Subgroups — C₂×C₃×#5071 #11873568

Order 1152.

1225 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#870	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#832	C₂	1	4	yes	maximal, complemented
4	C₃×#5071	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#490	—	3	1	no	maximal
7	C₂×#5071	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×D₄	—	2	2	no	—
9	C₂²×C₃²×D₄	C₂²	1	2	yes	—
10	C₃²×#127	—	2	2	no	—
11	C₃²×#127	C₂²	1	4	yes	—
12	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×Dic₃⋊C₄	—	2	2	no	—
15	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
16	C₃×#870	—	2	2	no	—
17	C₃×#870	C₂²	1	4	yes	complemented
18	C₃×#832	C₂²	1	16	yes	complemented
19	C₃×#803	C₂²	1	4	yes	complemented
20	C₂×C₃×#127	—	2	1	no	—
21	C₂×C₃×#127	D₃	1	1	yes	characteristic
22	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#125	—	3	1	no	—
24	C₂×#870	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×#490	—	3	8	no	—
26	C₂×C₃×#102	—	3	4	no	—
27	C₂×C₃×#97	—	3	1	no	—
28	C₂×#832	C₂×C₃	1	4	yes	complemented
29	#5071	C₂×C₃	1	8	yes	direct product factor
30	C₂×#803	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂⁴×C₃²	—	2	2	no	—
32	C₂×C₃²×D₄	—	2	8	no	—
33	C₂×C₃²×D₄	—	2	2	no	—
34	C₂×C₃²×D₄	—	2	8	no	—
35	C₂×C₃²×D₄	—	4	8	no	—
36	C₂×C₃²×D₄	C₂³	1	4	yes	—
37	C₂²×C₃²×C₄	D₄	1	2	yes	—
38	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₃×Dic₃	—	2	2	no	—
40	C₃²×C₄²	D₄	1	2	yes	—
41	C₃²×C₄²	C₂³	1	2	yes	—
42	C₃×C₄⋊Dic₃	—	2	2	no	—
43	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
44	C₃×Dic₃⋊C₄	—	2	8	no	—
45	C₂×C₃×C₃⋊C₈	—	2	4	no	—
46	C₂×C₃×C₃⋊C₈	C₂³	1	4	yes	—
47	C₂×#490	—	3	1	no	Sylow
48	C₂²×C₃×D₄	—	2	4	no	—
49	C₂²×C₃×D₄	—	2	2	no	—
50	C₂²×C₃×D₄	—	2	2	no	—
51	C₂²×C₃×D₄	—	2	2	no	—
52	C₂²×C₃×D₄	C₂×D₃	1	2	yes	—
53	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
54	C₂²×C₃×C₈	—	3	2	no	—
55	C₃×#127	—	2	4	no	—
56	C₃×#127	—	2	2	no	—
57	C₃×#127	—	2	2	no	—
58	C₃×#127	—	2	4	no	—
59	C₃×#127	C₂×D₃	1	4	yes	—
60	C₃×#127	C₂²×C₃	1	4	yes	—
61	C₃×#125	—	3	4	no	—
62	C₃×#125	—	6	2	no	—
63	C₂×C₃×C₄⋊C₄	—	3	2	no	—
64	C₂×C₃×C₄⋊C₄	—	6	2	no	—
65	C₂×C₃×C₄²	—	2	1	no	—
66	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
67	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
69	C₂×Dic₃⋊C₄	—	2	2	no	—
70	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
71	#870	—	2	2	no	—
72	#870	C₂²×C₃	1	4	yes	complemented
73	C₃×#102	—	3	16	no	—
74	C₃×#97	—	3	4	no	—
75	#832	C₂²×C₃	1	16	yes	complemented
76	#803	C₂²×C₃	1	4	yes	complemented
77	C₂³×C₃²	—	2	4	no	—
78	C₂³×C₃²	—	4	4	no	—
79	C₂³×C₃²	—	4	8	no	—
80	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
81	C₃²×D₄	—	2	8	no	—
82	C₃²×D₄	—	4	16	no	—
83	C₃²×D₄	—	4	4	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
86	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	#41	1	4	yes	—
88	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
89	C₂×C₃×Dic₃	—	2	4	no	—
90	C₂×C₃×Dic₃	—	4	4	no	—
91	C₃×C₃⋊C₈	—	2	4	no	—
92	C₂×#127	C₃×D₃	1	1	yes	characteristic
93	C₂×#125	—	3	1	no	—
94	#490	—	3	8	no	—
95	C₂×#102	—	3	4	no	—
96	C₂×#97	—	3	1	no	—
97	C₂⁴×C₃	—	2	4	no	—
98	C₂⁴×C₃	—	2	2	no	—
99	C₂⁴×C₃	—	2	2	no	—
100	C₂×C₃×D₄	—	2	4	no	—
101	C₂×C₃×D₄	—	2	16	no	—
102	C₂×C₃×D₄	—	2	8	no	—
