Subgroups — C₄₇×#32681 #26072379

Order 12032.

914 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2548	C₂	1	2	yes	maximal, complemented
2	C₄₇×#2054	C₂	1	2	yes	maximal, complemented
3	C₄₇×#2011	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1972	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1926	C₂	1	4	yes	maximal, complemented
7	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
8	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
9	C₂×#127×C₄₇	C₂²	1	2	yes	—
10	C₂×#124×C₄₇	—	2	2	no	—
11	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#106×C₄₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#102×C₄₇	C₂²	1	4	yes	—
15	C₂×#102×C₄₇	C₂²	1	4	yes	—
16	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#394	—	2	2	no	—
18	C₄₇×#394	—	2	2	no	—
19	C₄₇×#390	—	2	2	no	—
20	C₄₇×#390	C₂²	1	4	yes	—
21	C₄₇×#390	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₄₇×#340	C₂²	1	2	yes	—
24	C₄₇×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₄₇	—	2	2	no	—
27	C₂²×D₄×C₄₇	—	2	2	no	—
28	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
29	C₂×D₈×C₄₇	—	2	4	no	—
30	C₂×D₈×C₄₇	—	4	2	no	—
31	C₂×D₈×C₄₇	—	4	2	no	—
32	C₂×D₈×C₄₇	—	4	2	no	—
33	C₂×D₈×C₄₇	—	4	4	no	—
34	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
36	#127×C₄₇	—	4	2	no	—
37	#127×C₄₇	—	4	2	no	—
38	#124×C₄₇	—	4	8	no	—
39	C₄×Q₈×C₄₇	—	4	1	no	—
40	C₄×Q₈×C₄₇	—	4	1	no	—
41	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
45	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
46	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
48	C₂×C₄²×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	#106×C₄₇	—	4	2	no	—
52	C₄⋊C₈×C₄₇	—	4	1	no	—
53	#102×C₄₇	—	2	8	no	—
54	#102×C₄₇	—	4	4	no	—
55	#97×C₄₇	—	4	1	no	—
56	#95×C₄₇	—	2	2	no	—
57	C₂⁴×C₄₇	—	2	4	no	—
58	C₂×Q₈×C₄₇	—	2	4	no	—
59	C₂×Q₈×C₄₇	—	4	1	no	—
60	C₂×D₄×C₄₇	—	2	4	no	—
61	C₂×D₄×C₄₇	—	2	4	no	—
62	C₂×D₄×C₄₇	—	2	4	no	—
63	C₂×D₄×C₄₇	—	2	4	no	—
64	C₂×D₄×C₄₇	—	2	4	no	—
65	C₂×D₄×C₄₇	—	2	4	no	—
66	C₂×D₄×C₄₇	—	4	4	no	—
67	C₂×D₄×C₄₇	—	4	4	no	—
68	C₂×D₄×C₄₇	—	4	4	no	—
69	C₂×D₄×C₄₇	—	4	4	no	—
70	C₂×D₄×C₄₇	—	8	4	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
77	D₈×C₄₇	—	4	8	no	—
78	D₈×C₄₇	—	4	8	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	#41	1	4	yes	—
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	4	2	no	—
87	C₄⋊C₄×C₄₇	—	4	1	no	—
88	C₂²⋊C₄×C₄₇	—	4	8	no	—
89	C₂²⋊C₄×C₄₇	—	4	8	no	—
90	C₂²⋊C₄×C₄₇	—	8	4	no	—
91	C₂²⋊C₄×C₄₇	—	8	4	no	—
92	C₄²×C₄₇	—	2	1	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	—	2	1	no	—
96	C₄²×C₄₇	—	4	4	no	—
97	C₂³×C₄₇	—	4	4	no	—
98	C₂³×C₄₇	—	4	4	no	—
99	C₂³×C₄₇	—	4	4	no	—
100	C₂³×C₄₇	—	4	4	no	—
101	C₂³×C₄₇	—	4	4	no	—
102	C₂³×C₄₇	—	4	4	no	—
103	C₂³×C₄₇	—	4	4	no	—
104	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₄₇	—	4	4	no	—
106	D₄×C₄₇	—	4	4	no	—
107	D₄×C₄₇	—	4	4	no	—
108	D₄×C₄₇	—	4	4	no	—
109	D₄×C₄₇	—	4	4	no	—
110	D₄×C₄₇	—	8	8	no	—
111	D₄×C₄₇	—	8	8	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	—	4	4	no	—
122	C₂×C₄×C₄₇	—	4	4	no	—
123	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
124	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
125	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
128	C₈×C₄₇	—	2	4	no	—
129	C₈×C₄₇	—	4	2	no	—
130	#32681	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₄₇	—	8	4	no	—
132	C₂²×C₄₇	—	8	4	no	—
133	C₂²×C₄₇	—	8	4	no	—
134	C₂²×C₄₇	—	8	4	no	—
135	C₂²×C₄₇	—	8	4	no	—
136	C₂²×C₄₇	—	8	4	no	—
137	C₂²×C₄₇	—	8	4	no	—
138	C₂²×C₄₇	#534	1	1	yes	characteristic
139	C₂²×C₄₇	#550	1	2	yes	—
140	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
141	C₂²×C₄₇	#576	1	2	yes	—
142	C₄×C₄₇	—	2	4	no	—
143	C₄×C₄₇	—	4	1	no	—
144	C₄×C₄₇	—	4	1	no	—
145	C₄×C₄₇	—	4	2	no	—
146	C₄×C₄₇	—	8	4	no	—
147	C₄×C₄₇	#538	1	2	yes	—
148	C₄×C₄₇	#555	1	1	yes	characteristic
149	C₄×C₄₇	H(3,𝔽₄)	1	1	yes	characteristic
150	#2548	C₂×C₄₇	1	2	yes	complemented
151	#2054	C₂×C₄₇	1	2	yes	complemented
152	#2011	C₂×C₄₇	1	4	yes	complemented
153	#1985	C₂×C₄₇	1	1	yes	characteristic, complemented
154	#1972	C₂×C₄₇	1	2	yes	complemented
155	#1926	C₂×C₄₇	1	4	yes	complemented
156	C₂×C₄₇	—	16	4	no	—
157	C₂×C₄₇	#2733	1	1	yes	characteristic
158	C₂×C₄₇	#3192	1	1	yes	characteristic
159	C₂×C₄₇	#3199	1	1	yes	characteristic
