Subgroups — #20246317

Order 6016.

660 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13093983	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689016	C₂	1	2	yes	maximal, complemented
4	#15689009	C₂	1	2	yes	maximal, complemented
5	#15689003	C₂	1	2	yes	maximal
6	#15688989	C₂	1	1	yes	maximal, characteristic, complemented
7	#15688983	C₂	1	1	yes	maximal, characteristic, complemented
8	#15688983	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₄×Dic₄₇	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₄₇	C₂²	1	2	yes	—
13	#13093983	—	2	4	no	—
14	C₂×C₄⋊Dic₄₇	—	2	4	no	—
15	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
23	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
24	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
25	C₄₇⋊#95	C₂²	1	2	yes	—
26	C₄₇⋊#95	C₂²	1	2	yes	—
27	C₄₇⋊#95	C₂²	1	2	yes	—
28	C₄₇⋊#95	C₂²	1	2	yes	—
29	C₄₇⋊#95	C₂²	1	2	yes	—
30	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₄₇	C₄	1	8	yes	complemented
34	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
38	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
39	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×D₄×C₄₇	—	2	2	no	—
41	C₂×D₄×C₄₇	—	2	2	no	—
42	C₂×D₄×C₄₇	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₄₇	—	2	2	no	—
44	C₂²×C₄×C₄₇	—	2	2	no	—
45	C₂²×C₄×C₄₇	—	2	2	no	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	complemented
49	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	2	4	no	—
55	C₂²×Dic₄₇	—	2	4	no	—
56	C₂²×Dic₄₇	—	4	2	no	—
57	C₂²×Dic₄₇	C₂³	1	2	yes	—
58	C₂²×Dic₄₇	C₂³	1	2	yes	—
59	C₂²×Dic₄₇	C₂³	1	2	yes	—
60	C₂²×Dic₄₇	C₂³	1	2	yes	—
61	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	complemented
62	C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
63	C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	complemented
64	C₄²×C₄₇	C₂×C₄	1	4	yes	complemented
65	C₂²⋊Dic₄₇	—	2	4	no	—
66	C₂²⋊Dic₄₇	—	2	4	no	—
67	C₂²⋊Dic₄₇	—	4	2	no	—
68	C₄⋊Dic₄₇	—	2	2	no	—
69	C₄⋊Dic₄₇	—	2	2	no	—
70	C₄⋊Dic₄₇	—	2	2	no	—
71	C₄⋊Dic₄₇	—	2	4	no	—
72	C₄⋊Dic₄₇	D₄	1	4	yes	complemented
73	C₄×Dic₄₇	—	2	4	no	—
74	C₄×Dic₄₇	—	4	2	no	—
75	C₂³×C₄₇	—	2	1	no	—
76	C₂³×C₄₇	—	2	1	no	—
77	C₂³×C₄₇	—	2	1	no	—
78	C₂³×C₄₇	—	2	1	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	1	no	—
84	C₂³×C₄₇	—	2	1	no	—
85	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂³×C₄₇	#41	1	2	yes	—
88	C₂³×C₄₇	#41	1	2	yes	—
89	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
90	D₄×C₄₇	—	2	8	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	8	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₄₇	#41	1	2	yes	—
108	C₂×Dic₄₇	—	2	2	no	—
109	C₂×Dic₄₇	—	2	2	no	—
110	C₂×Dic₄₇	—	2	2	no	—
111	C₂×Dic₄₇	—	2	2	no	—
112	C₂×Dic₄₇	—	2	2	no	—
113	C₂×Dic₄₇	—	2	2	no	—
114	C₂×Dic₄₇	—	2	2	no	—
115	C₂×Dic₄₇	—	2	2	no	—
116	C₂×Dic₄₇	—	2	2	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	2	2	no	—
119	C₂×Dic₄₇	—	4	8	no	—
120	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	2	no	—
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	—	4	1	no	—
131	C₂²×C₄₇	—	4	1	no	—
132	C₂²×C₄₇	—	4	1	no	—
133	C₂²×C₄₇	—	4	1	no	—
134	C₂²×C₄₇	#117	1	4	yes	complemented
135	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
136	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
140	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	C₄×D₄	1	4	yes	—
147	Dic₄₇	—	2	4	no	—
148	Dic₄₇	—	4	2	no	—
149	Dic₄₇	—	4	2	no	—
150	Dic₄₇	—	4	2	no	—
151	#2352	—	47	1	no	maximal, Sylow
152	C₂×C₄₇	—	2	4	no	—
153	C₂×C₄₇	—	4	2	no	—
154	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
155	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂×C₄₇	#520	1	1	yes	characteristic
157	C₂×C₄₇	#546	1	1	yes	characteristic
158	C₂×C₄₇	#548	1	1	yes	characteristic
159	C₂×C₄₇	#552	1	1	yes	characteristic
160	C₂×C₄₇	#553	1	1	yes	characteristic
161	C₂×C₄×D₄	D₄₇	1	1	yes	characteristic
162	C₂×#117	—	47	1	no	—
163	C₂²×C₄⋊C₄	—	47	1	no	—
164	#388	—	47	2	no	—
165	#387	—	47	2	no	—
166	#386	—	47	2	no	—
167	#382	—	47	1	no	—
168	#380	—	47	1	no	—
169	#380	—	47	1	no	—
170	C₄×C₄⋊C₄	—	47	1	no	—
171	C₄×C₂²⋊C₄	—	47	2	no	—
172	C₄₇	#2352	1	1	yes	characteristic, complemented, Sylow
173	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
174	C₂³×C₄	—	47	2	no	—
