Subgroups — C₃×#90462 #13061113

Order 1344.

496 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	C₃×#6554	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6514	C₂	1	2	yes	maximal, complemented
4	C₃×D₇×Q₁₆	C₂	1	2	yes	maximal, complemented
5	C₃×#6478	C₂	1	4	yes	maximal, complemented
6	C₃×#6475	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×Dic₂₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#6464	C₂	1	1	yes	maximal, characteristic, complemented
9	#90462	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×C₇×#41	C₂²	1	2	yes	—
11	C₃×Q₈×D₇	C₂²	1	4	yes	—
12	C₃×#1159	—	2	4	no	—
13	C₃×#1159	C₂²	1	4	yes	—
14	C₃×#1157	—	2	2	no	—
15	C₃×#1157	C₂²	1	1	yes	characteristic
16	C₂×C₃×Dic₁₄	—	2	2	no	—
17	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
18	C₃×C₇×Q₁₆	C₂²	1	1	yes	characteristic
19	C₃×C₇×SD₁₆	C₂²	1	2	yes	—
20	C₃×C₇×D₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
22	C₃×#1144	C₂²	1	4	yes	—
23	C₃×#1142	C₂²	1	4	yes	—
24	C₃×#1136	C₂²	1	1	yes	characteristic
25	C₃×Dic₂₈	C₂²	1	2	yes	—
26	C₃×Dic₂₈	C₂²	1	2	yes	—
27	C₃×#1131	C₂²	1	2	yes	—
28	C₃×C₈×D₇	C₂²	1	2	yes	—
29	#6554	C₂×C₃	1	2	yes	complemented
30	C₇×#135	C₂×C₃	1	1	yes	characteristic, complemented
31	#6514	C₂×C₃	1	2	yes	complemented
32	D₇×Q₁₆	C₂×C₃	1	2	yes	complemented
33	#6478	C₂×C₃	1	4	yes	complemented
34	#6475	C₂×C₃	1	2	yes	complemented
35	C₂×Dic₂₈	C₂×C₃	1	1	yes	characteristic, complemented
36	#6464	C₂×C₃	1	1	yes	characteristic, complemented
37	C₃×#578	—	7	1	no	maximal, Hall
38	C₃×C₇×Q₈	C₂³	1	2	yes	—
39	C₃×C₇×D₄	—	2	2	no	—
40	C₃×C₇×D₄	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₇	—	2	2	no	—
42	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₃×C₂²⋊D₇	—	2	4	no	—
44	C₃×C₂²⋊D₇	D₄	1	2	yes	—
45	C₂×C₃×Dic₇	—	2	2	no	—
46	C₂×C₃×Dic₇	—	2	4	no	—
47	C₃×D₂₈	D₄	1	1	yes	characteristic
48	C₃×C₄×D₇	—	2	4	no	—
49	C₃×C₄×D₇	C₂³	1	2	yes	—
50	C₃×Dic₁₄	—	2	2	no	—
51	C₃×Dic₁₄	—	2	4	no	—
52	C₃×Dic₁₄	D₄	1	1	yes	characteristic
53	C₃×Dic₁₄	C₂³	1	4	yes	—
54	C₃×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₃×C₇×C₈	C₂³	1	1	yes	characteristic
56	C₃×C₇⋊C₈	C₂³	1	2	yes	—
57	C₇×#41	C₂²×C₃	1	2	yes	—
58	Q₈×D₇	C₂²×C₃	1	4	yes	—
59	#1159	—	2	4	no	—
60	#1159	C₂²×C₃	1	4	yes	—
61	#1157	—	2	2	no	—
62	#1157	C₂²×C₃	1	1	yes	characteristic
63	C₂×Dic₁₄	—	2	2	no	—
64	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
65	C₇×Q₁₆	C₂²×C₃	1	1	yes	characteristic
66	C₇×SD₁₆	C₂²×C₃	1	2	yes	—
67	C₇×D₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
69	#1144	C₂²×C₃	1	4	yes	—
70	#1142	C₂²×C₃	1	4	yes	—
71	#1136	C₂²×C₃	1	1	yes	characteristic
72	Dic₂₈	C₂²×C₃	1	2	yes	—
73	Dic₂₈	C₂²×C₃	1	2	yes	—
74	#1131	C₂²×C₃	1	2	yes	—
75	C₈×D₇	C₂²×C₃	1	2	yes	—
76	C₃×2₋¹⁺⁴	—	7	2	no	—
77	C₃×#137	—	7	2	no	—
78	C₃×#137	—	7	4	no	—
79	C₃×#135	—	7	2	no	—
80	C₃×#135	D₇	1	1	yes	characteristic, complemented
81	C₂×C₃×Q₁₆	—	7	2	no	—
82	C₂×C₃×Q₁₆	—	7	1	no	—
83	C₃×#131	—	7	1	no	—
84	C₂²×C₃×C₇	—	2	2	no	—
85	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×D₇	C₂×D₄	1	2	yes	—
87	C₃×C₄×C₇	—	2	2	no	—
88	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
89	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
90	C₃×Dic₇	—	2	4	no	—
91	C₃×Dic₇	C₂×D₄	1	2	yes	—
92	#578	—	7	1	no	Sylow
93	C₇×Q₈	C₂³×C₃	1	2	yes	—
94	C₇×D₄	—	2	2	no	—
95	C₇×D₄	C₂³×C₃	1	2	yes	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
98	C₂²⋊D₇	—	2	4	no	—
99	C₂²⋊D₇	C₃×D₄	1	2	yes	—
100	C₂×Dic₇	—	2	2	no	—
101	C₂×Dic₇	—	2	4	no	—
102	D₂₈	C₃×D₄	1	1	yes	characteristic
103	C₄×D₇	—	2	4	no	—
104	C₄×D₇	C₂³×C₃	1	2	yes	—
105	Dic₁₄	—	2	2	no	—
106	Dic₁₄	—	2	4	no	—
107	Dic₁₄	C₃×D₄	1	1	yes	characteristic
108	Dic₁₄	C₂³×C₃	1	4	yes	—
109	C₇×C₈	C₂³×C₃	1	1	yes	characteristic
110	C₇×C₈	C₂³×C₃	1	1	yes	characteristic
111	C₇⋊C₈	C₂³×C₃	1	2	yes	—
112	C₃×#41	—	7	1	no	—
113	C₃×#41	—	7	4	no	—
114	C₃×#41	—	14	2	no	—
115	C₃×#41	—	14	4	no	—
