Subgroups — C₂×C₄₇×#3586 #26102787

Order 12032.

10794 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#585	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×2₋¹⁺⁴×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#3586	C₂	1	32	yes	maximal, direct product factor
4	C₂×C₄₇×#578	C₂	1	8	yes	maximal, complemented
5	C₂×C₄₇×#577	C₂	1	8	yes	maximal, complemented
6	C₂×C₄₇×#575	C₂	1	6	yes	maximal, complemented
7	C₂²×#137×C₄₇	C₂	1	6	yes	maximal, complemented
8	C₂×C₄₇×#568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#585	—	2	8	no	—
10	C₄₇×#585	C₂²	1	16	yes	complemented
11	C₂×2₋¹⁺⁴×C₄₇	—	2	4	no	—
12	C₂×2₋¹⁺⁴×C₄₇	—	2	8	no	—
13	C₂×2₋¹⁺⁴×C₄₇	—	2	1	no	—
14	C₂×2₋¹⁺⁴×C₄₇	C₂²	1	8	yes	complemented
15	C₂×2₋¹⁺⁴×C₄₇	C₂²	1	4	yes	complemented
16	C₂×2₋¹⁺⁴×C₄₇	C₂²	1	16	yes	complemented
17	C₂×H(5,𝔽₂)×C₄₇	—	2	3	no	—
18	C₂×H(5,𝔽₂)×C₄₇	C₂²	1	4	yes	complemented
19	C₂²×#41×C₄₇	—	2	4	no	—
20	C₂²×#41×C₄₇	—	2	3	no	—
21	C₂²×#41×C₄₇	C₂²	1	6	yes	—
22	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×#41×C₄₇	C₂²	1	3	yes	complemented
24	C₂³×Q₈×C₄₇	—	2	1	no	—
25	C₂³×Q₈×C₄₇	C₂²	1	3	yes	complemented
26	C₄₇×#578	C₂²	1	128	yes	complemented
27	C₄₇×#577	C₂²	1	128	yes	complemented
28	C₄₇×#575	C₂²	1	96	yes	complemented
29	C₂×#137×C₄₇	C₂²	1	12	yes	complemented
30	C₂×#137×C₄₇	C₂²	1	24	yes	complemented
31	C₂×#137×C₄₇	C₂²	1	12	yes	complemented
32	C₂×#137×C₄₇	C₂²	1	96	yes	complemented
33	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	4	yes	complemented
34	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	12	yes	complemented
35	C₂×#135×C₄₇	C₂²	1	24	yes	complemented
36	C₂²×Q₁₆×C₄₇	C₂²	1	12	yes	complemented
37	C₂²×SD₁₆×C₄₇	C₂²	1	12	yes	complemented
38	C₄₇×#568	C₂²	1	16	yes	complemented
39	C₂×#131×C₄₇	C₂²	1	8	yes	complemented
40	C₂²×M₁₆×C₄₇	C₂²	1	6	yes	complemented
41	2₋¹⁺⁴×C₄₇	—	2	32	no	—
42	2₋¹⁺⁴×C₄₇	—	2	64	no	—
43	2₋¹⁺⁴×C₄₇	—	2	16	no	—
44	2₋¹⁺⁴×C₄₇	—	2	16	no	—
45	2₋¹⁺⁴×C₄₇	C₂³	1	64	yes	complemented
46	2₋¹⁺⁴×C₄₇	C₂³	1	32	yes	complemented
47	H(5,𝔽₂)×C₄₇	—	2	16	no	—
48	H(5,𝔽₂)×C₄₇	—	2	48	no	—
49	H(5,𝔽₂)×C₄₇	C₂³	1	32	yes	complemented
50	C₂×#41×C₄₇	—	2	64	no	—
51	C₂×#41×C₄₇	—	2	4	no	—
52	C₂×#41×C₄₇	—	2	12	no	—
53	C₂×#41×C₄₇	—	2	24	no	—
54	C₂×#41×C₄₇	—	2	48	no	—
55	C₂×#41×C₄₇	—	2	6	no	—
56	C₂×#41×C₄₇	—	2	6	no	—
57	C₂×#41×C₄₇	—	2	24	no	—
58	C₂×#41×C₄₇	—	2	6	no	—
59	C₂×#41×C₄₇	—	2	12	no	—
60	C₂×#41×C₄₇	D₄	1	4	yes	—
61	C₂×#41×C₄₇	D₄	1	8	yes	—
