Subgroups — C₄₇×#60431 #26100129

Order 12032.

2782 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#577	C₂	1	2	yes	maximal, complemented
2	C₄₇×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×2₋¹⁺⁴×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#3045	C₂	1	6	yes	maximal, complemented
5	C₄₇×#3043	C₂	1	6	yes	maximal, complemented
6	C₄₇×#3038	C₂	1	3	yes	maximal, complemented
7	C₄₇×#3037	C₂	1	3	yes	maximal, complemented
8	C₄₇×#3035	C₂	1	3	yes	maximal, complemented
9	C₂×C₄₇×#460	C₂	1	3	yes	maximal, complemented
10	C₄₇×#2906	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2897	C₂	1	2	yes	maximal, complemented
12	C₂×2₋¹⁺⁴×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×H(5,𝔽₂)×C₄₇	C₂²	1	2	yes	—
14	C₄₇×#577	—	2	16	no	—
15	C₂×#137×C₄₇	C₂²	1	6	yes	complemented
16	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	6	yes	complemented
17	C₂×#135×C₄₇	C₂²	1	6	yes	complemented
18	C₂²×SD₁₆×C₄₇	C₂²	1	6	yes	complemented
19	C₂×#131×C₄₇	C₂²	1	2	yes	—
20	C₄₇×#550	—	2	1	no	—
21	C₄₇×#550	—	2	1	no	—
22	D₄²×C₄₇	—	2	3	no	—
23	D₄²×C₄₇	—	2	3	no	—
24	C₄₇×#535	C₂²	1	6	yes	—
25	C₄₇×#532	C₂²	1	3	yes	—
26	C₂×#127×C₄₇	C₂²	1	3	yes	—
27	C₄₇×#520	—	2	3	no	—
28	C₄₇×#520	C₂²	1	1	yes	characteristic
29	C₄₇×#520	C₂²	1	3	yes	complemented
30	C₄₇×#519	—	2	1	no	—
31	C₄₇×#519	C₂²	1	3	yes	complemented
32	C₄×#41×C₄₇	—	2	3	no	—
33	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
34	C₄×#41×C₄₇	C₂²	1	3	yes	complemented
35	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
36	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	complemented
37	C₄₇×#464	C₂²	1	6	yes	complemented
38	C₄₇×#464	C₂²	1	12	yes	complemented
39	C₄₇×#463	C₂²	1	6	yes	complemented
40	C₄₇×#462	C₂²	1	3	yes	complemented
41	C₄₇×#461	C₂²	1	1	yes	characteristic, complemented
42	C₄₇×#460	C₂²	1	3	yes	complemented
43	C₄₇×#460	C₂²	1	6	yes	complemented
44	C₄₇×#460	C₂²	1	12	yes	complemented
45	C₄₇×#460	C₂²	1	6	yes	complemented
46	C₄₇×#459	C₂²	1	3	yes	complemented
47	C₄₇×#459	C₂²	1	6	yes	complemented
48	C₄₇×#424	C₂²	1	6	yes	complemented
49	C₄₇×#423	C₂²	1	3	yes	complemented
50	C₂×C₄⋊C₈×C₄₇	C₂²	1	3	yes	complemented
51	C₄₇×#418	C₂²	1	6	yes	complemented
52	C₄₇×#417	C₂²	1	6	yes	complemented
53	C₄₇×#416	C₂²	1	6	yes	complemented
54	C₂×#102×C₄₇	C₂²	1	6	yes	complemented
55	2₋¹⁺⁴×C₄₇	—	2	4	no	—
56	2₋¹⁺⁴×C₄₇	D₄	1	8	yes	complemented
57	H(5,𝔽₂)×C₄₇	—	2	8	no	—
58	H(5,𝔽₂)×C₄₇	—	2	8	no	—
59	C₂×#41×C₄₇	—	2	2	no	—
60	C₂×#41×C₄₇	—	2	3	no	—
61	C₂×#41×C₄₇	C₂³	1	6	yes	—
62	C₂×#41×C₄₇	C₂³	1	3	yes	—