160	C₂×C₄₇	#3399	1	4	yes	—
161	C₂²×D₈	C₂²×C₄₇	1	2	yes	—
162	C₂²×D₈	C₂²×C₄₇	1	2	yes	—
163	C₂×#127	C₂²×C₄₇	1	2	yes	—
164	C₂×#124	—	2	2	no	—
165	C₂×C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
166	C₂×#106	C₂²×C₄₇	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
168	C₂×#102	C₂²×C₄₇	1	4	yes	—
169	C₂×#102	C₂²×C₄₇	1	4	yes	—
170	C₂×#97	C₂²×C₄₇	1	1	yes	characteristic
171	#394	—	2	2	no	—
172	#394	—	2	2	no	—
173	#390	—	2	2	no	—
174	#390	C₂²×C₄₇	1	4	yes	—
175	#390	C₂²×C₄₇	1	4	yes	—
176	C₄×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
177	#340	C₂²×C₄₇	1	2	yes	—
178	#336	C₂²×C₄₇	1	4	yes	—
179	C₄₇	#32681	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²×Q₈	D₄×C₄₇	1	1	yes	characteristic
181	C₂²×D₄	—	2	2	no	—
182	C₂²×D₄	—	2	2	no	—
183	C₂²×D₄	C₂³×C₄₇	1	4	yes	—
184	C₂×D₈	—	2	4	no	—
185	C₂×D₈	—	4	2	no	—
186	C₂×D₈	—	4	2	no	—
187	C₂×D₈	—	4	4	no	—
188	C₂×D₈	—	4	2	no	—
189	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
190	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
191	#127	—	4	2	no	—
192	#127	—	4	2	no	—
193	#124	—	4	8	no	—
194	C₄×Q₈	—	4	1	no	—
195	C₄×Q₈	—	4	1	no	—
196	C₂×C₄⋊C₄	D₄×C₄₇	1	2	yes	—
197	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
198	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
199	C₂×C₂²⋊C₄	—	2	4	no	—
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₂²⋊C₄	—	2	4	no	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
205	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
206	#106	—	4	2	no	—
207	C₄⋊C₈	—	4	1	no	—
208	#102	—	2	8	no	—
209	#102	—	4	4	no	—
210	#97	—	4	1	no	—
211	#95	—	2	2	no	—
212	C₂⁴	—	2	4	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	—	4	4	no	—
223	C₂×D₄	—	4	4	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	—	8	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
231	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
232	D₈	—	4	8	no	—
233	D₈	—	4	8	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	#41×C₄₇	1	4	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₂²⋊C₄	—	4	8	no	—
244	C₂²⋊C₄	—	4	8	no	—
245	C₂²⋊C₄	—	8	4	no	—
246	C₂²⋊C₄	—	8	4	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	4	4	no	—
252	C₂³	—	4	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	4	4	no	—
255	C₂³	—	4	4	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	4	4	no	—
258	C₂³	—	4	4	no	—
259	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
260	Q₈	—	4	4	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	4	no	—
263	D₄	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	8	8	no	—
266	D₄	—	8	8	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	4	yes	—
279	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
280	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
281	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
282	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	4	2	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	8	4	no	—
288	C₂²	—	8	4	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	4	no	—
291	C₂²	—	8	4	no	—
292	C₂²	C₄₇×#534	1	1	yes	characteristic
293	C₂²	C₄₇×#550	1	2	yes	—
294	C₂²	C₂×AGL(1,ℤ/8)×C₄₇	1	2	yes	—
295	C₂²	C₄₇×#576	1	2	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	8	4	no	—
301	C₄	C₄₇×#538	1	2	yes	—
302	C₄	C₄₇×#555	1	1	yes	characteristic
303	C₄	C₄₇×H(3,𝔽₄)	1	1	yes	characteristic
304	C₂	—	16	4	no	—
305	C₂	C₄₇×#2733	1	1	yes	characteristic
306	C₂	C₄₇×#3192	1	1	yes	characteristic
307	C₂	C₄₇×#3199	1	1	yes	characteristic
308	C₂	C₄₇×#3399	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂×C₄×Q₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#2548	C₂	semidirect
2. C₄₇×#2054	C₂	semidirect
3. C₄₇×#2011	C₂	semidirect
4. C₄₇×#1985	C₂	semidirect
5. C₄₇×#1972	C₂	semidirect
6. C₄₇×#1926	C₂	semidirect
130. #32681	C₄₇	direct (N × H)
150. #2548	C₂×C₄₇	semidirect
151. #2054	C₂×C₄₇	semidirect
152. #2011	C₂×C₄₇	semidirect
153. #1985	C₂×C₄₇	semidirect
154. #1972	C₂×C₄₇	semidirect
155. #1926	C₂×C₄₇	semidirect
179. C₄₇	#32681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#2548	2
C₄₇×#2054	2
C₄₇×#2011	2
C₄₇×#1985	2
C₄₇×#1972	2
C₄₇×#1926	2
#32681	47

Subgroups — C47×#32681 #26072379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#32681 #26072379