Subgroups — C₂×C₄₇×#1555 #26057426

Order 12032.

1870 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#432	C₂	1	2	yes	maximal, complemented
3	C₄₇×#1555	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₇×#331	C₂	1	4	yes	maximal, complemented
5	C₂³×D₄×C₄₇	—	2	2	no	—
6	C₂³×D₄×C₄₇	C₄	1	2	yes	—
7	C₂×#127×C₄₇	—	2	2	no	—
8	C₂×#127×C₄₇	—	2	4	no	—
9	C₂×#127×C₄₇	C₄	1	4	yes	—
10	C₂×#127×C₄₇	C₄	1	2	yes	—
11	C₂×#127×C₄₇	C₂²	1	2	yes	—
12	C₂×#127×C₄₇	C₂²	1	4	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#432	—	2	4	no	—
15	C₄₇×#432	C₂²	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	complemented
17	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
18	C₂×#97×C₄₇	—	2	2	no	—
19	C₄₇×#331	C₂²	1	16	yes	complemented
20	C₂⁵×C₄₇	—	2	2	no	—
21	C₂²×D₄×C₄₇	—	2	4	no	—
22	C₂²×D₄×C₄₇	—	2	2	no	—
23	C₂²×D₄×C₄₇	—	2	8	no	—
24	C₂²×D₄×C₄₇	—	2	4	no	—
25	C₂²×D₄×C₄₇	—	2	8	no	—
26	C₂²×D₄×C₄₇	—	2	4	no	—
27	C₂²×D₄×C₄₇	—	4	16	no	—
28	C₂²×D₄×C₄₇	—	4	8	no	—
29	C₂²×D₄×C₄₇	C₂×C₄	1	8	yes	—
30	C₂²×D₄×C₄₇	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	D₄	1	2	yes	—
33	C₂²×C₈×C₄₇	—	2	4	no	—
34	#127×C₄₇	—	2	4	no	—
35	#127×C₄₇	—	2	8	no	—
36	#127×C₄₇	—	2	4	no	—
37	#127×C₄₇	—	2	8	no	—
38	#127×C₄₇	C₂×C₄	1	4	yes	—
39	#127×C₄₇	C₂×C₄	1	8	yes	—
40	#127×C₄₇	C₂³	1	4	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	D₄	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
47	C₄⋊C₈×C₄₇	—	2	4	no	—
48	C₄⋊C₈×C₄₇	C₂³	1	8	yes	complemented
49	C₂²⋊C₈×C₄₇	—	2	8	no	—
50	#97×C₄₇	—	2	8	no	—
51	C₂⁴×C₄₇	—	2	8	no	—
52	C₂⁴×C₄₇	—	2	4	no	—
53	C₂⁴×C₄₇	—	4	16	no	—
54	C₂⁴×C₄₇	—	4	8	no	—
55	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×D₄×C₄₇	—	2	4	no	—
57	C₂×D₄×C₄₇	—	2	8	no	—
58	C₂×D₄×C₄₇	—	2	16	no	—
59	C₂×D₄×C₄₇	—	2	8	no	—
60	C₂×D₄×C₄₇	—	2	8	no	—
61	C₂×D₄×C₄₇	—	4	16	no	—
62	C₂×D₄×C₄₇	—	4	8	no	—
63	C₂×D₄×C₄₇	—	4	16	no	—
64	C₂×D₄×C₄₇	—	4	32	no	—
65	C₂×D₄×C₄₇	—	4	32	no	—
66	C₂×D₄×C₄₇	—	4	16	no	—
67	C₂×D₄×C₄₇	—	4	8	no	—
68	C₂×D₄×C₄₇	—	4	16	no	—
69	C₂×D₄×C₄₇	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	—	2	8	no	—
72	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₄₇	—	2	8	no	—
81	C₂×C₈×C₄₇	—	4	8	no	—
82	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
84	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
85	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₄₇	C₂×D₄	1	2	yes	—
87	C₄²×C₄₇	C₂×D₄	1	4	yes	—
88	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	—	2	8	no	—
91	C₂³×C₄₇	—	4	32	no	—
92	C₂³×C₄₇	—	4	16	no	—
93	C₂³×C₄₇	—	4	8	no	—
94	C₂³×C₄₇	—	4	16	no	—
95	C₂³×C₄₇	—	8	32	no	—
96	C₂³×C₄₇	#100	1	4	yes	—
97	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	D₄×C₄₇	—	4	8	no	—
100	D₄×C₄₇	—	4	8	no	—
101	D₄×C₄₇	—	4	16	no	—
102	D₄×C₄₇	—	4	32	no	—
103	D₄×C₄₇	—	4	16	no	—