63	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₄₇	—	2	1	no	—
65	C₂²×Q₈×C₄₇	C₂³	1	3	yes	—
66	C₂²×D₄×C₄₇	—	2	6	no	—
67	C₂²×D₄×C₄₇	—	2	3	no	—
68	C₂²×D₄×C₄₇	—	2	3	no	—
69	C₂²×D₄×C₄₇	C₂³	1	6	yes	—
70	#137×C₄₇	—	2	24	no	—
71	AGL(1,ℤ/8)×C₄₇	—	2	24	no	—
72	#135×C₄₇	—	2	24	no	—
73	C₂×Q₁₆×C₄₇	C₂³	1	6	yes	—
74	C₂×SD₁₆×C₄₇	—	2	24	no	—
75	C₂×SD₁₆×C₄₇	C₂³	1	6	yes	—
76	C₂×SD₁₆×C₄₇	C₂³	1	12	yes	—
77	C₂×SD₁₆×C₄₇	C₂³	1	2	yes	—
78	C₂×D₈×C₄₇	C₂³	1	6	yes	—
79	#131×C₄₇	—	2	8	no	—
80	C₂×M₁₆×C₄₇	C₂³	1	6	yes	—
81	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
82	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
83	#127×C₄₇	—	2	3	no	—
84	#127×C₄₇	—	2	3	no	—
85	#127×C₄₇	C₂³	1	3	yes	—
86	#127×C₄₇	C₂³	1	6	yes	—
87	#124×C₄₇	C₂³	1	6	yes	—
88	#121×C₄₇	—	2	3	no	—
89	#121×C₄₇	—	2	3	no	—
90	#121×C₄₇	—	2	6	no	—
91	#121×C₄₇	—	2	6	no	—
92	#120×C₄₇	—	2	12	no	—
93	C₄×Q₈×C₄₇	—	2	1	no	—
94	C₄×Q₈×C₄₇	—	2	3	no	—
95	C₄×Q₈×C₄₇	—	2	3	no	—
96	C₄×Q₈×C₄₇	—	2	3	no	—
97	C₄×Q₈×C₄₇	—	2	3	no	—
98	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
99	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
100	C₄×Q₈×C₄₇	C₂³	1	3	yes	—
101	C₄×Q₈×C₄₇	C₂³	1	6	yes	—
102	C₄×Q₈×C₄₇	C₂³	1	3	yes	—
103	C₄×D₄×C₄₇	—	2	3	no	—
104	C₄×D₄×C₄₇	—	2	3	no	—
105	C₄×D₄×C₄₇	—	2	3	no	—
106	C₄×D₄×C₄₇	—	2	6	no	—
107	C₄×D₄×C₄₇	—	2	3	no	—
108	C₄×D₄×C₄₇	—	2	3	no	—
109	C₄×D₄×C₄₇	D₄	1	3	yes	—
110	C₄×D₄×C₄₇	D₄	1	3	yes	—
111	C₄×D₄×C₄₇	C₂³	1	1	yes	characteristic
112	C₄×D₄×C₄₇	C₂³	1	3	yes	—
113	#117×C₄₇	—	2	3	no	—
114	#117×C₄₇	—	2	6	no	—
115	#117×C₄₇	—	2	3	no	—
116	#117×C₄₇	C₂³	1	3	yes	—
117	#117×C₄₇	C₂³	1	3	yes	—
118	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
119	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
120	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
121	C₂×C₄²×C₄₇	—	2	3	no	—
122	C₂×C₄²×C₄₇	—	2	3	no	—
123	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
124	C₄⋊C₈×C₄₇	C₂³	1	3	yes	—
125	C₄⋊C₈×C₄₇	C₂³	1	6	yes	—
126	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
127	C₄⋊C₈×C₄₇	C₂³	1	6	yes	—
128	#103×C₄₇	C₂³	1	2	yes	—
129	#103×C₄₇	C₂³	1	6	yes	—
130	#102×C₄₇	C₂³	1	12	yes	—
131	#102×C₄₇	C₂³	1	6	yes	—
132	#102×C₄₇	C₂³	1	6	yes	—
133	C₂⁴×C₄₇	—	2	6	no	—
134	#41×C₄₇	—	2	24	no	—
135	#41×C₄₇	—	2	6	no	—
136	#41×C₄₇	—	2	12	no	—
137	#41×C₄₇	—	2	12	no	—
138	#41×C₄₇	—	4	8	no	—
139	#41×C₄₇	C₂×D₄	1	4	yes	—
140	#41×C₄₇	C₂×D₄	1	12	yes	complemented
141	#41×C₄₇	#41	1	4	yes	—
142	C₂×Q₈×C₄₇	—	2	8	no	—
143	C₂×Q₈×C₄₇	—	2	3	no	—
144	C₂×Q₈×C₄₇	—	2	6	no	—
