Subgroups — C₂×C₄₇×#3508 #26102496

Order 12032.

8622 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3508	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₄₇×#559	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₇×#550	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#546	C₂	1	8	yes	maximal, complemented
5	C₂×C₄₇×#540	C₂	1	4	yes	maximal, complemented
6	C₂×C₄₇×#535	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#532	C₂	1	4	yes	maximal, complemented
9	C₂²×#127×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄₇×#518	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×#41×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×#41×C₄₇	C₂²	1	8	yes	complemented
14	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
15	C₂³×D₄×C₄₇	C₂²	1	4	yes	—
16	C₄₇×#559	C₂²	1	64	yes	complemented
17	C₄₇×#550	C₂²	1	64	yes	complemented
18	C₄₇×#546	C₂²	1	128	yes	complemented
19	C₄₇×#540	C₂²	1	64	yes	complemented
20	C₄₇×#535	C₂²	1	32	yes	complemented
21	C₄₇×#533	C₂²	1	16	yes	complemented
22	C₄₇×#532	C₂²	1	64	yes	complemented
23	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	complemented
24	C₂×#127×C₄₇	C₂²	1	2	yes	—
25	C₂×#127×C₄₇	C₂²	1	2	yes	—
26	C₂×#127×C₄₇	C₂²	1	4	yes	—
27	C₂×#127×C₄₇	C₂²	1	16	yes	complemented
28	C₂×#127×C₄₇	C₂²	1	4	yes	complemented
29	C₂×#125×C₄₇	C₂²	1	2	yes	complemented
30	C₂×#124×C₄₇	C₂²	1	8	yes	complemented
31	C₂×#124×C₄₇	C₂²	1	4	yes	complemented
32	C₂×#123×C₄₇	C₂²	1	16	yes	complemented
33	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	4	yes	—
34	C₂×#121×C₄₇	C₂²	1	8	yes	complemented
35	C₂×#121×C₄₇	C₂²	1	8	yes	complemented
36	C₂×#121×C₄₇	C₂²	1	16	yes	complemented
37	C₂×#120×C₄₇	C₂²	1	8	yes	—
38	C₄₇×#518	C₂²	1	32	yes	complemented
39	C₄×#41×C₄₇	C₂²	1	16	yes	complemented
40	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	complemented
41	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	complemented
42	C₂×C₄×D₄×C₄₇	C₂²	1	4	yes	—
43	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	complemented
44	C₂×C₄×D₄×C₄₇	C₂²	1	8	yes	complemented
45	C₂×C₄×D₄×C₄₇	C₂²	1	8	yes	complemented
46	C₂×C₄×D₄×C₄₇	C₂²	1	8	yes	complemented
47	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
48	C₂×#117×C₄₇	C₂²	1	2	yes	—
49	C₂×#117×C₄₇	C₂²	1	2	yes	complemented
50	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	8	yes	—
51	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
52	C₂²×C₄²×C₄₇	C₂²	1	2	yes	complemented
53	C₂⁵×C₄₇	C₂³	1	4	yes	—
54	C₂×#41×C₄₇	—	2	8	no	—
55	C₂×#41×C₄₇	—	4	32	no	—
56	C₂×#41×C₄₇	C₂³	1	8	yes	—
57	C₂×#41×C₄₇	C₂³	1	64	yes	complemented
58	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
60	C₂²×D₄×C₄₇	—	2	16	no	—
