Subgroups — #26234718

Order 12032.

910 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×Dic₄₇	C₂	1	2	yes	maximal
2	#20246309	C₂	1	2	yes	maximal
3	#20244122	C₂	1	2	yes	maximal
4	#20243658	C₂	1	4	yes	maximal
5	#20243552	C₂	1	1	yes	maximal, characteristic
6	#20243403	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	—
10	Q₁₆×Dic₄₇	—	2	8	no	—
11	C₂×#13093958	C₂²	1	2	yes	—
12	C₂×#13093956	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#13093948	C₂²	1	4	yes	—
15	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
16	#15689022	—	2	4	no	—
17	#15689022	C₂²	1	4	yes	—
18	#15688989	—	2	2	no	—
19	#15688989	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₄₇	—	2	2	no	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	#15688885	C₂²	1	4	yes	—
23	#15688877	C₂²	1	2	yes	—
24	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₄₇	C₄	1	8	yes	complemented
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	Q₈×Dic₄₇	—	2	8	no	—
32	Q₈×Dic₄₇	—	4	4	no	—
33	C₂×C₄⋊Dic₄₇	—	2	4	no	—
34	C₂×C₄⋊Dic₄₇	—	2	4	no	—
35	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
37	C₂×C₄×Dic₄₇	—	2	4	no	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	#13093958	—	2	4	no	—
40	#13093956	—	4	2	no	—
41	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
42	#13093948	—	2	8	no	—
43	C₈×Dic₄₇	—	2	4	no	—
44	C₄₇⋊#95	—	2	2	no	—
45	C₄₇⋊#95	—	2	2	no	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₄₇⋊#95	D₄	1	1	yes	characteristic
48	C₄₇⋊#95	D₄	1	1	yes	characteristic
49	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×Q₁₆×C₄₇	D₄	1	4	yes	—
52	C₂×Q₁₆×C₄₇	Q₈	1	4	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₄×Q₈×C₄₇	—	2	4	no	—
56	C₄×Q₈×C₄₇	C₂×C₄	1	8	yes	—
57	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₄₇	—	2	2	no	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	#107×C₄₇	C₂×C₄	1	4	yes	—
62	#103×C₄₇	C₂×C₄	1	8	yes	—
63	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
64	C₂×Q₈×C₄₇	—	2	2	no	—
65	C₂×Q₈×C₄₇	—	2	2	no	—
66	C₂×Q₈×C₄₇	—	2	2	no	—
67	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
69	C₂×Q₈×C₄₇	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₄₇	—	2	4	no	—
76	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
77	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
78	Q₁₆×C₄₇	C₄⋊C₄	1	16	yes	complemented
79	C₂×C₈×C₄₇	—	2	1	no	—
80	C₂×C₈×C₄₇	—	2	1	no	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
85	C₄⋊C₄×C₄₇	—	2	8	no	—
86	C₄⋊C₄×C₄₇	—	2	2	no	—
87	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
88	C₄²×C₄₇	—	2	8	no	—
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₇	#41	1	2	yes	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	4	8	no	—
95	C₄⋊Dic₄₇	—	4	8	no	—
96	C₄×Dic₄₇	—	2	2	no	—
97	C₄×Dic₄₇	—	2	2	no	—
98	C₄×Dic₄₇	—	4	8	no	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₄₇	—	2	4	no	—
104	Q₈×C₄₇	—	2	4	no	—
105	Q₈×C₄₇	—	2	4	no	—
106	Q₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	2	2	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	#135	1	4	yes	—
131	C₈×C₄₇	—	2	2	no	—
132	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
133	C₄₇⋊C₈	—	4	2	no	—
134	#22685	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₄₇	D₄²	1	1	yes	characteristic
137	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#135	1	2	yes	—
139	C₂²×C₄₇	#578	1	2	yes	—
140	C₄×C₄₇	—	2	8	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	4	4	no	—
144	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₄₇	#520	1	1	yes	characteristic
146	C₄×C₄₇	#547	1	1	yes	characteristic
147	C₄×C₄₇	#554	1	1	yes	characteristic
148	Dic₄₇	—	2	4	no	—
149	Dic₄₇	—	4	2	no	—
150	Dic₄₇	—	8	4	no	—
151	C₂×C₄×Q₁₆	—	47	2	no	—
152	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
153	#2351	—	47	2	no	—
154	#1942	—	47	2	no	—
155	#1865	—	47	4	no	—
156	#1846	—	47	1	no	—
157	#1805	—	47	2	no	—
158	C₈×C₄⋊C₄	—	47	1	no	—
159	C₂×C₄₇	D₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄₇	C₄×#135	1	1	yes	characteristic
161	C₂×C₄₇	#2950	1	1	yes	characteristic
162	C₂×C₄₇	#3285	1	2	yes	—
163	C₂×C₄₇	#3383	1	2	yes	—
164	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
165	C₂×C₄×Q₈	—	47	4	no	—
166	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
167	C₄×Q₁₆	—	94	8	no	—
168	C₄×Q₁₆	Dic₄₇	1	8	yes	complemented
169	C₂×#107	—	47	2	no	—
170	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
