Subgroups — C₃×C₇×#547 #13061967

Order 1344.

584 conjugacy classes of subgroups.

The subgroup lattice has 240 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	2	yes	maximal, complemented
2	C₃×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#123	C₂	1	4	yes	maximal, complemented
4	C₃×C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₇×#121	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×C₇×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₇×#547	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×C₇×#41	—	2	8	no	—
11	C₂×C₃×C₇×Q₈	C₂²	1	2	yes	—
12	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₇×D₄	C₂²	1	4	yes	—
14	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
16	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
17	C₂²×C₃×C₄×C₇	C₂²	1	4	yes	—
18	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
22	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
23	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
27	C₂×C₇×#41	C₂×C₃	1	2	yes	complemented
28	C₇×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₇×#123	C₂×C₃	1	4	yes	complemented
30	C₇×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
31	C₇×#121	C₂×C₃	1	2	yes	complemented
32	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
33	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₇×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
35	C₃×#547	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
36	C₂³×C₃×C₇	C₂³	1	2	yes	—
37	C₂³×C₃×C₇	C₂³	1	2	yes	—
38	C₃×C₇×Q₈	—	2	4	no	—
39	C₃×C₇×D₄	—	2	8	no	—
40	C₃×C₇×D₄	—	2	2	no	—
41	C₃×C₇×D₄	D₄	1	4	yes	complemented
42	C₂×C₃×C₄×C₇	—	2	8	no	—
43	C₂×C₃×C₄×C₇	—	2	4	no	—
44	C₂×C₃×C₄×C₇	—	2	4	no	—
45	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
48	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
49	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₇×#41	—	2	8	no	—
52	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
53	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₇×D₄	C₂²×C₃	1	4	yes	—
55	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
57	C₂²×C₄×C₇	C₂²×C₃	1	4	yes	—
58	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
59	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
62	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
65	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
66	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
67	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×#41	C₂×C₇	1	2	yes	complemented
69	C₃×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
70	C₃×#123	C₂×C₇	1	4	yes	complemented
71	C₃×C₂²⋊Q₈	C₂×C₇	1	2	yes	complemented
72	C₃×#121	C₂×C₇	1	2	yes	complemented
73	C₃×C₄×D₄	C₂×C₇	1	1	yes	characteristic, complemented
74	C₃×C₄×D₄	C₂×C₇	1	1	yes	characteristic, complemented
75	C₂×C₃×C₄⋊C₄	C₂×C₇	1	2	yes	complemented
76	C₂²×C₃×C₇	—	2	2	no	—
77	C₂²×C₃×C₇	—	2	2	no	—
78	C₂²×C₃×C₇	—	2	2	no	—
79	C₂²×C₃×C₇	—	2	2	no	—
80	C₂²×C₃×C₇	—	2	2	no	—
81	C₂²×C₃×C₇	C₂×D₄	1	4	yes	—
82	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
83	C₃×C₄×C₇	—	2	2	no	—
84	C₃×C₄×C₇	—	2	4	no	—
85	C₃×C₄×C₇	—	2	2	no	—
86	C₃×C₄×C₇	—	2	1	no	—
87	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
88	C₃×C₄×C₇	#41	1	2	yes	complemented
89	#547	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
