Subgroups — #26234577

Order 12032.

1050 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689235	C₂	1	2	yes	maximal
2	#20246298	C₂	1	2	yes	maximal, complemented
3	#20244127	C₂	1	2	yes	maximal, complemented
4	#20243637	C₂	1	4	yes	maximal, complemented
5	#20243552	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243398	C₂	1	2	yes	maximal, complemented
7	#20243255	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×D₄×Dic₄₇	C₂²	1	4	yes	—
10	#15689235	—	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×#13093945	C₂²	1	4	yes	—
15	C₂×#13093934	C₂²	1	2	yes	—
16	#15689022	C₂²	1	4	yes	—
17	#15688998	—	2	4	no	—
18	#15688989	C₂²	1	2	yes	—
19	#15688983	—	2	2	no	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄₇⋊#95	—	2	2	no	—
22	#15688885	C₂²	1	4	yes	—
23	#15688877	C₂²	1	2	yes	—
24	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
26	C₄×D₈×C₄₇	C₄	1	8	yes	complemented
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂³×Dic₄₇	—	2	4	no	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	D₄×Dic₄₇	—	2	8	no	—
33	D₄×Dic₄₇	—	4	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×C₂²⋊Dic₄₇	—	2	4	no	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	#13093958	—	4	2	no	—
40	#13093956	—	2	4	no	—
41	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
42	#13093945	—	2	8	no	—
43	#13093934	—	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	2	yes	—
48	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₄₇	—	2	2	no	—
50	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×D₈×C₄₇	D₄	1	4	yes	—
52	C₂×D₈×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×D₄×C₄₇	—	2	4	no	—
56	C₄×D₄×C₄₇	C₂×C₄	1	8	yes	complemented
57	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	#107×C₄₇	C₂×C₄	1	4	yes	complemented
61	#102×C₄₇	C₂×C₄	1	8	yes	complemented
62	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
63	C₂⁴×C₄₇	—	2	2	no	—
64	C₂×D₄×C₄₇	—	2	2	no	—
65	C₂×D₄×C₄₇	—	2	2	no	—
66	C₂×D₄×C₄₇	—	2	2	no	—
67	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂×D₄×C₄₇	C₂×D₄	1	4	yes	—
69	C₂×D₄×C₄₇	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	8	no	—
72	C₂²×C₄×C₄₇	—	8	4	no	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	4	no	—
77	C₂²×Dic₄₇	—	4	8	no	—
78	C₂²×Dic₄₇	—	4	8	no	—
79	C₂²×Dic₄₇	—	8	4	no	—
80	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
81	D₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
82	C₂×C₈×C₄₇	—	2	1	no	—
83	C₂×C₈×C₄₇	—	2	1	no	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄×C₄₇	—	2	8	no	—
91	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	#41	1	2	yes	—
93	C₂²⋊Dic₄₇	—	4	8	no	—
94	C₄⋊Dic₄₇	—	2	4	no	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	—	4	2	no	—
97	C₄⋊Dic₄₇	—	8	4	no	—
98	C₄×Dic₄₇	—	2	2	no	—
99	C₄×Dic₄₇	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂³×C₄₇	—	2	4	no	—
104	C₂³×C₄₇	—	2	4	no	—
105	C₂³×C₄₇	—	2	4	no	—
106	C₂³×C₄₇	—	4	4	no	—
107	C₂³×C₄₇	—	4	4	no	—
108	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	D₄×C₄₇	—	2	4	no	—
110	D₄×C₄₇	—	2	4	no	—
111	D₄×C₄₇	—	2	4	no	—
112	D₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	8	8	no	—
118	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
126	C₂×Dic₄₇	—	2	4	no	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	8	16	no	—
133	C₈×C₄₇	—	2	2	no	—
134	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
135	C₄₇⋊C₈	—	4	2	no	—
136	#22672	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	2	8	no	—
138	C₂²×C₄₇	—	4	8	no	—
139	C₂²×C₄₇	—	4	4	no	—
140	C₂²×C₄₇	—	4	4	no	—
141	C₂²×C₄₇	—	4	8	no	—
142	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₄₇	D₄²	1	1	yes	characteristic
144	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
145	C₂²×C₄₇	#575	1	2	yes	—
146	C₂²×C₄₇	#576	1	2	yes	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	4	4	no	—
150	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₄₇	#520	1	1	yes	characteristic
152	C₄×C₄₇	#547	1	1	yes	characteristic
153	C₄×C₄₇	#554	1	1	yes	characteristic
154	Dic₄₇	—	4	4	no	—
155	Dic₄₇	—	8	4	no	—
156	C₂×#442	—	47	2	no	—
157	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic
158	#2349	—	47	2	no	—
159	#1943	—	47	2	no	—
160	#1862	—	47	4	no	—
161	#1846	—	47	1	no	—
162	#1804	—	47	2	no	—
163	#1770	—	47	1	no	—
164	C₂×C₄₇	—	4	8	no	—
165	C₂×C₄₇	D₄×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄₇	#2943	1	1	yes	characteristic
167	C₂×C₄₇	#2948	1	1	yes	characteristic
168	C₂×C₄₇	#3273	1	2	yes	—
169	C₂×C₄₇	#3389	1	2	yes	—
170	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
171	C₂×C₄×D₄	—	47	4	no	—
172	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
173	#442	—	94	8	no	—
174	C₄×D₈	Dic₄₇	1	8	yes	complemented
175	C₂×#107	—	47	2	no	—
176	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
177	C₂×#106	—	47	2	no	—
