Subgroups — #20246408

Order 6016.

788 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₄₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689026	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689022	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689016	C₂	1	2	yes	maximal, complemented
7	#15689003	C₂	1	2	yes	maximal, complemented
8	#15688998	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#121×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₄₇	C₂²	1	2	yes	—
13	C₂³×Dic₄₇	C₂²	1	2	yes	—
14	D₄×Dic₄₇	—	2	8	no	—
15	D₄×Dic₄₇	—	2	4	no	—
16	C₂×C₄⋊Dic₄₇	—	2	4	no	—
17	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
29	C₄₇⋊#95	C₂²	1	1	yes	characteristic
30	C₄₇⋊#95	C₂²	1	1	yes	characteristic
31	C₄₇⋊#95	C₂²	1	2	yes	—
32	C₄₇⋊#95	C₂²	1	2	yes	—
33	C₂²×D₄×C₄₇	C₂²	1	2	yes	—
34	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
36	#121×C₄₇	C₄	1	8	yes	complemented
37	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
39	C₂⁴×C₄₇	C₂³	1	2	yes	—
40	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₄₇	—	2	4	no	—
42	C₂×D₄×C₄₇	—	2	4	no	—
43	C₂×D₄×C₄₇	—	2	2	no	—
44	C₂×D₄×C₄₇	—	2	2	no	—
45	C₂×D₄×C₄₇	C₂×C₄	1	8	yes	complemented
46	C₂×D₄×C₄₇	C₂×C₄	1	4	yes	complemented
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	complemented
50	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×Dic₄₇	—	2	4	no	—
54	C₂²×Dic₄₇	—	2	4	no	—
55	C₂²×Dic₄₇	—	2	4	no	—
56	C₂²×Dic₄₇	—	2	4	no	—
57	C₂²×Dic₄₇	—	4	2	no	—
58	C₂²×Dic₄₇	—	4	2	no	—
59	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
60	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
61	C₂²×Dic₄₇	C₂³	1	2	yes	—
62	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
63	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
64	C₂²×Dic₄₇	C₂³	1	2	yes	—
65	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	complemented
66	C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	complemented
67	C₂²⋊Dic₄₇	—	2	2	no	—
68	C₂²⋊Dic₄₇	—	2	4	no	—
69	C₂²⋊Dic₄₇	—	2	4	no	—
70	C₂²⋊Dic₄₇	—	4	2	no	—
71	C₂²⋊Dic₄₇	D₄	1	4	yes	complemented
72	C₄⋊Dic₄₇	—	2	2	no	—
73	C₄⋊Dic₄₇	—	2	4	no	—
74	C₄⋊Dic₄₇	—	2	4	no	—
75	C₄⋊Dic₄₇	D₄	1	4	yes	complemented
76	C₄×Dic₄₇	—	2	2	no	—
77	C₄×Dic₄₇	—	2	2	no	—
78	C₄×Dic₄₇	—	2	4	no	—
79	C₂³×C₄₇	—	2	1	no	—
80	C₂³×C₄₇	—	2	1	no	—
81	C₂³×C₄₇	—	2	1	no	—
82	C₂³×C₄₇	—	2	1	no	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	—	2	2	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	C₂²×C₄	1	4	yes	—
90	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
91	C₂³×C₄₇	C₂×D₄	1	2	yes	—
92	C₂³×C₄₇	#41	1	2	yes	—
93	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
94	D₄×C₄₇	—	2	16	no	—
95	D₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
103	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₄₇	#41	1	2	yes	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	2	1	no	—
108	C₂×Dic₄₇	—	2	1	no	—
109	C₂×Dic₄₇	—	2	1	no	—
110	C₂×Dic₄₇	—	2	2	no	—
111	C₂×Dic₄₇	—	2	1	no	—
112	C₂×Dic₄₇	—	2	1	no	—
113	C₂×Dic₄₇	—	2	2	no	—
114	C₂×Dic₄₇	—	2	2	no	—
115	C₂×Dic₄₇	—	2	1	no	—
116	C₂×Dic₄₇	—	2	1	no	—
117	C₂×Dic₄₇	—	2	1	no	—
118	C₂×Dic₄₇	—	2	1	no	—