145	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
146	C₂×Q₈×C₄₇	C₂×D₄	1	12	yes	complemented
147	C₂×Q₈×C₄₇	C₂⁴	1	2	yes	—
148	C₂×Q₈×C₄₇	C₂⁴	1	3	yes	—
149	C₂×D₄×C₄₇	—	2	3	no	—
150	C₂×D₄×C₄₇	—	2	6	no	—
151	C₂×D₄×C₄₇	—	2	12	no	—
152	C₂×D₄×C₄₇	—	2	3	no	—
153	C₂×D₄×C₄₇	—	2	3	no	—
154	C₂×D₄×C₄₇	—	2	6	no	—
155	C₂×D₄×C₄₇	—	2	6	no	—
156	C₂×D₄×C₄₇	—	2	6	no	—
157	C₂×D₄×C₄₇	—	2	12	no	—
158	C₂×D₄×C₄₇	—	4	24	no	—
159	C₂×D₄×C₄₇	—	8	6	no	—
160	C₂×D₄×C₄₇	—	8	6	no	—
161	C₂×D₄×C₄₇	C₂×D₄	1	3	yes	—
162	C₂×D₄×C₄₇	C₂⁴	1	6	yes	—
163	C₂×D₄×C₄₇	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₄₇	—	2	3	no	—
165	C₂²×C₄×C₄₇	—	2	3	no	—
166	C₂²×C₄×C₄₇	—	2	3	no	—
167	C₂²×C₄×C₄₇	—	2	1	no	—
168	C₂²×C₄×C₄₇	—	2	6	no	—
169	C₂²×C₄×C₄₇	—	2	3	no	—
170	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
171	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
172	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
173	Q₁₆×C₄₇	—	2	12	no	—
174	SD₁₆×C₄₇	—	2	12	no	—
175	SD₁₆×C₄₇	—	2	24	no	—
176	SD₁₆×C₄₇	—	2	4	no	—
177	D₈×C₄₇	—	2	12	no	—
178	M₁₆×C₄₇	—	2	12	no	—
179	C₂×C₈×C₄₇	—	2	12	no	—
180	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
181	C₂×C₈×C₄₇	C₂⁴	1	6	yes	—
182	C₄⋊C₄×C₄₇	—	2	6	no	—
183	C₄⋊C₄×C₄₇	—	2	3	no	—
184	C₄⋊C₄×C₄₇	—	2	3	no	—
185	C₄⋊C₄×C₄₇	—	2	3	no	—
186	C₄⋊C₄×C₄₇	—	2	1	no	—
187	C₄⋊C₄×C₄₇	—	2	1	no	—
188	C₄⋊C₄×C₄₇	C₂×D₄	1	3	yes	—
189	C₄⋊C₄×C₄₇	C₂×D₄	1	3	yes	—
190	C₄⋊C₄×C₄₇	C₂⁴	1	3	yes	—
191	C₄⋊C₄×C₄₇	C₂⁴	1	1	yes	characteristic
192	C₂²⋊C₄×C₄₇	—	2	3	no	—
193	C₂²⋊C₄×C₄₇	—	2	1	no	—
194	C₂²⋊C₄×C₄₇	—	2	3	no	—
195	C₂²⋊C₄×C₄₇	—	2	6	no	—
196	C₂²⋊C₄×C₄₇	—	2	6	no	—
197	C₂²⋊C₄×C₄₇	C₂×D₄	1	6	yes	—
198	C₄²×C₄₇	—	2	3	no	—
199	C₄²×C₄₇	—	2	3	no	—
200	C₄²×C₄₇	—	2	6	no	—
201	C₄²×C₄₇	—	2	3	no	—
202	C₄²×C₄₇	C₂×D₄	1	3	yes	—
203	C₄²×C₄₇	C₂×D₄	1	3	yes	—
204	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
205	C₄²×C₄₇	C₂⁴	1	3	yes	—
206	C₂³×C₄₇	—	2	1	no	—
207	C₂³×C₄₇	—	2	6	no	—
208	C₂³×C₄₇	—	4	12	no	—
209	C₂³×C₄₇	—	8	6	no	—
210	C₂³×C₄₇	—	8	6	no	—
211	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
212	Q₈×C₄₇	—	2	12	no	—
213	Q₈×C₄₇	—	2	3	no	—
214	Q₈×C₄₇	—	2	2	no	—
215	Q₈×C₄₇	—	2	2	no	—
216	Q₈×C₄₇	#121	1	4	yes	—
217	Q₈×C₄₇	C₂²×D₄	1	6	yes	—
218	D₄×C₄₇	—	2	12	no	—
219	D₄×C₄₇	—	2	6	no	—
220	D₄×C₄₇	—	2	1	no	—
221	D₄×C₄₇	—	2	6	no	—
222	D₄×C₄₇	—	4	6	no	—
223	D₄×C₄₇	—	4	12	no	—
224	D₄×C₄₇	—	4	12	no	—
225	D₄×C₄₇	—	4	6	no	—
226	D₄×C₄₇	—	8	6	no	—
227	D₄×C₄₇	—	8	6	no	—
228	D₄×C₄₇	#121	1	12	yes	—
229	D₄×C₄₇	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₄₇	—	2	12	no	—