61	C₂²×D₄×C₄₇	—	4	16	no	—
62	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
63	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
64	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
65	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₄₇	C₂³	1	8	yes	—
67	C₂²×D₄×C₄₇	C₂³	1	16	yes	—
68	C₂²×D₄×C₄₇	C₂³	1	16	yes	—
69	C₂²×D₄×C₄₇	C₂³	1	8	yes	—
70	C₂²×D₄×C₄₇	C₂³	1	32	yes	—
71	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
72	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
73	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
74	C₂³×C₄×C₄₇	C₂³	1	16	yes	—
75	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
76	C₄⋊Q₈×C₄₇	C₂³	1	16	yes	complemented
77	#127×C₄₇	C₂³	1	16	yes	—
78	#127×C₄₇	C₂³	1	16	yes	—
79	#127×C₄₇	C₂³	1	32	yes	—
80	#127×C₄₇	C₂³	1	32	yes	complemented
81	#125×C₄₇	C₂³	1	16	yes	complemented
82	#124×C₄₇	C₂³	1	64	yes	complemented
83	#124×C₄₇	C₂³	1	32	yes	complemented
84	#123×C₄₇	C₂³	1	128	yes	complemented
85	C₂²⋊Q₈×C₄₇	C₂³	1	32	yes	—
86	#121×C₄₇	C₂³	1	64	yes	complemented
87	#121×C₄₇	C₂³	1	64	yes	complemented
88	#121×C₄₇	C₂³	1	128	yes	complemented
89	#120×C₄₇	C₂³	1	64	yes	—
90	C₄×Q₈×C₄₇	C₂³	1	16	yes	complemented
91	C₄×Q₈×C₄₇	C₂³	1	32	yes	complemented
92	C₄×D₄×C₄₇	C₂³	1	32	yes	—
93	C₄×D₄×C₄₇	C₂³	1	16	yes	complemented
94	C₄×D₄×C₄₇	C₂³	1	64	yes	complemented
95	C₄×D₄×C₄₇	C₂³	1	64	yes	complemented
96	C₄×D₄×C₄₇	C₂³	1	64	yes	complemented
97	#117×C₄₇	C₂³	1	8	yes	—
98	#117×C₄₇	C₂³	1	16	yes	—
99	#117×C₄₇	C₂³	1	16	yes	complemented
100	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
105	C₂×C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₄₇	C₂³	1	16	yes	—
108	C₂×C₂²⋊C₄×C₄₇	C₂³	1	64	yes	—
109	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₄₇	C₂³	1	16	yes	—
111	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
114	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
115	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
116	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
117	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
118	C₂×C₄²×C₄₇	C₂³	1	16	yes	complemented
119	C₂⁴×C₄₇	—	2	16	no	—
120	C₂⁴×C₄₇	—	4	16	no	—
121	C₂⁴×C₄₇	C₂⁴	1	8	yes	—
122	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
123	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
124	C₂⁴×C₄₇	C₂⁴	1	16	yes	—
125	#41×C₄₇	—	2	32	no	—
126	#41×C₄₇	—	4	128	no	—
127	C₂×Q₈×C₄₇	—	2	4	no	—
128	C₂×Q₈×C₄₇	—	2	16	no	—
129	C₂×Q₈×C₄₇	C₂⁴	1	4	yes	—
130	C₂×Q₈×C₄₇	C₂⁴	1	16	yes	—
131	C₂×D₄×C₄₇	—	2	8	no	—
132	C₂×D₄×C₄₇	—	2	16	no	—
133	C₂×D₄×C₄₇	—	2	16	no	—
134	C₂×D₄×C₄₇	—	2	4	no	—
135	C₂×D₄×C₄₇	—	2	64	no	—
136	C₂×D₄×C₄₇	—	2	32	no	—
137	C₂×D₄×C₄₇	—	4	64	no	—
138	C₂×D₄×C₄₇	—	4	16	no	—