103	C₂×C₃×D₄	—	2	2	no	—
104	C₂×C₃×D₄	—	2	8	no	—
105	C₂×C₃×D₄	—	2	2	no	—
106	C₂×C₃×D₄	—	2	8	no	—
107	C₂×C₃×D₄	—	2	8	no	—
108	C₂×C₃×D₄	—	2	4	no	—
109	C₂×C₃×D₄	—	4	8	no	—
110	C₂×C₃×D₄	—	4	16	no	—
111	C₂×C₃×D₄	—	4	8	no	—
112	C₂×C₃×D₄	—	4	8	no	—
113	C₂×C₃×D₄	C₂²×D₃	1	4	yes	—
114	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
115	C₂²×C₃×C₄	—	2	2	no	—
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₂²×C₃×C₄	—	6	2	no	—
118	C₂²×C₃×C₄	#67	1	2	yes	—
119	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
120	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₂²×Dic₃	—	2	2	no	—
123	C₂×C₃×C₈	—	3	4	no	—
124	C₂×C₃×C₈	—	6	4	no	—
125	C₃×C₄⋊C₄	—	3	4	no	—
126	C₃×C₄⋊C₄	—	6	8	no	—
127	C₃×C₄⋊C₄	—	6	2	no	—
128	C₃×C₄²	—	2	2	no	—
129	C₃×C₄²	—	2	2	no	—
130	C₃×C₄²	#67	1	2	yes	—
131	C₃×C₄²	C₃×D₄	1	2	yes	—
132	C₃×C₄²	C₂²×D₃	1	2	yes	—
133	C₃×C₄²	C₂³×C₃	1	2	yes	—
134	C₄⋊Dic₃	—	2	2	no	—
135	C₄⋊Dic₃	C₂³×C₃	1	4	yes	—
136	Dic₃⋊C₄	—	2	8	no	—
137	C₂×C₃⋊C₈	—	2	4	no	—
138	C₂×C₃⋊C₈	C₂³×C₃	1	4	yes	—
139	C₂²×C₃²	—	4	4	no	—
140	C₂²×C₃²	—	4	8	no	—
141	C₂²×C₃²	—	8	8	no	—
142	C₂²×C₃²	#124	1	2	yes	—
143	C₂²×C₃²	AGL(1,ℤ/8)	1	4	yes	—
144	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	#124	1	2	yes	—
147	C₃²×C₄	C₂×#41	1	2	yes	—
148	C₃×Dic₃	—	4	4	no	—
149	C₂²×D₄	—	2	2	no	—
150	C₂²×D₄	C₂×C₃×D₃	1	2	yes	—
151	C₂²×C₈	—	3	2	no	—
152	#127	—	2	2	no	—
153	#127	C₂×C₃×D₃	1	4	yes	—
154	#125	—	3	4	no	—
155	#125	—	6	2	no	—
156	C₂×C₄⋊C₄	—	3	2	no	—
157	C₂×C₄⋊C₄	—	6	2	no	—
158	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
159	#102	—	3	16	no	—
160	#97	—	3	4	no	—
161	C₂³×C₃	—	2	8	no	—
162	C₂³×C₃	—	2	4	no	—
163	C₂³×C₃	—	2	4	no	—
164	C₂³×C₃	—	2	1	no	—
165	C₂³×C₃	—	4	16	no	—
166	C₂³×C₃	—	4	8	no	—
167	C₂³×C₃	—	4	8	no	—
168	C₂³×C₃	—	4	4	no	—
169	C₂³×C₃	—	4	4	no	—
170	C₂³×C₃	—	4	8	no	—
171	C₂³×C₃	C₂×#67	1	1	yes	characteristic
172	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
173	C₃×D₄	—	2	8	no	—
174	C₃×D₄	—	2	8	no	—
175	C₃×D₄	—	4	8	no	—
176	C₃×D₄	—	4	32	no	—
177	C₃×D₄	—	4	16	no	—
178	C₃×D₄	—	4	16	no	—
179	C₃×D₄	—	4	8	no	—
180	C₃×D₄	—	4	4	no	—
181	C₃×D₄	—	4	4	no	—
182	C₂×C₃×C₄	—	2	8	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	6	4	no	—
190	C₂×C₃×C₄	—	12	4	no	—
191	C₂×C₃×C₄	#237	1	4	yes	—
192	C₂×C₃×C₄	C₂×#67	1	4	yes	—
193	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
195	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₃×#41	1	4	yes	—
197	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
199	C₂×Dic₃	—	2	4	no	—
200	C₂×Dic₃	—	4	4	no	—
201	C₃×C₈	—	6	4	no	—
202	C₃⋊C₈	—	2	4	no	—
203	C₂×C₃²	—	8	4	no	—
204	C₂×C₃²	#490	1	4	yes	complemented
205	C₂×C₃²	C₂×#124	1	1	yes	characteristic
206	C₂×C₃²	C₂×AGL(1,ℤ/8)	1	2	yes	—
207	C₂⁴	—	2	2	no	—
208	C₂×D₄	—	2	8	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	8	no	—
211	C₂×D₄	—	4	8	no	—
212	C₂×D₄	C₂²×C₃×D₃	1	4	yes	—
213	C₂²×C₄	—	6	2	no	—
214	C₂²×C₄	C₃×#67	1	2	yes	—
215	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
216	C₂×C₈	—	3	4	no	—
217	C₂×C₈	—	6	4	no	—
218	C₄⋊C₄	—	3	4	no	—
219	C₄⋊C₄	—	6	2	no	—
220	C₄⋊C₄	—	6	8	no	—
221	C₄²	C₃×#67	1	2	yes	—
222	C₄²	C₂²×C₃×D₃	1	2	yes	—
223	C₂²×C₃	—	2	4	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	4	16	no	—
227	C₂²×C₃	—	4	8	no	—
