Subgroups — C₄₇×#33348 #26073046

Order 12032.

1410 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3484	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#2888	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#2883	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2122	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2121	C₂	1	4	yes	maximal, complemented
6	C₄₇×#2120	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×AΣL(1,𝔽₁₆)	C₂	1	4	yes	maximal, complemented
8	C₂×H(5,𝔽₂)×C₄₇	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₄₇	C₄	1	1	yes	characteristic, complemented
10	C₄₇×#568	—	2	1	no	—
11	C₄₇×#546	—	2	1	no	—
12	C₄₇×#546	—	2	2	no	—
13	C₄₇×#546	—	2	1	no	—
14	D₄²×C₄₇	—	2	2	no	—
15	D₄²×C₄₇	—	2	1	no	—
16	D₄²×C₄₇	—	2	1	no	—
17	C₄₇×#535	C₄	1	1	yes	characteristic, complemented
18	C₄₇×#535	C₂²	1	1	yes	characteristic
19	C₄₇×#534	C₄	1	2	yes	complemented
20	C₄₇×#534	C₂²	1	2	yes	—
21	C₄₇×#525	C₄	1	1	yes	characteristic, complemented
22	C₄₇×#525	C₂²	1	1	yes	characteristic
23	C₂×#120×C₄₇	C₄	1	2	yes	complemented
24	C₂×#120×C₄₇	C₂²	1	2	yes	—
25	C₄₇×#518	—	2	1	no	—
26	C₄₇×#518	C₄	1	1	yes	characteristic, complemented
27	C₄₇×#413	—	2	2	no	—
28	C₄₇×#413	C₂²	1	2	yes	complemented
29	C₂×#100×C₄₇	—	2	1	no	—
30	C₂×#100×C₄₇	C₂²	1	4	yes	complemented
31	C₄₇×#410	—	2	1	no	—
32	C₄₇×#410	C₂²	1	2	yes	complemented
33	C₂×#99×C₄₇	—	2	2	no	—
34	C₂×#99×C₄₇	C₂²	1	4	yes	complemented
35	C₄₇×#352	C₂²	1	4	yes	complemented
36	C₄₇×AΣL(1,𝔽₁₆)	C₂²	1	4	yes	complemented
37	C₄₇×AΣL(1,𝔽₁₆)	C₂²	1	8	yes	complemented
38	C₂⁵×C₄₇	C₂×C₄	1	2	yes	complemented
39	H(5,𝔽₂)×C₄₇	—	4	4	no	—
40	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×#41×C₄₇	D₄	1	2	yes	—
42	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
43	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
44	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₄₇	—	2	2	no	—
46	C₂²×D₄×C₄₇	—	2	2	no	—
47	C₂²×D₄×C₄₇	—	2	2	no	—
48	C₂²×D₄×C₄₇	—	2	1	no	—
49	C₂²×D₄×C₄₇	—	2	1	no	—
50	C₂²×D₄×C₄₇	—	4	4	no	—
51	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
52	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	complemented
54	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂²×D₄×C₄₇	D₄	1	2	yes	—
56	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
57	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
58	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
60	#131×C₄₇	—	4	4	no	—
61	C₂×M₁₆×C₄₇	—	2	2	no	—
62	C₂×M₁₆×C₄₇	—	2	4	no	—
63	#127×C₄₇	—	2	1	no	—
64	#127×C₄₇	—	2	1	no	—
65	#124×C₄₇	—	2	1	no	—
66	#124×C₄₇	—	2	1	no	—
67	#123×C₄₇	—	2	2	no	—
68	#123×C₄₇	—	2	2	no	—
69	#123×C₄₇	C₂×C₄	1	2	yes	complemented
70	#123×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
71	#123×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²⋊Q₈×C₄₇	—	2	1	no	—
73	C₂²⋊Q₈×C₄₇	—	2	1	no	—
74	#121×C₄₇	—	2	1	no	—