178	C₂×C₄⋊C₈	—	47	1	no	—
179	C₂×#102	—	47	4	no	—
180	C₂×#102	C₂×D₄₇	1	2	yes	—
181	C₂×#97	—	47	2	no	—
182	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
183	#389	—	47	4	no	—
184	#385	—	94	4	no	—
185	#382	—	47	2	no	—
186	#380	—	94	2	no	—
187	C₄×C₄⋊C₄	—	47	1	no	—
188	C₂×#95	—	94	2	no	—
189	#340	—	47	4	no	—
190	#336	—	47	2	no	—
191	C₄₇	#22672	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	—	94	4	no	—
195	C₂×D₈	Dic₉₄	1	4	yes	—
196	C₂×D₈	D₁₈₈	1	4	yes	—
197	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
198	C₂²×C₈	—	47	2	no	—
199	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
201	C₄×D₄	—	2	4	no	—
202	C₄×D₄	—	94	8	no	—
203	C₄×D₄	—	188	4	no	—
204	C₄×D₄	C₂×Dic₄₇	1	8	yes	complemented
205	C₂×C₄⋊C₄	—	47	4	no	—
206	C₂×C₄⋊C₄	—	47	2	no	—
207	C₂×C₄⋊C₄	—	94	4	no	—
208	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₂²⋊C₄	—	94	4	no	—
211	C₂×C₄²	—	47	2	no	—
212	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
213	#107	—	188	2	no	—
214	#107	C₂×Dic₄₇	1	4	yes	complemented
215	#106	—	94	4	no	—
216	C₄⋊C₈	—	94	2	no	—
217	#102	—	94	8	no	—
218	#102	C₂×Dic₄₇	1	8	yes	complemented
219	#97	—	94	4	no	—
220	C₄×C₈	C₂×Dic₄₇	1	4	yes	complemented
221	#95	—	47	2	no	—
222	#95	—	94	4	no	—
223	#95	—	94	2	no	—
224	#95	—	94	2	no	—
225	C₂⁴	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂×Dic₉₄	1	4	yes	—
230	C₂×D₄	C₂×D₁₈₈	1	4	yes	—
231	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	8	4	no	—
235	C₂²×C₄	—	47	4	no	—
236	C₂²×C₄	—	94	4	no	—
237	C₂²×C₄	—	376	4	no	—
238	C₂²×C₄	—	188	8	no	—
239	C₂²×C₄	—	188	8	no	—
240	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
243	D₈	C₄⋊Dic₄₇	1	16	yes	complemented
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	—	94	2	no	—
248	C₂×C₈	—	94	2	no	—
249	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
250	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
251	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
252	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	94	4	no	—
255	C₄⋊C₄	—	94	4	no	—
256	C₄⋊C₄	—	376	4	no	—
257	C₄⋊C₄	—	188	2	no	—
258	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₂²⋊C₄	—	188	8	no	—
261	C₄²	—	94	2	no	—
262	C₄²	—	94	2	no	—
263	C₄²	#10625352	1	2	yes	—
264	C₄²	C₂²×Dic₄₇	1	2	yes	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	4	no	—
269	C₂³	—	4	4	no	—
270	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
271	D₄	—	2	4	no	—
272	D₄	—	2	4	no	—
273	D₄	—	2	4	no	—
274	D₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	8	8	no	—
280	C₂×C₄	—	94	4	no	—
281	C₂×C₄	—	94	4	no	—
282	C₂×C₄	—	94	4	no	—
283	C₂×C₄	—	376	16	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	—	188	4	no	—
287	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
288	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
289	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
291	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
292	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
293	C₂×C₄	#13094070	1	1	yes	characteristic
294	C₂×C₄	#13094071	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	188	2	no	—
297	C₈	C₂×C₄⋊Dic₄₇	1	4	yes	—
298	C₂²	—	2	8	no	—
299	C₂²	—	4	8	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	8	no	—
303	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
304	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
305	C₂²	D₄×Dic₉₄	1	1	yes	characteristic
306	C₂²	#15689920	1	2	yes	—
307	C₂²	#15689924	1	2	yes	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	4	4	no	—
311	C₄	—	376	4	no	—
312	C₄	—	188	4	no	—
313	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
314	C₄	#15689606	1	1	yes	characteristic
315	C₄	#15689777	1	1	yes	characteristic
316	C₄	#15689830	1	1	yes	characteristic
317	C₂	—	4	8	no	—
318	C₂	D₄×C₄⋊Dic₄₇	1	1	yes	characteristic
319	C₂	#20252189	1	1	yes	characteristic
320	C₂	#20252240	1	1	yes	characteristic
321	C₂	#20254998	1	2	yes	—
322	C₂	#20256344	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×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	#22672
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #20246298	C₂	semidirect
3. #20244127	C₂	semidirect
4. #20243637	C₂	semidirect
5. #20243552	C₂	semidirect
6. #20243398	C₂	semidirect
7. #20243255	C₂	semidirect
26. C₄×D₈×C₄₇	C₄	semidirect
56. C₄×D₄×C₄₇	C₂×C₄	semidirect
60. #107×C₄₇	C₂×C₄	semidirect
61. #102×C₄₇	C₂×C₄	semidirect
62. C₄×C₈×C₄₇	C₂×C₄	semidirect
81. D₈×C₄₇	C₄⋊C₄	semidirect
174. C₄×D₈	Dic₄₇	semidirect
191. C₄₇	#22672	semidirect
204. C₄×D₄	C₂×Dic₄₇	semidirect
214. #107	C₂×Dic₄₇	semidirect
218. #102	C₂×Dic₄₇	semidirect
220. C₄×C₈	C₂×Dic₄₇	semidirect
243. D₈	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×#15689235	2
#20246298	2
#20244127	2
#20243637	2
#20243552	2
#20243398	2
#20243255	2
C₂×C₄×D₈×C₄₇	2
#22672	47