139	C₂×D₄×C₄₇	—	4	32	no	—
140	C₂×D₄×C₄₇	C₂⁴	1	8	yes	—
141	C₂×D₄×C₄₇	C₂⁴	1	16	yes	—
142	C₂×D₄×C₄₇	C₂⁴	1	16	yes	—
143	C₂×D₄×C₄₇	C₂⁴	1	4	yes	—
144	C₂×D₄×C₄₇	C₂⁴	1	32	yes	—
145	C₂×D₄×C₄₇	C₂⁴	1	64	yes	—
146	C₂×D₄×C₄₇	C₂⁴	1	32	yes	—
147	C₂²×C₄×C₄₇	—	2	8	no	—
148	C₂²×C₄×C₄₇	—	2	8	no	—
149	C₂²×C₄×C₄₇	—	2	16	no	—
150	C₂²×C₄×C₄₇	—	4	32	no	—
151	C₂²×C₄×C₄₇	#41	1	8	yes	complemented
152	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
153	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
154	C₂²×C₄×C₄₇	C₂⁴	1	16	yes	—
155	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
158	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
159	C₂²×C₄×C₄₇	C₂⁴	1	64	yes	—
160	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
161	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
164	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
165	C₄⋊C₄×C₄₇	C₂⁴	1	16	yes	—
166	C₄⋊C₄×C₄₇	C₂⁴	1	16	yes	—
167	C₄⋊C₄×C₄₇	C₂⁴	1	16	yes	—
168	C₄⋊C₄×C₄₇	C₂⁴	1	32	yes	—
169	C₂²⋊C₄×C₄₇	C₂⁴	1	8	yes	—
170	C₂²⋊C₄×C₄₇	C₂⁴	1	16	yes	—
171	C₂²⋊C₄×C₄₇	C₂⁴	1	64	yes	—
172	C₂²⋊C₄×C₄₇	C₂⁴	1	16	yes	—
173	C₂²⋊C₄×C₄₇	C₂⁴	1	64	yes	—
174	C₄²×C₄₇	C₂⁴	1	4	yes	—
175	C₄²×C₄₇	C₂⁴	1	4	yes	—
176	C₄²×C₄₇	C₂⁴	1	8	yes	—
177	C₄²×C₄₇	C₂⁴	1	8	yes	—
178	C₄²×C₄₇	C₂⁴	1	16	yes	—
179	C₄²×C₄₇	C₂⁴	1	16	yes	—
180	C₂³×C₄₇	—	2	4	no	—
181	C₂³×C₄₇	—	2	4	no	—
182	C₂³×C₄₇	—	2	16	no	—
183	C₂³×C₄₇	—	2	8	no	—
184	C₂³×C₄₇	—	2	32	no	—
185	C₂³×C₄₇	—	2	32	no	—
186	C₂³×C₄₇	—	4	32	no	—
187	C₂³×C₄₇	—	8	32	no	—
188	C₂³×C₄₇	C₂×#41	1	4	yes	—
189	C₂³×C₄₇	C₂⁵	1	16	yes	—
190	C₂³×C₄₇	C₂⁵	1	4	yes	—
191	C₂³×C₄₇	C₂⁵	1	2	yes	—
192	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
193	Q₈×C₄₇	—	2	8	no	—
194	Q₈×C₄₇	—	2	32	no	—
195	D₄×C₄₇	—	2	16	no	—
196	D₄×C₄₇	—	2	32	no	—
197	D₄×C₄₇	—	2	32	no	—
198	D₄×C₄₇	—	2	8	no	—
199	D₄×C₄₇	—	2	64	no	—
200	D₄×C₄₇	—	4	32	no	—
201	D₄×C₄₇	—	4	64	no	—
202	C₂×C₄×C₄₇	—	2	4	no	—
203	C₂×C₄×C₄₇	—	2	8	no	—
204	C₂×C₄×C₄₇	—	2	4	no	—
205	C₂×C₄×C₄₇	—	2	16	no	—
206	C₂×C₄×C₄₇	—	2	16	no	—
207	C₂×C₄×C₄₇	—	2	16	no	—
208	C₂×C₄×C₄₇	—	2	32	no	—
209	C₂×C₄×C₄₇	—	4	64	no	—
210	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
211	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
212	C₂×C₄×C₄₇	C₂×#41	1	16	yes	complemented
213	C₂×C₄×C₄₇	H(5,𝔽₂)	1	8	yes	—
214	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
215	C₂×C₄×C₄₇	C₂⁵	1	8	yes	—
216	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
217	C₂×C₄×C₄₇	C₂⁵	1	16	yes	—
218	C₂×C₄×C₄₇	C₂⁵	1	2	yes	—
219	C₂×C₄×C₄₇	C₂⁵	1	2	yes	—
220	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
221	C₂×#3508	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂²×C₄₇	—	2	4	no	—
223	C₂²×C₄₇	—	2	8	no	—
224	C₂²×C₄₇	—	2	4	no	—