116	C₃×#41	C₂×D₇	1	2	yes	—
117	C₂×C₃×Q₈	—	7	4	no	—
118	C₂×C₃×Q₈	—	7	2	no	—
119	C₂×C₃×Q₈	—	14	2	no	—
120	C₃×Q₁₆	—	7	4	no	—
121	C₃×Q₁₆	—	7	2	no	—
122	C₃×Q₁₆	—	7	2	no	—
123	C₃×Q₁₆	C₂×D₇	1	1	yes	characteristic
124	C₃×SD₁₆	—	7	4	no	—
125	C₃×SD₁₆	C₂×D₇	1	2	yes	—
126	C₃×D₈	C₂×D₇	1	1	yes	characteristic
127	C₃×M₁₆	—	7	1	no	—
128	C₃×M₁₆	—	7	2	no	—
129	C₂×C₃×C₈	—	7	2	no	—
130	C₂×C₃×C₈	C₂×D₇	1	1	yes	characteristic
131	C₂×C₃×C₇	—	2	1	no	—
132	C₂×C₃×C₇	—	4	2	no	—
133	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
134	C₃×D₇	—	2	2	no	—
135	2₋¹⁺⁴	—	7	2	no	—
136	#137	—	7	2	no	—
137	#137	—	7	4	no	—
138	#135	—	7	2	no	—
139	#135	C₃×D₇	1	1	yes	characteristic, complemented
140	C₂×Q₁₆	—	7	2	no	—
141	C₂×Q₁₆	—	7	1	no	—
142	#131	—	7	1	no	—
143	C₂²×C₇	—	2	2	no	—
144	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
146	C₄×C₇	—	2	2	no	—
147	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
148	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
149	Dic₇	—	2	4	no	—
150	Dic₇	C₂×C₃×D₄	1	2	yes	—
151	C₃×Q₈	—	7	1	no	—
152	C₃×Q₈	—	7	4	no	—
153	C₃×Q₈	—	14	4	no	—
154	C₃×Q₈	—	14	2	no	—
155	C₃×Q₈	C₂²×D₇	1	2	yes	—
156	C₃×D₄	—	2	2	no	—
157	C₃×D₄	—	7	2	no	—
158	C₃×D₄	—	7	1	no	—
159	C₃×D₄	—	14	4	no	—
160	C₃×D₄	C₂²×D₇	1	2	yes	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	7	2	no	—
163	C₂×C₃×C₄	—	14	4	no	—
164	C₂×C₃×C₄	—	14	4	no	—
165	C₂×C₃×C₄	—	14	2	no	—
166	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
167	C₃×C₈	—	7	2	no	—
168	C₃×C₈	C₂²×D₇	1	1	yes	characteristic
169	C₃×C₈	C₂²×D₇	1	1	yes	characteristic
170	C₃×C₇	#578	1	1	yes	characteristic, complemented, Hall
171	#41	—	7	1	no	—
172	#41	—	7	4	no	—
173	#41	—	14	2	no	—
174	#41	—	14	4	no	—
175	#41	C₂×C₃×D₇	1	2	yes	—
176	C₂×Q₈	—	7	4	no	—
177	C₂×Q₈	—	7	2	no	—
178	C₂×Q₈	—	14	2	no	—
179	Q₁₆	—	7	4	no	—
180	Q₁₆	—	7	2	no	—
181	Q₁₆	—	7	2	no	—
182	Q₁₆	C₂×C₃×D₇	1	1	yes	characteristic
183	SD₁₆	—	7	4	no	—
184	SD₁₆	C₂×C₃×D₇	1	2	yes	—
185	D₈	C₂×C₃×D₇	1	1	yes	characteristic
186	M₁₆	—	7	1	no	—
187	M₁₆	—	7	2	no	—
188	C₂×C₈	—	7	2	no	—
189	C₂×C₈	C₂×C₃×D₇	1	1	yes	characteristic
190	C₂×C₇	—	2	1	no	—
191	C₂×C₇	—	4	2	no	—
192	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
193	D₇	—	2	2	no	—
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	—	7	2	no	—
196	C₂²×C₃	D₄×D₇	1	1	yes	characteristic
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	7	2	no	—
199	C₃×C₄	—	14	4	no	—
200	C₃×C₄	D₄×D₇	1	1	yes	characteristic
201	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
202	Q₈	—	7	1	no	—
203	Q₈	—	7	4	no	—
204	Q₈	—	14	4	no	—
205	Q₈	—	14	2	no	—
206	Q₈	C₂²×C₃×D₇	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	7	2	no	—
209	D₄	—	7	1	no	—
210	D₄	—	14	4	no	—
211	D₄	C₂²×C₃×D₇	1	2	yes	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	7	2	no	—
214	C₂×C₄	—	14	4	no	—
215	C₂×C₄	—	14	4	no	—
216	C₂×C₄	—	14	2	no	—
217	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
218	C₈	—	7	2	no	—
219	C₈	C₂²×C₃×D₇	1	1	yes	characteristic
220	C₈	C₂²×C₃×D₇	1	1	yes	characteristic
221	C₇	C₃×#578	1	1	yes	characteristic, complemented, Sylow
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	4	2	no	—
224	C₂×C₃	—	14	2	no	—
225	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
226	C₂²	—	2	2	no	—
227	C₂²	—	7	2	no	—
228	C₂²	C₃×D₄×D₇	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	7	2	no	—
231	C₄	—	14	4	no	—
232	C₄	C₃×D₄×D₇	1	1	yes	characteristic
233	C₄	C₂³×C₃×D₇	1	1	yes	characteristic
234	C₃	#90462	1	1	yes	characteristic, direct product factor, Sylow
235	C₂	—	2	1	no	—
236	C₂	—	4	2	no	—
237	C₂	—	14	2	no	—
238	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₄×D₇
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₇
Fitting	C₃×C₇×#135	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₃×#135	D₇
hypocenter	C₇	C₃×#578
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#135