175	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
176	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
177	C₄×D₄	Dic₄₇	1	8	yes	complemented
178	#117	—	94	4	no	—
179	C₂×C₄⋊C₄	—	47	1	no	—
180	C₂×C₄⋊C₄	—	47	1	no	—
181	C₂×C₄⋊C₄	—	47	2	no	—
182	C₂×C₄⋊C₄	—	47	1	no	—
183	C₂×C₄⋊C₄	—	47	1	no	—
184	C₂×C₄⋊C₄	—	94	4	no	—
185	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	47	2	no	—
187	C₂×C₂²⋊C₄	—	47	2	no	—
188	C₂×C₂²⋊C₄	—	47	2	no	—
189	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
191	C₂×C₄²	—	47	2	no	—
192	C₂×C₄²	—	47	2	no	—
193	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
194	#95	—	47	2	no	—
195	#95	—	47	2	no	—
196	#95	—	47	2	no	—
197	#95	—	47	2	no	—
198	#95	—	47	2	no	—
199	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
200	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	C₂×Dic₄₇	1	4	yes	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	47	2	no	—
209	C₂²×C₄	—	47	2	no	—
210	C₂²×C₄	—	47	2	no	—
211	C₂²×C₄	—	47	2	no	—
212	C₂²×C₄	—	94	4	no	—
213	C₂²×C₄	—	94	4	no	—
214	C₂²×C₄	—	188	2	no	—
215	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
216	C₂²×C₄	C₂×Dic₄₇	1	4	yes	complemented
217	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
222	C₄⋊C₄	—	47	4	no	—
223	C₄⋊C₄	—	94	4	no	—
224	C₄⋊C₄	—	94	2	no	—
225	C₄⋊C₄	—	94	2	no	—
226	C₄⋊C₄	—	94	2	no	—
227	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	complemented
228	C₂²⋊C₄	—	94	4	no	—
229	C₂²⋊C₄	—	94	4	no	—
230	C₂²⋊C₄	—	188	2	no	—
231	C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
232	C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	complemented
233	C₄²	—	94	4	no	—
234	C₄²	—	188	2	no	—
235	C₄²	C₂×Dic₄₇	1	4	yes	complemented
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	#10625350	1	2	yes	—
247	C₂³	#10625350	1	2	yes	—
248	C₂³	C₂²×Dic₄₇	1	2	yes	—
249	C₂³	C₂²×Dic₄₇	1	2	yes	—
250	C₂³	C₂³×D₄₇	1	1	yes	characteristic
251	D₄	—	2	8	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	47	4	no	—
263	C₂×C₄	—	94	2	no	—
264	C₂×C₄	—	94	2	no	—
265	C₂×C₄	—	94	2	no	—
266	C₂×C₄	—	94	2	no	—
267	C₂×C₄	—	94	2	no	—
268	C₂×C₄	—	94	2	no	—
269	C₂×C₄	—	94	2	no	—
270	C₂×C₄	—	94	2	no	—
271	C₂×C₄	—	94	2	no	—
272	C₂×C₄	—	94	2	no	—
273	C₂×C₄	—	94	2	no	—
274	C₂×C₄	—	188	8	no	—
275	C₂×C₄	D₄×D₄₇	1	2	yes	—
276	C₂×C₄	#10625352	1	2	yes	—
277	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
278	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
279	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
280	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
281	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	#13093983	1	4	yes	complemented
296	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
297	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
298	C₂²	C₂×#10625352	1	1	yes	characteristic
299	C₂²	C₂×#10625350	1	1	yes	characteristic
300	C₂²	C₂×#10625350	1	1	yes	characteristic
301	C₂²	#13094070	1	1	yes	characteristic
302	C₂²	#13094071	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	94	4	no	—
308	C₄	—	188	2	no	—
309	C₄	—	188	2	no	—
310	C₄	—	188	2	no	—
311	C₄	D₄×Dic₄₇	1	4	yes	—
312	C₂	—	2	4	no	—
313	C₂	—	4	2	no	—
314	C₂	C₂×#13093983	1	1	yes	characteristic
315	C₂	C₂×D₄×Dic₄₇	1	1	yes	characteristic
316	C₂	#15689606	1	1	yes	characteristic
317	C₂	#15689769	1	1	yes	characteristic
318	C₂	#15689782	1	1	yes	characteristic
319	C₂	#15689813	1	1	yes	characteristic
320	C₂	#15689819	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×C₄×D₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×C₄×D₄	D₄₇
hypocenter	C₄₇	#2352
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13093983	C₂	semidirect
2. C₂²×C₄⋊Dic₄₇	C₂	semidirect
3. #15689016	C₂	semidirect
4. #15689009	C₂	semidirect
6. #15688989	C₂	semidirect
7. #15688983	C₂	semidirect
8. #15688983	C₂	semidirect
9. C₄×C₄⋊Dic₄₇	C₂	semidirect
10. C₂²⋊C₄×Dic₄₇	C₂	semidirect
33. C₄×D₄×C₄₇	C₄	semidirect
48. C₂²×C₄×C₄₇	C₂×C₄	semidirect
61. C₄⋊C₄×C₄₇	C₂×C₄	semidirect
63. C₂²⋊C₄×C₄₇	C₂×C₄	semidirect
64. C₄²×C₄₇	C₂×C₄	semidirect
72. C₄⋊Dic₄₇	D₄	semidirect
134. C₂²×C₄₇	#117	semidirect
172. C₄₇	#2352	semidirect
177. C₄×D₄	Dic₄₇	semidirect
216. C₂²×C₄	C₂×Dic₄₇	semidirect
227. C₄⋊C₄	C₂×Dic₄₇	semidirect
232. C₂²⋊C₄	C₂×Dic₄₇	semidirect
235. C₄²	C₂×Dic₄₇	semidirect
295. C₂²	#13093983	semidirect