225	C₂²×C₄₇	—	2	16	no	—
226	C₂²×C₄₇	—	2	8	no	—
227	C₂²×C₄₇	—	2	32	no	—
228	C₂²×C₄₇	—	4	8	no	—
229	C₂²×C₄₇	—	4	32	no	—
230	C₂²×C₄₇	—	8	32	no	—
231	C₂²×C₄₇	C₂²×#41	1	4	yes	—
232	C₂²×C₄₇	C₂²×#41	1	2	yes	—
233	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	4	yes	—
234	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
235	C₄×C₄₇	—	2	4	no	—
236	C₄×C₄₇	—	2	8	no	—
237	C₄×C₄₇	—	2	4	no	—
238	C₄×C₄₇	—	2	16	no	—
239	C₄×C₄₇	C₂²×#41	1	4	yes	—
240	C₄×C₄₇	C₂²×#41	1	4	yes	—
241	C₄×C₄₇	C₂×H(5,𝔽₂)	1	8	yes	—
242	#3508	C₂×C₄₇	1	32	yes	direct product factor
243	C₂×#559	C₂×C₄₇	1	4	yes	complemented
244	C₂×#550	C₂×C₄₇	1	4	yes	complemented
245	C₂×#546	C₂×C₄₇	1	8	yes	complemented
246	C₂×#540	C₂×C₄₇	1	4	yes	complemented
247	C₂×#535	C₂×C₄₇	1	2	yes	complemented
248	C₂×#533	C₂×C₄₇	1	1	yes	characteristic, complemented
249	C₂×#532	C₂×C₄₇	1	4	yes	complemented
250	C₂²×#127	C₂×C₄₇	1	1	yes	characteristic, complemented
251	C₂×#518	C₂×C₄₇	1	2	yes	complemented
252	C₂×C₄×#41	C₂×C₄₇	1	1	yes	characteristic, complemented
253	C₂×C₄₇	—	2	4	no	—
254	C₂×C₄₇	—	4	4	no	—
255	C₂×C₄₇	—	4	16	no	—
256	C₂×C₄₇	#3508	1	4	yes	direct product factor
257	C₂×C₄₇	C₂³×#41	1	1	yes	characteristic
258	C₂×C₄₇	C₂²×H(5,𝔽₂)	1	2	yes	—
259	C₂²×#41	C₂²×C₄₇	1	1	yes	characteristic
260	C₂²×#41	C₂²×C₄₇	1	8	yes	complemented
261	C₂³×D₄	C₂²×C₄₇	1	2	yes	—
262	C₂³×D₄	C₂²×C₄₇	1	4	yes	—
263	#559	C₂²×C₄₇	1	64	yes	complemented
264	#550	C₂²×C₄₇	1	64	yes	complemented
265	#546	C₂²×C₄₇	1	128	yes	complemented
266	#540	C₂²×C₄₇	1	64	yes	complemented
267	#535	C₂²×C₄₇	1	32	yes	complemented
268	#533	C₂²×C₄₇	1	16	yes	complemented
269	#532	C₂²×C₄₇	1	64	yes	complemented
270	C₂×C₄⋊Q₈	C₂²×C₄₇	1	2	yes	complemented
271	C₂×#127	C₂²×C₄₇	1	2	yes	—
272	C₂×#127	C₂²×C₄₇	1	2	yes	—
273	C₂×#127	C₂²×C₄₇	1	4	yes	—
274	C₂×#127	C₂²×C₄₇	1	16	yes	complemented
275	C₂×#127	C₂²×C₄₇	1	4	yes	complemented
276	C₂×#125	C₂²×C₄₇	1	2	yes	complemented
277	C₂×#124	C₂²×C₄₇	1	8	yes	complemented
278	C₂×#124	C₂²×C₄₇	1	4	yes	complemented
279	C₂×#123	C₂²×C₄₇	1	16	yes	complemented
280	C₂×C₂²⋊Q₈	C₂²×C₄₇	1	4	yes	—
281	C₂×#121	C₂²×C₄₇	1	8	yes	complemented
282	C₂×#121	C₂²×C₄₇	1	8	yes	complemented
283	C₂×#121	C₂²×C₄₇	1	16	yes	complemented
284	C₂×#120	C₂²×C₄₇	1	8	yes	—
285	#518	C₂²×C₄₇	1	32	yes	complemented
286	C₄×#41	C₂²×C₄₇	1	16	yes	complemented
287	C₂×C₄×Q₈	C₂²×C₄₇	1	2	yes	complemented
288	C₂×C₄×Q₈	C₂²×C₄₇	1	4	yes	complemented
289	C₂×C₄×D₄	C₂²×C₄₇	1	4	yes	—
290	C₂×C₄×D₄	C₂²×C₄₇	1	2	yes	complemented
291	C₂×C₄×D₄	C₂²×C₄₇	1	8	yes	complemented
292	C₂×C₄×D₄	C₂²×C₄₇	1	8	yes	complemented
293	C₂×C₄×D₄	C₂²×C₄₇	1	8	yes	complemented
294	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
295	C₂×#117	C₂²×C₄₇	1	2	yes	—