231	C₂×C₄×C₄₇	—	2	3	no	—
232	C₂×C₄×C₄₇	—	2	12	no	—
233	C₂×C₄×C₄₇	—	2	6	no	—
234	C₂×C₄×C₄₇	—	2	1	no	—
235	C₂×C₄×C₄₇	—	2	3	no	—
236	C₂×C₄×C₄₇	—	2	2	no	—
237	C₂×C₄×C₄₇	—	4	3	no	—
238	C₂×C₄×C₄₇	—	4	12	no	—
239	C₂×C₄×C₄₇	—	4	2	no	—
240	C₂×C₄×C₄₇	—	8	3	no	—
241	C₂×C₄×C₄₇	—	8	3	no	—
242	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
243	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
244	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
246	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
247	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
249	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
250	C₈×C₄₇	—	2	2	no	—
251	C₈×C₄₇	—	2	6	no	—
252	#60431	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂²×C₄₇	—	2	3	no	—
254	C₂²×C₄₇	—	2	3	no	—
255	C₂²×C₄₇	—	2	2	no	—
256	C₂²×C₄₇	—	4	1	no	—
257	C₂²×C₄₇	—	4	1	no	—
258	C₂²×C₄₇	—	4	6	no	—
259	C₂²×C₄₇	—	4	6	no	—
260	C₂²×C₄₇	—	4	6	no	—
261	C₂²×C₄₇	—	16	12	no	—
262	C₂²×C₄₇	C₂×#121	1	6	yes	—
263	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
264	C₄×C₄₇	—	2	1	no	—
265	C₄×C₄₇	—	2	3	no	—
266	C₄×C₄₇	—	2	6	no	—
267	C₄×C₄₇	—	2	3	no	—
268	C₄×C₄₇	—	4	3	no	—
269	C₄×C₄₇	—	4	2	no	—
270	C₄×C₄₇	—	4	1	no	—
271	C₄×C₄₇	C₂×#121	1	6	yes	—
272	C₄×C₄₇	#577	1	2	yes	complemented
273	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
274	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
275	C₂×#577	C₂×C₄₇	1	2	yes	complemented
276	#3498	C₂×C₄₇	1	1	yes	characteristic, complemented
277	C₄×2₋¹⁺⁴	C₂×C₄₇	1	1	yes	characteristic, complemented
278	#3045	C₂×C₄₇	1	6	yes	complemented
279	#3043	C₂×C₄₇	1	6	yes	complemented
280	#3038	C₂×C₄₇	1	3	yes	complemented
281	#3037	C₂×C₄₇	1	3	yes	complemented
282	#3035	C₂×C₄₇	1	3	yes	complemented
283	C₂×#460	C₂×C₄₇	1	3	yes	complemented
284	#2906	C₂×C₄₇	1	1	yes	characteristic, complemented
285	#2897	C₂×C₄₇	1	2	yes	complemented
286	C₂×C₄₇	—	2	6	no	—
287	C₂×C₄₇	—	4	2	no	—
288	C₂×C₄₇	—	8	6	no	—
289	C₂×C₄₇	C₂²×#121	1	1	yes	characteristic
290	C₂×C₄₇	C₂×#577	1	1	yes	characteristic
291	C₂×C₄₇	#3585	1	1	yes	characteristic
292	C₂×2₋¹⁺⁴	C₂²×C₄₇	1	1	yes	characteristic
293	C₂×H(5,𝔽₂)	C₂²×C₄₇	1	2	yes	—
294	#577	—	2	16	no	—
295	C₂×#137	C₂²×C₄₇	1	6	yes	complemented
296	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	6	yes	complemented
297	C₂×#135	C₂²×C₄₇	1	6	yes	complemented
298	C₂²×SD₁₆	C₂²×C₄₇	1	6	yes	complemented
299	C₂×#131	C₂²×C₄₇	1	2	yes	—
300	#550	—	2	1	no	—
301	#550	—	2	1	no	—
302	D₄²	—	2	3	no	—