90	C₂³×C₇	C₂³×C₃	1	2	yes	—
91	C₂³×C₇	C₂³×C₃	1	2	yes	—
92	C₇×Q₈	—	2	4	no	—
93	C₇×D₄	—	2	8	no	—
94	C₇×D₄	—	2	2	no	—
95	C₇×D₄	C₃×D₄	1	4	yes	complemented
96	C₂×C₄×C₇	—	2	8	no	—
97	C₂×C₄×C₇	—	2	4	no	—
98	C₂×C₄×C₇	—	2	4	no	—
99	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
101	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
102	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
103	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
105	C₃×#41	—	2	8	no	—
106	C₂×C₃×Q₈	C₂²×C₇	1	2	yes	—
107	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₃×D₄	C₂²×C₇	1	4	yes	—
109	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
111	C₂²×C₃×C₄	C₂²×C₇	1	4	yes	—
112	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
113	C₃×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
114	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
115	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
116	C₃×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
117	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
118	C₃×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
119	C₃×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
120	C₃×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
121	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
122	C₂×C₃×C₇	—	2	4	no	—
123	C₂×C₃×C₇	—	4	2	no	—
124	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂×#41	C₂×C₃×C₇	1	2	yes	complemented
128	C₄⋊Q₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
129	#123	C₂×C₃×C₇	1	4	yes	complemented
130	C₂²⋊Q₈	C₂×C₃×C₇	1	2	yes	complemented
131	#121	C₂×C₃×C₇	1	2	yes	complemented
132	C₄×D₄	C₂×C₃×C₇	1	1	yes	characteristic, complemented
133	C₄×D₄	C₂×C₃×C₇	1	1	yes	characteristic, complemented
134	C₂×C₄⋊C₄	C₂×C₃×C₇	1	2	yes	complemented
135	C₂²×C₇	—	2	2	no	—
136	C₂²×C₇	—	2	2	no	—
137	C₂²×C₇	—	2	2	no	—
138	C₂²×C₇	—	2	2	no	—
139	C₂²×C₇	—	2	2	no	—
140	C₂²×C₇	C₂×C₃×D₄	1	4	yes	—
141	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
142	C₄×C₇	—	2	2	no	—
143	C₄×C₇	—	2	4	no	—
144	C₄×C₇	—	2	2	no	—
145	C₄×C₇	—	2	1	no	—
146	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
147	C₄×C₇	C₃×#41	1	2	yes	complemented
148	C₂³×C₃	C₂³×C₇	1	2	yes	—
149	C₂³×C₃	C₂³×C₇	1	2	yes	—
150	C₃×Q₈	—	2	4	no	—
151	C₃×D₄	—	2	8	no	—
152	C₃×D₄	—	2	2	no	—
153	C₃×D₄	C₇×D₄	1	4	yes	complemented
154	C₂×C₃×C₄	—	2	8	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
159	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
160	C₂×C₃×C₄	C₂³×C₇	1	4	yes	—
161	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
163	C₃×C₇	#547	1	1	yes	characteristic, direct product factor, Hall
164	#41	—	2	8	no	—
165	C₂×Q₈	C₂²×C₃×C₇	1	2	yes	—
166	C₂×D₄	C₂²×C₃×C₇	1	1	yes	characteristic
167	C₂×D₄	C₂²×C₃×C₇	1	4	yes	—
168	C₂×D₄	C₂²×C₃×C₇	1	1	yes	characteristic
169	C₂²×C₄	C₂²×C₃×C₇	1	2	yes	—
170	C₂²×C₄	C₂²×C₃×C₇	1	4	yes	—
171	C₂²×C₄	C₂²×C₃×C₇	1	2	yes	—
172	C₄⋊C₄	C₂²×C₃×C₇	1	1	yes	characteristic
173	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
174	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
175	C₄⋊C₄	C₂²×C₃×C₇	1	1	yes	characteristic
176	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
177	C₂²⋊C₄	C₂²×C₃×C₇	1	2	yes	—
178	C₂²⋊C₄	C₂²×C₃×C₇	1	2	yes	—
179	C₂²⋊C₄	C₂²×C₃×C₇	1	4	yes	—