296	C₂×#117	C₂²×C₄₇	1	2	yes	complemented
297	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	8	yes	—
298	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
299	C₂²×C₄²	C₂²×C₄₇	1	2	yes	complemented
300	C₄₇	C₂×#3508	1	1	yes	characteristic, direct product factor, Sylow
301	C₂⁵	C₂³×C₄₇	1	4	yes	—
302	C₂×#41	—	2	8	no	—
303	C₂×#41	—	4	32	no	—
304	C₂×#41	C₂³×C₄₇	1	8	yes	—
305	C₂×#41	C₂³×C₄₇	1	64	yes	complemented
306	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
307	C₂²×Q₈	C₂³×C₄₇	1	4	yes	—
308	C₂²×D₄	—	2	16	no	—
309	C₂²×D₄	—	4	16	no	—
310	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
311	C₂²×D₄	C₂³×C₄₇	1	4	yes	—
312	C₂²×D₄	C₂³×C₄₇	1	4	yes	—
313	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
314	C₂²×D₄	C₂³×C₄₇	1	8	yes	—
315	C₂²×D₄	C₂³×C₄₇	1	16	yes	—
316	C₂²×D₄	C₂³×C₄₇	1	16	yes	—
317	C₂²×D₄	C₂³×C₄₇	1	8	yes	—
318	C₂²×D₄	C₂³×C₄₇	1	32	yes	—
319	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
320	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
321	C₂³×C₄	C₂³×C₄₇	1	4	yes	—
322	C₂³×C₄	C₂³×C₄₇	1	16	yes	—
323	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
324	C₄⋊Q₈	C₂³×C₄₇	1	16	yes	complemented
325	#127	C₂³×C₄₇	1	16	yes	—
326	#127	C₂³×C₄₇	1	16	yes	—
327	#127	C₂³×C₄₇	1	32	yes	—
328	#127	C₂³×C₄₇	1	32	yes	complemented
329	#125	C₂³×C₄₇	1	16	yes	complemented
330	#124	C₂³×C₄₇	1	64	yes	complemented
331	#124	C₂³×C₄₇	1	32	yes	complemented
332	#123	C₂³×C₄₇	1	128	yes	complemented
333	C₂²⋊Q₈	C₂³×C₄₇	1	32	yes	—
334	#121	C₂³×C₄₇	1	64	yes	complemented
335	#121	C₂³×C₄₇	1	64	yes	complemented
336	#121	C₂³×C₄₇	1	128	yes	complemented
337	#120	C₂³×C₄₇	1	64	yes	—
338	C₄×Q₈	C₂³×C₄₇	1	16	yes	complemented
339	C₄×Q₈	C₂³×C₄₇	1	32	yes	complemented
340	C₄×D₄	C₂³×C₄₇	1	32	yes	—
341	C₄×D₄	C₂³×C₄₇	1	16	yes	complemented
342	C₄×D₄	C₂³×C₄₇	1	64	yes	complemented
343	C₄×D₄	C₂³×C₄₇	1	64	yes	complemented
344	C₄×D₄	C₂³×C₄₇	1	64	yes	complemented
345	#117	C₂³×C₄₇	1	8	yes	—
346	#117	C₂³×C₄₇	1	16	yes	—
347	#117	C₂³×C₄₇	1	16	yes	complemented
348	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
349	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
350	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
351	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
352	C₂×C₄⋊C₄	C₂³×C₄₇	1	8	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₄₇	1	2	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₄₇	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₄₇	1	16	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₄₇	1	64	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₄₇	1	4	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₄₇	1	16	yes	—
359	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