62	C₂×#41×C₄₇	C₂³	1	12	yes	—
63	C₂×#41×C₄₇	C₂³	1	12	yes	—
64	C₂×#41×C₄₇	C₂³	1	4	yes	—
65	C₂×#41×C₄₇	C₂³	1	48	yes	—
66	C₂×#41×C₄₇	C₂³	1	4	yes	—
67	C₂×#41×C₄₇	C₂³	1	8	yes	—
68	C₂×#41×C₄₇	C₂³	1	24	yes	complemented
69	C₂²×Q₈×C₄₇	—	2	8	no	—
70	C₂²×Q₈×C₄₇	—	2	16	no	—
71	C₂²×Q₈×C₄₇	—	2	4	no	—
72	C₂²×Q₈×C₄₇	—	2	3	no	—
73	C₂²×Q₈×C₄₇	—	2	6	no	—
74	C₂²×Q₈×C₄₇	—	2	12	no	—
75	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
76	C₂²×Q₈×C₄₇	C₂³	1	12	yes	—
77	C₂²×Q₈×C₄₇	C₂³	1	6	yes	—
78	C₂²×Q₈×C₄₇	C₂³	1	3	yes	—
79	C₂²×Q₈×C₄₇	C₂³	1	24	yes	complemented
80	C₂²×D₄×C₄₇	—	2	3	no	—
81	C₂²×D₄×C₄₇	—	2	6	no	—
82	C₂²×D₄×C₄₇	—	2	12	no	—
83	C₂²×D₄×C₄₇	D₄	1	3	yes	—
84	C₂²×D₄×C₄₇	C₂³	1	6	yes	—
85	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₄₇	—	2	3	no	—
87	C₂³×C₄×C₄₇	—	2	3	no	—
88	C₂³×C₄×C₄₇	—	2	6	no	—
89	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
90	#137×C₄₇	C₂³	1	96	yes	complemented
91	#137×C₄₇	C₂³	1	192	yes	complemented
92	#137×C₄₇	C₂³	1	96	yes	complemented
93	AGL(1,ℤ/8)×C₄₇	C₂³	1	32	yes	complemented
94	AGL(1,ℤ/8)×C₄₇	C₂³	1	96	yes	complemented
95	#135×C₄₇	C₂³	1	192	yes	complemented
96	C₂×Q₁₆×C₄₇	C₂³	1	24	yes	—
97	C₂×Q₁₆×C₄₇	C₂³	1	96	yes	complemented
98	C₂×SD₁₆×C₄₇	C₂³	1	24	yes	—
99	C₂×SD₁₆×C₄₇	C₂³	1	8	yes	—
100	C₂×SD₁₆×C₄₇	C₂³	1	96	yes	complemented
101	C₂×D₈×C₄₇	C₂³	1	8	yes	—
102	#131×C₄₇	C₂³	1	64	yes	complemented
103	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
104	C₂×M₁₆×C₄₇	C₂³	1	12	yes	—
105	C₂×M₁₆×C₄₇	C₂³	1	48	yes	complemented
106	C₂²×C₈×C₄₇	C₂³	1	12	yes	—
107	C₂⁴×C₄₇	—	2	1	no	—
108	C₂⁴×C₄₇	—	2	6	no	—
109	C₂⁴×C₄₇	C₂×D₄	1	3	yes	—
110	#41×C₄₇	—	2	32	no	—
111	#41×C₄₇	—	2	96	no	—
112	#41×C₄₇	—	2	192	no	—
113	#41×C₄₇	—	2	48	no	—
114	#41×C₄₇	—	2	8	no	—
115	#41×C₄₇	—	2	48	no	—
116	#41×C₄₇	—	2	24	no	—
117	#41×C₄₇	—	2	24	no	—
118	#41×C₄₇	—	2	48	no	—
119	#41×C₄₇	C₂×D₄	1	32	yes	—
120	#41×C₄₇	C₂×D₄	1	16	yes	—
121	#41×C₄₇	C₂⁴	1	48	yes	—
122	#41×C₄₇	C₂⁴	1	48	yes	—
123	#41×C₄₇	C₂⁴	1	16	yes	—
124	#41×C₄₇	C₂⁴	1	16	yes	—
125	C₂×Q₈×C₄₇	—	2	64	no	—
126	C₂×Q₈×C₄₇	—	2	32	no	—
127	C₂×Q₈×C₄₇	—	2	6	no	—
128	C₂×Q₈×C₄₇	—	2	3	no	—
129	C₂×Q₈×C₄₇	—	2	12	no	—
130	C₂×Q₈×C₄₇	—	2	24	no	—
131	C₂×Q₈×C₄₇	—	2	6	no	—
132	C₂×Q₈×C₄₇	—	2	24	no	—
133	C₂×Q₈×C₄₇	—	2	48	no	—
134	C₂×Q₈×C₄₇	—	2	12	no	—
135	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
136	C₂×Q₈×C₄₇	C₂×D₄	1	8	yes	—
137	C₂×Q₈×C₄₇	C₂⁴	1	6	yes	—
138	C₂×Q₈×C₄₇	C₂⁴	1	4	yes	—