119	C₂×Dic₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	2	1	no	—
121	C₂×Dic₄₇	—	4	8	no	—
122	C₂×Dic₄₇	—	4	8	no	—
123	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
124	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
125	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	4	no	—
133	C₂²×C₄₇	—	2	4	no	—
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	—	4	2	no	—
136	C₂²×C₄₇	—	4	2	no	—
137	C₂²×C₄₇	—	4	2	no	—
138	C₂²×C₄₇	—	4	2	no	—
139	C₂²×C₄₇	C₄×D₄	1	4	yes	complemented
140	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
141	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	4	no	—
149	C₄×C₄₇	C₄×D₄	1	4	yes	complemented
150	Dic₄₇	—	2	2	no	—
151	Dic₄₇	—	2	4	no	—
152	Dic₄₇	—	2	2	no	—
153	Dic₄₇	—	4	2	no	—
154	Dic₄₇	—	4	1	no	—
155	Dic₄₇	—	4	1	no	—
156	#2361	—	47	1	no	maximal, Sylow
157	C₂×C₄₇	—	2	4	no	—
158	C₂×C₄₇	—	4	4	no	—
159	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂×C₄₇	#520	1	1	yes	characteristic
162	C₂×C₄₇	D₄²	1	1	yes	characteristic
163	C₂×C₄₇	#547	1	1	yes	characteristic
164	C₂×C₄₇	#548	1	1	yes	characteristic
165	C₂×C₄₇	#553	1	1	yes	characteristic
166	C₂×#121	D₄₇	1	1	yes	characteristic
167	C₂×C₄×D₄	—	47	1	no	—
168	C₂×C₄×D₄	—	47	2	no	—
169	C₂²×C₄⋊C₄	—	47	1	no	—
170	#390	—	47	1	no	—
171	#389	—	47	1	no	—
172	#388	—	47	2	no	—
173	#386	—	47	2	no	—
174	#385	—	47	2	no	—
175	C₄×C₄⋊C₄	—	47	1	no	—
176	C₄×C₂²⋊C₄	—	47	1	no	—
177	C₄₇	#2361	1	1	yes	characteristic, complemented, Sylow
178	C₂²×D₄	C₂×D₄₇	1	2	yes	—
179	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
180	C₂³×C₄	—	47	2	no	—
181	C₂³×C₄	—	47	2	no	—
182	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
183	#121	Dic₄₇	1	8	yes	complemented
184	C₄×D₄	—	94	8	no	—
185	C₄×D₄	—	94	4	no	—
186	C₂×C₄⋊C₄	—	47	1	no	—
187	C₂×C₄⋊C₄	—	47	2	no	—
188	C₂×C₄⋊C₄	—	47	1	no	—
189	C₂×C₄⋊C₄	—	47	2	no	—
190	C₂×C₄⋊C₄	—	94	4	no	—
191	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	—	47	1	no	—
193	C₂×C₂²⋊C₄	—	47	2	no	—
194	C₂×C₂²⋊C₄	—	47	2	no	—
195	C₂×C₂²⋊C₄	—	47	1	no	—
196	C₂×C₂²⋊C₄	—	47	2	no	—
197	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
198	C₂×C₄²	—	47	1	no	—
199	C₂×C₄²	—	47	2	no	—
200	C₂×C₄²	—	47	1	no	—
201	#95	—	47	1	no	—
202	#95	—	47	1	no	—
203	#95	—	47	2	no	—
204	#95	—	47	2	no	—
205	C₂⁴	C₂²×D₄₇	1	2	yes	—
206	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂×Dic₄₇	1	8	yes	complemented
212	C₂×D₄	C₂×Dic₄₇	1	4	yes	complemented
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	47	1	no	—
216	C₂²×C₄	—	47	1	no	—
217	C₂²×C₄	—	47	2	no	—
218	C₂²×C₄	—	47	1	no	—
219	C₂²×C₄	—	47	1	no	—
220	C₂²×C₄	—	47	2	no	—
221	C₂²×C₄	—	94	4	no	—
222	C₂²×C₄	—	94	4	no	—
223	C₂²×C₄	—	94	4	no	—
224	C₂²×C₄	—	94	4	no	—
225	C₂²×C₄	—	188	2	no	—
226	C₂²×C₄	—	188	2	no	—
227	C₂²×C₄	C₂×Dic₄₇	1	4	yes	complemented
228	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
229	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
231	C₄⋊C₄	—	47	4	no	—
232	C₄⋊C₄	—	94	4	no	—
233	C₄⋊C₄	—	94	4	no	—
234	C₄⋊C₄	—	94	2	no	—
235	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	complemented
236	C₂²⋊C₄	—	47	4	no	—