360	C₂×C₄²	C₂³×C₄₇	1	4	yes	—
361	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
362	C₂×C₄²	C₂³×C₄₇	1	4	yes	—
363	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
364	C₂×C₄²	C₂³×C₄₇	1	8	yes	—
365	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
366	C₂×C₄²	C₂³×C₄₇	1	16	yes	complemented
367	C₂⁴	—	2	16	no	—
368	C₂⁴	—	4	16	no	—
369	C₂⁴	C₂⁴×C₄₇	1	8	yes	—
370	C₂⁴	C₂⁴×C₄₇	1	2	yes	—
371	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
372	C₂⁴	C₂⁴×C₄₇	1	16	yes	—
373	#41	—	2	32	no	—
374	#41	—	4	128	no	—
375	C₂×Q₈	—	2	4	no	—
376	C₂×Q₈	—	2	16	no	—
377	C₂×Q₈	C₂⁴×C₄₇	1	4	yes	—
378	C₂×Q₈	C₂⁴×C₄₇	1	16	yes	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	16	no	—
381	C₂×D₄	—	2	16	no	—
382	C₂×D₄	—	2	4	no	—
383	C₂×D₄	—	2	64	no	—
384	C₂×D₄	—	2	32	no	—
385	C₂×D₄	—	4	64	no	—
386	C₂×D₄	—	4	16	no	—
387	C₂×D₄	—	4	32	no	—
388	C₂×D₄	C₂⁴×C₄₇	1	8	yes	—
389	C₂×D₄	C₂⁴×C₄₇	1	16	yes	—
390	C₂×D₄	C₂⁴×C₄₇	1	16	yes	—
391	C₂×D₄	C₂⁴×C₄₇	1	4	yes	—
392	C₂×D₄	C₂⁴×C₄₇	1	32	yes	—
393	C₂×D₄	C₂⁴×C₄₇	1	64	yes	—
394	C₂×D₄	C₂⁴×C₄₇	1	32	yes	—
395	C₂²×C₄	—	2	8	no	—
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	16	no	—
398	C₂²×C₄	—	4	32	no	—
399	C₂²×C₄	#41×C₄₇	1	8	yes	complemented
400	C₂²×C₄	C₂⁴×C₄₇	1	2	yes	—
401	C₂²×C₄	C₂⁴×C₄₇	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₄₇	1	2	yes	—
403	C₂²×C₄	C₂⁴×C₄₇	1	8	yes	—
404	C₂²×C₄	C₂⁴×C₄₇	1	8	yes	—
405	C₂²×C₄	C₂⁴×C₄₇	1	8	yes	—
406	C₂²×C₄	C₂⁴×C₄₇	1	16	yes	—
407	C₂²×C₄	C₂⁴×C₄₇	1	64	yes	—
408	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
409	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
410	C₂²×C₄	C₂⁴×C₄₇	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₄₇	1	2	yes	—
412	C₄⋊C₄	C₂⁴×C₄₇	1	8	yes	—
413	C₄⋊C₄	C₂⁴×C₄₇	1	16	yes	—
414	C₄⋊C₄	C₂⁴×C₄₇	1	16	yes	—
415	C₄⋊C₄	C₂⁴×C₄₇	1	16	yes	—
416	C₄⋊C₄	C₂⁴×C₄₇	1	32	yes	—
417	C₂²⋊C₄	C₂⁴×C₄₇	1	8	yes	—
418	C₂²⋊C₄	C₂⁴×C₄₇	1	16	yes	—
419	C₂²⋊C₄	C₂⁴×C₄₇	1	64	yes	—
420	C₂²⋊C₄	C₂⁴×C₄₇	1	16	yes	—
421	C₂²⋊C₄	C₂⁴×C₄₇	1	64	yes	—
422	C₄²	C₂⁴×C₄₇	1	4	yes	—
423	C₄²	C₂⁴×C₄₇	1	4	yes	—
424	C₄²	C₂⁴×C₄₇	1	8	yes	—
425	C₄²	C₂⁴×C₄₇	1	8	yes	—
426	C₄²	C₂⁴×C₄₇	1	16	yes	—
427	C₄²	C₂⁴×C₄₇	1	16	yes	—
428	C₂³	—	2	4	no	—
429	C₂³	—	2	4	no	—
430	C₂³	—	2	16	no	—
431	C₂³	—	2	8	no	—
432	C₂³	—	2	32	no	—
433	C₂³	—	2	32	no	—
434	C₂³	—	4	32	no	—
435	C₂³	—	8	32	no	—
436	C₂³	C₂×#41×C₄₇	1	4	yes	—
437	C₂³	C₂⁵×C₄₇	1	16	yes	—
438	C₂³	C₂⁵×C₄₇	1	4	yes	—
439	C₂³	C₂⁵×C₄₇	1	2	yes	—
440	C₂³	C₂⁵×C₄₇	1	1	yes	characteristic
441	Q₈	—	2	8	no	—
442	Q₈	—	2	32	no	—
443	D₄	—	2	16	no	—
444	D₄	—	2	32	no	—
445	D₄	—	2	32	no	—
446	D₄	—	2	8	no	—
447	D₄	—	2	64	no	—
448	D₄	—	4	32	no	—
