Subgroups — C₂×C₃×C₈×#157 #13173234

Order 1728.

520 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×#157	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3787	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
4	C₃×C₈×#157	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₈×#157	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃×#157	C₄	1	2	yes	—
7	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
8	C₂×C₃×#622	C₄	1	2	yes	complemented
9	C₃×C₄×#157	C₄	1	2	yes	—
10	C₃×C₄×#157	C₂²	1	2	yes	—
11	C₃×#3787	C₄	1	2	yes	complemented
12	C₃×#3787	C₂²	1	2	yes	—
13	C₂×C₃³×C₈	C₄	1	1	yes	characteristic, complemented
14	C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic, complemented
15	C₃×C₈×C₃⋊D₃	C₄	1	2	yes	complemented
16	C₃×C₈×C₃⋊D₃	C₂²	1	2	yes	—
17	C₂×C₄×#157	C₂×C₃	1	1	yes	characteristic
18	C₂×#3787	C₂×C₃	1	1	yes	characteristic
19	C₂×C₈×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×D₃×C₈	—	6	2	no	—
21	C₈×#157	C₂×C₃	1	4	yes	direct product factor
22	C₂²×C₃×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×#157	C₈	1	8	yes	direct product factor
24	C₂×C₃×#157	C₂×C₄	1	4	yes	—
25	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
31	C₃×#622	C₂×C₄	1	4	yes	complemented
32	C₃³×C₈	C₂×C₄	1	2	yes	complemented
33	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	complemented
34	C₂×C₃×C₄×C₈	—	9	1	no	maximal
35	C₂²×#157	C₃×C₄	1	2	yes	—
36	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
37	C₂×C₃×C₄×D₃	—	6	2	no	—
38	C₂×#622	C₃×C₄	1	2	yes	complemented
39	C₄×#157	C₃×C₄	1	2	yes	—
40	C₄×#157	C₂²×C₃	1	2	yes	—
41	#3787	C₃×C₄	1	2	yes	complemented
42	#3787	C₂²×C₃	1	2	yes	—
43	C₂×C₃²×C₈	—	2	2	no	—
44	C₂×C₃²×C₈	—	4	2	no	—
45	C₂×C₃²×C₈	C₃×C₄	1	1	yes	characteristic, complemented
46	C₂×C₃⋊(C₃⋊C₈)	C₃×C₄	1	1	yes	characteristic, complemented
47	C₈×C₃⋊D₃	C₃×C₄	1	2	yes	complemented
48	C₈×C₃⋊D₃	C₂²×C₃	1	2	yes	—
49	C₂×C₃×C₃⋊C₈	—	6	2	no	—
50	C₃×D₃×C₈	—	6	4	no	—
51	C₃×D₃×C₈	—	6	4	no	—
52	C₂²×C₃³	C₄²	1	1	yes	characteristic
53	C₂×C₃×C₃⋊D₃	C₄²	1	1	yes	characteristic
54	C₂×C₃×C₃⋊D₃	C₂×C₈	1	4	yes	—
55	C₂×C₃×C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
56	C₃×#157	C₂×C₈	1	8	yes	direct product factor
57	C₃³×C₄	C₄²	1	1	yes	characteristic
58	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₃⋊Dic₃	C₄²	1	1	yes	characteristic