139	C₂×Q₈×C₄₇	C₂⁴	1	48	yes	—
140	C₂×Q₈×C₄₇	C₂⁴	1	24	yes	—
141	C₂×Q₈×C₄₇	C₂⁴	1	12	yes	—
142	C₂×D₄×C₄₇	—	2	4	no	—
143	C₂×D₄×C₄₇	—	2	12	no	—
144	C₂×D₄×C₄₇	—	2	12	no	—
145	C₂×D₄×C₄₇	—	2	24	no	—
146	C₂×D₄×C₄₇	—	2	48	no	—
147	C₂×D₄×C₄₇	—	2	96	no	—
148	C₂×D₄×C₄₇	—	2	6	no	—
149	C₂×D₄×C₄₇	—	2	2	no	—
150	C₂×D₄×C₄₇	—	2	1	no	—
151	C₂×D₄×C₄₇	—	2	12	no	—
152	C₂×D₄×C₄₇	C₂×D₄	1	12	yes	—
153	C₂×D₄×C₄₇	C₂×D₄	1	24	yes	—
154	C₂×D₄×C₄₇	C₂⁴	1	2	yes	—
155	C₂×D₄×C₄₇	C₂⁴	1	4	yes	—
156	C₂×D₄×C₄₇	C₂⁴	1	24	yes	—
157	C₂×D₄×C₄₇	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₄₇	—	2	3	no	—
159	C₂²×C₄×C₄₇	—	2	6	no	—
160	C₂²×C₄×C₄₇	—	2	12	no	—
161	C₂²×C₄×C₄₇	—	2	24	no	—
162	C₂²×C₄×C₄₇	—	2	4	no	—
163	C₂²×C₄×C₄₇	—	2	12	no	—
164	C₂²×C₄×C₄₇	—	2	12	no	—
165	C₂²×C₄×C₄₇	—	2	24	no	—
166	C₂²×C₄×C₄₇	—	2	48	no	—
167	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
168	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
169	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
170	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
171	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
173	Q₁₆×C₄₇	C₂⁴	1	96	yes	—
174	SD₁₆×C₄₇	C₂⁴	1	96	yes	—
175	SD₁₆×C₄₇	C₂⁴	1	32	yes	—
176	D₈×C₄₇	C₂⁴	1	32	yes	—
177	M₁₆×C₄₇	C₂⁴	1	16	yes	—
178	M₁₆×C₄₇	C₂⁴	1	48	yes	—
179	C₂×C₈×C₄₇	C₂⁴	1	8	yes	—
180	C₂×C₈×C₄₇	C₂⁴	1	48	yes	—
181	C₂³×C₄₇	—	2	4	no	—
182	C₂³×C₄₇	—	2	2	no	—
183	C₂³×C₄₇	—	2	4	no	—
184	C₂³×C₄₇	—	2	24	no	—
185	C₂³×C₄₇	—	4	6	no	—
186	C₂³×C₄₇	—	4	4	no	—
187	C₂³×C₄₇	—	8	12	no	—
188	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
189	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
190	C₂³×C₄₇	C₂²×D₄	1	12	yes	—
191	Q₈×C₄₇	—	2	12	no	—
192	Q₈×C₄₇	—	2	48	no	—
193	Q₈×C₄₇	—	2	6	no	—
194	Q₈×C₄₇	—	2	24	no	—
195	Q₈×C₄₇	—	2	4	no	—
196	Q₈×C₄₇	C₂²×D₄	1	16	yes	—
197	Q₈×C₄₇	C₂⁵	1	12	yes	—
198	Q₈×C₄₇	C₂⁵	1	8	yes	—
199	D₄×C₄₇	—	2	16	no	—
200	D₄×C₄₇	—	2	48	no	—
201	D₄×C₄₇	—	2	48	no	—
202	D₄×C₄₇	—	2	24	no	—
203	D₄×C₄₇	—	2	4	no	—
204	D₄×C₄₇	—	2	4	no	—
205	D₄×C₄₇	—	2	2	no	—
206	D₄×C₄₇	C₂²×D₄	1	48	yes	—
207	D₄×C₄₇	C₂⁵	1	4	yes	—
208	D₄×C₄₇	C₂⁵	1	8	yes	—
209	C₂×C₄×C₄₇	—	2	12	no	—
210	C₂×C₄×C₄₇	—	2	24	no	—
211	C₂×C₄×C₄₇	—	2	48	no	—
212	C₂×C₄×C₄₇	—	2	4	no	—
213	C₂×C₄×C₄₇	—	2	6	no	—
214	C₂×C₄×C₄₇	—	2	16	no	—
215	C₂×C₄×C₄₇	—	2	48	no	—
216	C₂×C₄×C₄₇	—	2	48	no	—
217	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
218	C₂×C₄×C₄₇	C₂²×D₄	1	24	yes	—
219	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
