Subgroups — #20247238

Order 6016.

620 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094024	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094002	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic
4	#15689045	C₂	1	2	yes	maximal, complemented
5	#15689036	C₂	1	2	yes	maximal, complemented
6	#15689025	C₂	1	1	yes	maximal, characteristic
7	#15689025	C₂	1	1	yes	maximal, characteristic
8	#15689021	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689021	C₂	1	1	yes	maximal, characteristic, complemented
10	#15688997	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₉₄	C₂²	1	2	yes	—
13	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
15	#13094024	—	2	4	no	—
16	#13094002	—	2	8	no	—
17	C₄×D₁₈₈	—	2	4	no	—
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×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
23	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
25	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
26	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
27	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
28	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
29	C₄₇⋊#95	C₂²	1	2	yes	—
30	C₄₇⋊#95	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
36	#95×C₄₇	C₂²	1	1	yes	characteristic
37	#95×C₄₇	C₂²	1	1	yes	characteristic
38	C₂³×D₄₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×Dic₄₇	C₂³	1	2	yes	—
47	C₂²×Dic₄₇	C₂³	1	2	yes	—
48	C₂²×Dic₄₇	C₂³	1	2	yes	—
49	C₂×D₁₈₈	—	2	2	no	—
50	C₂×D₁₈₈	—	2	2	no	—
51	C₂×D₁₈₈	Q₈	1	4	yes	complemented
52	C₂×C₄×D₄₇	—	2	4	no	—
53	C₂×C₄×D₄₇	—	2	4	no	—
54	C₂×C₄×D₄₇	—	4	2	no	—
55	C₂×Dic₉₄	—	2	4	no	—
56	C₂×Dic₉₄	—	2	4	no	—
57	C₂×Dic₉₄	—	4	2	no	—
58	C₄⋊C₄×C₄₇	—	2	2	no	—
59	C₄⋊C₄×C₄₇	D₄	1	4	yes	complemented
60	C₄²×C₄₇	—	2	2	no	—
61	C₄²×C₄₇	—	2	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	(C₂×D₄₇)⋊C₄	—	2	4	no	—
64	(C₂×D₄₇)⋊C₄	—	2	4	no	—
65	(C₂×D₄₇)⋊C₄	—	2	4	no	—
66	(C₂×D₄₇)⋊C₄	—	2	4	no	—
67	(C₂×D₄₇)⋊C₄	—	4	2	no	—
68	C₄⋊Dic₄₇	—	2	2	no	—
69	C₄⋊Dic₄₇	—	2	2	no	—
70	C₄⋊Dic₄₇	—	2	4	no	—
71	C₄⋊Dic₄₇	D₄	1	4	yes	—
72	Dic₄₇⋊C₄	—	2	4	no	—
73	Dic₄₇⋊C₄	—	2	4	no	—
74	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂²×D₄₇	—	2	2	no	—
76	C₂²×D₄₇	—	2	2	no	—
77	C₂²×D₄₇	—	2	2	no	—
78	C₂²×D₄₇	—	2	2	no	—
79	C₂²×D₄₇	—	2	2	no	—
80	C₂²×D₄₇	—	2	2	no	—
81	C₂²×D₄₇	C₂×Q₈	1	4	yes	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	1	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	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₄₇	#41	1	2	yes	—
102	C₂×C₄×C₄₇	#41	1	2	yes	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	—	2	2	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	—	2	2	no	—
109	C₂×Dic₄₇	—	2	2	no	—
110	C₂×Dic₄₇	—	2	2	no	—
111	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
112	D₁₈₈	—	2	8	no	—
113	C₄×D₄₇	—	4	8	no	—
114	Dic₉₄	—	4	8	no	—
115	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
120	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×D₄₇	—	2	4	no	—
123	C₂×D₄₇	—	2	4	no	—
124	C₂×D₄₇	—	2	4	no	—
125	C₂×D₄₇	—	4	2	no	—
126	C₂×D₄₇	—	4	2	no	—
127	C₂×D₄₇	—	4	2	no	—
128	C₂×D₄₇	—	4	2	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	4	1	no	—
134	C₄×C₄₇	—	4	1	no	—
135	C₄×C₄₇	C₂²⋊Q₈	1	4	yes	complemented
136	Dic₄₇	—	2	4	no	—
137	Dic₄₇	—	4	2	no	—
138	Dic₄₇	—	4	2	no	—
139	#2452	—	47	1	no	maximal, Sylow
140	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄₇	#532	1	1	yes	characteristic
142	C₂×C₄₇	#542	1	1	yes	characteristic
143	C₂×C₄₇	D₄²	1	1	yes	characteristic
144	C₂×C₄₇	#548	1	1	yes	characteristic
145	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
146	C₂×C₄₇	#554	1	1	yes	characteristic
147	D₄₇	—	4	4	no	—
148	C₂×C₄⋊Q₈	—	47	1	no	—
149	C₂×C₂²⋊Q₈	—	47	2	no	—
150	C₂×C₄×D₄	—	47	1	no	—
151	#396	—	47	2	no	—
152	#393	—	47	2	no	—
153	#390	—	47	1	no	—
154	#390	—	47	1	no	—
155	#389	—	47	1	no	—
156	#389	—	47	1	no	—