303	D₄²	—	2	3	no	—
304	#535	C₂²×C₄₇	1	6	yes	—
305	#532	C₂²×C₄₇	1	3	yes	—
306	C₂×#127	C₂²×C₄₇	1	3	yes	—
307	#520	—	2	3	no	—
308	#520	C₂²×C₄₇	1	1	yes	characteristic
309	#520	C₂²×C₄₇	1	3	yes	complemented
310	#519	—	2	1	no	—
311	#519	C₂²×C₄₇	1	3	yes	complemented
312	C₄×#41	—	2	3	no	—
313	C₄×#41	C₂²×C₄₇	1	1	yes	characteristic
314	C₄×#41	C₂²×C₄₇	1	3	yes	complemented
315	C₂×C₄×Q₈	—	2	1	no	—
316	C₂×C₄×Q₈	C₂²×C₄₇	1	3	yes	complemented
317	#464	C₂²×C₄₇	1	6	yes	complemented
318	#464	C₂²×C₄₇	1	12	yes	complemented
319	#463	C₂²×C₄₇	1	6	yes	complemented
320	#462	C₂²×C₄₇	1	3	yes	complemented
321	#461	C₂²×C₄₇	1	1	yes	characteristic, complemented
322	#460	C₂²×C₄₇	1	3	yes	complemented
323	#460	C₂²×C₄₇	1	6	yes	complemented
324	#460	C₂²×C₄₇	1	12	yes	complemented
325	#460	C₂²×C₄₇	1	6	yes	complemented
326	#459	C₂²×C₄₇	1	3	yes	complemented
327	#459	C₂²×C₄₇	1	6	yes	complemented
328	#424	C₂²×C₄₇	1	6	yes	complemented
329	#423	C₂²×C₄₇	1	3	yes	complemented
330	C₂×C₄⋊C₈	C₂²×C₄₇	1	3	yes	complemented
331	#418	C₂²×C₄₇	1	6	yes	complemented
332	#417	C₂²×C₄₇	1	6	yes	complemented
333	#416	C₂²×C₄₇	1	6	yes	complemented
334	C₂×#102	C₂²×C₄₇	1	6	yes	complemented
335	C₄₇	#60431	1	1	yes	characteristic, direct product factor, Sylow
336	2₋¹⁺⁴	—	2	4	no	—
337	2₋¹⁺⁴	D₄×C₄₇	1	8	yes	complemented
338	H(5,𝔽₂)	—	2	8	no	—
339	H(5,𝔽₂)	—	2	8	no	—
340	C₂×#41	—	2	2	no	—
341	C₂×#41	—	2	3	no	—
342	C₂×#41	C₂³×C₄₇	1	6	yes	—
343	C₂×#41	C₂³×C₄₇	1	3	yes	—
344	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
345	C₂²×Q₈	—	2	1	no	—
346	C₂²×Q₈	C₂³×C₄₇	1	3	yes	—
347	C₂²×D₄	—	2	6	no	—
348	C₂²×D₄	—	2	3	no	—
349	C₂²×D₄	—	2	3	no	—
350	C₂²×D₄	C₂³×C₄₇	1	6	yes	—
351	#137	—	2	24	no	—
352	AGL(1,ℤ/8)	—	2	24	no	—
353	#135	—	2	24	no	—
354	C₂×Q₁₆	C₂³×C₄₇	1	6	yes	—
355	C₂×SD₁₆	—	2	24	no	—
356	C₂×SD₁₆	C₂³×C₄₇	1	6	yes	—
357	C₂×SD₁₆	C₂³×C₄₇	1	12	yes	—
358	C₂×SD₁₆	C₂³×C₄₇	1	2	yes	—
359	C₂×D₈	C₂³×C₄₇	1	6	yes	—
360	#131	—	2	8	no	—
361	C₂×M₁₆	C₂³×C₄₇	1	6	yes	—
362	C₂²×C₈	C₂³×C₄₇	1	6	yes	—
363	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
364	#127	—	2	3	no	—
365	#127	—	2	3	no	—
366	#127	C₂³×C₄₇	1	3	yes	—
367	#127	C₂³×C₄₇	1	6	yes	—
368	#124	C₂³×C₄₇	1	6	yes	—
369	#121	—	2	3	no	—
370	#121	—	2	3	no	—
371	#121	—	2	6	no	—
372	#121	—	2	6	no	—
373	#120	—	2	12	no	—
374	C₄×Q₈	—	2	1	no	—
375	C₄×Q₈	—	2	3	no	—
376	C₄×Q₈	—	2	3	no	—
377	C₄×Q₈	—	2	3	no	—
378	C₄×Q₈	—	2	3	no	—
379	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
380	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
381	C₄×Q₈	C₂³×C₄₇	1	3	yes	—