221	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
223	C₂×C₄×C₄₇	C₂⁵	1	12	yes	—
224	C₂×C₄×C₄₇	C₂⁵	1	2	yes	—
225	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
226	C₈×C₄₇	C₂⁵	1	16	yes	—
227	C₂×#3586	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂²×C₄₇	—	2	2	no	—
229	C₂²×C₄₇	—	2	12	no	—
230	C₂²×C₄₇	—	2	4	no	—
231	C₂²×C₄₇	—	2	8	no	—
232	C₂²×C₄₇	—	4	12	no	—
233	C₂²×C₄₇	—	4	8	no	—
234	C₂²×C₄₇	—	4	4	no	—
235	C₂²×C₄₇	—	4	8	no	—
236	C₂²×C₄₇	—	8	24	no	—
237	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
238	C₂²×C₄₇	C₂³×D₄	1	12	yes	—
239	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
240	C₄×C₄₇	—	2	8	no	—
241	C₄×C₄₇	—	2	12	no	—
242	C₄×C₄₇	C₂³×D₄	1	2	yes	—
243	C₄×C₄₇	C₂³×D₄	1	12	yes	—
244	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
245	C₄×C₄₇	C₂⁶	1	1	yes	characteristic
246	C₂×#585	C₂×C₄₇	1	1	yes	characteristic, complemented
247	C₂²×2₋¹⁺⁴	C₂×C₄₇	1	1	yes	characteristic, complemented
248	#3586	C₂×C₄₇	1	32	yes	direct product factor
249	C₂×#578	C₂×C₄₇	1	8	yes	complemented
250	C₂×#577	C₂×C₄₇	1	8	yes	complemented
251	C₂×#575	C₂×C₄₇	1	6	yes	complemented
252	C₂²×#137	C₂×C₄₇	1	6	yes	complemented
253	C₂×#568	C₂×C₄₇	1	1	yes	characteristic, complemented
254	C₂×C₄₇	—	2	2	no	—
255	C₂×C₄₇	—	2	12	no	—
256	C₂×C₄₇	—	4	4	no	—
257	C₂×C₄₇	—	4	8	no	—
258	C₂×C₄₇	#3586	1	2	yes	direct product factor
259	C₂×C₄₇	C₂⁴×D₄	1	1	yes	characteristic
260	#585	—	2	8	no	—
261	#585	C₂²×C₄₇	1	16	yes	complemented
262	C₂×2₋¹⁺⁴	—	2	4	no	—
263	C₂×2₋¹⁺⁴	—	2	8	no	—
264	C₂×2₋¹⁺⁴	—	2	1	no	—
265	C₂×2₋¹⁺⁴	C₂²×C₄₇	1	8	yes	complemented
266	C₂×2₋¹⁺⁴	C₂²×C₄₇	1	4	yes	complemented
267	C₂×2₋¹⁺⁴	C₂²×C₄₇	1	16	yes	complemented
268	C₂×H(5,𝔽₂)	—	2	3	no	—
269	C₂×H(5,𝔽₂)	C₂²×C₄₇	1	4	yes	complemented
270	C₂²×#41	—	2	4	no	—
271	C₂²×#41	—	2	3	no	—
272	C₂²×#41	C₂²×C₄₇	1	6	yes	—
273	C₂²×#41	C₂²×C₄₇	1	1	yes	characteristic
274	C₂²×#41	C₂²×C₄₇	1	3	yes	complemented
275	C₂³×Q₈	—	2	1	no	—
276	C₂³×Q₈	C₂²×C₄₇	1	3	yes	complemented
277	#578	C₂²×C₄₇	1	128	yes	complemented
278	#577	C₂²×C₄₇	1	128	yes	complemented
279	#575	C₂²×C₄₇	1	96	yes	complemented
280	C₂×#137	C₂²×C₄₇	1	12	yes	complemented
281	C₂×#137	C₂²×C₄₇	1	24	yes	complemented
282	C₂×#137	C₂²×C₄₇	1	12	yes	complemented
283	C₂×#137	C₂²×C₄₇	1	96	yes	complemented
284	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	4	yes	complemented
285	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	12	yes	complemented
