Subgroups — #427×Dic₄₇ #26214330

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#117×Dic₄₇	C₂	1	2	yes	maximal
4	#107×Dic₄₇	C₂	1	4	yes	maximal, complemented
5	#106×Dic₄₇	C₂	1	4	yes	maximal, complemented
6	#20243104	C₂	1	2	yes	maximal
7	C₂×C₄₇×#427	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#13093983	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15689161	C₄	1	8	yes	complemented
12	C₂×#13093958	C₂²	1	2	yes	—
13	C₂×#13093956	C₂²	1	2	yes	—
14	C₂×C₈×Dic₄₇	C₄	1	8	yes	complemented
15	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
17	C₄⋊C₄×Dic₄₇	C₂²	1	8	yes	—
18	C₂²⋊C₄×Dic₄₇	—	2	4	no	—
19	#15688969	—	2	2	no	—
20	C₄²×Dic₄₇	—	2	2	no	—
21	#15688885	C₂²	1	8	yes	—
22	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₄₇	C₂²	1	2	yes	—
24	C₄₇×#427	C₄	1	8	yes	complemented
25	C₂×#107×C₄₇	C₂²	1	2	yes	—
26	C₂×#106×C₄₇	C₂²	1	2	yes	—
27	C₂³×Dic₄₇	D₄	1	2	yes	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	#13093983	—	2	4	no	—
33	#13093983	C₂×C₄	1	8	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	—	2	8	no	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
42	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
43	#13093958	C₂×C₄	1	8	yes	complemented
44	#13093956	C₂×C₄	1	8	yes	complemented
45	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
46	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
47	C₄₇⋊#95	—	2	8	no	—
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	#117×C₄₇	—	2	4	no	—
55	#117×C₄₇	C₂×C₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
57	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄²×C₄₇	—	2	2	no	—
59	#107×C₄₇	C₂×C₄	1	8	yes	complemented
60	#106×C₄₇	C₂×C₄	1	8	yes	complemented
61	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	8	no	—
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	C₄⋊C₄	1	8	yes	—
74	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
75	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₄²	1	16	yes	complemented
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄⋊C₄×C₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
85	C₂²⋊C₄×C₄₇	—	2	8	no	—
86	C₄²×C₄₇	—	2	8	no	—
87	C₂²⋊Dic₄₇	—	2	8	no	—
88	C₄⋊Dic₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
90	C₄×Dic₄₇	—	2	8	no	—
91	C₄×Dic₄₇	—	4	16	no	—
92	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
93	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
94	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
95	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
96	C₂×C₄₇⋊C₈	—	2	8	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
100	C₂³×C₄₇	—	2	4	no	—
101	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂³×C₄₇	C₄×D₄	1	2	yes	—
103	C₂³×C₄₇	C₄×D₄	1	2	yes	—
104	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	—	2	8	no	—
106	C₂×C₄×C₄₇	—	4	8	no	—
107	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₇	#117	1	4	yes	—
110	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₄₇	—	2	4	no	—
121	C₂×Dic₄₇	—	2	16	no	—
122	C₂×Dic₄₇	—	2	8	no	—
123	C₂×Dic₄₇	—	4	8	no	—
124	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×Dic₄₇	#135	1	8	yes	—
127	C₈×C₄₇	C₂×C₄²	1	4	yes	—
128	C₈×C₄₇	C₂×C₄²	1	4	yes	—
129	C₄₇⋊C₈	—	2	4	no	—
130	C₄×#427	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	—	2	4	no	—
133	C₂²×C₄₇	—	2	4	no	—
134	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
135	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#135	1	4	yes	—
139	C₄×C₄₇	—	4	8	no	—
140	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
141	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
143	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
145	Dic₄₇	—	2	4	no	—
146	Dic₄₇	—	4	8	no	—
147	Dic₄₇	#427	1	8	yes	direct product factor
148	C₂×#427	—	47	1	no	—
149	C₂×#427	D₄₇	1	1	yes	characteristic
150	C₂²×C₄×C₈	—	47	1	no	—
151	C₄×#117	—	47	2	no	—
152	C₄×#107	—	47	4	no	—
153	C₄×#106	—	47	4	no	—
154	#1735	—	47	2	no	—
155	C₂×C₄₇	—	2	4	no	—
156	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#427	1	2	yes	—
158	C₂×C₄₇	C₄×#135	1	2	yes	—
159	C₂×C₄₇	C₄×#135	1	2	yes	—
160	C₂³×C₈	—	47	1	no	—
161	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
162	C₂×#117	—	47	2	no	—
163	C₂×#117	C₂×D₄₇	1	2	yes	—
164	C₂²×C₄²	—	47	1	no	—
165	#427	—	47	8	no	—
166	#427	Dic₄₇	1	8	yes	direct product factor
167	C₂×#107	—	47	2	no	—
168	C₂×#107	C₂×D₄₇	1	2	yes	—
169	C₂×#106	—	47	2	no	—
170	C₂×#106	C₂×D₄₇	1	2	yes	—
171	C₂×C₄×C₈	—	47	8	no	—
172	C₂×C₄×C₈	—	47	2	no	—
173	C₂×C₄×C₈	—	47	2	no	—
174	C₄×C₄⋊C₄	—	47	8	no	—
175	C₄×C₂²⋊C₄	—	94	4	no	—
176	#376	—	94	2	no	—
177	C₄³	—	94	2	no	—
178	#340	—	47	8	no	—