60	C₃×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₃×C₈	—	9	1	no	—
62	C₂²×C₃×C₈	—	9	1	no	—
63	C₂×C₃×C₄²	—	9	1	no	—
64	C₂×D₃×C₈	—	6	2	no	—
65	C₃×C₄×C₈	—	9	4	no	—
66	C₂²×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
67	C₂²×C₃×D₃	—	6	2	no	—
68	C₂×#157	C₃×C₈	1	8	yes	direct product factor
69	C₂×#157	C₂×C₃×C₄	1	4	yes	—
70	C₂×C₃²×C₄	—	2	2	no	—
71	C₂×C₃²×C₄	—	4	2	no	—
72	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
73	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
74	C₄×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
75	C₄×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
76	C₄×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
77	C₄×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
78	C₂×C₃×Dic₃	—	6	2	no	—
79	C₃×C₄×D₃	—	6	2	no	—
80	C₃×C₄×D₃	—	6	2	no	—
81	C₃×C₄×D₃	—	6	2	no	—
82	C₃×C₄×D₃	—	6	2	no	—
83	#622	C₂×C₃×C₄	1	4	yes	complemented
84	C₃²×C₈	—	2	4	no	—
85	C₃²×C₈	—	4	4	no	—
86	C₃²×C₈	C₂×C₃×C₄	1	2	yes	complemented
87	C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	1	2	yes	complemented
88	C₃×C₃⋊C₈	—	6	4	no	—
89	C₂×C₄×C₈	—	9	1	no	Sylow
90	C₂×C₃³	C₄×C₈	1	2	yes	complemented
91	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
92	C₃×C₃⋊D₃	C₄×C₈	1	2	yes	complemented
93	C₃×C₃⋊D₃	C₂²×C₈	1	2	yes	—
94	C₂²×C₃×C₄	—	9	2	no	—
95	C₂²×C₃×C₄	—	9	1	no	—
96	C₂×C₄×D₃	—	6	2	no	—
97	C₂×C₃×C₈	—	2	2	no	—
98	C₂×C₃×C₈	—	4	2	no	—
99	C₂×C₃×C₈	—	9	2	no	—
100	C₂×C₃×C₈	—	9	2	no	—
101	C₂×C₃×C₈	—	9	2	no	—
102	C₂×C₃×C₈	—	9	1	no	—
103	C₂×C₃×C₈	—	9	2	no	—
104	C₂×C₃×C₈	—	9	2	no	—
105	C₂×C₃×C₈	#157	1	1	yes	characteristic, direct product factor
106	C₃×C₄²	—	9	2	no	—
107	C₃×C₄²	—	9	2	no	—
108	C₂×C₃⋊C₈	—	6	2	no	—
109	D₃×C₈	—	6	4	no	—
110	D₃×C₈	—	6	4	no	—
111	C₂²×C₃²	—	2	2	no	—
112	C₂²×C₃²	—	4	2	no	—
113	C₂²×C₃²	C₃×C₄²	1	1	yes	characteristic
114	C₂×C₃⋊D₃	C₃×C₄²	1	1	yes	characteristic
115	C₂×C₃⋊D₃	C₂×C₃×C₈	1	4	yes	—
116	C₂×C₃⋊D₃	C₂²×C₃×C₄	1	1	yes	characteristic
117	C₂×C₃×D₃	—	6	2	no	—
118	C₂×C₃×D₃	—	6	8	no	—
119	C₂×C₃×D₃	—	6	2	no	—
120	#157	C₂×C₃×C₈	1	8	yes	direct product factor
121	C₃²×C₄	—	2	2	no	—
122	C₃²×C₄	—	2	2	no	—
123	C₃²×C₄	—	4	2	no	—
124	C₃²×C₄	—	4	2	no	—
125	C₃²×C₄	C₃×C₄²	1	1	yes	characteristic
126	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
127	C₃⋊Dic₃	C₃×C₄²	1	1	yes	characteristic
128	C₃⋊Dic₃	C₂²×C₃×C₄	1	1	yes	characteristic
129	C₃×Dic₃	—	6	2	no	—
