Subgroups — C₄₇×#34141 #26073839

Order 12032.

3298 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₄₇	C₂	1	2	yes	maximal, complemented
3	C₄₇×#2367	C₂	1	4	yes	maximal, complemented
4	C₄₇×#2366	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2365	C₂	1	2	yes	maximal, complemented
6	C₄₇×#2364	C₂	1	4	yes	maximal, complemented
7	C₄₇×#2361	C₂	1	4	yes	maximal, complemented
8	C₄₇×#2297	C₂	1	2	yes	maximal, complemented
9	C₄₇×#2293	C₂	1	2	yes	maximal, complemented
10	C₄₇×#2290	C₂	1	4	yes	maximal, complemented
11	C₄²×D₄×C₄₇	C₂	1	2	yes	maximal, complemented
12	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₄₇	C₂²	1	2	yes	—
15	C₄₇×#532	C₄	1	16	yes	complemented
16	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#124×C₄₇	C₂²	1	2	yes	—
19	C₂×#121×C₄₇	C₂²	1	2	yes	—
20	C₂×#121×C₄₇	C₂²	1	2	yes	—
21	C₄×#41×C₄₇	—	2	16	no	—
22	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₄₇	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₄₇	C₂²	1	8	yes	—
25	C₂×C₄×D₄×C₄₇	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
28	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#117×C₄₇	C₂²	1	4	yes	—
30	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
32	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
33	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
34	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#391	C₂²	1	4	yes	—
36	C₄₇×#390	C₂²	1	4	yes	—
37	C₄₇×#390	C₂²	1	4	yes	—
38	C₄₇×#389	C₂²	1	4	yes	—
39	C₄₇×#388	C₂²	1	8	yes	—
40	C₄₇×#388	C₂²	1	8	yes	—
41	C₄₇×#387	C₂²	1	8	yes	—
42	C₄₇×#386	C₂²	1	8	yes	—
43	C₄₇×#386	C₂²	1	8	yes	—
44	C₄₇×#385	C₂²	1	8	yes	—
45	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
49	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
50	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
53	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
54	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
55	C₄×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
56	C₄×C₂²⋊C₄×C₄₇	C₂²	1	8	yes	—
57	C₄×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
58	C₄₇×#376	C₂²	1	2	yes	—
59	C₂×#95×C₄₇	C₂²	1	2	yes	—
60	C₄³×C₄₇	C₂²	1	2	yes	—
61	C₂×#41×C₄₇	—	2	8	no	—
62	C₂×#41×C₄₇	—	2	8	no	—
63	C₂×#41×C₄₇	C₂×C₄	1	16	yes	complemented
64	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
65	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
66	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
67	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
68	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
69	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
70	C₂³×C₄×C₄₇	C₂³	1	8	yes	—
71	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
73	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	complemented
76	#127×C₄₇	C₂×C₄	1	8	yes	complemented
77	#124×C₄₇	C₂×C₄	1	16	yes	complemented
78	#121×C₄₇	C₂×C₄	1	16	yes	complemented