286	C₂×#135	C₂²×C₄₇	1	24	yes	complemented
287	C₂²×Q₁₆	C₂²×C₄₇	1	12	yes	complemented
288	C₂²×SD₁₆	C₂²×C₄₇	1	12	yes	complemented
289	#568	C₂²×C₄₇	1	16	yes	complemented
290	C₂×#131	C₂²×C₄₇	1	8	yes	complemented
291	C₂²×M₁₆	C₂²×C₄₇	1	6	yes	complemented
292	C₄₇	C₂×#3586	1	1	yes	characteristic, direct product factor, Sylow
293	2₋¹⁺⁴	—	2	32	no	—
294	2₋¹⁺⁴	—	2	64	no	—
295	2₋¹⁺⁴	—	2	16	no	—
296	2₋¹⁺⁴	—	2	16	no	—
297	2₋¹⁺⁴	C₂³×C₄₇	1	64	yes	complemented
298	2₋¹⁺⁴	C₂³×C₄₇	1	32	yes	complemented
299	H(5,𝔽₂)	—	2	16	no	—
300	H(5,𝔽₂)	—	2	48	no	—
301	H(5,𝔽₂)	C₂³×C₄₇	1	32	yes	complemented
302	C₂×#41	—	2	64	no	—
303	C₂×#41	—	2	4	no	—
304	C₂×#41	—	2	12	no	—
305	C₂×#41	—	2	24	no	—
306	C₂×#41	—	2	48	no	—
307	C₂×#41	—	2	6	no	—
308	C₂×#41	—	2	6	no	—
309	C₂×#41	—	2	24	no	—
310	C₂×#41	—	2	6	no	—
311	C₂×#41	—	2	12	no	—
312	C₂×#41	D₄×C₄₇	1	4	yes	—
313	C₂×#41	D₄×C₄₇	1	8	yes	—
314	C₂×#41	C₂³×C₄₇	1	12	yes	—
315	C₂×#41	C₂³×C₄₇	1	12	yes	—
316	C₂×#41	C₂³×C₄₇	1	4	yes	—
317	C₂×#41	C₂³×C₄₇	1	48	yes	—
318	C₂×#41	C₂³×C₄₇	1	4	yes	—
319	C₂×#41	C₂³×C₄₇	1	8	yes	—
320	C₂×#41	C₂³×C₄₇	1	24	yes	complemented
321	C₂²×Q₈	—	2	8	no	—
322	C₂²×Q₈	—	2	16	no	—
323	C₂²×Q₈	—	2	4	no	—
324	C₂²×Q₈	—	2	3	no	—
325	C₂²×Q₈	—	2	6	no	—
326	C₂²×Q₈	—	2	12	no	—
327	C₂²×Q₈	D₄×C₄₇	1	1	yes	characteristic
328	C₂²×Q₈	C₂³×C₄₇	1	12	yes	—
329	C₂²×Q₈	C₂³×C₄₇	1	6	yes	—
330	C₂²×Q₈	C₂³×C₄₇	1	3	yes	—
331	C₂²×Q₈	C₂³×C₄₇	1	24	yes	complemented
332	C₂²×D₄	—	2	3	no	—
333	C₂²×D₄	—	2	6	no	—
334	C₂²×D₄	—	2	12	no	—
335	C₂²×D₄	D₄×C₄₇	1	3	yes	—
336	C₂²×D₄	C₂³×C₄₇	1	6	yes	—
337	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
338	C₂³×C₄	—	2	3	no	—
339	C₂³×C₄	—	2	3	no	—
340	C₂³×C₄	—	2	6	no	—
341	C₂³×C₄	C₂³×C₄₇	1	3	yes	—
342	#137	C₂³×C₄₇	1	96	yes	complemented
343	#137	C₂³×C₄₇	1	192	yes	complemented
344	#137	C₂³×C₄₇	1	96	yes	complemented
345	AGL(1,ℤ/8)	C₂³×C₄₇	1	32	yes	complemented
346	AGL(1,ℤ/8)	C₂³×C₄₇	1	96	yes	complemented
347	#135	C₂³×C₄₇	1	192	yes	complemented
348	C₂×Q₁₆	C₂³×C₄₇	1	24	yes	—
349	C₂×Q₁₆	C₂³×C₄₇	1	96	yes	complemented
350	C₂×SD₁₆	C₂³×C₄₇	1	24	yes	—
351	C₂×SD₁₆	C₂³×C₄₇	1	8	yes	—
352	C₂×SD₁₆	C₂³×C₄₇	1	96	yes	complemented
353	C₂×D₈	C₂³×C₄₇	1	8	yes	—
354	#131	C₂³×C₄₇	1	64	yes	complemented
355	C₂×M₁₆	C₂³×C₄₇	1	4	yes	—
356	C₂×M₁₆	C₂³×C₄₇	1	12	yes	—
357	C₂×M₁₆	C₂³×C₄₇	1	48	yes	complemented
358	C₂²×C₈	C₂³×C₄₇	1	12	yes	—
359	C₂⁴	—	2	1	no	—
360	C₂⁴	—	2	6	no	—