75	#121×C₄₇	—	2	2	no	—
76	#121×C₄₇	—	2	2	no	—
77	#121×C₄₇	—	2	1	no	—
78	#121×C₄₇	—	2	2	no	—
79	#121×C₄₇	—	2	2	no	—
80	#121×C₄₇	—	2	2	no	—
81	#121×C₄₇	—	2	1	no	—
82	#121×C₄₇	—	2	1	no	—
83	#120×C₄₇	—	2	2	no	—
84	#120×C₄₇	—	2	2	no	—
85	#120×C₄₇	—	2	2	no	—
86	#120×C₄₇	—	2	2	no	—
87	#120×C₄₇	—	2	2	no	—
88	#120×C₄₇	—	2	2	no	—
89	#120×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
90	#120×C₄₇	C₂×C₄	1	2	yes	complemented
91	#120×C₄₇	C₂×C₄	1	2	yes	complemented
92	#120×C₄₇	C₂×C₄	1	2	yes	complemented
93	#120×C₄₇	C₂×C₄	1	2	yes	complemented
94	#120×C₄₇	C₂³	1	1	yes	characteristic
95	#120×C₄₇	C₂³	1	2	yes	—
96	C₄×D₄×C₄₇	—	2	2	no	—
97	C₄×D₄×C₄₇	—	2	1	no	—
98	C₄×D₄×C₄₇	—	2	1	no	—
99	C₄×D₄×C₄₇	—	2	2	no	—
100	C₄×D₄×C₄₇	—	2	1	no	—
101	C₄×D₄×C₄₇	—	2	1	no	—
102	C₄×D₄×C₄₇	—	4	4	no	—
103	#117×C₄₇	—	2	2	no	—
104	#117×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
105	#117×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
106	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
107	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
108	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
109	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	complemented
110	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	complemented
111	#101×C₄₇	—	2	2	no	—
112	#100×C₄₇	—	2	4	no	—
113	#100×C₄₇	C₂³	1	4	yes	—
114	#99×C₄₇	—	2	2	no	—
115	#99×C₄₇	—	2	4	no	—
116	#99×C₄₇	C₂³	1	4	yes	—
117	C₂⁴×C₄₇	—	2	2	no	—
118	C₂⁴×C₄₇	—	2	2	no	—
119	C₂⁴×C₄₇	—	4	4	no	—
120	C₂⁴×C₄₇	—	8	4	no	—
121	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
122	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
123	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
124	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
125	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
126	#41×C₄₇	—	2	4	no	—
127	#41×C₄₇	—	4	2	no	—
128	#41×C₄₇	—	4	4	no	—
129	#41×C₄₇	—	4	2	no	—
130	#41×C₄₇	—	4	2	no	—
131	C₂×Q₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
132	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
133	C₂×D₄×C₄₇	—	2	4	no	—
134	C₂×D₄×C₄₇	—	2	4	no	—
135	C₂×D₄×C₄₇	—	2	2	no	—
136	C₂×D₄×C₄₇	—	2	1	no	—
137	C₂×D₄×C₄₇	—	2	2	no	—
138	C₂×D₄×C₄₇	—	2	2	no	—
139	C₂×D₄×C₄₇	—	2	1	no	—
140	C₂×D₄×C₄₇	—	2	2	no	—
141	C₂×D₄×C₄₇	—	2	2	no	—
142	C₂×D₄×C₄₇	—	2	2	no	—
143	C₂×D₄×C₄₇	—	2	2	no	—
144	C₂×D₄×C₄₇	—	2	1	no	—
145	C₂×D₄×C₄₇	—	2	1	no	—
146	C₂×D₄×C₄₇	—	4	2	no	—
147	C₂×D₄×C₄₇	—	4	4	no	—
148	C₂×D₄×C₄₇	—	4	2	no	—
149	C₂×D₄×C₄₇	—	4	4	no	—
150	C₂×D₄×C₄₇	—	4	4	no	—
151	C₂×D₄×C₄₇	—	4	8	no	—
152	C₂×D₄×C₄₇	—	4	4	no	—
153	C₂×D₄×C₄₇	—	4	4	no	—
154	C₂×D₄×C₄₇	—	4	4	no	—
155	C₂×D₄×C₄₇	—	8	4	no	—
156	C₂×D₄×C₄₇	—	8	4	no	—
157	C₂×D₄×C₄₇	C₂²⋊C₄	1	2	yes	—
158	C₂×D₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
159	C₂×D₄×C₄₇	C₂²⋊C₄	1	2	yes	—
160	C₂×D₄×C₄₇	C₂²⋊C₄	1	2	yes	—