79	#121×C₄₇	C₂×C₄	1	16	yes	complemented
80	C₄×Q₈×C₄₇	—	2	16	no	—
81	C₄×D₄×C₄₇	—	2	16	no	—
82	C₄×D₄×C₄₇	—	2	16	no	—
83	C₄×D₄×C₄₇	—	2	32	no	—
84	C₄×D₄×C₄₇	C₂×C₄	1	16	yes	complemented
85	C₄×D₄×C₄₇	C₂×C₄	1	16	yes	complemented
86	#117×C₄₇	—	2	4	no	—
87	#117×C₄₇	—	2	16	no	—
88	#117×C₄₇	D₄	1	8	yes	complemented
89	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
90	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
93	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
99	C₂×C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
105	C₂×C₄²×C₄₇	—	2	8	no	—
106	C₂×C₄²×C₄₇	—	2	16	no	—
107	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	complemented
108	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
109	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
110	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
111	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
112	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
113	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
114	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
115	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
120	#95×C₄₇	C₂³	1	2	yes	—
121	#95×C₄₇	C₂³	1	4	yes	—
122	#95×C₄₇	C₂³	1	2	yes	—
123	#95×C₄₇	C₂³	1	4	yes	—
124	#95×C₄₇	C₂³	1	4	yes	—
125	#95×C₄₇	C₂³	1	8	yes	—
126	#95×C₄₇	C₂³	1	8	yes	—
127	C₂⁴×C₄₇	C₂⁴	1	4	yes	—
128	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
129	#41×C₄₇	—	2	64	no	—
130	C₂×Q₈×C₄₇	—	2	4	no	—
131	C₂×Q₈×C₄₇	—	2	4	no	—
132	C₂×Q₈×C₄₇	C₂²×C₄	1	8	yes	—
133	C₂×D₄×C₄₇	—	2	8	no	—
134	C₂×D₄×C₄₇	—	2	8	no	—
135	C₂×D₄×C₄₇	—	2	4	no	—
136	C₂×D₄×C₄₇	—	2	8	no	—
137	C₂×D₄×C₄₇	—	2	4	no	—
138	C₂×D₄×C₄₇	—	4	4	no	—
139	C₂×D₄×C₄₇	C₂²×C₄	1	8	yes	—
140	C₂×D₄×C₄₇	C₂²×C₄	1	16	yes	—
141	C₂×D₄×C₄₇	C₂²×C₄	1	8	yes	—
142	C₂×D₄×C₄₇	C₂²×C₄	1	8	yes	—
143	C₂²×C₄×C₄₇	—	2	8	no	—
144	C₂²×C₄×C₄₇	—	2	4	no	—
145	C₂²×C₄×C₄₇	—	2	8	no	—
146	C₂²×C₄×C₄₇	—	2	8	no	—
147	C₂²×C₄×C₄₇	—	2	16	no	—
148	C₂²×C₄×C₄₇	—	2	8	no	—
149	C₂²×C₄×C₄₇	—	2	16	no	—
150	C₂²×C₄×C₄₇	—	4	2	no	—
151	C₂²×C₄×C₄₇	—	4	8	no	—
152	C₂²×C₄×C₄₇	—	4	4	no	—
153	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
154	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
155	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₄₇	#41	1	4	yes	—
159	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
170	C₄⋊C₄×C₄₇	—	2	16	no	—
171	C₄⋊C₄×C₄₇	—	2	8	no	—
172	C₄⋊C₄×C₄₇	—	2	4	no	—
173	C₄⋊C₄×C₄₇	—	2	4	no	—
174	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
175	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
176	C₄⋊C₄×C₄₇	C₂×D₄	1	8	yes	—
177	C₄⋊C₄×C₄₇	#41	1	8	yes	complemented
178	C₂²⋊C₄×C₄₇	—	2	4	no	—
179	C₂²⋊C₄×C₄₇	—	2	16	no	—
180	C₂²⋊C₄×C₄₇	—	2	16	no	—
181	C₂²⋊C₄×C₄₇	—	4	8	no	—
182	C₂²⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
183	C₂²⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