179	C₄₇	C₄×#427	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	—	47	2	no	—
181	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
182	C₂²×C₈	—	47	4	no	—
183	C₂²×C₈	—	47	2	no	—
184	C₂²×C₈	—	94	2	no	—
185	C₂²×C₈	—	94	2	no	—
186	C₂²×C₈	C₄×D₄₇	1	4	yes	—
187	C₂²×C₈	C₄×D₄₇	1	4	yes	—
188	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
189	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
191	#117	—	2	4	no	—
192	#117	—	47	8	no	—
193	#117	—	94	4	no	—
194	#117	C₂×Dic₄₇	1	8	yes	—
195	C₂×C₄⋊C₄	—	47	4	no	—
196	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	94	2	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	47	4	no	—
201	C₂×C₄²	—	47	1	no	—
202	C₂×C₄²	—	47	1	no	—
203	C₂×C₄²	—	47	4	no	—
204	C₂×C₄²	—	47	2	no	—
205	C₂×C₄²	—	94	2	no	—
206	C₂×C₄²	—	94	8	no	—
207	#107	—	47	8	no	—
208	#107	C₂×Dic₄₇	1	8	yes	complemented
209	#106	—	47	8	no	—
210	#106	C₂×Dic₄₇	1	8	yes	complemented
211	C₄×C₈	—	47	8	no	—
212	C₄×C₈	—	47	8	no	—
213	#95	—	94	8	no	—
214	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	47	2	no	—
217	C₂²×C₄	—	47	8	no	—
218	C₂²×C₄	—	47	2	no	—
219	C₂²×C₄	—	94	8	no	—
220	C₂²×C₄	—	94	4	no	—
221	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
223	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
224	C₂²×C₄	#10625352	1	2	yes	—
225	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
226	C₂²×C₄	#10625354	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	C₂×C₈	—	47	4	no	—
230	C₂×C₈	—	94	8	no	—
231	C₂×C₈	—	94	2	no	—
232	C₂×C₈	—	94	2	no	—
233	C₂×C₈	C₄×Dic₄₇	1	16	yes	complemented
234	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
237	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
238	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
239	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	47	8	no	—
242	C₄⋊C₄	—	94	4	no	—
243	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
244	C₂²⋊C₄	—	2	8	no	—
245	C₂²⋊C₄	—	94	8	no	—
246	C₄²	—	2	8	no	—
247	C₄²	—	47	4	no	—
248	C₄²	—	47	4	no	—
249	C₄²	—	47	4	no	—
250	C₄²	—	47	4	no	—
251	C₄²	—	94	8	no	—
252	C₄²	—	188	16	no	—
253	C₂³	—	2	4	no	—
254	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
255	C₂³	#13093986	1	2	yes	—
256	C₂³	D₄×Dic₄₇	1	2	yes	—
257	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	47	4	no	—
261	C₂×C₄	—	47	8	no	—
262	C₂×C₄	—	47	4	no	—
263	C₂×C₄	—	94	16	no	—
264	C₂×C₄	—	94	4	no	—
265	C₂×C₄	—	94	8	no	—
266	C₂×C₄	—	188	8	no	—
267	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
268	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
269	C₂×C₄	#13093982	1	4	yes	—
270	C₂×C₄	#13093986	1	2	yes	—
271	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
272	C₂×C₄	#13093989	1	4	yes	—
273	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
274	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
278	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
279	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
280	C₈	—	94	4	no	—
281	C₈	C₂×C₄×Dic₄₇	1	4	yes	—
282	C₈	C₂×C₄×Dic₄₇	1	4	yes	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	C₄⋊C₄×Dic₄₇	1	4	yes	—
287	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
288	C₂²	C₂×#13093986	1	1	yes	characteristic
289	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
290	C₂²	#135×D₄₇	1	4	yes	—
291	C₄	—	4	8	no	—
292	C₄	—	47	8	no	—
293	C₄	—	94	4	no	—
294	C₄	—	188	8	no	—
295	C₄	C₄⋊C₄×Dic₄₇	1	4	yes	—
296	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
297	C₄	C₂×#13093982	1	1	yes	characteristic
298	C₄	C₂×#13093989	1	1	yes	characteristic
299	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
300	C₂	—	2	4	no	—
301	C₂	C₂×C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
302	C₂	#427×D₄₇	1	2	yes	—
303	C₂	#20252159	1	2	yes	—
304	C₂	#135×Dic₄₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#427	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#427	D₄₇
hypocenter	C₄₇	C₄×#427
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #107×Dic₄₇	C₂	semidirect
5. #106×Dic₄₇	C₂	semidirect
11. #15689161	C₄	semidirect
14. C₂×C₈×Dic₄₇	C₄	semidirect
24. C₄₇×#427	C₄	semidirect
43. #13093958	C₂×C₄	semidirect
44. #13093956	C₂×C₄	semidirect
59. #107×C₄₇	C₂×C₄	semidirect
60. #106×C₄₇	C₂×C₄	semidirect
76. C₂×C₈×C₄₇	C₄²	semidirect
147. Dic₄₇	#427	direct (N × H)
166. #427	Dic₄₇	direct (N × H)
179. C₄₇	C₄×#427	semidirect
208. #107	C₂×Dic₄₇	semidirect
210. #106	C₂×Dic₄₇	semidirect
233. C₂×C₈	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
C₂²×C₈×Dic₄₇	2
#117×Dic₄₇	2
#107×Dic₄₇	2
#106×Dic₄₇	2
#20243104	2
C₂×C₄₇×#427	2
C₄×#427	47

Subgroups — #427×Dic47 #26214330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #427×Dic₄₇ #26214330