237	C₂²⋊C₄	—	94	4	no	—
238	C₂²⋊C₄	—	94	4	no	—
239	C₂²⋊C₄	—	94	2	no	—
240	C₂²⋊C₄	—	188	2	no	—
241	C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	complemented
242	C₄²	—	94	4	no	—
243	C₄²	—	94	2	no	—
244	C₄²	—	94	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	D₄×D₄₇	1	2	yes	—
256	C₂³	#10625352	1	2	yes	—
257	C₂³	C₂²×Dic₄₇	1	4	yes	—
258	C₂³	C₂²×Dic₄₇	1	2	yes	—
259	C₂³	C₂³×D₄₇	1	1	yes	characteristic
260	D₄	—	2	16	no	—
261	D₄	—	2	8	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	47	2	no	—
268	C₂×C₄	—	47	2	no	—
269	C₂×C₄	—	47	4	no	—
270	C₂×C₄	—	94	2	no	—
271	C₂×C₄	—	94	1	no	—
272	C₂×C₄	—	94	1	no	—
273	C₂×C₄	—	94	1	no	—
274	C₂×C₄	—	94	2	no	—
275	C₂×C₄	—	94	1	no	—
276	C₂×C₄	—	94	1	no	—
277	C₂×C₄	—	94	2	no	—
278	C₂×C₄	—	94	2	no	—
279	C₂×C₄	—	94	1	no	—
280	C₂×C₄	—	94	1	no	—
281	C₂×C₄	—	94	1	no	—
282	C₂×C₄	—	94	1	no	—
283	C₂×C₄	—	94	2	no	—
284	C₂×C₄	—	94	1	no	—
285	C₂×C₄	—	188	8	no	—
286	C₂×C₄	—	188	8	no	—
287	C₂×C₄	D₄×D₄₇	1	2	yes	—
288	C₂×C₄	#10625352	1	2	yes	—
289	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
290	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
291	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	D₄×Dic₄₇	1	4	yes	complemented
306	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
307	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
308	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
309	C₂²	C₂×#10625352	1	1	yes	characteristic
310	C₂²	C₂×#10625352	1	1	yes	characteristic
311	C₂²	#13094070	1	1	yes	characteristic
312	C₂²	#13094071	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	4	no	—
315	C₄	—	94	2	no	—
316	C₄	—	94	4	no	—
317	C₄	—	94	2	no	—
318	C₄	—	188	2	no	—
319	C₄	—	188	1	no	—
320	C₄	—	188	1	no	—
321	C₄	D₄×Dic₄₇	1	4	yes	complemented
322	C₂	—	2	4	no	—
323	C₂	—	4	4	no	—
324	C₂	C₂×D₄×Dic₄₇	1	1	yes	characteristic
325	C₂	C₂×D₄×Dic₄₇	1	1	yes	characteristic
326	C₂	#15689606	1	1	yes	characteristic
327	C₂	#15689757	1	1	yes	characteristic
328	C₂	#15689773	1	1	yes	characteristic
329	C₂	#15689779	1	1	yes	characteristic
330	C₂	#15689825	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×#121×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×#121	D₄₇
hypocenter	C₄₇	#2361
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Dic₄₇	C₂	semidirect
2. C₂×D₄×Dic₄₇	C₂	semidirect
3. C₂²×C₄⋊Dic₄₇	C₂	semidirect
4. #15689026	C₂	semidirect
5. #15689022	C₂	semidirect
6. #15689016	C₂	semidirect
7. #15689003	C₂	semidirect
8. #15688998	C₂	semidirect
9. C₄⋊C₄×Dic₄₇	C₂	semidirect
10. C₄×C₂²⋊Dic₄₇	C₂	semidirect
36. #121×C₄₇	C₄	semidirect
45. C₂×D₄×C₄₇	C₂×C₄	semidirect
46. C₂×D₄×C₄₇	C₂×C₄	semidirect
49. C₂²×C₄×C₄₇	C₂×C₄	semidirect
65. C₄⋊C₄×C₄₇	C₂×C₄	semidirect
66. C₂²⋊C₄×C₄₇	C₂×C₄	semidirect
71. C₂²⋊Dic₄₇	D₄	semidirect
75. C₄⋊Dic₄₇	D₄	semidirect
139. C₂²×C₄₇	C₄×D₄	semidirect
149. C₄×C₄₇	C₄×D₄	semidirect
177. C₄₇	#2361	semidirect
183. #121	Dic₄₇	semidirect
211. C₂×D₄	C₂×Dic₄₇	semidirect
212. C₂×D₄	C₂×Dic₄₇	semidirect
227. C₂²×C₄	C₂×Dic₄₇	semidirect
235. C₄⋊C₄	C₂×Dic₄₇	semidirect
241. C₂²⋊C₄	C₂×Dic₄₇	semidirect
305. C₂²	D₄×Dic₄₇	semidirect
321. C₄	D₄×Dic₄₇	semidirect