104	D₄×C₄₇	—	4	16	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	—	2	8	no	—
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	8	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
119	C₈×C₄₇	—	4	8	no	—
120	C₂×#1555	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₄₇	—	2	8	no	—
122	C₂²×C₄₇	—	2	4	no	—
123	C₂²×C₄₇	—	4	8	no	—
124	C₂²×C₄₇	—	4	16	no	—
125	C₂²×C₄₇	—	8	32	no	—
126	C₂²×C₄₇	—	8	16	no	—
127	C₂²×C₄₇	C₂×#100	1	4	yes	—
128	C₂²×C₄₇	C₂×#100	1	2	yes	—
129	C₂²×C₄₇	#418	1	4	yes	—
130	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₄₇	—	2	1	no	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	—	2	4	no	—
135	C₄×C₄₇	#418	1	4	yes	—
136	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
137	C₂²×#127	C₂×C₄₇	1	1	yes	characteristic
138	C₂×#432	C₂×C₄₇	1	2	yes	complemented
139	#1555	C₂×C₄₇	1	8	yes	direct product factor
140	C₂×#331	C₂×C₄₇	1	4	yes	complemented
141	C₂×C₄₇	—	2	4	no	—
142	C₂×C₄₇	—	8	8	no	—
143	C₂×C₄₇	#1555	1	4	yes	direct product factor
144	C₂×C₄₇	C₂²×#100	1	1	yes	characteristic
145	C₂×C₄₇	C₂×#418	1	2	yes	—
146	C₂³×D₄	—	2	2	no	—
147	C₂³×D₄	C₄×C₄₇	1	2	yes	—
148	C₂×#127	—	2	2	no	—
149	C₂×#127	—	2	4	no	—
150	C₂×#127	C₄×C₄₇	1	4	yes	—
151	C₂×#127	C₄×C₄₇	1	2	yes	—
152	C₂×#127	C₂²×C₄₇	1	2	yes	—
153	C₂×#127	C₂²×C₄₇	1	4	yes	—
154	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
155	#432	—	2	4	no	—
156	#432	C₂²×C₄₇	1	8	yes	complemented
157	C₂×C₄⋊C₈	C₂²×C₄₇	1	4	yes	complemented
158	C₂×C₂²⋊C₈	—	2	2	no	—
159	C₂×#97	—	2	2	no	—
160	#331	C₂²×C₄₇	1	16	yes	complemented
161	C₄₇	C₂×#1555	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁵	—	2	2	no	—
163	C₂²×D₄	—	2	4	no	—
164	C₂²×D₄	—	2	2	no	—
165	C₂²×D₄	—	2	4	no	—
166	C₂²×D₄	—	2	8	no	—
167	C₂²×D₄	—	2	8	no	—
168	C₂²×D₄	—	2	4	no	—
169	C₂²×D₄	—	4	8	no	—
170	C₂²×D₄	—	4	16	no	—
171	C₂²×D₄	C₂×C₄×C₄₇	1	4	yes	—
172	C₂²×D₄	C₂×C₄×C₄₇	1	8	yes	—
173	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₄₇	1	2	yes	—
175	C₂²×C₈	—	2	4	no	—
176	#127	—	2	4	no	—
177	#127	—	2	8	no	—
178	#127	—	2	8	no	—
179	#127	—	2	4	no	—
180	#127	C₂×C₄×C₄₇	1	4	yes	—
181	#127	C₂×C₄×C₄₇	1	8	yes	—
182	#127	C₂³×C₄₇	1	4	yes	—
183	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
184	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
185	C₂×C₄²	D₄×C₄₇	1	2	yes	—
186	C₂×C₄²	D₄×C₄₇	1	4	yes	—
187	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
189	C₄⋊C₈	—	2	4	no	—
190	C₄⋊C₈	C₂³×C₄₇	1	8	yes	complemented
191	C₂²⋊C₈	—	2	8	no	—
192	#97	—	2	8	no	—
193	C₂⁴	—	2	8	no	—
194	C₂⁴	—	2	4	no	—
195	C₂⁴	—	4	16	no	—
196	C₂⁴	—	4	8	no	—
197	C₂⁴	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
198	C₂×D₄	—	2	8	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	16	no	—
201	C₂×D₄	—	2	8	no	—
202	C₂×D₄	—	2	8	no	—
203	C₂×D₄	—	4	16	no	—
