Subgroups — C₃×#90523 #13061173

Order 1344.

620 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#1157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6510	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6507	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6503	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6502	C₂	1	4	yes	maximal, complemented
7	C₃×#6499	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×C₂²⋊D₇	C₂	1	4	yes	maximal, complemented
9	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#90523	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₇×#41	—	2	4	no	—
12	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
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	2	yes	—
16	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₂²⋊D₇	C₂²	1	4	yes	—
18	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
19	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
20	C₃×#1157	—	2	4	no	—
21	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
23	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
24	C₃×C₂²⋊Dic₇	C₂²	1	4	yes	—
25	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
26	C₃×Dic₇⋊C₄	C₂²	1	4	yes	—
27	C₃×C₄×Dic₇	C₂²	1	2	yes	—
28	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
29	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
30	C₂×C₇×#41	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#1157	C₂×C₃	1	1	yes	characteristic, complemented
32	#6510	C₂×C₃	1	1	yes	characteristic, complemented
33	#6507	C₂×C₃	1	1	yes	characteristic, complemented
34	#6503	C₂×C₃	1	1	yes	characteristic, complemented
35	#6502	C₂×C₃	1	4	yes	complemented
36	#6499	C₂×C₃	1	1	yes	characteristic, complemented
37	C₄×C₂²⋊D₇	C₂×C₃	1	4	yes	complemented
38	C₂×C₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
39	C₃×#532	—	7	1	no	maximal, Hall
40	C₂³×C₃×C₇	C₂³	1	2	yes	—
41	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
42	C₂²×C₃×D₇	C₂³	1	2	yes	—
43	C₃×C₇×Q₈	—	2	2	no	—
44	C₃×C₇×D₄	—	2	4	no	—
45	C₃×C₇×D₄	—	2	2	no	—
46	C₂×C₃×C₄×C₇	—	2	4	no	—
47	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
48	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₃×C₂²⋊D₇	—	2	4	no	—
52	C₃×C₂²⋊D₇	—	2	8	no	—
53	C₂×C₃×Dic₇	—	2	4	no	—
54	C₂×C₃×Dic₇	C₂³	1	2	yes	—
55	C₂×C₃×Dic₇	C₂³	1	2	yes	—
56	C₂×C₃×Dic₇	C₂³	1	2	yes	—
57	C₃×D₂₈	—	2	2	no	—
58	C₃×C₄×D₇	—	2	4	no	—
59	C₃×Dic₁₄	—	2	2	no	—
60	C₇×#41	—	2	4	no	—
61	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
64	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
65	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	—
67	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
68	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
69	#1157	—	2	4	no	—
70	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
72	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
73	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	—
74	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
75	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
76	C₄×Dic₇	C₂²×C₃	1	2	yes	—
77	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
78	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×#41	—	7	1	no	—
80	C₂×C₃×#41	D₇	1	1	yes	characteristic, complemented
81	C₃×C₄⋊Q₈	—	7	1	no	—
82	C₃×#127	—	7	1	no	—
83	C₃×#124	—	7	1	no	—
84	C₃×#124	—	7	1	no	—
85	C₃×#121	—	7	4	no	—
86	C₃×C₄×D₄	—	7	4	no	—