161	C₂×D₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
162	C₂×D₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
163	C₂×D₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
164	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
165	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
166	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
167	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
168	C₂×D₄×C₄₇	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₄₇	—	2	2	no	—
170	C₂²×C₄×C₄₇	—	2	2	no	—
171	C₂²×C₄×C₄₇	—	2	1	no	—
172	C₂²×C₄×C₄₇	—	2	1	no	—
173	C₂²×C₄×C₄₇	—	4	2	no	—
174	C₂²×C₄×C₄₇	—	4	4	no	—
175	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
176	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
177	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
178	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
179	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
182	M₁₆×C₄₇	—	2	4	no	—
183	M₁₆×C₄₇	—	4	4	no	—
184	M₁₆×C₄₇	—	4	2	no	—
185	C₂×C₈×C₄₇	—	4	4	no	—
186	C₂×C₈×C₄₇	—	4	2	no	—
187	C₄⋊C₄×C₄₇	—	2	1	no	—
188	C₄⋊C₄×C₄₇	—	2	1	no	—
189	C₄⋊C₄×C₄₇	—	2	1	no	—
190	C₄⋊C₄×C₄₇	—	2	2	no	—
191	C₄⋊C₄×C₄₇	—	2	1	no	—
192	C₄⋊C₄×C₄₇	—	4	2	no	—
193	C₂²⋊C₄×C₄₇	—	2	2	no	—
194	C₂²⋊C₄×C₄₇	—	2	2	no	—
195	C₂²⋊C₄×C₄₇	—	2	2	no	—
196	C₂²⋊C₄×C₄₇	—	2	1	no	—
197	C₂²⋊C₄×C₄₇	—	2	1	no	—
198	C₂²⋊C₄×C₄₇	—	2	1	no	—
199	C₂²⋊C₄×C₄₇	—	2	2	no	—
200	C₂²⋊C₄×C₄₇	—	2	2	no	—
201	C₂²⋊C₄×C₄₇	—	2	2	no	—
202	C₂²⋊C₄×C₄₇	—	2	2	no	—
203	C₂²⋊C₄×C₄₇	—	2	1	no	—
204	C₂²⋊C₄×C₄₇	—	2	2	no	—
205	C₂²⋊C₄×C₄₇	—	2	4	no	—
206	C₂²⋊C₄×C₄₇	—	4	4	no	—
207	C₂²⋊C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
208	C₂²⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
209	C₂²⋊C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
210	C₄²×C₄₇	—	2	1	no	—
211	C₄²×C₄₇	—	2	1	no	—
212	C₄²×C₄₇	—	4	2	no	—
213	C₂³×C₄₇	—	2	1	no	—
214	C₂³×C₄₇	—	2	4	no	—
215	C₂³×C₄₇	—	2	4	no	—
216	C₂³×C₄₇	—	2	2	no	—
217	C₂³×C₄₇	—	4	2	no	—
218	C₂³×C₄₇	—	4	1	no	—
219	C₂³×C₄₇	—	4	2	no	—
220	C₂³×C₄₇	—	4	4	no	—
221	C₂³×C₄₇	—	4	4	no	—
222	C₂³×C₄₇	—	4	4	no	—
223	C₂³×C₄₇	—	4	2	no	—
224	C₂³×C₄₇	—	4	4	no	—
225	C₂³×C₄₇	—	8	2	no	—
226	C₂³×C₄₇	—	8	2	no	—
227	C₂³×C₄₇	—	8	2	no	—
228	C₂³×C₄₇	—	8	2	no	—
229	C₂³×C₄₇	—	8	2	no	—
230	C₂³×C₄₇	—	8	2	no	—
231	C₂³×C₄₇	—	8	8	no	—
232	C₂³×C₄₇	—	8	8	no	—
233	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
234	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
235	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
237	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
238	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
239	Q₈×C₄₇	—	2	2	no	—
240	Q₈×C₄₇	—	4	1	no	—
241	D₄×C₄₇	—	2	4	no	—
242	D₄×C₄₇	—	2	4	no	—
243	D₄×C₄₇	—	2	4	no	—
244	D₄×C₄₇	—	2	2	no	—
245	D₄×C₄₇	—	2	2	no	—
246	D₄×C₄₇	—	4	1	no	—
247	D₄×C₄₇	—	4	2	no	—
248	D₄×C₄₇	—	4	2	no	—
249	D₄×C₄₇	—	4	1	no	—
250	D₄×C₄₇	—	4	2	no	—