184	C₂²⋊C₄×C₄₇	C₂×D₄	1	8	yes	—
185	C₄²×C₄₇	—	2	4	no	—
186	C₄²×C₄₇	—	2	4	no	—
187	C₄²×C₄₇	—	2	16	no	—
188	C₄²×C₄₇	—	2	8	no	—
189	C₄²×C₄₇	—	2	4	no	—
190	C₄²×C₄₇	—	2	8	no	—
191	C₄²×C₄₇	—	4	4	no	—
192	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
193	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
194	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
195	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
196	C₄²×C₄₇	C₂×D₄	1	8	yes	—
197	C₂³×C₄₇	—	2	2	no	—
198	C₂³×C₄₇	—	2	2	no	—
199	C₂³×C₄₇	—	2	4	no	—
200	C₂³×C₄₇	—	2	8	no	—
201	C₂³×C₄₇	—	2	4	no	—
202	C₂³×C₄₇	—	2	8	no	—
203	C₂³×C₄₇	C₂³×C₄	1	8	yes	—
204	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
205	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
206	C₂³×C₄₇	C₂×#41	1	2	yes	—
207	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
208	Q₈×C₄₇	—	2	16	no	—
209	D₄×C₄₇	—	2	32	no	—
210	D₄×C₄₇	—	2	16	no	—
211	D₄×C₄₇	—	2	16	no	—
212	D₄×C₄₇	—	4	8	no	—
213	C₂×C₄×C₄₇	—	2	4	no	—
214	C₂×C₄×C₄₇	—	2	4	no	—
215	C₂×C₄×C₄₇	—	2	4	no	—
216	C₂×C₄×C₄₇	—	2	2	no	—
217	C₂×C₄×C₄₇	—	2	4	no	—
218	C₂×C₄×C₄₇	—	2	2	no	—
219	C₂×C₄×C₄₇	—	2	4	no	—
220	C₂×C₄×C₄₇	—	2	4	no	—
221	C₂×C₄×C₄₇	—	2	2	no	—
222	C₂×C₄×C₄₇	—	2	32	no	—
223	C₂×C₄×C₄₇	—	2	8	no	—
224	C₂×C₄×C₄₇	—	2	4	no	—
225	C₂×C₄×C₄₇	—	2	4	no	—
226	C₂×C₄×C₄₇	—	2	4	no	—
227	C₂×C₄×C₄₇	—	2	4	no	—
228	C₂×C₄×C₄₇	—	2	4	no	—
229	C₂×C₄×C₄₇	—	2	4	no	—
230	C₂×C₄×C₄₇	—	4	4	no	—
231	C₂×C₄×C₄₇	—	4	32	no	—
232	C₂×C₄×C₄₇	—	4	16	no	—
233	C₂×C₄×C₄₇	C₄×D₄	1	16	yes	complemented
234	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
235	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
236	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
237	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
238	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
239	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
240	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
241	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
242	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
245	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
246	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
247	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
248	#34141	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₂²×C₄₇	—	2	2	no	—
250	C₂²×C₄₇	—	2	2	no	—
251	C₂²×C₄₇	—	2	4	no	—
252	C₂²×C₄₇	—	2	4	no	—
253	C₂²×C₄₇	—	2	8	no	—
254	C₂²×C₄₇	—	2	8	no	—
255	C₂²×C₄₇	—	2	8	no	—
256	C₂²×C₄₇	—	4	8	no	—
257	C₂²×C₄₇	—	4	8	no	—
258	C₂²×C₄₇	C₂×C₄×D₄	1	4	yes	—
259	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
260	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
261	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
262	C₂²×C₄₇	C₂²×#41	1	2	yes	—
263	C₂²×C₄₇	#585	1	2	yes	—
264	C₄×C₄₇	—	2	8	no	—
265	C₄×C₄₇	—	2	2	no	—
266	C₄×C₄₇	—	2	2	no	—
267	C₄×C₄₇	—	2	2	no	—