228	C₂²×C₃	—	4	8	no	—
229	C₂²×C₃	—	4	8	no	—
230	C₂²×C₃	—	4	4	no	—
231	C₂²×C₃	—	4	4	no	—
232	C₂²×C₃	—	8	16	no	—
233	C₂²×C₃	—	8	8	no	—
234	C₂²×C₃	—	8	8	no	—
235	C₂²×C₃	#932	1	4	yes	—
236	C₂²×C₃	#936	1	2	yes	—
237	C₂²×C₃	C₃×#124	1	2	yes	—
238	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	4	yes	—
239	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
240	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
241	C₃×C₄	—	2	8	no	—
242	C₃×C₄	—	2	4	no	—
243	C₃×C₄	—	2	4	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	12	4	no	—
247	C₃×C₄	#936	1	2	yes	—
248	C₃×C₄	C₃×#124	1	2	yes	—
249	C₃×C₄	C₂×#237	1	2	yes	—
250	C₃×C₄	C₂×C₃×#41	1	2	yes	—
251	Dic₃	—	4	4	no	—
252	C₃²	C₂×#490	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	2	4	no	—
254	C₂³	—	4	8	no	—
255	C₂³	—	4	4	no	—
256	C₂³	C₂×C₃×#67	1	1	yes	characteristic
257	D₄	—	2	8	no	—
258	D₄	—	4	16	no	—
259	D₄	—	4	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	6	4	no	—
262	C₂×C₄	—	12	4	no	—
263	C₂×C₄	C₃×#237	1	4	yes	—
264	C₂×C₄	C₂×C₃×#67	1	4	yes	—
265	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
266	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
267	C₈	—	6	4	no	—
268	C₂×C₃	—	2	2	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	4	no	—
271	C₂×C₃	—	8	8	no	—
272	C₂×C₃	—	8	4	no	—
273	C₂×C₃	—	8	4	no	—
274	C₂×C₃	#5071	1	4	yes	direct product factor
275	C₂×C₃	C₃×#490	1	4	yes	complemented
276	C₂×C₃	C₂×#932	1	2	yes	—
277	C₂×C₃	C₂×#936	1	1	yes	characteristic
278	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
279	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	8	8	no	—
283	C₂²	C₃×#932	1	4	yes	—
284	C₂²	C₃×#936	1	2	yes	—
285	C₂²	C₂²×C₃×#67	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	12	4	no	—
288	C₄	C₃×#936	1	2	yes	—
289	C₄	C₂×C₃×#237	1	2	yes	—
290	C₃	—	2	1	no	—
291	C₃	C₂×#5071	1	1	yes	characteristic, direct product factor
292	C₃	C₂×C₃×#490	1	1	yes	characteristic, complemented
293	C₂	—	8	4	no	—
294	C₂	C₃×#5071	1	4	yes	direct product factor
295	C₂	C₂×C₃×#932	1	2	yes	—
296	C₂	C₂×C₃×#936	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting, prime radical	C₂×C₃²×#127	C₂
socle	C₂³×C₃²	C₂×D₄
hypercenter	C₂×C₃×#127	D₃
hypocenter	C₃	C₂×C₃×#490
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#127

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#870	C₂	semidirect
3. C₂×C₃×#832	C₂	semidirect
4. C₃×#5071	C₂	direct (N × H)
5. C₂×C₃×#803	C₂	semidirect
7. C₂×#5071	C₃	direct (N × H)
17. C₃×#870	C₂²	semidirect
18. C₃×#832	C₂²	semidirect
19. C₃×#803	C₂²	semidirect
24. C₂×#870	C₂×C₃	semidirect
28. C₂×#832	C₂×C₃	semidirect
29. #5071	C₂×C₃	direct (N × H)
30. C₂×#803	C₂×C₃	semidirect
72. #870	C₂²×C₃	semidirect
75. #832	C₂²×C₃	semidirect
76. #803	C₂²×C₃	semidirect
204. C₂×C₃²	#490	semidirect
252. C₃²	C₂×#490	semidirect
274. C₂×C₃	#5071	direct (N × H)
275. C₂×C₃	C₃×#490	semidirect
291. C₃	C₂×#5071	direct (N × H)
292. C₃	C₂×C₃×#490	semidirect
294. C₂	C₃×#5071	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×C₃²×#127	2
C₂×C₃×#870	2
C₂×C₃×#832	2
C₃×#5071	2
C₂×C₃×#803	2
C₂×C₃×#490	3
C₂×#5071	3

Subgroups — C2×C3×#5071 #11873568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#5071 #11873568