361	C₂⁴	C₂×D₄×C₄₇	1	3	yes	—
362	#41	—	2	32	no	—
363	#41	—	2	96	no	—
364	#41	—	2	192	no	—
365	#41	—	2	48	no	—
366	#41	—	2	8	no	—
367	#41	—	2	48	no	—
368	#41	—	2	24	no	—
369	#41	—	2	24	no	—
370	#41	—	2	48	no	—
371	#41	C₂×D₄×C₄₇	1	32	yes	—
372	#41	C₂×D₄×C₄₇	1	16	yes	—
373	#41	C₂⁴×C₄₇	1	48	yes	—
374	#41	C₂⁴×C₄₇	1	48	yes	—
375	#41	C₂⁴×C₄₇	1	16	yes	—
376	#41	C₂⁴×C₄₇	1	16	yes	—
377	C₂×Q₈	—	2	64	no	—
378	C₂×Q₈	—	2	32	no	—
379	C₂×Q₈	—	2	3	no	—
380	C₂×Q₈	—	2	12	no	—
381	C₂×Q₈	—	2	6	no	—
382	C₂×Q₈	—	2	24	no	—
383	C₂×Q₈	—	2	6	no	—
384	C₂×Q₈	—	2	24	no	—
385	C₂×Q₈	—	2	48	no	—
386	C₂×Q₈	—	2	12	no	—
387	C₂×Q₈	C₂×D₄×C₄₇	1	4	yes	—
388	C₂×Q₈	C₂×D₄×C₄₇	1	8	yes	—
389	C₂×Q₈	C₂⁴×C₄₇	1	6	yes	—
390	C₂×Q₈	C₂⁴×C₄₇	1	4	yes	—
391	C₂×Q₈	C₂⁴×C₄₇	1	48	yes	—
392	C₂×Q₈	C₂⁴×C₄₇	1	24	yes	—
393	C₂×Q₈	C₂⁴×C₄₇	1	12	yes	—
394	C₂×D₄	—	2	4	no	—
395	C₂×D₄	—	2	12	no	—
396	C₂×D₄	—	2	12	no	—
397	C₂×D₄	—	2	24	no	—
398	C₂×D₄	—	2	48	no	—
399	C₂×D₄	—	2	96	no	—
400	C₂×D₄	—	2	6	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	12	no	—
404	C₂×D₄	C₂×D₄×C₄₇	1	12	yes	—
405	C₂×D₄	C₂×D₄×C₄₇	1	24	yes	—
406	C₂×D₄	C₂⁴×C₄₇	1	2	yes	—
407	C₂×D₄	C₂⁴×C₄₇	1	4	yes	—
408	C₂×D₄	C₂⁴×C₄₇	1	24	yes	—
409	C₂×D₄	C₂⁴×C₄₇	1	4	yes	—
410	C₂²×C₄	—	2	3	no	—
411	C₂²×C₄	—	2	6	no	—
412	C₂²×C₄	—	2	12	no	—
413	C₂²×C₄	—	2	24	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	12	no	—
416	C₂²×C₄	—	2	12	no	—
417	C₂²×C₄	—	2	24	no	—
418	C₂²×C₄	—	2	48	no	—
419	C₂²×C₄	C₂×D₄×C₄₇	1	3	yes	—
420	C₂²×C₄	C₂×D₄×C₄₇	1	6	yes	—
421	C₂²×C₄	C₂×D₄×C₄₇	1	12	yes	—
422	C₂²×C₄	C₂⁴×C₄₇	1	6	yes	—
423	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₄₇	1	12	yes	—
425	Q₁₆	C₂⁴×C₄₇	1	96	yes	—
426	SD₁₆	C₂⁴×C₄₇	1	96	yes	—
427	SD₁₆	C₂⁴×C₄₇	1	32	yes	—
428	D₈	C₂⁴×C₄₇	1	32	yes	—
429	M₁₆	C₂⁴×C₄₇	1	16	yes	—
430	M₁₆	C₂⁴×C₄₇	1	48	yes	—
431	C₂×C₈	C₂⁴×C₄₇	1	8	yes	—
432	C₂×C₈	C₂⁴×C₄₇	1	48	yes	—
433	C₂³	—	2	4	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	4	no	—
436	C₂³	—	2	24	no	—
437	C₂³	—	4	6	no	—
438	C₂³	—	4	4	no	—
439	C₂³	—	8	12	no	—
440	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
441	C₂³	C₂²×D₄×C₄₇	1	6	yes	—
442	C₂³	C₂²×D₄×C₄₇	1	12	yes	—
443	Q₈	—	2	12	no	—
444	Q₈	—	2	48	no	—
445	Q₈	—	2	6	no	—
446	Q₈	—	2	24	no	—
447	Q₈	—	2	4	no	—
448	Q₈	C₂²×D₄×C₄₇	1	16	yes	—
449	Q₈	C₂⁵×C₄₇	1	12	yes	—
450	Q₈	C₂⁵×C₄₇	1	8	yes	—
451	D₄	—	2	16	no	—
452	D₄	—	2	48	no	—
453	D₄	—	2	48	no	—
454	D₄	—	2	24	no	—
455	D₄	—	2	4	no	—
456	D₄	—	2	4	no	—