449	D₄	—	4	64	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	8	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	16	no	—
454	C₂×C₄	—	2	16	no	—
455	C₂×C₄	—	2	16	no	—
456	C₂×C₄	—	2	32	no	—
457	C₂×C₄	—	4	64	no	—
458	C₂×C₄	C₂×#41×C₄₇	1	4	yes	—
459	C₂×C₄	C₂×#41×C₄₇	1	4	yes	—
460	C₂×C₄	C₂×#41×C₄₇	1	16	yes	complemented
461	C₂×C₄	H(5,𝔽₂)×C₄₇	1	8	yes	—
462	C₂×C₄	C₂⁵×C₄₇	1	4	yes	—
463	C₂×C₄	C₂⁵×C₄₇	1	8	yes	—
464	C₂×C₄	C₂⁵×C₄₇	1	4	yes	—
465	C₂×C₄	C₂⁵×C₄₇	1	16	yes	—
466	C₂×C₄	C₂⁵×C₄₇	1	2	yes	—
467	C₂×C₄	C₂⁵×C₄₇	1	2	yes	—
468	C₂×C₄	C₂⁵×C₄₇	1	4	yes	—
469	C₂²	—	2	4	no	—
470	C₂²	—	2	8	no	—
471	C₂²	—	2	4	no	—
472	C₂²	—	2	16	no	—
473	C₂²	—	2	8	no	—
474	C₂²	—	2	32	no	—
475	C₂²	—	4	32	no	—
476	C₂²	—	4	8	no	—
477	C₂²	—	8	32	no	—
478	C₂²	C₂²×#41×C₄₇	1	4	yes	—
479	C₂²	C₂²×#41×C₄₇	1	2	yes	—
480	C₂²	C₂×H(5,𝔽₂)×C₄₇	1	4	yes	—
481	C₂²	C₂⁶×C₄₇	1	1	yes	characteristic
482	C₄	—	2	4	no	—
483	C₄	—	2	8	no	—
484	C₄	—	2	4	no	—
485	C₄	—	2	16	no	—
486	C₄	C₂²×#41×C₄₇	1	4	yes	—
487	C₄	C₂²×#41×C₄₇	1	4	yes	—
488	C₄	C₂×H(5,𝔽₂)×C₄₇	1	8	yes	—
489	C₂	—	2	4	no	—
490	C₂	—	4	16	no	—
491	C₂	—	4	4	no	—
492	C₂	C₄₇×#3508	1	4	yes	direct product factor
493	C₂	C₂³×#41×C₄₇	1	1	yes	characteristic
494	C₂	C₂²×H(5,𝔽₂)×C₄₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₄₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#3508	C₂	direct (N × H)
2. C₂×C₄₇×#559	C₂	semidirect
3. C₂×C₄₇×#550	C₂	semidirect
4. C₂×C₄₇×#546	C₂	semidirect
5. C₂×C₄₇×#540	C₂	semidirect
6. C₂×C₄₇×#535	C₂	semidirect
7. C₂×C₄₇×#533	C₂	semidirect
8. C₂×C₄₇×#532	C₂	semidirect
9. C₂²×#127×C₄₇	C₂	semidirect
10. C₂×C₄₇×#518	C₂	semidirect
11. C₂×C₄×#41×C₄₇	C₂	semidirect
13. C₂²×#41×C₄₇	C₂²	semidirect
16. C₄₇×#559	C₂²	semidirect
17. C₄₇×#550	C₂²	semidirect
18. C₄₇×#546	C₂²	semidirect
19. C₄₇×#540	C₂²	semidirect
20. C₄₇×#535	C₂²	semidirect
21. C₄₇×#533	C₂²	semidirect
22. C₄₇×#532	C₂²	semidirect
23. C₂×C₄⋊Q₈×C₄₇	C₂²	semidirect
27. C₂×#127×C₄₇	C₂²	semidirect
28. C₂×#127×C₄₇	C₂²	semidirect
29. C₂×#125×C₄₇	C₂²	semidirect
30. C₂×#124×C₄₇	C₂²	semidirect
31. C₂×#124×C₄₇	C₂²	semidirect
32. C₂×#123×C₄₇	C₂²	semidirect
34. C₂×#121×C₄₇	C₂²	semidirect
35. C₂×#121×C₄₇	C₂²	semidirect
36. C₂×#121×C₄₇	C₂²	semidirect
38. C₄₇×#518	C₂²	semidirect
39. C₄×#41×C₄₇	C₂²	semidirect
40. C₂×C₄×Q₈×C₄₇	C₂²	semidirect
41. C₂×C₄×Q₈×C₄₇	C₂²	semidirect
43. C₂×C₄×D₄×C₄₇	C₂²	semidirect
44. C₂×C₄×D₄×C₄₇	C₂²	semidirect
45. C₂×C₄×D₄×C₄₇	C₂²	semidirect
46. C₂×C₄×D₄×C₄₇	C₂²	semidirect
49. C₂×#117×C₄₇	C₂²	semidirect
52. C₂²×C₄²×C₄₇	C₂²	semidirect
57. C₂×#41×C₄₇	C₂³	semidirect
76. C₄⋊Q₈×C₄₇	C₂³	semidirect
80. #127×C₄₇	C₂³	semidirect
81. #125×C₄₇	C₂³	semidirect
82. #124×C₄₇	C₂³	semidirect
83. #124×C₄₇	C₂³	semidirect
84. #123×C₄₇	C₂³	semidirect