87	C₂×C₃×C₄²	—	7	1	no	—
88	C₂²×C₃×C₇	—	2	2	no	—
89	C₂²×C₃×C₇	—	2	2	no	—
90	C₂²×C₃×C₇	—	2	4	no	—
91	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
92	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
93	C₂×C₃×D₇	—	2	2	no	—
94	C₂×C₃×D₇	—	2	4	no	—
95	C₃×C₄×C₇	—	2	2	no	—
96	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
97	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
98	C₃×Dic₇	—	2	2	no	—
99	C₃×Dic₇	—	2	2	no	—
100	C₃×Dic₇	#41	1	4	yes	complemented
101	#532	—	7	1	no	Sylow
102	C₂³×C₇	C₂³×C₃	1	2	yes	—
103	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
104	C₂²×D₇	C₂³×C₃	1	2	yes	—
105	C₇×Q₈	—	2	2	no	—
106	C₇×D₄	—	2	4	no	—
107	C₇×D₄	—	2	2	no	—
108	C₂×C₄×C₇	—	2	4	no	—
109	C₂×C₄×C₇	C₃×D₄	1	4	yes	complemented
110	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
111	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
113	C₂²⋊D₇	—	2	4	no	—
114	C₂²⋊D₇	—	2	8	no	—
115	C₂×Dic₇	—	2	4	no	—
116	C₂×Dic₇	C₂³×C₃	1	2	yes	—
117	C₂×Dic₇	C₂³×C₃	1	2	yes	—
118	C₂×Dic₇	C₂³×C₃	1	2	yes	—
119	D₂₈	—	2	2	no	—
120	C₄×D₇	—	2	4	no	—
121	Dic₁₄	—	2	2	no	—
122	C₃×#41	—	2	4	no	—
123	C₃×#41	—	14	4	no	—
124	C₂×C₃×Q₈	—	7	1	no	—
125	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
126	C₂×C₃×D₄	—	7	4	no	—
127	C₂×C₃×D₄	—	7	2	no	—
128	C₂×C₃×D₄	—	7	1	no	—
129	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
130	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
131	C₂²×C₃×C₄	—	7	2	no	—
132	C₂²×C₃×C₄	—	7	2	no	—
133	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
134	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
135	C₃×C₄⋊C₄	—	7	4	no	—
136	C₃×C₂²⋊C₄	—	7	4	no	—
137	C₃×C₂²⋊C₄	—	7	4	no	—
138	C₃×C₄²	—	7	1	no	—
139	C₃×C₄²	—	7	1	no	—
140	C₃×C₄²	—	7	2	no	—
141	C₂×C₃×C₇	—	2	2	no	—
142	C₂×C₃×C₇	—	4	2	no	—
143	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₃×C₇	C₂×#41	1	2	yes	—
145	C₃×D₇	—	4	2	no	—
146	C₂×#41	—	7	1	no	—
147	C₂×#41	C₃×D₇	1	1	yes	characteristic, complemented
148	C₄⋊Q₈	—	7	1	no	—
149	#127	—	7	1	no	—
150	#124	—	7	1	no	—
151	#124	—	7	1	no	—
152	#121	—	7	4	no	—
153	C₄×D₄	—	7	4	no	—
154	C₂×C₄²	—	7	1	no	—
155	C₂²×C₇	—	2	2	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	2	4	no	—
158	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
159	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
160	C₂×D₇	—	2	2	no	—
161	C₂×D₇	—	2	4	no	—
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
164	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
165	Dic₇	—	2	2	no	—
166	Dic₇	—	2	2	no	—
167	Dic₇	C₃×#41	1	4	yes	complemented
168	C₂³×C₃	—	7	2	no	—
169	C₂³×C₃	C₂²×D₇	1	2	yes	—
170	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
171	C₃×Q₈	—	2	2	no	—
172	C₃×Q₈	—	14	2	no	—
173	C₃×D₄	—	2	4	no	—
174	C₃×D₄	—	2	2	no	—
175	C₃×D₄	—	14	8	no	—
176	C₃×D₄	—	14	4	no	—
177	C₃×D₄	—	14	2	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	7	2	no	—
180	C₂×C₃×C₄	—	7	2	no	—
181	C₂×C₃×C₄	—	7	2	no	—
182	C₂×C₃×C₄	—	14	4	no	—
183	C₂×C₃×C₄	—	14	4	no	—
184	C₂×C₃×C₄	C₂²⋊D₇	1	4	yes	complemented
185	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
186	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
188	C₃×C₇	#532	1	1	yes	characteristic, complemented, Hall
189	#41	—	2	4	no	—
190	#41	—	14	4	no	—
191	C₂×Q₈	—	7	1	no	—
192	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
193	C₂×D₄	—	7	4	no	—
194	C₂×D₄	—	7	2	no	—
195	C₂×D₄	—	7	1	no	—