268	C₄×C₄₇	—	2	8	no	—
269	C₄×C₄₇	—	2	8	no	—
270	C₄×C₄₇	—	4	4	no	—
271	C₄×C₄₇	—	4	4	no	—
272	C₄×C₄₇	C₂×C₄×D₄	1	4	yes	—
273	C₄×C₄₇	C₂×C₄×D₄	1	4	yes	—
274	C₄×C₄₇	C₄×#41	1	4	yes	complemented
275	C₂×#532	C₂×C₄₇	1	1	yes	characteristic
276	C₂×C₄×#41	C₂×C₄₇	1	2	yes	complemented
277	#2367	C₂×C₄₇	1	4	yes	complemented
278	#2366	C₂×C₄₇	1	2	yes	complemented
279	#2365	C₂×C₄₇	1	2	yes	complemented
280	#2364	C₂×C₄₇	1	4	yes	complemented
281	#2361	C₂×C₄₇	1	4	yes	complemented
282	#2297	C₂×C₄₇	1	2	yes	complemented
283	#2293	C₂×C₄₇	1	2	yes	complemented
284	#2290	C₂×C₄₇	1	4	yes	complemented
285	C₄²×D₄	C₂×C₄₇	1	2	yes	complemented
286	C₄×#117	C₂×C₄₇	1	1	yes	characteristic, complemented
287	C₂×C₄×C₄⋊C₄	C₂×C₄₇	1	1	yes	characteristic, complemented
288	C₂×C₄₇	—	2	4	no	—
289	C₂×C₄₇	—	4	8	no	—
290	C₂×C₄₇	C₂²×C₄×D₄	1	1	yes	characteristic
291	C₂×C₄₇	C₂×C₄×#41	1	1	yes	characteristic
292	C₂×C₄₇	#3428	1	1	yes	characteristic
293	C₂×C₄₇	D₄×#41	1	2	yes	—
294	C₂×C₄₇	#3480	1	2	yes	—
295	C₂²×#41	C₂²×C₄₇	1	2	yes	—
296	#532	C₄×C₄₇	1	16	yes	complemented
297	C₂×C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
298	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic
299	C₂×#124	C₂²×C₄₇	1	2	yes	—
300	C₂×#121	C₂²×C₄₇	1	2	yes	—
301	C₂×#121	C₂²×C₄₇	1	2	yes	—
302	C₄×#41	—	2	16	no	—
303	C₂×C₄×Q₈	C₂²×C₄₇	1	4	yes	—
304	C₂×C₄×D₄	C₂²×C₄₇	1	8	yes	—
305	C₂×C₄×D₄	C₂²×C₄₇	1	4	yes	—
306	C₂×C₄×D₄	C₂²×C₄₇	1	4	yes	—
307	C₂×C₄×D₄	C₂²×C₄₇	1	2	yes	—
308	C₂×C₄×D₄	C₂²×C₄₇	1	2	yes	—
309	C₂×#117	C₂²×C₄₇	1	4	yes	—
310	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
311	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
312	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
313	C₂²×C₄²	C₂²×C₄₇	1	2	yes	—
314	C₂²×C₄²	C₂²×C₄₇	1	4	yes	—
315	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
316	#391	C₂²×C₄₇	1	4	yes	—
317	#390	C₂²×C₄₇	1	4	yes	—
318	#390	C₂²×C₄₇	1	4	yes	—
319	#389	C₂²×C₄₇	1	4	yes	—
320	#388	C₂²×C₄₇	1	8	yes	—
321	#388	C₂²×C₄₇	1	8	yes	—
322	#387	C₂²×C₄₇	1	8	yes	—
323	#386	C₂²×C₄₇	1	8	yes	—
324	#386	C₂²×C₄₇	1	8	yes	—
325	#385	C₂²×C₄₇	1	8	yes	—
326	C₄×C₄⋊C₄	C₂²×C₄₇	1	4	yes	—
327	C₄×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
328	C₄×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
329	C₄×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
330	C₄×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
331	C₄×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
332	C₄×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
333	C₄×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
334	C₄×C₂²⋊C₄	C₂²×C₄₇	1	2	yes	—
335	C₄×C₂²⋊C₄	C₂²×C₄₇	1	2	yes	—
336	C₄×C₂²⋊C₄	C₂²×C₄₇	1	8	yes	—
337	C₄×C₂²⋊C₄	C₂²×C₄₇	1	4	yes	—
338	C₄×C₂²⋊C₄	C₂²×C₄₇	1	4	yes	—
339	#376	C₂²×C₄₇	1	2	yes	—
340	C₂×#95	C₂²×C₄₇	1	2	yes	—
341	C₄³	C₂²×C₄₇	1	2	yes	—
342	C₄₇	#34141	1	1	yes	characteristic, direct product factor, Sylow