86. #121×C₄₇	C₂³	semidirect
87. #121×C₄₇	C₂³	semidirect
88. #121×C₄₇	C₂³	semidirect
90. C₄×Q₈×C₄₇	C₂³	semidirect
91. C₄×Q₈×C₄₇	C₂³	semidirect
93. C₄×D₄×C₄₇	C₂³	semidirect
94. C₄×D₄×C₄₇	C₂³	semidirect
95. C₄×D₄×C₄₇	C₂³	semidirect
96. C₄×D₄×C₄₇	C₂³	semidirect
99. #117×C₄₇	C₂³	semidirect
118. C₂×C₄²×C₄₇	C₂³	semidirect
151. C₂²×C₄×C₄₇	#41	semidirect
212. C₂×C₄×C₄₇	C₂×#41	semidirect
221. C₂×#3508	C₄₇	direct (N × H)
242. #3508	C₂×C₄₇	direct (N × H)
243. C₂×#559	C₂×C₄₇	semidirect
244. C₂×#550	C₂×C₄₇	semidirect
245. C₂×#546	C₂×C₄₇	semidirect
246. C₂×#540	C₂×C₄₇	semidirect
247. C₂×#535	C₂×C₄₇	semidirect
248. C₂×#533	C₂×C₄₇	semidirect
249. C₂×#532	C₂×C₄₇	semidirect
250. C₂²×#127	C₂×C₄₇	semidirect
251. C₂×#518	C₂×C₄₇	semidirect
252. C₂×C₄×#41	C₂×C₄₇	semidirect
256. C₂×C₄₇	#3508	direct (N × H)
260. C₂²×#41	C₂²×C₄₇	semidirect
263. #559	C₂²×C₄₇	semidirect
264. #550	C₂²×C₄₇	semidirect
265. #546	C₂²×C₄₇	semidirect
266. #540	C₂²×C₄₇	semidirect
267. #535	C₂²×C₄₇	semidirect
268. #533	C₂²×C₄₇	semidirect
269. #532	C₂²×C₄₇	semidirect
270. C₂×C₄⋊Q₈	C₂²×C₄₇	semidirect
274. C₂×#127	C₂²×C₄₇	semidirect
275. C₂×#127	C₂²×C₄₇	semidirect
276. C₂×#125	C₂²×C₄₇	semidirect
277. C₂×#124	C₂²×C₄₇	semidirect
278. C₂×#124	C₂²×C₄₇	semidirect
279. C₂×#123	C₂²×C₄₇	semidirect
281. C₂×#121	C₂²×C₄₇	semidirect
282. C₂×#121	C₂²×C₄₇	semidirect
283. C₂×#121	C₂²×C₄₇	semidirect
285. #518	C₂²×C₄₇	semidirect
286. C₄×#41	C₂²×C₄₇	semidirect
287. C₂×C₄×Q₈	C₂²×C₄₇	semidirect
288. C₂×C₄×Q₈	C₂²×C₄₇	semidirect
290. C₂×C₄×D₄	C₂²×C₄₇	semidirect
291. C₂×C₄×D₄	C₂²×C₄₇	semidirect
292. C₂×C₄×D₄	C₂²×C₄₇	semidirect
293. C₂×C₄×D₄	C₂²×C₄₇	semidirect
296. C₂×#117	C₂²×C₄₇	semidirect
299. C₂²×C₄²	C₂²×C₄₇	semidirect
300. C₄₇	C₂×#3508	direct (N × H)
305. C₂×#41	C₂³×C₄₇	semidirect
324. C₄⋊Q₈	C₂³×C₄₇	semidirect
328. #127	C₂³×C₄₇	semidirect
329. #125	C₂³×C₄₇	semidirect
330. #124	C₂³×C₄₇	semidirect
331. #124	C₂³×C₄₇	semidirect
332. #123	C₂³×C₄₇	semidirect
334. #121	C₂³×C₄₇	semidirect
335. #121	C₂³×C₄₇	semidirect
336. #121	C₂³×C₄₇	semidirect
338. C₄×Q₈	C₂³×C₄₇	semidirect
339. C₄×Q₈	C₂³×C₄₇	semidirect
341. C₄×D₄	C₂³×C₄₇	semidirect
342. C₄×D₄	C₂³×C₄₇	semidirect
343. C₄×D₄	C₂³×C₄₇	semidirect
344. C₄×D₄	C₂³×C₄₇	semidirect
347. #117	C₂³×C₄₇	semidirect
366. C₂×C₄²	C₂³×C₄₇	semidirect
399. C₂²×C₄	#41×C₄₇	semidirect
460. C₂×C₄	C₂×#41×C₄₇	semidirect
492. C₂	C₄₇×#3508	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×#3508	2
C₂×C₄₇×#559	2
C₂×C₄₇×#550	2
C₂×C₄₇×#546	2
C₂×C₄₇×#540	2
C₂×C₄₇×#535	2
C₂×C₄₇×#533	2
C₂×C₄₇×#532	2
C₂²×#127×C₄₇	2
C₂×C₄₇×#518	2
C₂×C₄×#41×C₄₇	2
C₂×#3508	47

Subgroups — C2×C47×#3508 #26102496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#3508 #26102496