204	C₂×D₄	—	4	8	no	—
205	C₂×D₄	—	4	16	no	—
206	C₂×D₄	—	4	8	no	—
207	C₂×D₄	—	4	16	no	—
208	C₂×D₄	—	4	32	no	—
209	C₂×D₄	—	4	32	no	—
210	C₂×D₄	—	4	16	no	—
211	C₂×D₄	C₂²×C₄×C₄₇	1	8	yes	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
215	C₂²×C₄	C₂²⋊C₄×C₄₇	1	4	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
221	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	4	8	no	—
224	C₄²	C₂²⋊C₄×C₄₇	1	2	yes	—
225	C₄²	C₂²⋊C₄×C₄₇	1	4	yes	—
226	C₄²	C₂²×C₄×C₄₇	1	2	yes	—
227	C₄²	C₂²×C₄×C₄₇	1	1	yes	characteristic
228	C₄²	C₂×D₄×C₄₇	1	4	yes	—
229	C₄²	C₂×D₄×C₄₇	1	2	yes	—
230	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
231	C₂³	—	2	4	no	—
232	C₂³	—	2	8	no	—
233	C₂³	—	4	32	no	—
234	C₂³	—	4	16	no	—
235	C₂³	—	4	8	no	—
236	C₂³	—	4	16	no	—
237	C₂³	—	8	32	no	—
238	C₂³	#100×C₄₇	1	4	yes	—
239	C₂³	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
240	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
241	D₄	—	4	8	no	—
242	D₄	—	4	8	no	—
243	D₄	—	4	16	no	—
244	D₄	—	4	32	no	—
245	D₄	—	4	16	no	—
246	D₄	—	4	16	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
258	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	8	yes	—
259	C₂×C₄	C₂³×C₄×C₄₇	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
261	C₈	—	4	8	no	—
262	C₂²	—	2	8	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	8	no	—
265	C₂²	—	4	16	no	—
266	C₂²	—	8	32	no	—
267	C₂²	—	8	16	no	—
268	C₂²	C₂×#100×C₄₇	1	4	yes	—
269	C₂²	C₂×#100×C₄₇	1	2	yes	—
270	C₂²	C₄₇×#418	1	4	yes	—
271	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	4	no	—
276	C₄	C₄₇×#418	1	4	yes	—
277	C₄	C₂×AGL(1,ℤ/8)×C₄₇	1	4	yes	—
278	C₂	—	2	4	no	—
279	C₂	—	8	8	no	—
280	C₂	C₄₇×#1555	1	4	yes	direct product factor
281	C₂	C₂²×#100×C₄₇	1	1	yes	characteristic
282	C₂	C₂×C₄₇×#418	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₄₇
Frattini, cosocle	C₄²	C₂⁴×C₄₇
prime radical	C₂²×#127×C₄₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂²×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇×#432	C₂	semidirect
3. C₄₇×#1555	C₂	direct (N × H)
4. C₂×C₄₇×#331	C₂	semidirect
15. C₄₇×#432	C₂²	semidirect
16. C₂×C₄⋊C₈×C₄₇	C₂²	semidirect
19. C₄₇×#331	C₂²	semidirect
48. C₄⋊C₈×C₄₇	C₂³	semidirect
120. C₂×#1555	C₄₇	direct (N × H)
138. C₂×#432	C₂×C₄₇	semidirect
139. #1555	C₂×C₄₇	direct (N × H)
140. C₂×#331	C₂×C₄₇	semidirect
143. C₂×C₄₇	#1555	direct (N × H)
156. #432	C₂²×C₄₇	semidirect
157. C₂×C₄⋊C₈	C₂²×C₄₇	semidirect
160. #331	C₂²×C₄₇	semidirect
161. C₄₇	C₂×#1555	direct (N × H)
190. C₄⋊C₈	C₂³×C₄₇	semidirect
280. C₂	C₄₇×#1555	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#127×C₄₇	2
C₂×C₄₇×#432	2
C₄₇×#1555	2
C₂×C₄₇×#331	2
C₂×#1555	47

Subgroups — C2×C47×#1555 #26057426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#1555 #26057426