382	C₄×Q₈	C₂³×C₄₇	1	6	yes	—
383	C₄×Q₈	C₂³×C₄₇	1	3	yes	—
384	C₄×D₄	—	2	3	no	—
385	C₄×D₄	—	2	3	no	—
386	C₄×D₄	—	2	3	no	—
387	C₄×D₄	—	2	6	no	—
388	C₄×D₄	—	2	3	no	—
389	C₄×D₄	—	2	3	no	—
390	C₄×D₄	D₄×C₄₇	1	3	yes	—
391	C₄×D₄	D₄×C₄₇	1	3	yes	—
392	C₄×D₄	C₂³×C₄₇	1	1	yes	characteristic
393	C₄×D₄	C₂³×C₄₇	1	3	yes	—
394	#117	—	2	3	no	—
395	#117	—	2	6	no	—
396	#117	—	2	3	no	—
397	#117	C₂³×C₄₇	1	3	yes	—
398	#117	C₂³×C₄₇	1	3	yes	—
399	C₂×C₄⋊C₄	—	2	3	no	—
400	C₂×C₄⋊C₄	—	2	3	no	—
401	C₂×C₄⋊C₄	C₂³×C₄₇	1	3	yes	—
402	C₂×C₄²	—	2	3	no	—
403	C₂×C₄²	—	2	3	no	—
404	C₂×C₄²	C₂³×C₄₇	1	3	yes	—
405	C₄⋊C₈	C₂³×C₄₇	1	1	yes	characteristic
406	C₄⋊C₈	C₂³×C₄₇	1	3	yes	—
407	C₄⋊C₈	C₂³×C₄₇	1	6	yes	—
408	C₄⋊C₈	C₂³×C₄₇	1	6	yes	—
409	#103	C₂³×C₄₇	1	2	yes	—
410	#103	C₂³×C₄₇	1	6	yes	—
411	#102	C₂³×C₄₇	1	6	yes	—
412	#102	C₂³×C₄₇	1	12	yes	—
413	#102	C₂³×C₄₇	1	6	yes	—
414	C₂⁴	—	2	6	no	—
415	#41	—	2	24	no	—
416	#41	—	2	6	no	—
417	#41	—	2	12	no	—
418	#41	—	2	12	no	—
419	#41	—	4	8	no	—
420	#41	C₂×D₄×C₄₇	1	4	yes	—
421	#41	C₂×D₄×C₄₇	1	12	yes	complemented
422	#41	#41×C₄₇	1	4	yes	—
423	C₂×Q₈	—	2	8	no	—
424	C₂×Q₈	—	2	3	no	—
425	C₂×Q₈	—	2	6	no	—
426	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
427	C₂×Q₈	C₂×D₄×C₄₇	1	12	yes	complemented
428	C₂×Q₈	C₂⁴×C₄₇	1	2	yes	—
429	C₂×Q₈	C₂⁴×C₄₇	1	3	yes	—
430	C₂×D₄	—	2	3	no	—
431	C₂×D₄	—	2	6	no	—
432	C₂×D₄	—	2	12	no	—
433	C₂×D₄	—	2	3	no	—
434	C₂×D₄	—	2	3	no	—
435	C₂×D₄	—	2	6	no	—
436	C₂×D₄	—	2	6	no	—
437	C₂×D₄	—	2	6	no	—
438	C₂×D₄	—	2	12	no	—
439	C₂×D₄	—	4	24	no	—
440	C₂×D₄	—	8	6	no	—
441	C₂×D₄	—	8	6	no	—
442	C₂×D₄	C₂×D₄×C₄₇	1	3	yes	—
443	C₂×D₄	C₂⁴×C₄₇	1	6	yes	—
444	C₂×D₄	C₂⁴×C₄₇	1	1	yes	characteristic
445	C₂²×C₄	—	2	3	no	—
446	C₂²×C₄	—	2	3	no	—
447	C₂²×C₄	—	2	3	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	6	no	—
450	C₂²×C₄	—	2	3	no	—
451	C₂²×C₄	C₂×D₄×C₄₇	1	3	yes	—
452	C₂²×C₄	C₂×D₄×C₄₇	1	3	yes	—
453	C₂²×C₄	C₂⁴×C₄₇	1	3	yes	—
454	Q₁₆	—	2	12	no	—
455	SD₁₆	—	2	12	no	—
456	SD₁₆	—	2	24	no	—
457	SD₁₆	—	2	4	no	—
458	D₈	—	2	12	no	—
459	M₁₆	—	2	12	no	—
460	C₂×C₈	—	2	12	no	—
461	C₂×C₈	C₂⁴×C₄₇	1	2	yes	—
462	C₂×C₈	C₂⁴×C₄₇	1	6	yes	—
463	C₄⋊C₄	—	2	6	no	—
464	C₄⋊C₄	—	2	3	no	—
465	C₄⋊C₄	—	2	3	no	—
466	C₄⋊C₄	—	2	3	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	C₂×D₄×C₄₇	1	3	yes	—
470	C₄⋊C₄	C₂×D₄×C₄₇	1	3	yes	—
471	C₄⋊C₄	C₂⁴×C₄₇	1	3	yes	—
472	C₄⋊C₄	C₂⁴×C₄₇	1	1	yes	characteristic
473	C₂²⋊C₄	—	2	3	no	—
474	C₂²⋊C₄	—	2	1	no	—
475	C₂²⋊C₄	—	2	3	no	—
476	C₂²⋊C₄	—	2	6	no	—
477	C₂²⋊C₄	—	2	6	no	—
478	C₂²⋊C₄	C₂×D₄×C₄₇	1	6	yes	—