251	D₄×C₄₇	—	4	1	no	—
252	D₄×C₄₇	—	4	2	no	—
253	D₄×C₄₇	—	4	4	no	—
254	D₄×C₄₇	—	4	4	no	—
255	D₄×C₄₇	—	4	2	no	—
256	D₄×C₄₇	—	4	2	no	—
257	D₄×C₄₇	—	4	4	no	—
258	D₄×C₄₇	—	4	2	no	—
259	D₄×C₄₇	—	8	2	no	—
260	D₄×C₄₇	—	8	2	no	—
261	D₄×C₄₇	—	8	2	no	—
262	D₄×C₄₇	—	8	2	no	—
263	D₄×C₄₇	#99	1	4	yes	—
264	C₂×C₄×C₄₇	—	2	4	no	—
265	C₂×C₄×C₄₇	—	2	2	no	—
266	C₂×C₄×C₄₇	—	2	2	no	—
267	C₂×C₄×C₄₇	—	2	2	no	—
268	C₂×C₄×C₄₇	—	2	2	no	—
269	C₂×C₄×C₄₇	—	2	1	no	—
270	C₂×C₄×C₄₇	—	2	1	no	—
271	C₂×C₄×C₄₇	—	2	1	no	—
272	C₂×C₄×C₄₇	—	4	2	no	—
273	C₂×C₄×C₄₇	—	4	2	no	—
274	C₂×C₄×C₄₇	—	4	2	no	—
275	C₂×C₄×C₄₇	—	4	2	no	—
276	C₂×C₄×C₄₇	—	4	1	no	—
277	C₂×C₄×C₄₇	—	4	4	no	—
278	C₂×C₄×C₄₇	—	8	2	no	—
279	C₂×C₄×C₄₇	—	8	2	no	—
280	C₂×C₄×C₄₇	—	16	4	no	—
281	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
284	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
285	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
286	C₈×C₄₇	—	4	4	no	—
287	#33348	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
288	C₂²×C₄₇	—	2	2	no	—
289	C₂²×C₄₇	—	2	1	no	—
290	C₂²×C₄₇	—	2	1	no	—
291	C₂²×C₄₇	—	2	1	no	—
292	C₂²×C₄₇	—	2	2	no	—
293	C₂²×C₄₇	—	2	2	no	—
294	C₂²×C₄₇	—	4	2	no	—
295	C₂²×C₄₇	—	4	2	no	—
296	C₂²×C₄₇	—	4	2	no	—
297	C₂²×C₄₇	—	4	1	no	—
298	C₂²×C₄₇	—	4	1	no	—
299	C₂²×C₄₇	—	4	1	no	—
300	C₂²×C₄₇	—	4	1	no	—
301	C₂²×C₄₇	—	4	2	no	—
302	C₂²×C₄₇	—	4	2	no	—
303	C₂²×C₄₇	—	4	2	no	—
304	C₂²×C₄₇	—	4	2	no	—
305	C₂²×C₄₇	—	4	1	no	—
306	C₂²×C₄₇	—	4	2	no	—
307	C₂²×C₄₇	—	8	4	no	—
308	C₂²×C₄₇	—	8	4	no	—
309	C₂²×C₄₇	—	8	8	no	—
310	C₂²×C₄₇	—	8	8	no	—
311	C₂²×C₄₇	—	8	4	no	—
312	C₂²×C₄₇	—	16	2	no	—
313	C₂²×C₄₇	C₂×#99	1	4	yes	—
314	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
315	C₄×C₄₇	—	2	1	no	—
316	C₄×C₄₇	—	2	1	no	—
317	C₄×C₄₇	—	2	1	no	—
318	C₄×C₄₇	—	2	2	no	—
319	C₄×C₄₇	—	4	2	no	—
320	C₄×C₄₇	—	4	1	no	—
321	C₄×C₄₇	—	4	1	no	—
322	C₄×C₄₇	—	4	1	no	—
323	C₄×C₄₇	—	8	4	no	—
324	C₄×C₄₇	C₂×#99	1	2	yes	—
325	#3484	C₂×C₄₇	1	1	yes	characteristic
326	#2888	C₂×C₄₇	1	1	yes	characteristic, complemented
327	#2883	C₂×C₄₇	1	1	yes	characteristic, complemented
328	#2122	C₂×C₄₇	1	2	yes	complemented
329	#2121	C₂×C₄₇	1	4	yes	complemented
330	#2120	C₂×C₄₇	1	2	yes	complemented
331	C₂×AΣL(1,𝔽₁₆)	C₂×C₄₇	1	4	yes	complemented
332	C₂×C₄₇	—	2	1	no	—
333	C₂×C₄₇	—	2	4	no	—
334	C₂×C₄₇	—	4	1	no	—
335	C₂×C₄₇	—	4	2	no	—
336	C₂×C₄₇	—	4	1	no	—
337	C₂×C₄₇	—	4	2	no	—
338	C₂×C₄₇	—	4	2	no	—
339	C₂×C₄₇	—	4	1	no	—
340	C₂×C₄₇	—	8	2	no	—
341	C₂×C₄₇	C₂²×#99	1	1	yes	characteristic
342	C₂×H(5,𝔽₂)	C₂²×C₄₇	1	1	yes	characteristic
343	C₂³×D₄	C₄×C₄₇	1	1	yes	characteristic, complemented
344	#568	—	2	1	no	—
345	#546	—	2	1	no	—
346	#546	—	2	2	no	—
347	#546	—	2	1	no	—
348	D₄²	—	2	2	no	—
349	D₄²	—	2	1	no	—
350	D₄²	—	2	1	no	—
351	#535	C₄×C₄₇	1	1	yes	characteristic, complemented
352	#535	C₂²×C₄₇	1	1	yes	characteristic
353	#534	C₄×C₄₇	1	2	yes	complemented