130	C₃×Dic₃	—	6	2	no	—
131	C₂²×C₈	—	9	1	no	—
132	C₂²×C₈	—	9	1	no	—
133	C₂×C₄²	—	9	1	no	—
134	C₄×C₈	—	9	4	no	—
135	C₃³	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₃	—	9	1	no	—
137	C₂²×D₃	—	6	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	—	9	8	no	—
141	C₂×C₃×C₄	—	9	4	no	—
142	C₂×C₃×C₄	—	9	1	no	—
143	C₂×C₃×C₄	—	9	1	no	—
144	C₂×C₃×C₄	—	9	1	no	—
145	C₂×C₃×C₄	—	9	1	no	—
146	C₂×C₃×C₄	—	9	1	no	—
147	C₂×C₃×C₄	C₂×#157	1	1	yes	characteristic
148	C₂×Dic₃	—	6	2	no	—
149	C₄×D₃	—	6	2	no	—
150	C₄×D₃	—	6	2	no	—
151	C₄×D₃	—	6	2	no	—
152	C₄×D₃	—	6	2	no	—
153	C₃×C₈	—	2	4	no	—
154	C₃×C₈	—	4	4	no	—
155	C₃×C₈	—	9	4	no	—
156	C₃×C₈	—	9	2	no	—
157	C₃×C₈	C₂×#157	1	2	yes	direct product factor
158	C₃⋊C₈	—	6	4	no	—
159	C₂×C₃²	—	2	4	no	—
160	C₂×C₃²	—	2	2	no	—
161	C₂×C₃²	—	4	4	no	—
162	C₂×C₃²	—	4	2	no	—
163	C₂×C₃²	C₃×C₄×C₈	1	2	yes	complemented
164	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
165	C₃⋊D₃	C₃×C₄×C₈	1	2	yes	complemented
166	C₃⋊D₃	C₂²×C₃×C₈	1	2	yes	—
167	C₃×D₃	—	6	4	no	—
168	C₃×D₃	—	6	4	no	—
169	C₂²×C₄	—	9	2	no	—
170	C₂²×C₄	—	9	1	no	—
171	C₂×C₈	—	9	2	no	—
172	C₂×C₈	—	9	2	no	—
173	C₂×C₈	—	9	2	no	—
174	C₂×C₈	—	9	1	no	—
175	C₂×C₈	—	9	2	no	—
176	C₂×C₈	—	9	2	no	—
177	C₂×C₈	C₃×#157	1	1	yes	characteristic, direct product factor
178	C₄²	—	9	2	no	—
179	C₄²	—	9	2	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	4	2	no	—
182	C₂²×C₃	—	9	4	no	—
183	C₂²×C₃	—	9	1	no	—
184	C₂²×C₃	—	9	1	no	—
185	C₂²×C₃	C₄×#157	1	1	yes	characteristic
186	C₂×D₃	—	6	2	no	—
187	C₂×D₃	—	6	2	no	—
188	C₂×D₃	—	6	8	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	4	2	no	—
192	C₃×C₄	—	4	2	no	—
193	C₃×C₄	—	9	8	no	—
194	C₃×C₄	—	9	1	no	—
195	C₃×C₄	—	9	1	no	—
196	C₃×C₄	C₄×#157	1	1	yes	characteristic
197	C₃×C₄	C₂²×#157	1	1	yes	characteristic
198	Dic₃	—	6	2	no	—
199	Dic₃	—	6	2	no	—
200	C₃²	—	2	2	no	—
201	C₃²	—	4	2	no	—
202	C₃²	C₂×C₃×C₄×C₈	1	1	yes	characteristic, complemented
203	C₂³	—	9	1	no	—
204	C₂×C₄	—	9	4	no	—
205	C₂×C₄	—	9	8	no	—
206	C₂×C₄	—	9	1	no	—
207	C₂×C₄	—	9	1	no	—
208	C₂×C₄	—	9	1	no	—
209	C₂×C₄	—	9	1	no	—
210	C₂×C₄	—	9	1	no	—
211	C₂×C₄	C₂×C₃×#157	1	1	yes	characteristic
212	C₈	—	9	4	no	—
213	C₈	—	9	2	no	—
214	C₈	C₂×C₃×#157	1	2	yes	direct product factor
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	—	2	2	no	—
217	C₂×C₃	—	4	4	no	—
218	C₂×C₃	—	4	2	no	—
219	C₂×C₃	—	9	2	no	—
220	C₂×C₃	—	9	2	no	—
