₃×C₇	—	2	2	no	—
89	C₂²×C₃×C₇	—	2	4	no	—
90	C₂²×C₃×C₇	—	4	4	no	—
91	C₂²×C₃×C₇	—	4	4	no	—
92	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
93	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
94	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
95	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
96	C₂×C₃×D₇	—	2	2	no	—
97	C₂×C₃×D₇	—	2	4	no	—
98	C₂×C₃×D₇	—	4	4	no	—
99	C₃×C₄×C₇	—	2	2	no	—
100	C₃×Dic₇	—	2	2	no	—
101	C₃×Dic₇	—	2	4	no	—
102	#534	—	7	1	no	Sylow
103	C₂³×C₇	—	2	2	no	—
104	C₂³×C₇	—	4	2	no	—
105	C₂³×C₇	—	4	2	no	—
106	C₂³×C₇	C₃×D₄	1	4	yes	complemented
107	C₂³×C₇	C₂³×C₃	1	2	yes	—
108	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
109	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
110	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
111	C₂²×D₇	—	2	2	no	—
112	C₂²×D₇	—	4	2	no	—
113	C₂²×D₇	C₂³×C₃	1	2	yes	—
114	C₇×D₄	—	2	4	no	—
115	C₇×D₄	—	2	4	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
118	C₂²⋊D₇	—	2	4	no	—
119	C₂²⋊D₇	—	2	4	no	—
120	C₂²⋊D₇	—	4	4	no	—
121	C₂×Dic₇	—	2	2	no	—
122	C₂×Dic₇	—	4	2	no	—
123	C₂×Dic₇	C₂³×C₃	1	2	yes	—
124	C₂×Dic₇	C₂³×C₃	1	4	yes	—
125	C₂⁴×C₃	—	7	1	no	—
126	C₂⁴×C₃	C₂×D₇	1	1	yes	characteristic
127	C₂⁴×C₃	C₂×D₇	1	1	yes	characteristic, complemented
128	C₂×C₃×D₄	—	2	4	no	—
129	C₂×C₃×D₄	—	7	2	no	—
130	C₂×C₃×D₄	—	7	2	no	—
131	C₂×C₃×D₄	—	7	4	no	—
132	C₂×C₃×D₄	—	14	4	no	—
133	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
134	C₂×C₃×D₄	C₂×D₇	1	2	yes	complemented
135	C₂²×C₃×C₄	—	7	1	no	—
136	C₂²×C₃×C₄	—	7	2	no	—
137	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic, complemented
138	C₃×C₄⋊C₄	—	7	4	no	—
139	C₃×C₂²⋊C₄	—	7	2	no	—
140	C₃×C₂²⋊C₄	—	7	2	no	—
141	C₃×C₂²⋊C₄	—	7	4	no	—
142	C₃×C₂²⋊C₄	—	7	4	no	—
143	C₂×C₃×C₇	—	2	2	no	—
144	C₂×C₃×C₇	—	2	2	no	—
145	C₂×C₃×C₇	—	2	2	no	—
146	C₂×C₃×C₇	—	4	2	no	—
147	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	—
149	C₃×D₇	—	4	2	no	—
150	C₂²×D₄	—	7	1	no	—
151	C₂²×D₄	C₃×D₇	1	1	yes	characteristic, complemented
152	#123	—	7	2	no	—
153	#123	—	7	2	no	—
154	#121	—	7	4	no	—
155	#120	—	7	2	no	—
156	#120	—	7	2	no	—
157	C₂×C₂²⋊C₄	—	7	1	no	—
158	C₂²×C₇	—	2	2	no	—
159	C₂²×C₇	—	2	2	no	—
160	C₂²×C₇	—	2	2	no	—
161	C₂²×C₇	—	2	2	no	—
162	C₂²×C₇	—	2	4	no	—
163	C₂²×C₇	—	4	4	no	—
164	C₂²×C₇	—	4	4	no	—
165	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
166	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
167	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
168	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
169	C₂×D₇	—	2	2	no	—
170	C₂×D₇	—	2	4	no	—
171	C₂×D₇	—	4	4	no	—
172	C₄×C₇	—	2	2	no	—
173	Dic₇	—	2	2	no	—
174	Dic₇	—	2	4	no	—
175	C₂³×C₃	—	2	2	no	—
176	C₂³×C₃	—	4	2	no	—
177	C₂³×C₃	—	4	2	no	—