457	D₄	—	2	2	no	—
458	D₄	C₂²×D₄×C₄₇	1	48	yes	—
459	D₄	C₂⁵×C₄₇	1	4	yes	—
460	D₄	C₂⁵×C₄₇	1	8	yes	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	12	no	—
463	C₂×C₄	—	2	6	no	—
464	C₂×C₄	—	2	24	no	—
465	C₂×C₄	—	2	48	no	—
466	C₂×C₄	—	2	16	no	—
467	C₂×C₄	—	2	48	no	—
468	C₂×C₄	—	2	48	no	—
469	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
470	C₂×C₄	C₂²×D₄×C₄₇	1	6	yes	—
471	C₂×C₄	C₂²×D₄×C₄₇	1	12	yes	—
472	C₂×C₄	C₂²×D₄×C₄₇	1	24	yes	—
473	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
474	C₂×C₄	C₂²×D₄×C₄₇	1	12	yes	—
475	C₂×C₄	C₂⁵×C₄₇	1	1	yes	characteristic
476	C₂×C₄	C₂⁵×C₄₇	1	12	yes	—
477	C₂×C₄	C₂⁵×C₄₇	1	2	yes	—
478	C₈	C₂⁵×C₄₇	1	16	yes	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	12	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	8	no	—
483	C₂²	—	4	12	no	—
484	C₂²	—	4	8	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	8	no	—
487	C₂²	—	8	24	no	—
488	C₂²	C₂³×D₄×C₄₇	1	2	yes	—
489	C₂²	C₂³×D₄×C₄₇	1	12	yes	—
490	C₂²	C₂³×D₄×C₄₇	1	1	yes	characteristic
491	C₄	—	2	8	no	—
492	C₄	—	2	12	no	—
493	C₄	C₂³×D₄×C₄₇	1	2	yes	—
494	C₄	C₂³×D₄×C₄₇	1	12	yes	—
495	C₄	C₂³×D₄×C₄₇	1	1	yes	characteristic
496	C₄	C₂⁶×C₄₇	1	1	yes	characteristic
497	C₂	—	2	2	no	—
498	C₂	—	2	12	no	—
499	C₂	—	4	4	no	—
500	C₂	—	4	8	no	—
501	C₂	C₄₇×#3586	1	2	yes	direct product factor
502	C₂	C₂⁴×D₄×C₄₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂³×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₄₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂²×#41×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₇×#585	C₂	semidirect
2. C₂²×2₋¹⁺⁴×C₄₇	C₂	semidirect
3. C₄₇×#3586	C₂	direct (N × H)
4. C₂×C₄₇×#578	C₂	semidirect
5. C₂×C₄₇×#577	C₂	semidirect
6. C₂×C₄₇×#575	C₂	semidirect
7. C₂²×#137×C₄₇	C₂	semidirect
8. C₂×C₄₇×#568	C₂	semidirect
10. C₄₇×#585	C₂²	semidirect
14. C₂×2₋¹⁺⁴×C₄₇	C₂²	semidirect
15. C₂×2₋¹⁺⁴×C₄₇	C₂²	semidirect
16. C₂×2₋¹⁺⁴×C₄₇	C₂²	semidirect
18. C₂×H(5,𝔽₂)×C₄₇	C₂²	semidirect
23. C₂²×#41×C₄₇	C₂²	semidirect
25. C₂³×Q₈×C₄₇	C₂²	semidirect
26. C₄₇×#578	C₂²	semidirect
27. C₄₇×#577	C₂²	semidirect
28. C₄₇×#575	C₂²	semidirect
29. C₂×#137×C₄₇	C₂²	semidirect
30. C₂×#137×C₄₇	C₂²	semidirect
31. C₂×#137×C₄₇	C₂²	semidirect
32. C₂×#137×C₄₇	C₂²	semidirect
33. C₂×AGL(1,ℤ/8)×C₄₇	C₂²	semidirect
34. C₂×AGL(1,ℤ/8)×C₄₇	C₂²	semidirect
35. C₂×#135×C₄₇	C₂²	semidirect
36. C₂²×Q₁₆×C₄₇	C₂²	semidirect
37. C₂²×SD₁₆×C₄₇	C₂²	semidirect