157	#385	—	47	2	no	—
158	C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic, complemented
159	C₄₇	#2452	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	47	2	no	—
161	C₂²×D₄	—	47	1	no	—
162	C₂³×C₄	—	47	2	no	—
163	C₄⋊Q₈	—	94	4	no	—
164	C₂²⋊Q₈	—	94	8	no	—
165	C₄×D₄	—	94	4	no	—
166	C₂×C₄⋊C₄	—	47	1	no	—
167	C₂×C₄⋊C₄	—	47	2	no	—
168	C₂×C₄⋊C₄	—	47	2	no	—
169	C₂×C₄⋊C₄	—	47	2	no	—
170	C₂×C₄⋊C₄	—	47	1	no	—
171	C₂×C₄⋊C₄	—	47	1	no	—
172	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	—	47	2	no	—
175	C₂×C₂²⋊C₄	—	47	2	no	—
176	C₂×C₂²⋊C₄	—	47	2	no	—
177	C₂×C₂²⋊C₄	—	47	2	no	—
178	C₂×C₂²⋊C₄	—	47	2	no	—
179	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
180	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
182	#95	—	47	2	no	—
183	#95	—	47	2	no	—
184	#95	C₂×D₄₇	1	1	yes	characteristic
185	#95	C₂×D₄₇	1	1	yes	characteristic
186	C₂⁴	—	47	2	no	—
187	C₂×Q₈	—	94	4	no	—
188	C₂×Q₈	—	94	4	no	—
189	C₂×Q₈	—	188	2	no	—
190	C₂×D₄	—	47	4	no	—
191	C₂×D₄	—	94	2	no	—
192	C₂×D₄	—	94	2	no	—
193	C₂²×C₄	—	47	2	no	—
194	C₂²×C₄	—	47	2	no	—
195	C₂²×C₄	—	47	2	no	—
196	C₂²×C₄	—	94	4	no	—
197	C₂²×C₄	—	94	4	no	—
198	C₂²×C₄	—	188	2	no	—
199	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	47	4	no	—
208	C₄⋊C₄	—	94	4	no	—
209	C₄⋊C₄	—	94	4	no	—
210	C₄⋊C₄	—	94	4	no	—
211	C₄⋊C₄	—	94	2	no	—
212	C₄⋊C₄	—	94	2	no	—
213	C₄⋊C₄	D₁₈₈	1	4	yes	complemented
214	C₂²⋊C₄	—	94	4	no	—
215	C₂²⋊C₄	—	94	4	no	—
216	C₂²⋊C₄	—	94	4	no	—
217	C₂²⋊C₄	—	94	4	no	—
218	C₂²⋊C₄	—	188	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₂³	—	47	4	no	—
223	C₂³	—	94	2	no	—
224	C₂³	—	94	2	no	—
225	C₂³	—	94	2	no	—
226	C₂³	—	94	2	no	—
227	C₂³	—	94	2	no	—
228	C₂³	—	94	2	no	—
229	C₂³	C₂³×D₄₇	1	1	yes	characteristic
230	Q₈	—	188	8	no	—
231	D₄	—	94	8	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	47	4	no	—
247	C₂×C₄	—	94	2	no	—
248	C₂×C₄	—	94	2	no	—
249	C₂×C₄	—	94	2	no	—
250	C₂×C₄	—	94	2	no	—
251	C₂×C₄	—	94	2	no	—
252	C₂×C₄	—	94	2	no	—
253	C₂×C₄	—	94	2	no	—
254	C₂×C₄	—	94	2	no	—
255	C₂×C₄	—	188	8	no	—
256	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
257	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
258	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
259	C₂×C₄	#10625350	1	2	yes	—
260	C₂×C₄	#10625350	1	2	yes	—
261	C₂×C₄	D₄×D₄₇	1	2	yes	—
262	C₂×C₄	Q₈×D₄₇	1	2	yes	—
263	C₂²	—	94	4	no	—
264	C₂²	—	94	4	no	—
265	C₂²	—	94	4	no	—
266	C₂²	—	188	2	no	—
267	C₂²	—	188	2	no	—
268	C₂²	—	188	2	no	—
269	C₂²	—	188	2	no	—
270	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
271	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
272	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
273	C₂²	C₂×#10625350	1	1	yes	characteristic
274	C₂²	C₂×#10625350	1	1	yes	characteristic
275	C₂²	#13094070	1	1	yes	characteristic
276	C₂²	#13094071	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	94	4	no	—
284	C₄	—	188	2	no	—
285	C₄	—	188	2	no	—
286	C₄	#13094002	1	4	yes	complemented
287	C₂	—	188	4	no	—
288	C₂	C₂×#13094002	1	1	yes	characteristic
289	C₂	#15689673	1	1	yes	characteristic
290	C₂	#15689744	1	1	yes	characteristic
291	C₂	D₄×D₁₈₈	1	1	yes	characteristic
292	C₂	#15689782	1	1	yes	characteristic
293	C₂	Q₈×D₁₈₈	1	1	yes	characteristic
294	C₂	#15689833	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄×C₄⋊C₄×C₄₇	C₂
hypercenter	C₄×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#2452
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13094024	C₂	semidirect
2. C₂×#13094002	C₂	semidirect
4. #15689045	C₂	semidirect
5. #15689036	C₂	semidirect
8. #15689021	C₂	semidirect
9. #15689021	C₂	semidirect
10. #15688997	C₂	semidirect
11. C₄×C₄⋊C₄×C₄₇	C₂	semidirect
51. C₂×D₁₈₈	Q₈	semidirect
59. C₄⋊C₄×C₄₇	D₄	semidirect
135. C₄×C₄₇	C₂²⋊Q₈	semidirect
158. C₄×C₄⋊C₄	D₄₇	semidirect
159. C₄₇	#2452	semidirect
213. C₄⋊C₄	D₁₈₈	semidirect
286. C₄	#13094002	semidirect