221	C₂×C₃	C₈×#157	1	2	yes	direct product factor
222	C₂×C₃	C₂×C₄×#157	1	1	yes	characteristic
223	D₃	—	6	4	no	—
224	D₃	—	6	4	no	—
225	C₂²	—	9	4	no	—
226	C₂²	—	9	1	no	—
227	C₂²	—	9	1	no	—
228	C₂²	C₃×C₄×#157	1	1	yes	characteristic
229	C₄	—	9	8	no	—
230	C₄	—	9	1	no	—
231	C₄	—	9	1	no	—
232	C₄	C₃×C₄×#157	1	1	yes	characteristic
233	C₄	C₂²×C₃×#157	1	1	yes	characteristic
234	C₃	—	2	2	no	—
235	C₃	—	4	2	no	—
236	C₃	C₂×C₈×#157	1	1	yes	characteristic, direct product factor
237	C₂	—	9	2	no	—
238	C₂	—	9	2	no	—
239	C₂	C₃×C₈×#157	1	2	yes	direct product factor
240	C₂	C₂×C₃×C₄×#157	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₈	#157
commutator, hypocenter	C₃²	C₂×C₃×C₄×C₈
Frattini	C₄	C₂²×C₃×#157
Fitting	C₂×C₃³×C₈	C₄
socle	C₂²×C₃³	C₄²
prime radical	C₂²×C₃×C₃⋊D₃	C₂×C₄
cosocle	C₄×C₃⋊D₃	C₂³×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×C₈×#157	C₂	direct (N × H)
5. C₂×C₈×#157	C₃	direct (N × H)
8. C₂×C₃×#622	C₄	semidirect
11. C₃×#3787	C₄	semidirect
13. C₂×C₃³×C₈	C₄	semidirect
14. C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	semidirect
15. C₃×C₈×C₃⋊D₃	C₄	semidirect
21. C₈×#157	C₂×C₃	direct (N × H)
23. C₂×C₃×#157	C₈	direct (N × H)
31. C₃×#622	C₂×C₄	semidirect
32. C₃³×C₈	C₂×C₄	semidirect
33. C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
38. C₂×#622	C₃×C₄	semidirect
41. #3787	C₃×C₄	semidirect
45. C₂×C₃²×C₈	C₃×C₄	semidirect
46. C₂×C₃⋊(C₃⋊C₈)	C₃×C₄	semidirect
47. C₈×C₃⋊D₃	C₃×C₄	semidirect
56. C₃×#157	C₂×C₈	direct (N × H)
68. C₂×#157	C₃×C₈	direct (N × H)
83. #622	C₂×C₃×C₄	semidirect
86. C₃²×C₈	C₂×C₃×C₄	semidirect
87. C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	semidirect
90. C₂×C₃³	C₄×C₈	semidirect
92. C₃×C₃⋊D₃	C₄×C₈	semidirect
105. C₂×C₃×C₈	#157	direct (N × H)
120. #157	C₂×C₃×C₈	direct (N × H)
135. C₃³	C₂×C₄×C₈	semidirect
157. C₃×C₈	C₂×#157	direct (N × H)
163. C₂×C₃²	C₃×C₄×C₈	semidirect
165. C₃⋊D₃	C₃×C₄×C₈	semidirect
177. C₂×C₈	C₃×#157	direct (N × H)
202. C₃²	C₂×C₃×C₄×C₈	semidirect
214. C₈	C₂×C₃×#157	direct (N × H)
221. C₂×C₃	C₈×#157	direct (N × H)
236. C₃	C₂×C₈×#157	direct (N × H)
239. C₂	C₃×C₈×#157	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×C₄×#157	2
C₂×C₃×#3787	2
C₂×C₃×C₈×C₃⋊D₃	2
C₃×C₈×#157	2
C₂×C₈×#157	3
C₂×C₃×C₄×C₈	9

Subgroups — C2×C3×C8×#157 #13173234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₈×#157 #13173234