479	C₄²	—	2	3	no	—
480	C₄²	—	2	3	no	—
481	C₄²	—	2	6	no	—
482	C₄²	—	2	3	no	—
483	C₄²	C₂×D₄×C₄₇	1	3	yes	—
484	C₄²	C₂×D₄×C₄₇	1	3	yes	—
485	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
486	C₄²	C₂⁴×C₄₇	1	3	yes	—
487	C₂³	—	2	1	no	—
488	C₂³	—	2	6	no	—
489	C₂³	—	4	12	no	—
490	C₂³	—	8	6	no	—
491	C₂³	—	8	6	no	—
492	C₂³	C₂²×D₄×C₄₇	1	3	yes	—
493	Q₈	—	2	12	no	—
494	Q₈	—	2	3	no	—
495	Q₈	—	2	2	no	—
496	Q₈	—	2	2	no	—
497	Q₈	#121×C₄₇	1	4	yes	—
498	Q₈	C₂²×D₄×C₄₇	1	6	yes	—
499	D₄	—	2	12	no	—
500	D₄	—	2	6	no	—
501	D₄	—	2	1	no	—
502	D₄	—	2	6	no	—
503	D₄	—	4	6	no	—
504	D₄	—	4	12	no	—
505	D₄	—	4	12	no	—
506	D₄	—	4	6	no	—
507	D₄	—	8	6	no	—
508	D₄	—	8	6	no	—
509	D₄	#121×C₄₇	1	12	yes	—
510	D₄	C₂²×D₄×C₄₇	1	2	yes	—
511	C₂×C₄	—	2	12	no	—
512	C₂×C₄	—	2	3	no	—
513	C₂×C₄	—	2	12	no	—
514	C₂×C₄	—	2	6	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	3	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	4	3	no	—
519	C₂×C₄	—	4	12	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	8	3	no	—
522	C₂×C₄	—	8	3	no	—
523	C₂×C₄	C₂²×D₄×C₄₇	1	3	yes	—
524	C₂×C₄	C₂²×D₄×C₄₇	1	3	yes	—
525	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₄₇	1	3	yes	—
527	C₂×C₄	C₂²×D₄×C₄₇	1	6	yes	—
528	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
529	C₂×C₄	C₂×#41×C₄₇	1	6	yes	—
530	C₂×C₄	C₂⁵×C₄₇	1	1	yes	characteristic
531	C₈	—	2	2	no	—
532	C₈	—	2	6	no	—
533	C₂²	—	2	3	no	—
534	C₂²	—	2	3	no	—
535	C₂²	—	2	2	no	—
536	C₂²	—	4	1	no	—
537	C₂²	—	4	1	no	—
538	C₂²	—	4	6	no	—
539	C₂²	—	4	6	no	—
540	C₂²	—	4	6	no	—
541	C₂²	—	16	12	no	—
542	C₂²	C₂×#121×C₄₇	1	6	yes	—
543	C₂²	C₂³×D₄×C₄₇	1	1	yes	characteristic
544	C₄	—	2	3	no	—
545	C₄	—	2	1	no	—
546	C₄	—	2	3	no	—
547	C₄	—	2	6	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	3	no	—
550	C₄	—	4	2	no	—
551	C₄	C₂×#121×C₄₇	1	6	yes	—
552	C₄	C₄₇×#577	1	2	yes	complemented
553	C₄	C₂³×D₄×C₄₇	1	1	yes	characteristic
554	C₄	C₂²×#41×C₄₇	1	1	yes	characteristic
555	C₂	—	2	6	no	—
556	C₂	—	4	2	no	—
557	C₂	—	8	6	no	—
558	C₂	C₂²×#121×C₄₇	1	1	yes	characteristic
559	C₂	C₂×C₄₇×#577	1	1	yes	characteristic
560	C₂	C₄₇×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₄×#41×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₇×#577	C₂	semidirect
2. C₄₇×#3498	C₂	semidirect
3. C₄×2₋¹⁺⁴×C₄₇	C₂	semidirect
4. C₄₇×#3045	C₂	semidirect
5. C₄₇×#3043	C₂	semidirect
6. C₄₇×#3038	C₂	semidirect
7. C₄₇×#3037	C₂	semidirect
8. C₄₇×#3035	C₂	semidirect
9. C₂×C₄₇×#460	C₂	semidirect
10. C₄₇×#2906	C₂	semidirect
11. C₄₇×#2897	C₂	semidirect