178	C₂³×C₃	—	7	2	no	—
179	C₂³×C₃	—	14	2	no	—
180	C₂³×C₃	—	28	2	no	—
181	C₂³×C₃	C₂²⋊D₇	1	4	yes	complemented
182	C₂³×C₃	C₂²×D₇	1	2	yes	—
183	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
184	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
185	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
186	C₃×D₄	—	2	4	no	—
187	C₃×D₄	—	2	4	no	—
188	C₃×D₄	—	14	4	no	—
189	C₃×D₄	—	14	4	no	—
190	C₃×D₄	—	28	4	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	7	4	no	—
193	C₂×C₃×C₄	—	7	2	no	—
194	C₂×C₃×C₄	—	14	2	no	—
195	C₂×C₃×C₄	—	28	2	no	—
196	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
197	C₃×C₇	#534	1	1	yes	characteristic, complemented, Hall
198	C₂⁴	—	7	1	no	—
199	C₂⁴	C₂×C₃×D₇	1	1	yes	characteristic
200	C₂⁴	C₂×C₃×D₇	1	1	yes	characteristic, complemented
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	7	2	no	—
203	C₂×D₄	—	7	2	no	—
204	C₂×D₄	—	7	4	no	—
205	C₂×D₄	—	14	4	no	—
206	C₂×D₄	C₂×C₃×D₇	1	2	yes	—
207	C₂×D₄	C₂×C₃×D₇	1	2	yes	complemented
208	C₂²×C₄	—	7	1	no	—
209	C₂²×C₄	—	7	2	no	—
210	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic, complemented
211	C₄⋊C₄	—	7	4	no	—
212	C₂²⋊C₄	—	7	2	no	—
213	C₂²⋊C₄	—	7	2	no	—
214	C₂²⋊C₄	—	7	4	no	—
215	C₂²⋊C₄	—	7	4	no	—
216	C₂×C₇	—	2	2	no	—
217	C₂×C₇	—	2	2	no	—
218	C₂×C₇	—	2	2	no	—
219	C₂×C₇	—	4	2	no	—
220	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
221	C₂×C₇	C₃×H(5,𝔽₂)	1	2	yes	—
222	D₇	—	4	2	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	4	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	4	4	no	—
229	C₂²×C₃	—	4	4	no	—
230	C₂²×C₃	—	14	4	no	—
231	C₂²×C₃	—	14	2	no	—
232	C₂²×C₃	—	28	4	no	—
233	C₂²×C₃	C₂×C₂²⋊D₇	1	2	yes	—
234	C₂²×C₃	C₂×C₂²⋊D₇	1	2	yes	—
235	C₂²×C₃	C₂×C₂²⋊D₇	1	2	yes	—
236	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	14	4	no	—
239	C₃×C₄	—	14	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	7	2	no	—
244	C₂³	—	14	2	no	—
245	C₂³	—	28	2	no	—
246	C₂³	C₃×C₂²⋊D₇	1	4	yes	complemented
247	C₂³	C₂²×C₃×D₇	1	2	yes	—
248	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
249	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
250	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
251	D₄	—	2	4	no	—
252	D₄	—	2	4	no	—
253	D₄	—	14	4	no	—
254	D₄	—	14	4	no	—
255	D₄	—	28	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	7	4	no	—
258	C₂×C₄	—	7	2	no	—
259	C₂×C₄	—	14	2	no	—
260	C₂×C₄	—	28	2	no	—
261	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
262	C₇	C₃×#534	1	1	yes	characteristic, complemented, Sylow
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	2	2	no	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	—	4	2	no	—
267	C₂×C₃	—	28	2	no	—
268	C₂×C₃	#6548	1	2	yes	—
269	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	14	4	no	—
