Subgroups — #20252465

Order 6016.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689670	C₂	1	1	yes	maximal, characteristic
2	#15689202	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689192	C₂	1	2	yes	maximal, complemented
4	#15689191	C₂	1	2	yes	maximal, complemented
5	D₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689188	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689179	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689174	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689091	C₂	1	2	yes	maximal, complemented
10	#15689088	C₂	1	2	yes	maximal, complemented
11	C₄₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	C₂²	1	2	yes	—
13	C₂×#10625324	C₂²	1	2	yes	—
14	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×C₈×D₄₇	C₂²	1	2	yes	—
16	#13094023	C₄	1	1	yes	characteristic
17	#13094020	C₄	1	1	yes	characteristic
18	#13094010	C₄	1	2	yes	—
19	#13093999	C₄	1	1	yes	characteristic
20	#13093997	C₄	1	2	yes	—
21	#13093995	C₄	1	1	yes	characteristic
22	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₄₇	C₂²	1	2	yes	—
24	C₄²×D₄₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
26	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
29	(C₂×D₄₇)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₄₇)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
34	#13093934	C₂²	1	2	yes	—
35	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
37	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
40	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
46	#10625350	—	4	2	no	—
47	#10625350	—	4	2	no	—
48	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
49	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₄₇	C₂³	1	2	yes	—
54	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
55	M₁₆×C₄₇	—	4	2	no	—
56	C₂×C₈×C₄₇	C₂³	1	2	yes	—
57	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
61	C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
63	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
64	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
65	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
66	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
67	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
70	C₂×C₄₇⋊C₈	—	4	2	no	—
71	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
74	#10625324	—	4	2	no	—
75	C₈×D₄₇	—	4	2	no	—
76	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×D₄₇	C₂²×C₄	1	1	yes	characteristic
79	D₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₄₇	—	4	2	no	—
87	C₂²⋊D₄₇	—	4	2	no	—
88	C₂²⋊D₄₇	—	4	2	no	—
89	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
93	D₁₈₈	—	2	2	no	—
94	D₁₈₈	—	4	2	no	—
95	C₄×D₄₇	—	2	2	no	—
96	C₄×D₄₇	—	2	2	no	—
97	C₄×D₄₇	—	2	2	no	—
98	C₄×D₄₇	—	2	1	no	—
99	C₄×D₄₇	—	2	1	no	—
100	C₄×D₄₇	—	4	1	no	—
101	Dic₉₄	—	4	2	no	—
102	C₈×C₄₇	—	2	2	no	—
103	C₈×C₄₇	—	2	1	no	—
104	C₈×C₄₇	—	2	1	no	—
105	C₄₇⋊C₈	—	2	1	no	—
106	C₄₇⋊C₈	—	2	2	no	—
107	C₄₇⋊C₈	—	2	1	no	—
108	C₂²×C₄₇	—	2	2	no	—
109	C₂²×C₄₇	—	2	2	no	—
110	C₂²×C₄₇	—	2	2	no	—
111	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₄₇	—	2	1	no	—
113	C₂×D₄₇	—	2	2	no	—