196	C₂×D₄	C₂×C₃×D₇	1	2	yes	—
197	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
198	C₂²×C₄	—	7	2	no	—
199	C₂²×C₄	—	7	2	no	—
200	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
201	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
202	C₄⋊C₄	—	7	4	no	—
203	C₂²⋊C₄	—	7	4	no	—
204	C₂²⋊C₄	—	7	4	no	—
205	C₄²	—	7	2	no	—
206	C₄²	—	7	1	no	—
207	C₄²	—	7	1	no	—
208	C₂×C₇	—	2	2	no	—
209	C₂×C₇	—	4	2	no	—
210	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
211	C₂×C₇	C₂×C₃×#41	1	2	yes	—
212	D₇	—	4	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	4	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	14	4	no	—
217	C₂²×C₃	—	14	2	no	—
218	C₂²×C₃	C₂×C₂²⋊D₇	1	2	yes	—
219	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	7	4	no	—
222	C₃×C₄	—	14	2	no	—
223	C₃×C₄	—	14	2	no	—
224	C₃×C₄	C₂×C₂²⋊D₇	1	2	yes	—
225	C₃×C₄	C₂×C₂²⋊D₇	1	2	yes	—
226	C₂³	—	7	2	no	—
227	C₂³	C₂²×C₃×D₇	1	2	yes	—
228	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	—	14	2	no	—
231	D₄	—	2	4	no	—
232	D₄	—	2	2	no	—
233	D₄	—	14	8	no	—
234	D₄	—	14	4	no	—
235	D₄	—	14	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	7	2	no	—
238	C₂×C₄	—	7	2	no	—
239	C₂×C₄	—	7	2	no	—
240	C₂×C₄	—	14	4	no	—
241	C₂×C₄	—	14	4	no	—
242	C₂×C₄	C₃×C₂²⋊D₇	1	4	yes	complemented
243	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
244	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
246	C₇	C₃×#532	1	1	yes	characteristic, complemented, Sylow
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	4	2	no	—
249	C₂×C₃	—	28	2	no	—
250	C₂×C₃	D₇×#41	1	2	yes	—
251	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	14	4	no	—
256	C₂²	—	14	2	no	—
257	C₂²	C₂×C₃×C₂²⋊D₇	1	2	yes	—
258	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	7	4	no	—
261	C₄	—	14	2	no	—
262	C₄	—	14	2	no	—
263	C₄	C₂×C₃×C₂²⋊D₇	1	2	yes	—
264	C₄	C₂×C₃×C₂²⋊D₇	1	2	yes	—
265	C₃	#90523	1	1	yes	characteristic, direct product factor, Sylow
266	C₂	—	2	2	no	—
267	C₂	—	4	2	no	—
268	C₂	—	28	2	no	—
269	C₂	C₃×D₇×#41	1	2	yes	—
270	C₂	C₂²×C₃×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₂×C₃×C₇×#41	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂×C₃×#41	D₇
hypocenter	C₇	C₃×#532
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×#41	C₂	semidirect
2. C₂×C₃×#1157	C₂	semidirect
3. C₃×#6510	C₂	semidirect
4. C₃×#6507	C₂	semidirect
5. C₃×#6503	C₂	semidirect
6. C₃×#6502	C₂	semidirect
7. C₃×#6499	C₂	semidirect
8. C₃×C₄×C₂²⋊D₇	C₂	semidirect
9. C₂×C₃×C₄×Dic₇	C₂	semidirect
10. #90523	C₃	direct (N × H)
30. C₂×C₇×#41	C₂×C₃	semidirect
31. C₂×#1157	C₂×C₃	semidirect
32. #6510	C₂×C₃	semidirect
33. #6507	C₂×C₃	semidirect
34. #6503	C₂×C₃	semidirect
35. #6502	C₂×C₃	semidirect
36. #6499	C₂×C₃	semidirect
37. C₄×C₂²⋊D₇	C₂×C₃	semidirect
38. C₂×C₄×Dic₇	C₂×C₃	semidirect
47. C₂×C₃×C₄×C₇	D₄	semidirect
80. C₂×C₃×#41	D₇	semidirect
100. C₃×Dic₇	#41	semidirect
109. C₂×C₄×C₇	C₃×D₄	semidirect
147. C₂×#41	C₃×D₇	semidirect
167. Dic₇	C₃×#41	semidirect
184. C₂×C₃×C₄	C₂²⋊D₇	semidirect
188. C₃×C₇	#532	semidirect
242. C₂×C₄	C₃×C₂²⋊D₇	semidirect
246. C₇	C₃×#532	semidirect
265. C₃	#90523	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×C₇×#41	2
C₂×C₃×#1157	2
C₃×#6510	2
C₃×#6507	2
C₃×#6503	2
C₃×#6502	2
C₃×#6499	2
C₃×C₄×C₂²⋊D₇	2
C₂×C₃×C₄×Dic₇	2
#90523	3
C₃×#532	7

Subgroups — C3×#90523 #13061173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90523 #13061173