171	C₂×#106	—	47	2	no	—
172	C₂×C₄⋊C₈	—	47	1	no	—
173	C₂×#103	—	47	4	no	—
174	C₂×#103	C₂×D₄₇	1	2	yes	—
175	C₂×C₄×C₈	—	47	2	no	—
176	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
177	#389	—	47	4	no	—
178	#389	—	94	4	no	—
179	#382	—	47	2	no	—
180	#382	—	94	2	no	—
181	C₄×C₄⋊C₄	—	47	1	no	—
182	C₄×C₄⋊C₄	—	94	2	no	—
183	#340	—	47	4	no	—
184	#336	—	47	2	no	—
185	C₄₇	#22685	1	1	yes	characteristic, complemented, Sylow
186	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
187	C₂×Q₁₆	Dic₉₄	1	4	yes	—
188	C₂×Q₁₆	D₁₈₈	1	4	yes	—
189	C₂×Q₁₆	C₂×Dic₄₇	1	4	yes	—
190	C₂²×C₈	—	47	2	no	—
191	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
193	C₄×Q₈	—	2	4	no	—
194	C₄×Q₈	—	94	8	no	—
195	C₄×Q₈	—	188	4	no	—
196	C₄×Q₈	C₂×Dic₄₇	1	8	yes	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	47	2	no	—
199	C₂×C₄⋊C₄	—	47	4	no	—
200	C₂×C₄⋊C₄	—	94	4	no	—
201	C₂×C₄⋊C₄	—	94	4	no	—
202	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	47	2	no	—
205	C₂×C₄²	—	94	4	no	—
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	#107	—	94	4	no	—
208	#107	C₂×Dic₄₇	1	4	yes	—
209	#106	—	188	2	no	—
210	C₄⋊C₈	—	94	2	no	—
211	#103	—	94	8	no	—
212	#103	C₂×Dic₄₇	1	8	yes	—
213	C₄×C₈	—	94	4	no	—
214	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
215	#95	—	47	1	no	—
216	#95	—	47	1	no	—
217	#95	—	94	4	no	—
218	#95	—	94	2	no	—
219	#95	—	94	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	C₂×Dic₉₄	1	4	yes	—
224	C₂×Q₈	C₂×D₁₈₈	1	4	yes	—
225	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	47	2	no	—
229	C₂²×C₄	—	47	2	no	—
230	C₂²×C₄	—	94	4	no	—
231	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
234	Q₁₆	C₄⋊Dic₄₇	1	16	yes	complemented
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	94	2	no	—
238	C₂×C₈	—	94	2	no	—
239	C₂×C₈	—	94	2	no	—
240	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
241	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
242	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
243	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	94	4	no	—
247	C₄⋊C₄	—	94	4	no	—
248	C₄⋊C₄	—	94	4	no	—
249	C₄⋊C₄	—	188	8	no	—
250	C₄⋊C₄	—	188	8	no	—
251	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
252	C₄²	—	2	8	no	—
253	C₄²	—	94	2	no	—
254	C₄²	—	94	2	no	—
255	C₄²	—	188	8	no	—
256	C₄²	#10625352	1	2	yes	—
257	C₄²	C₂²×Dic₄₇	1	2	yes	—
258	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
259	Q₈	—	2	4	no	—
260	Q₈	—	2	4	no	—
261	Q₈	—	2	4	no	—
262	Q₈	C₂×C₄⋊Dic₄₇	1	8	yes	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	47	4	no	—
271	C₂×C₄	—	94	2	no	—
272	C₂×C₄	—	94	2	no	—
273	C₂×C₄	—	94	2	no	—
274	C₂×C₄	—	94	2	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	—	188	4	no	—
277	C₂×C₄	—	188	4	no	—
278	C₂×C₄	—	188	4	no	—
279	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
280	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
281	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
283	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
284	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
285	C₂×C₄	#13094070	1	1	yes	characteristic
286	C₂×C₄	#13094071	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	188	2	no	—
289	C₈	C₂×C₄⋊Dic₄₇	1	4	yes	—
290	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
291	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
292	C₂²	D₄×Dic₉₄	1	1	yes	characteristic
293	C₂²	#135×D₄₇	1	2	yes	—
294	C₂²	#15689936	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	8	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	94	4	no	—
300	C₄	—	376	4	no	—
301	C₄	—	188	2	no	—
302	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
303	C₄	#15689606	1	1	yes	characteristic
304	C₄	#15689777	1	1	yes	characteristic
305	C₄	#15689830	1	1	yes	characteristic
306	C₂	D₄×C₄⋊Dic₄₇	1	1	yes	characteristic
307	C₂	#135×Dic₄₇	1	1	yes	characteristic
308	C₂	#20252256	1	1	yes	characteristic
309	C₂	#20255130	1	2	yes	—
310	C₂	#20256279	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	#22685
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×Q₁₆×C₄₇	C₄	semidirect
78. Q₁₆×C₄₇	C₄⋊C₄	semidirect
168. C₄×Q₁₆	Dic₄₇	semidirect
185. C₄₇	#22685	semidirect
234. Q₁₆	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×Q₁₆×Dic₄₇	2
#20246309	2
#20244122	2
#20243658	2
#20243552	2
#20243403	2
C₈×C₄⋊Dic₄₇	2
C₂×C₄×Q₁₆×C₄₇	2
#22685	47