354	#534	C₂²×C₄₇	1	2	yes	—
355	#525	C₄×C₄₇	1	1	yes	characteristic, complemented
356	#525	C₂²×C₄₇	1	1	yes	characteristic
357	C₂×#120	C₄×C₄₇	1	2	yes	complemented
358	C₂×#120	C₂²×C₄₇	1	2	yes	—
359	#518	—	2	1	no	—
360	#518	C₄×C₄₇	1	1	yes	characteristic, complemented
361	#413	—	2	2	no	—
362	#413	C₂²×C₄₇	1	2	yes	complemented
363	C₂×#100	—	2	1	no	—
364	C₂×#100	C₂²×C₄₇	1	4	yes	complemented
365	#410	—	2	1	no	—
366	#410	C₂²×C₄₇	1	2	yes	complemented
367	C₂×#99	—	2	2	no	—
368	C₂×#99	C₂²×C₄₇	1	4	yes	complemented
369	#352	C₂²×C₄₇	1	4	yes	complemented
370	AΣL(1,𝔽₁₆)	C₂²×C₄₇	1	4	yes	complemented
371	AΣL(1,𝔽₁₆)	C₂²×C₄₇	1	8	yes	complemented
372	C₄₇	#33348	1	1	yes	characteristic, direct product factor, Sylow
373	C₂⁵	C₂×C₄×C₄₇	1	2	yes	complemented
374	H(5,𝔽₂)	—	4	4	no	—
375	C₂×#41	C₂×C₄×C₄₇	1	1	yes	characteristic
376	C₂×#41	D₄×C₄₇	1	2	yes	—
377	C₂×#41	D₄×C₄₇	1	1	yes	characteristic
378	C₂×#41	D₄×C₄₇	1	1	yes	characteristic
379	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
380	C₂²×D₄	—	2	2	no	—
381	C₂²×D₄	—	2	2	no	—
382	C₂²×D₄	—	2	2	no	—
383	C₂²×D₄	—	2	1	no	—
384	C₂²×D₄	—	2	1	no	—
385	C₂²×D₄	—	4	4	no	—
386	C₂²×D₄	C₂×C₄×C₄₇	1	2	yes	—
387	C₂²×D₄	C₂×C₄×C₄₇	1	1	yes	characteristic
388	C₂²×D₄	C₂×C₄×C₄₇	1	2	yes	complemented
389	C₂²×D₄	C₂×C₄×C₄₇	1	1	yes	characteristic, complemented
390	C₂²×D₄	D₄×C₄₇	1	2	yes	—
391	C₂²×D₄	D₄×C₄₇	1	1	yes	characteristic
392	C₂²×D₄	D₄×C₄₇	1	1	yes	characteristic
393	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
394	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic, complemented
395	#131	—	4	4	no	—
396	C₂×M₁₆	—	2	2	no	—
397	C₂×M₁₆	—	2	4	no	—
398	#127	—	2	1	no	—
399	#127	—	2	1	no	—
400	#124	—	2	1	no	—
401	#124	—	2	1	no	—
402	#123	—	2	2	no	—
403	#123	—	2	2	no	—
404	#123	C₂×C₄×C₄₇	1	2	yes	complemented
405	#123	C₂×C₄×C₄₇	1	1	yes	characteristic, complemented
406	#123	C₂³×C₄₇	1	1	yes	characteristic
407	C₂²⋊Q₈	—	2	1	no	—
408	C₂²⋊Q₈	—	2	1	no	—
409	#121	—	2	1	no	—
410	#121	—	2	2	no	—
411	#121	—	2	2	no	—
412	#121	—	2	1	no	—
413	#121	—	2	2	no	—
414	#121	—	2	2	no	—
415	#121	—	2	2	no	—
416	#121	—	2	1	no	—
417	#121	—	2	1	no	—
418	#120	—	2	2	no	—
419	#120	—	2	2	no	—
420	#120	—	2	2	no	—
421	#120	—	2	2	no	—
422	#120	—	2	2	no	—
423	#120	—	2	2	no	—
424	#120	C₂×C₄×C₄₇	1	1	yes	characteristic, complemented
425	#120	C₂×C₄×C₄₇	1	2	yes	complemented
426	#120	C₂×C₄×C₄₇	1	2	yes	complemented
427	#120	C₂×C₄×C₄₇	1	2	yes	complemented
428	#120	C₂×C₄×C₄₇	1	2	yes	complemented
429	#120	C₂³×C₄₇	1	1	yes	characteristic
430	#120	C₂³×C₄₇	1	2	yes	—
431	C₄×D₄	—	2	2	no	—
432	C₄×D₄	—	2	1	no	—
433	C₄×D₄	—	2	1	no	—
434	C₄×D₄	—	2	2	no	—
435	C₄×D₄	—	2	1	no	—
436	C₄×D₄	—	2	1	no	—
437	C₄×D₄	—	4	4	no	—
438	#117	—	2	2	no	—
439	#117	C₂×C₄×C₄₇	1	1	yes	characteristic, complemented
440	#117	C₂×C₄×C₄₇	1	1	yes	characteristic, complemented
441	C₂×C₂²⋊C₄	—	2	2	no	—
442	C₂×C₂²⋊C₄	—	2	2	no	—