278	C₂²	—	14	2	no	—
279	C₂²	—	28	4	no	—
280	C₂²	C₂×C₃×C₂²⋊D₇	1	2	yes	—
281	C₂²	C₂×C₃×C₂²⋊D₇	1	2	yes	—
282	C₂²	C₂×C₃×C₂²⋊D₇	1	2	yes	—
283	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	14	4	no	—
286	C₄	—	14	2	no	—
287	C₃	#90495	1	1	yes	characteristic, direct product factor, Sylow
288	C₂	—	2	2	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	—	4	2	no	—
292	C₂	—	28	2	no	—
293	C₂	C₃×#6548	1	2	yes	—
294	C₂	C₂²×C₃×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₂²×C₃×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂²×C₃×D₄	D₇
hypocenter	C₇	C₃×#534
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇×D₄	C₂	semidirect
2. C₂²×C₃×C₂²⋊D₇	C₂	semidirect
3. C₃×#6516	C₂	semidirect
4. C₂×C₃×C₂²⋊Dic₇	C₂	semidirect
5. C₃×#6502	C₂	semidirect
6. C₃×#6500	C₂	semidirect
7. C₃×#6498	C₂	semidirect
8. C₃×#6492	C₂	semidirect
9. #90495	C₃	direct (N × H)
11. C₂⁴×C₃×C₇	C₂²	semidirect
12. C₂³×C₃×D₇	C₂²	semidirect
15. C₂×C₃×C₇×D₄	C₂²	semidirect
16. C₂²×C₃×C₄×C₇	C₂²	semidirect
19. C₂×C₃×C₂²⋊D₇	C₂²	semidirect
20. C₂×C₃×C₂²⋊D₇	C₂²	semidirect
21. C₂²×C₃×Dic₇	C₂²	semidirect
22. C₂²×C₃×Dic₇	C₂²	semidirect
23. C₃×C₂²⋊Dic₇	C₂²	semidirect
24. C₃×C₂²⋊Dic₇	C₂²	semidirect
25. C₃×C₂²⋊Dic₇	C₂²	semidirect
26. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
27. C₃×Dic₇⋊C₄	C₂²	semidirect
28. C₂²×C₇×D₄	C₂×C₃	semidirect
29. C₂²×C₂²⋊D₇	C₂×C₃	semidirect
30. #6516	C₂×C₃	semidirect
31. C₂×C₂²⋊Dic₇	C₂×C₃	semidirect
32. #6502	C₂×C₃	semidirect
33. #6500	C₂×C₃	semidirect
34. #6498	C₂×C₃	semidirect
35. #6492	C₂×C₃	semidirect
40. C₂³×C₃×C₇	D₄	semidirect
60. C₂⁴×C₇	C₂²×C₃	semidirect
61. C₂³×D₇	C₂²×C₃	semidirect
64. C₂×C₇×D₄	C₂²×C₃	semidirect
65. C₂²×C₄×C₇	C₂²×C₃	semidirect
68. C₂×C₂²⋊D₇	C₂²×C₃	semidirect
69. C₂×C₂²⋊D₇	C₂²×C₃	semidirect
70. C₂²×Dic₇	C₂²×C₃	semidirect
71. C₂²×Dic₇	C₂²×C₃	semidirect
72. C₂²⋊Dic₇	C₂²×C₃	semidirect
73. C₂²⋊Dic₇	C₂²×C₃	semidirect
74. C₂²⋊Dic₇	C₂²×C₃	semidirect
75. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
76. Dic₇⋊C₄	C₂²×C₃	semidirect
78. C₂²×C₃×D₄	D₇	semidirect
106. C₂³×C₇	C₃×D₄	semidirect
127. C₂⁴×C₃	C₂×D₇	semidirect
134. C₂×C₃×D₄	C₂×D₇	semidirect
137. C₂²×C₃×C₄	C₂×D₇	semidirect
151. C₂²×D₄	C₃×D₇	semidirect
181. C₂³×C₃	C₂²⋊D₇	semidirect
197. C₃×C₇	#534	semidirect
200. C₂⁴	C₂×C₃×D₇	semidirect
207. C₂×D₄	C₂×C₃×D₇	semidirect
210. C₂²×C₄	C₂×C₃×D₇	semidirect
246. C₂³	C₃×C₂²⋊D₇	semidirect
262. C₇	C₃×#534	semidirect
287. C₃	#90495	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₇×D₄	2
C₂²×C₃×C₂²⋊D₇	2
C₃×#6516	2
C₂×C₃×C₂²⋊Dic₇	2
C₃×#6502	2
C₃×#6500	2
C₃×#6498	2
C₃×#6492	2
#90495	3
C₃×#534	7

Subgroups — C3×#90495 #13061146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90495 #13061146