343	C₂×#41	—	2	8	no	—
344	C₂×#41	—	2	8	no	—
345	C₂×#41	C₂×C₄×C₄₇	1	16	yes	complemented
346	C₂²×Q₈	C₂³×C₄₇	1	2	yes	—
347	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
348	C₂²×D₄	C₂³×C₄₇	1	4	yes	—
349	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
350	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
351	C₂³×C₄	C₂³×C₄₇	1	4	yes	—
352	C₂³×C₄	C₂³×C₄₇	1	8	yes	—
353	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₄₇	1	4	yes	—
355	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
357	C₄⋊Q₈	C₂×C₄×C₄₇	1	8	yes	complemented
358	#127	C₂×C₄×C₄₇	1	8	yes	complemented
359	#124	C₂×C₄×C₄₇	1	16	yes	complemented
360	#121	C₂×C₄×C₄₇	1	16	yes	complemented
361	#121	C₂×C₄×C₄₇	1	16	yes	complemented
362	C₄×Q₈	—	2	16	no	—
363	C₄×D₄	—	2	16	no	—
364	C₄×D₄	—	2	16	no	—
365	C₄×D₄	—	2	32	no	—
366	C₄×D₄	C₂×C₄×C₄₇	1	16	yes	complemented
367	C₄×D₄	C₂×C₄×C₄₇	1	16	yes	complemented
368	#117	—	2	4	no	—
369	#117	—	2	16	no	—
370	#117	D₄×C₄₇	1	8	yes	complemented
371	C₂×C₄⋊C₄	—	2	8	no	—
372	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
373	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
374	C₂×C₄⋊C₄	C₂³×C₄₇	1	8	yes	—
375	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
379	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₄₇	1	8	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₄₇	1	2	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₄₇	1	8	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₄₇	1	8	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₄₇	1	4	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₄₇	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₄₇	1	2	yes	—
387	C₂×C₄²	—	2	8	no	—
388	C₂×C₄²	—	2	16	no	—
389	C₂×C₄²	C₂×C₄×C₄₇	1	8	yes	complemented
390	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
391	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
392	C₂×C₄²	C₂³×C₄₇	1	8	yes	—
393	C₂×C₄²	C₂³×C₄₇	1	4	yes	—
394	C₂×C₄²	C₂³×C₄₇	1	4	yes	—
395	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
396	C₂×C₄²	C₂³×C₄₇	1	4	yes	—
397	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
400	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
402	#95	C₂³×C₄₇	1	2	yes	—
403	#95	C₂³×C₄₇	1	4	yes	—
404	#95	C₂³×C₄₇	1	2	yes	—
405	#95	C₂³×C₄₇	1	4	yes	—
406	#95	C₂³×C₄₇	1	4	yes	—
407	#95	C₂³×C₄₇	1	8	yes	—
408	#95	C₂³×C₄₇	1	8	yes	—
409	C₂⁴	C₂⁴×C₄₇	1	4	yes	—
410	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
411	#41	—	2	64	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	4	no	—
414	C₂×Q₈	C₂²×C₄×C₄₇	1	8	yes	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	4	4	no	—
421	C₂×D₄	C₂²×C₄×C₄₇	1	8	yes	—
422	C₂×D₄	C₂²×C₄×C₄₇	1	16	yes	—
423	C₂×D₄	C₂²×C₄×C₄₇	1	8	yes	—
424	C₂×D₄	C₂²×C₄×C₄₇	1	8	yes	—
425	C₂²×C₄	—	2	8	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	8	no	—
428	C₂²×C₄	—	2	8	no	—
429	C₂²×C₄	—	2	16	no	—
430	C₂²×C₄	—	2	8	no	—