443	C₂×C₂²⋊C₄	—	2	4	no	—
444	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	2	yes	complemented
445	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	2	yes	complemented
446	#101	—	2	2	no	—
447	#100	—	2	4	no	—
448	#100	C₂³×C₄₇	1	4	yes	—
449	#99	—	2	2	no	—
450	#99	—	2	4	no	—
451	#99	C₂³×C₄₇	1	4	yes	—
452	C₂⁴	—	2	2	no	—
453	C₂⁴	—	2	2	no	—
454	C₂⁴	—	4	4	no	—
455	C₂⁴	—	8	4	no	—
456	C₂⁴	C₂²⋊C₄×C₄₇	1	2	yes	—
457	C₂⁴	C₂²×C₄×C₄₇	1	2	yes	—
458	C₂⁴	C₂²×C₄×C₄₇	1	1	yes	characteristic
459	C₂⁴	C₂²×C₄×C₄₇	1	2	yes	—
460	C₂⁴	C₂×D₄×C₄₇	1	2	yes	—
461	#41	—	2	4	no	—
462	#41	—	4	2	no	—
463	#41	—	4	4	no	—
464	#41	—	4	2	no	—
465	#41	—	4	2	no	—
466	C₂×Q₈	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
467	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	—	2	2	no	—
473	C₂×D₄	—	2	2	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	2	no	—
476	C₂×D₄	—	2	2	no	—
477	C₂×D₄	—	2	2	no	—
478	C₂×D₄	—	2	2	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	1	no	—
481	C₂×D₄	—	4	2	no	—
482	C₂×D₄	—	4	4	no	—
483	C₂×D₄	—	4	2	no	—
484	C₂×D₄	—	4	4	no	—
485	C₂×D₄	—	4	4	no	—
486	C₂×D₄	—	4	8	no	—
487	C₂×D₄	—	4	4	no	—
488	C₂×D₄	—	4	4	no	—
489	C₂×D₄	—	4	4	no	—
490	C₂×D₄	—	8	4	no	—
491	C₂×D₄	—	8	4	no	—
492	C₂×D₄	C₂²⋊C₄×C₄₇	1	2	yes	—
493	C₂×D₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
494	C₂×D₄	C₂²⋊C₄×C₄₇	1	2	yes	—
495	C₂×D₄	C₂²⋊C₄×C₄₇	1	2	yes	—
496	C₂×D₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
497	C₂×D₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
498	C₂×D₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
499	C₂×D₄	C₂²×C₄×C₄₇	1	2	yes	—
500	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
501	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
502	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
503	C₂×D₄	C₂⁴×C₄₇	1	1	yes	characteristic
504	C₂²×C₄	—	2	2	no	—
505	C₂²×C₄	—	2	2	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	4	2	no	—
509	C₂²×C₄	—	4	4	no	—
510	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
511	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
512	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
513	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
514	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
515	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
516	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
517	M₁₆	—	2	4	no	—
518	M₁₆	—	4	4	no	—
519	M₁₆	—	4	2	no	—
520	C₂×C₈	—	4	4	no	—
521	C₂×C₈	—	4	2	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	4	2	no	—
528	C₂²⋊C₄	—	2	2	no	—
529	C₂²⋊C₄	—	2	2	no	—
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	1	no	—
532	C₂²⋊C₄	—	2	1	no	—
533	C₂²⋊C₄	—	2	1	no	—
534	C₂²⋊C₄	—	2	2	no	—
535	C₂²⋊C₄	—	2	2	no	—
536	C₂²⋊C₄	—	2	2	no	—
537	C₂²⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	2	1	no	—
539	C₂²⋊C₄	—	2	2	no	—