Complements

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

N	H ≅ G/N	type
1. C₃×#6554	C₂	semidirect
2. C₃×C₇×#135	C₂	semidirect
3. C₃×#6514	C₂	semidirect
4. C₃×D₇×Q₁₆	C₂	semidirect
5. C₃×#6478	C₂	semidirect
6. C₃×#6475	C₂	semidirect
7. C₂×C₃×Dic₂₈	C₂	semidirect
8. C₃×#6464	C₂	semidirect
9. #90462	C₃	direct (N × H)
29. #6554	C₂×C₃	semidirect
30. C₇×#135	C₂×C₃	semidirect
31. #6514	C₂×C₃	semidirect
32. D₇×Q₁₆	C₂×C₃	semidirect
33. #6478	C₂×C₃	semidirect
34. #6475	C₂×C₃	semidirect
35. C₂×Dic₂₈	C₂×C₃	semidirect
36. #6464	C₂×C₃	semidirect
80. C₃×#135	D₇	semidirect
139. #135	C₃×D₇	semidirect
170. C₃×C₇	#578	semidirect
221. C₇	C₃×#578	semidirect
234. C₃	#90462	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×#6554	2
C₃×C₇×#135	2
C₃×#6514	2
C₃×D₇×Q₁₆	2
C₃×#6478	2
C₃×#6475	2
C₂×C₃×Dic₂₈	2
C₃×#6464	2
#90462	3
C₃×#578	7

Subgroups — C3×#90462 #13061113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90462 #13061113