38. C₄₇×#568	C₂²	semidirect
39. C₂×#131×C₄₇	C₂²	semidirect
40. C₂²×M₁₆×C₄₇	C₂²	semidirect
45. 2₋¹⁺⁴×C₄₇	C₂³	semidirect
46. 2₋¹⁺⁴×C₄₇	C₂³	semidirect
49. H(5,𝔽₂)×C₄₇	C₂³	semidirect
68. C₂×#41×C₄₇	C₂³	semidirect
79. C₂²×Q₈×C₄₇	C₂³	semidirect
90. #137×C₄₇	C₂³	semidirect
91. #137×C₄₇	C₂³	semidirect
92. #137×C₄₇	C₂³	semidirect
93. AGL(1,ℤ/8)×C₄₇	C₂³	semidirect
94. AGL(1,ℤ/8)×C₄₇	C₂³	semidirect
95. #135×C₄₇	C₂³	semidirect
97. C₂×Q₁₆×C₄₇	C₂³	semidirect
100. C₂×SD₁₆×C₄₇	C₂³	semidirect
102. #131×C₄₇	C₂³	semidirect
105. C₂×M₁₆×C₄₇	C₂³	semidirect
227. C₂×#3586	C₄₇	direct (N × H)
246. C₂×#585	C₂×C₄₇	semidirect
247. C₂²×2₋¹⁺⁴	C₂×C₄₇	semidirect
248. #3586	C₂×C₄₇	direct (N × H)
249. C₂×#578	C₂×C₄₇	semidirect
250. C₂×#577	C₂×C₄₇	semidirect
251. C₂×#575	C₂×C₄₇	semidirect
252. C₂²×#137	C₂×C₄₇	semidirect
253. C₂×#568	C₂×C₄₇	semidirect
258. C₂×C₄₇	#3586	direct (N × H)
261. #585	C₂²×C₄₇	semidirect
265. C₂×2₋¹⁺⁴	C₂²×C₄₇	semidirect
266. C₂×2₋¹⁺⁴	C₂²×C₄₇	semidirect
267. C₂×2₋¹⁺⁴	C₂²×C₄₇	semidirect
269. C₂×H(5,𝔽₂)	C₂²×C₄₇	semidirect
274. C₂²×#41	C₂²×C₄₇	semidirect
276. C₂³×Q₈	C₂²×C₄₇	semidirect
277. #578	C₂²×C₄₇	semidirect
278. #577	C₂²×C₄₇	semidirect
279. #575	C₂²×C₄₇	semidirect
280. C₂×#137	C₂²×C₄₇	semidirect
281. C₂×#137	C₂²×C₄₇	semidirect
282. C₂×#137	C₂²×C₄₇	semidirect
283. C₂×#137	C₂²×C₄₇	semidirect
284. C₂×AGL(1,ℤ/8)	C₂²×C₄₇	semidirect
285. C₂×AGL(1,ℤ/8)	C₂²×C₄₇	semidirect
286. C₂×#135	C₂²×C₄₇	semidirect
287. C₂²×Q₁₆	C₂²×C₄₇	semidirect
288. C₂²×SD₁₆	C₂²×C₄₇	semidirect
289. #568	C₂²×C₄₇	semidirect
290. C₂×#131	C₂²×C₄₇	semidirect
291. C₂²×M₁₆	C₂²×C₄₇	semidirect
292. C₄₇	C₂×#3586	direct (N × H)
297. 2₋¹⁺⁴	C₂³×C₄₇	semidirect
298. 2₋¹⁺⁴	C₂³×C₄₇	semidirect
301. H(5,𝔽₂)	C₂³×C₄₇	semidirect
320. C₂×#41	C₂³×C₄₇	semidirect
331. C₂²×Q₈	C₂³×C₄₇	semidirect
342. #137	C₂³×C₄₇	semidirect
343. #137	C₂³×C₄₇	semidirect
344. #137	C₂³×C₄₇	semidirect
345. AGL(1,ℤ/8)	C₂³×C₄₇	semidirect
346. AGL(1,ℤ/8)	C₂³×C₄₇	semidirect
347. #135	C₂³×C₄₇	semidirect
349. C₂×Q₁₆	C₂³×C₄₇	semidirect
352. C₂×SD₁₆	C₂³×C₄₇	semidirect
354. #131	C₂³×C₄₇	semidirect
357. C₂×M₁₆	C₂³×C₄₇	semidirect
501. C₂	C₄₇×#3586	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×C₄₇×#585	2
C₂²×2₋¹⁺⁴×C₄₇	2
C₄₇×#3586	2
C₂×C₄₇×#578	2
C₂×C₄₇×#577	2
C₂×C₄₇×#575	2
C₂²×#137×C₄₇	2
C₂×C₄₇×#568	2
C₂×#3586	47

Subgroups — C2×C47×#3586 #26102787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#3586 #26102787