540	C₂²⋊C₄	—	2	4	no	—
541	C₂²⋊C₄	—	4	4	no	—
542	C₂²⋊C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
543	C₂²⋊C₄	C₂²×C₄×C₄₇	1	2	yes	—
544	C₂²⋊C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	4	2	no	—
548	C₂³	—	2	1	no	—
549	C₂³	—	2	4	no	—
550	C₂³	—	2	4	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	4	2	no	—
553	C₂³	—	4	1	no	—
554	C₂³	—	4	2	no	—
555	C₂³	—	4	4	no	—
556	C₂³	—	4	4	no	—
557	C₂³	—	4	4	no	—
558	C₂³	—	4	2	no	—
559	C₂³	—	4	4	no	—
560	C₂³	—	8	2	no	—
561	C₂³	—	8	2	no	—
562	C₂³	—	8	2	no	—
563	C₂³	—	8	2	no	—
564	C₂³	—	8	2	no	—
565	C₂³	—	8	2	no	—
566	C₂³	—	8	8	no	—
567	C₂³	—	8	8	no	—
568	C₂³	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
569	C₂³	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
570	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
571	C₂³	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
572	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
573	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
574	Q₈	—	2	2	no	—
575	Q₈	—	4	1	no	—
576	D₄	—	2	4	no	—
577	D₄	—	2	4	no	—
578	D₄	—	2	4	no	—
579	D₄	—	2	2	no	—
580	D₄	—	2	2	no	—
581	D₄	—	4	1	no	—
582	D₄	—	4	2	no	—
583	D₄	—	4	2	no	—
584	D₄	—	4	1	no	—
585	D₄	—	4	2	no	—
586	D₄	—	4	1	no	—
587	D₄	—	4	2	no	—
588	D₄	—	4	4	no	—
589	D₄	—	4	4	no	—
590	D₄	—	4	2	no	—
591	D₄	—	4	2	no	—
592	D₄	—	4	4	no	—
593	D₄	—	4	2	no	—
594	D₄	—	8	2	no	—
595	D₄	—	8	2	no	—
596	D₄	—	8	2	no	—
597	D₄	—	8	2	no	—
598	D₄	#99×C₄₇	1	4	yes	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	2	no	—
609	C₂×C₄	—	4	2	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	4	no	—
613	C₂×C₄	—	8	2	no	—
614	C₂×C₄	—	8	2	no	—
615	C₂×C₄	—	16	4	no	—
616	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
617	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
618	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
619	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
620	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
621	C₈	—	4	4	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	2	1	no	—
624	C₂²	—	2	1	no	—
625	C₂²	—	2	1	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	2	2	no	—
628	C₂²	—	4	2	no	—
629	C₂²	—	4	2	no	—
630	C₂²	—	4	2	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	1	no	—
633	C₂²	—	4	1	no	—
634	C₂²	—	4	1	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	4	2	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	4	2	no	—
641	C₂²	—	8	4	no	—
642	C₂²	—	8	4	no	—
643	C₂²	—	8	8	no	—
644	C₂²	—	8	8	no	—
645	C₂²	—	8	4	no	—
646	C₂²	—	16	2	no	—
647	C₂²	C₂×#99×C₄₇	1	4	yes	—
648	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
649	C₄	—	2	1	no	—
650	C₄	—	2	1	no	—
651	C₄	—	2	1	no	—
652	C₄	—	2	2	no	—
653	C₄	—	4	2	no	—
654	C₄	—	4	1	no	—
655	C₄	—	4	1	no	—
656	C₄	—	4	1	no	—
657	C₄	—	8	4	no	—
658	C₄	C₂×#99×C₄₇	1	2	yes	—
659	C₂	—	2	1	no	—
660	C₂	—	2	4	no	—