431	C₂²×C₄	—	2	16	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	8	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
436	C₂²×C₄	C₂²×C₄×C₄₇	1	16	yes	—
437	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
438	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
439	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
440	C₂²×C₄	#41×C₄₇	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₄₇	1	4	yes	—
442	C₂²×C₄	C₂⁴×C₄₇	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
444	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₄₇	1	4	yes	—
447	C₂²×C₄	C₂⁴×C₄₇	1	4	yes	—
448	C₂²×C₄	C₂⁴×C₄₇	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
450	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
452	C₄⋊C₄	—	2	16	no	—
453	C₄⋊C₄	—	2	8	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	C₂²×C₄×C₄₇	1	8	yes	—
457	C₄⋊C₄	C₂²×C₄×C₄₇	1	8	yes	—
458	C₄⋊C₄	C₂×D₄×C₄₇	1	8	yes	—
459	C₄⋊C₄	#41×C₄₇	1	8	yes	complemented
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	16	no	—
462	C₂²⋊C₄	—	2	16	no	—
463	C₂²⋊C₄	—	4	8	no	—
464	C₂²⋊C₄	C₂²×C₄×C₄₇	1	16	yes	—
465	C₂²⋊C₄	C₂²×C₄×C₄₇	1	16	yes	—
466	C₂²⋊C₄	C₂×D₄×C₄₇	1	8	yes	—
467	C₄²	—	2	4	no	—
468	C₄²	—	2	4	no	—
469	C₄²	—	2	16	no	—
470	C₄²	—	2	8	no	—
471	C₄²	—	2	4	no	—
472	C₄²	—	2	8	no	—
473	C₄²	—	4	4	no	—
474	C₄²	C₂²×C₄×C₄₇	1	4	yes	—
475	C₄²	C₂²×C₄×C₄₇	1	4	yes	—
476	C₄²	C₂²×C₄×C₄₇	1	4	yes	—
477	C₄²	C₂²×C₄×C₄₇	1	4	yes	—
478	C₄²	C₂×D₄×C₄₇	1	8	yes	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	4	no	—
482	C₂³	—	2	8	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	8	no	—
485	C₂³	C₂³×C₄×C₄₇	1	8	yes	—
486	C₂³	C₂³×C₄×C₄₇	1	2	yes	—
487	C₂³	C₂²×D₄×C₄₇	1	2	yes	—
488	C₂³	C₂×#41×C₄₇	1	2	yes	—
489	C₂³	C₂⁵×C₄₇	1	1	yes	characteristic
490	Q₈	—	2	16	no	—
491	D₄	—	2	32	no	—
492	D₄	—	2	16	no	—
493	D₄	—	2	16	no	—
494	D₄	—	4	8	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	32	no	—
505	C₂×C₄	—	2	8	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	4	4	no	—
513	C₂×C₄	—	4	32	no	—
514	C₂×C₄	—	4	16	no	—
515	C₂×C₄	C₄×D₄×C₄₇	1	16	yes	complemented
516	C₂×C₄	C₂³×C₄×C₄₇	1	2	yes	—
517	C₂×C₄	C₂³×C₄×C₄₇	1	2	yes	—
518	C₂×C₄	C₂³×C₄×C₄₇	1	2	yes	—
519	C₂×C₄	C₂³×C₄×C₄₇	1	8	yes	—
520	C₂×C₄	C₂³×C₄×C₄₇	1	2	yes	—
521	C₂×C₄	C₂³×C₄×C₄₇	1	2	yes	—
522	C₂×C₄	C₂³×C₄×C₄₇	1	2	yes	—
523	C₂×C₄	C₂²×D₄×C₄₇	1	8	yes	—
524	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
525	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
526	C₂×C₄	C₂×#41×C₄₇	1	8	yes	—
527	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
528	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
529	C₂×C₄	C₂×#41×C₄₇	1	4	yes	—
530	C₂²	—	2	2	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	2	4	no	—
534	C₂²	—	2	8	no	—
535	C₂²	—	2	8	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	4	8	no	—
538	C₂²	—	4	8	no	—
539	C₂²	C₂×C₄×D₄×C₄₇	1	4	yes	—