180	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
181	C₂×C₇	—	2	4	no	—
182	C₂×C₇	—	4	2	no	—
183	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
184	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
185	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	C₂×C₇×D₄	1	4	yes	—
192	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	2	4	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
198	C₃×C₄	C₇×#41	1	2	yes	complemented
199	C₂³	C₂³×C₃×C₇	1	2	yes	—
200	C₂³	C₂³×C₃×C₇	1	2	yes	—
201	Q₈	—	2	4	no	—
202	D₄	—	2	8	no	—
203	D₄	—	2	2	no	—
204	D₄	C₃×C₇×D₄	1	4	yes	complemented
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
209	C₂×C₄	C₂³×C₃×C₇	1	2	yes	—
210	C₂×C₄	C₂³×C₃×C₇	1	2	yes	—
211	C₂×C₄	C₂³×C₃×C₇	1	4	yes	—
212	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
213	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
214	C₇	C₃×#547	1	1	yes	characteristic, direct product factor, Sylow
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	—	4	2	no	—
217	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
218	C₂×C₃	C₂×C₇×#41	1	1	yes	characteristic
219	C₂×C₃	C₇×2₋¹⁺⁴	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	C₂×C₃×C₇×D₄	1	4	yes	—
226	C₂²	C₂⁴×C₃×C₇	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	C₂×C₃×C₇×D₄	1	2	yes	—
232	C₄	C₃×C₇×#41	1	2	yes	complemented
233	C₃	C₇×#547	1	1	yes	characteristic, direct product factor, Sylow
234	C₂	—	2	4	no	—
235	C₂	—	4	2	no	—
236	C₂	C₂²×C₃×C₇×D₄	1	1	yes	characteristic
237	C₂	C₂×C₃×C₇×#41	1	1	yes	characteristic
238	C₂	C₃×C₇×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×#41	C₂	semidirect
2. C₃×C₇×C₄⋊Q₈	C₂	semidirect
3. C₃×C₇×#123	C₂	semidirect
4. C₃×C₇×C₂²⋊Q₈	C₂	semidirect
5. C₃×C₇×#121	C₂	semidirect
6. C₃×C₄×C₇×D₄	C₂	semidirect
7. C₃×C₄×C₇×D₄	C₂	semidirect
8. C₂×C₃×C₇×C₄⋊C₄	C₂	semidirect
9. C₇×#547	C₃	direct (N × H)
27. C₂×C₇×#41	C₂×C₃	semidirect
28. C₇×C₄⋊Q₈	C₂×C₃	semidirect
29. C₇×#123	C₂×C₃	semidirect
30. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
31. C₇×#121	C₂×C₃	semidirect
32. C₄×C₇×D₄	C₂×C₃	semidirect
33. C₄×C₇×D₄	C₂×C₃	semidirect
34. C₂×C₇×C₄⋊C₄	C₂×C₃	semidirect
35. C₃×#547	C₇	direct (N × H)
41. C₃×C₇×D₄	D₄	semidirect
68. C₂×C₃×#41	C₂×C₇	semidirect
69. C₃×C₄⋊Q₈	C₂×C₇	semidirect
70. C₃×#123	C₂×C₇	semidirect
71. C₃×C₂²⋊Q₈	C₂×C₇	semidirect
72. C₃×#121	C₂×C₇	semidirect
73. C₃×C₄×D₄	C₂×C₇	semidirect
74. C₃×C₄×D₄	C₂×C₇	semidirect
75. C₂×C₃×C₄⋊C₄	C₂×C₇	semidirect
88. C₃×C₄×C₇	#41	semidirect
89. #547	C₃×C₇	direct (N × H)
95. C₇×D₄	C₃×D₄	semidirect
127. C₂×#41	C₂×C₃×C₇	semidirect
128. C₄⋊Q₈	C₂×C₃×C₇	semidirect
129. #123	C₂×C₃×C₇	semidirect
130. C₂²⋊Q₈	C₂×C₃×C₇	semidirect
131. #121	C₂×C₃×C₇	semidirect
132. C₄×D₄	C₂×C₃×C₇	semidirect
133. C₄×D₄	C₂×C₃×C₇	semidirect
134. C₂×C₄⋊C₄	C₂×C₃×C₇	semidirect
147. C₄×C₇	C₃×#41	semidirect
153. C₃×D₄	C₇×D₄	semidirect
163. C₃×C₇	#547	direct (N × H)
198. C₃×C₄	C₇×#41	semidirect
204. D₄	C₃×C₇×D₄	semidirect
214. C₇	C₃×#547	direct (N × H)
232. C₄	C₃×C₇×#41	semidirect
233. C₃	C₇×#547	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
C₃×C₇×C₄⋊Q₈	2
C₃×C₇×#123	2
C₃×C₇×C₂²⋊Q₈	2
C₃×C₇×#121	2
C₃×C₄×C₇×D₄	2
C₂×C₃×C₇×C₄⋊C₄	2
C₇×#547	3
C₃×#547	7

Subgroups — C3×C7×#547 #13061967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#547 #13061967