661	C₂	—	4	1	no	—
662	C₂	—	4	2	no	—
663	C₂	—	4	1	no	—
664	C₂	—	4	2	no	—
665	C₂	—	4	2	no	—
666	C₂	—	4	1	no	—
667	C₂	—	8	2	no	—
668	C₂	C₂²×#99×C₄₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₇	C₂²×#99
commutator	C₂³	C₂³×C₄×C₄₇
Frattini, cosocle	C₂×D₄	C₂⁴×C₄₇
prime radical	C₄₇×#3484	C₂
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₇ ◁ C₂×D₄×C₄₇ ◁ C₂×H(5,𝔽₂)×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₇×#2888	C₂	semidirect
3. C₄₇×#2883	C₂	semidirect
4. C₄₇×#2122	C₂	semidirect
5. C₄₇×#2121	C₂	semidirect
6. C₄₇×#2120	C₂	semidirect
7. C₂×C₄₇×AΣL(1,𝔽₁₆)	C₂	semidirect
9. C₂³×D₄×C₄₇	C₄	semidirect
17. C₄₇×#535	C₄	semidirect
19. C₄₇×#534	C₄	semidirect
21. C₄₇×#525	C₄	semidirect
23. C₂×#120×C₄₇	C₄	semidirect
26. C₄₇×#518	C₄	semidirect
28. C₄₇×#413	C₂²	semidirect
30. C₂×#100×C₄₇	C₂²	semidirect
32. C₄₇×#410	C₂²	semidirect
34. C₂×#99×C₄₇	C₂²	semidirect
35. C₄₇×#352	C₂²	semidirect
36. C₄₇×AΣL(1,𝔽₁₆)	C₂²	semidirect
37. C₄₇×AΣL(1,𝔽₁₆)	C₂²	semidirect
38. C₂⁵×C₄₇	C₂×C₄	semidirect
53. C₂²×D₄×C₄₇	C₂×C₄	semidirect
54. C₂²×D₄×C₄₇	C₂×C₄	semidirect
59. C₂³×C₄×C₄₇	C₂×C₄	semidirect
69. #123×C₄₇	C₂×C₄	semidirect
70. #123×C₄₇	C₂×C₄	semidirect
89. #120×C₄₇	C₂×C₄	semidirect
90. #120×C₄₇	C₂×C₄	semidirect
91. #120×C₄₇	C₂×C₄	semidirect
92. #120×C₄₇	C₂×C₄	semidirect
93. #120×C₄₇	C₂×C₄	semidirect
104. #117×C₄₇	C₂×C₄	semidirect
105. #117×C₄₇	C₂×C₄	semidirect
109. C₂×C₂²⋊C₄×C₄₇	C₂×C₄	semidirect
110. C₂×C₂²⋊C₄×C₄₇	C₂×C₄	semidirect
287. #33348	C₄₇	direct (N × H)
326. #2888	C₂×C₄₇	semidirect
327. #2883	C₂×C₄₇	semidirect
328. #2122	C₂×C₄₇	semidirect
329. #2121	C₂×C₄₇	semidirect
330. #2120	C₂×C₄₇	semidirect
331. C₂×AΣL(1,𝔽₁₆)	C₂×C₄₇	semidirect
343. C₂³×D₄	C₄×C₄₇	semidirect
351. #535	C₄×C₄₇	semidirect
353. #534	C₄×C₄₇	semidirect
355. #525	C₄×C₄₇	semidirect
357. C₂×#120	C₄×C₄₇	semidirect
360. #518	C₄×C₄₇	semidirect
362. #413	C₂²×C₄₇	semidirect
364. C₂×#100	C₂²×C₄₇	semidirect
366. #410	C₂²×C₄₇	semidirect
368. C₂×#99	C₂²×C₄₇	semidirect
369. #352	C₂²×C₄₇	semidirect
370. AΣL(1,𝔽₁₆)	C₂²×C₄₇	semidirect
371. AΣL(1,𝔽₁₆)	C₂²×C₄₇	semidirect
372. C₄₇	#33348	direct (N × H)
373. C₂⁵	C₂×C₄×C₄₇	semidirect
388. C₂²×D₄	C₂×C₄×C₄₇	semidirect
389. C₂²×D₄	C₂×C₄×C₄₇	semidirect
394. C₂³×C₄	C₂×C₄×C₄₇	semidirect
404. #123	C₂×C₄×C₄₇	semidirect
405. #123	C₂×C₄×C₄₇	semidirect
424. #120	C₂×C₄×C₄₇	semidirect
425. #120	C₂×C₄×C₄₇	semidirect
426. #120	C₂×C₄×C₄₇	semidirect
427. #120	C₂×C₄×C₄₇	semidirect
428. #120	C₂×C₄×C₄₇	semidirect
439. #117	C₂×C₄×C₄₇	semidirect
440. #117	C₂×C₄×C₄₇	semidirect
444. C₂×C₂²⋊C₄	C₂×C₄×C₄₇	semidirect
445. C₂×C₂²⋊C₄	C₂×C₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#3484	2
C₄₇×#2888	2
C₄₇×#2883	2
C₄₇×#2122	2
C₄₇×#2121	2
C₄₇×#2120	2
C₂×C₄₇×AΣL(1,𝔽₁₆)	2
#33348	47

Subgroups — C47×#33348 #26073046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#33348 #26073046