15. C₂×#137×C₄₇	C₂²	semidirect
16. C₂×AGL(1,ℤ/8)×C₄₇	C₂²	semidirect
17. C₂×#135×C₄₇	C₂²	semidirect
18. C₂²×SD₁₆×C₄₇	C₂²	semidirect
29. C₄₇×#520	C₂²	semidirect
31. C₄₇×#519	C₂²	semidirect
34. C₄×#41×C₄₇	C₂²	semidirect
36. C₂×C₄×Q₈×C₄₇	C₂²	semidirect
37. C₄₇×#464	C₂²	semidirect
38. C₄₇×#464	C₂²	semidirect
39. C₄₇×#463	C₂²	semidirect
40. C₄₇×#462	C₂²	semidirect
41. C₄₇×#461	C₂²	semidirect
42. C₄₇×#460	C₂²	semidirect
43. C₄₇×#460	C₂²	semidirect
44. C₄₇×#460	C₂²	semidirect
45. C₄₇×#460	C₂²	semidirect
46. C₄₇×#459	C₂²	semidirect
47. C₄₇×#459	C₂²	semidirect
48. C₄₇×#424	C₂²	semidirect
49. C₄₇×#423	C₂²	semidirect
50. C₂×C₄⋊C₈×C₄₇	C₂²	semidirect
51. C₄₇×#418	C₂²	semidirect
52. C₄₇×#417	C₂²	semidirect
53. C₄₇×#416	C₂²	semidirect
54. C₂×#102×C₄₇	C₂²	semidirect
56. 2₋¹⁺⁴×C₄₇	D₄	semidirect
140. #41×C₄₇	C₂×D₄	semidirect
146. C₂×Q₈×C₄₇	C₂×D₄	semidirect
252. #60431	C₄₇	direct (N × H)
272. C₄×C₄₇	#577	semidirect
275. C₂×#577	C₂×C₄₇	semidirect
276. #3498	C₂×C₄₇	semidirect
277. C₄×2₋¹⁺⁴	C₂×C₄₇	semidirect
278. #3045	C₂×C₄₇	semidirect
279. #3043	C₂×C₄₇	semidirect
280. #3038	C₂×C₄₇	semidirect
281. #3037	C₂×C₄₇	semidirect
282. #3035	C₂×C₄₇	semidirect
283. C₂×#460	C₂×C₄₇	semidirect
284. #2906	C₂×C₄₇	semidirect
285. #2897	C₂×C₄₇	semidirect
295. C₂×#137	C₂²×C₄₇	semidirect
296. C₂×AGL(1,ℤ/8)	C₂²×C₄₇	semidirect
297. C₂×#135	C₂²×C₄₇	semidirect
298. C₂²×SD₁₆	C₂²×C₄₇	semidirect
309. #520	C₂²×C₄₇	semidirect
311. #519	C₂²×C₄₇	semidirect
314. C₄×#41	C₂²×C₄₇	semidirect
316. C₂×C₄×Q₈	C₂²×C₄₇	semidirect
317. #464	C₂²×C₄₇	semidirect
318. #464	C₂²×C₄₇	semidirect
319. #463	C₂²×C₄₇	semidirect
320. #462	C₂²×C₄₇	semidirect
321. #461	C₂²×C₄₇	semidirect
322. #460	C₂²×C₄₇	semidirect
323. #460	C₂²×C₄₇	semidirect
324. #460	C₂²×C₄₇	semidirect
325. #460	C₂²×C₄₇	semidirect
326. #459	C₂²×C₄₇	semidirect
327. #459	C₂²×C₄₇	semidirect
328. #424	C₂²×C₄₇	semidirect
329. #423	C₂²×C₄₇	semidirect
330. C₂×C₄⋊C₈	C₂²×C₄₇	semidirect
331. #418	C₂²×C₄₇	semidirect
332. #417	C₂²×C₄₇	semidirect
333. #416	C₂²×C₄₇	semidirect
334. C₂×#102	C₂²×C₄₇	semidirect
335. C₄₇	#60431	direct (N × H)
337. 2₋¹⁺⁴	D₄×C₄₇	semidirect
421. #41	C₂×D₄×C₄₇	semidirect
427. C₂×Q₈	C₂×D₄×C₄₇	semidirect
552. C₄	C₄₇×#577	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#577	2
C₄₇×#3498	2
C₄×2₋¹⁺⁴×C₄₇	2
C₄₇×#3045	2
C₄₇×#3043	2
C₄₇×#3038	2
C₄₇×#3037	2
C₄₇×#3035	2
C₂×C₄₇×#460	2
C₄₇×#2906	2
C₄₇×#2897	2
#60431	47

Subgroups — C47×#60431 #26100129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#60431 #26100129