540	C₂²	C₂⁴×C₄×C₄₇	1	1	yes	characteristic
541	C₂²	C₂³×D₄×C₄₇	1	1	yes	characteristic
542	C₂²	C₂²×#41×C₄₇	1	1	yes	characteristic
543	C₂²	C₂²×#41×C₄₇	1	2	yes	—
544	C₂²	C₄₇×#585	1	2	yes	—
545	C₄	—	2	8	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	2	no	—
548	C₄	—	2	2	no	—
549	C₄	—	2	8	no	—
550	C₄	—	2	8	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	4	no	—
553	C₄	C₂×C₄×D₄×C₄₇	1	4	yes	—
554	C₄	C₂×C₄×D₄×C₄₇	1	4	yes	—
555	C₄	C₄×#41×C₄₇	1	4	yes	complemented
556	C₂	—	2	4	no	—
557	C₂	—	4	8	no	—
558	C₂	C₂²×C₄×D₄×C₄₇	1	1	yes	characteristic
559	C₂	C₂×C₄×#41×C₄₇	1	1	yes	characteristic
560	C₂	C₄₇×#3428	1	1	yes	characteristic
561	C₂	D₄×#41×C₄₇	1	2	yes	—
562	C₂	C₄₇×#3480	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₄₇
Frattini, cosocle	C₂³	C₂⁵×C₄₇
socle	C₂³×C₄₇	C₂⁵
prime radical	C₂×C₄₇×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#41×C₄₇	C₂	semidirect
3. C₄₇×#2367	C₂	semidirect
4. C₄₇×#2366	C₂	semidirect
5. C₄₇×#2365	C₂	semidirect
6. C₄₇×#2364	C₂	semidirect
7. C₄₇×#2361	C₂	semidirect
8. C₄₇×#2297	C₂	semidirect
9. C₄₇×#2293	C₂	semidirect
10. C₄₇×#2290	C₂	semidirect
11. C₄²×D₄×C₄₇	C₂	semidirect
12. C₄×#117×C₄₇	C₂	semidirect
13. C₂×C₄×C₄⋊C₄×C₄₇	C₂	semidirect
15. C₄₇×#532	C₄	semidirect
63. C₂×#41×C₄₇	C₂×C₄	semidirect
75. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
76. #127×C₄₇	C₂×C₄	semidirect
77. #124×C₄₇	C₂×C₄	semidirect
78. #121×C₄₇	C₂×C₄	semidirect
79. #121×C₄₇	C₂×C₄	semidirect
84. C₄×D₄×C₄₇	C₂×C₄	semidirect
85. C₄×D₄×C₄₇	C₂×C₄	semidirect
88. #117×C₄₇	D₄	semidirect
107. C₂×C₄²×C₄₇	C₂×C₄	semidirect
177. C₄⋊C₄×C₄₇	#41	semidirect
233. C₂×C₄×C₄₇	C₄×D₄	semidirect
248. #34141	C₄₇	direct (N × H)
274. C₄×C₄₇	C₄×#41	semidirect
276. C₂×C₄×#41	C₂×C₄₇	semidirect
277. #2367	C₂×C₄₇	semidirect
278. #2366	C₂×C₄₇	semidirect
279. #2365	C₂×C₄₇	semidirect
280. #2364	C₂×C₄₇	semidirect
281. #2361	C₂×C₄₇	semidirect
282. #2297	C₂×C₄₇	semidirect
283. #2293	C₂×C₄₇	semidirect
284. #2290	C₂×C₄₇	semidirect
285. C₄²×D₄	C₂×C₄₇	semidirect
286. C₄×#117	C₂×C₄₇	semidirect
287. C₂×C₄×C₄⋊C₄	C₂×C₄₇	semidirect
296. #532	C₄×C₄₇	semidirect
342. C₄₇	#34141	direct (N × H)
345. C₂×#41	C₂×C₄×C₄₇	semidirect
357. C₄⋊Q₈	C₂×C₄×C₄₇	semidirect
358. #127	C₂×C₄×C₄₇	semidirect
359. #124	C₂×C₄×C₄₇	semidirect
360. #121	C₂×C₄×C₄₇	semidirect
361. #121	C₂×C₄×C₄₇	semidirect
366. C₄×D₄	C₂×C₄×C₄₇	semidirect
367. C₄×D₄	C₂×C₄×C₄₇	semidirect
370. #117	D₄×C₄₇	semidirect
389. C₂×C₄²	C₂×C₄×C₄₇	semidirect
459. C₄⋊C₄	#41×C₄₇	semidirect
515. C₂×C₄	C₄×D₄×C₄₇	semidirect
555. C₄	C₄×#41×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#532	2
C₂×C₄×#41×C₄₇	2
C₄₇×#2367	2
C₄₇×#2366	2
C₄₇×#2365	2
C₄₇×#2364	2
C₄₇×#2361	2
C₄₇×#2297	2
C₄₇×#2293	2
C₄₇×#2290	2
C₄²×D₄×C₄₇	2
C₄×#117×C₄₇	2
C₂×C₄×C₄⋊C₄×C₄₇	2
#34141	47

Subgroups — C47×#34141 #26073839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#34141 #26073839