114	C₂×D₄₇	—	2	1	no	—
115	C₂×D₄₇	—	2	2	no	—
116	C₂×D₄₇	—	2	1	no	—
117	C₂×D₄₇	—	2	2	no	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	#131	1	2	yes	—
121	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₄₇	—	2	1	no	—
124	Dic₄₇	—	2	2	no	—
125	Dic₄₇	—	2	1	no	—
126	Dic₄₇	—	2	1	no	—
127	#2981	—	47	1	no	maximal, Sylow
128	C₂×C₄₇	—	4	2	no	—
129	C₂×C₄₇	#518	1	1	yes	characteristic
130	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
131	C₂×C₄₇	#568	1	1	yes	characteristic
132	D₄₇	—	4	1	no	—
133	D₄₇	—	4	2	no	—
134	#532	—	47	1	no	—
135	#436	—	47	1	no	—
136	#436	D₄₇	1	1	yes	characteristic, complemented
137	#435	—	47	2	no	—
138	#435	—	47	2	no	—
139	C₈×D₄	—	47	1	no	—
140	C₈×D₄	—	47	1	no	—
141	#432	—	47	1	no	—
142	#431	—	47	1	no	—
143	#408	—	47	2	no	—
144	#408	—	47	2	no	—
145	C₄₇	#2981	1	1	yes	characteristic, complemented, Sylow
146	C₂×#41	—	47	2	no	—
147	C₂×M₁₆	—	47	2	no	—
148	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
149	C₂²×C₈	—	47	2	no	—
150	C₂²×C₈	—	47	2	no	—
151	C₄⋊Q₈	—	47	1	no	—
152	#127	—	47	1	no	—
153	#124	—	47	2	no	—
154	#121	—	47	1	no	—
155	#121	—	47	1	no	—
156	#121	—	47	2	no	—
157	C₄×D₄	—	47	2	no	—
158	C₄×D₄	—	47	1	no	—
159	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
160	C₂×C₄²	—	47	1	no	—
161	C₄⋊C₈	—	47	1	no	—
162	C₄⋊C₈	—	47	1	no	—
163	C₄⋊C₈	—	47	1	no	—
164	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
165	C₂²⋊C₈	—	47	2	no	—
166	C₂²⋊C₈	—	47	1	no	—
167	C₂²⋊C₈	—	47	1	no	—
168	C₂²⋊C₈	—	47	2	no	—
169	C₂²⋊C₈	—	47	2	no	—
170	C₂²⋊C₈	—	47	2	no	—
171	C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
172	#97	—	47	2	no	—
173	C₄×C₈	—	47	1	no	—
174	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
175	#41	—	188	2	no	—
176	#41	—	188	2	no	—
177	C₂×Q₈	—	47	2	no	—
178	C₂×D₄	—	47	2	no	—
179	C₂×D₄	—	47	2	no	—
180	C₂×D₄	—	47	2	no	—
181	C₂×D₄	—	47	1	no	—
182	C₂×D₄	—	47	2	no	—
183	C₂×D₄	C₄×D₄₇	1	1	yes	characteristic
184	C₂²×C₄	—	47	1	no	—
185	C₂²×C₄	—	47	1	no	—
186	C₂²×C₄	—	47	2	no	—
187	C₂²×C₄	—	47	1	no	—
188	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
189	M₁₆	—	4	2	no	—
190	M₁₆	—	188	2	no	—
191	C₂×C₈	—	47	1	no	—
192	C₂×C₈	—	47	1	no	—
193	C₂×C₈	—	47	2	no	—
194	C₂×C₈	—	188	2	no	—
195	C₂×C₈	—	188	2	no	—
196	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₄₇	1	2	yes	—
199	C₄⋊C₄	—	47	2	no	—
200	C₄⋊C₄	—	47	1	no	—
201	C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
202	C₂²⋊C₄	—	47	2	no	—
203	C₂²⋊C₄	—	47	2	no	—
204	C₂²⋊C₄	—	47	2	no	—
205	C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
206	C₄²	—	47	1	no	—
207	C₄²	—	47	1	no	—
208	C₄²	—	47	1	no	—
209	C₄²	C₂²×D₄₇	1	1	yes	characteristic
210	C₂³	—	47	2	no	—
211	C₂³	—	47	1	no	—
212	C₂³	C₂×C₄×D₄₇	1	2	yes	—
213	Q₈	—	188	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	94	2	no	—
216	D₄	—	188	2	no	—
217	D₄	—	188	2	no	—
218	D₄	—	188	2	no	—
219	D₄	—	188	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	47	1	no	—
223	C₂×C₄	—	47	1	no	—
224	C₂×C₄	—	47	1	no	—
225	C₂×C₄	—	47	2	no	—
226	C₂×C₄	—	94	2	no	—
227	C₂×C₄	—	94	2	no	—
228	C₂×C₄	—	94	1	no	—
229	C₂×C₄	—	94	2	no	—
230	C₂×C₄	—	94	1	no	—
231	C₂×C₄	—	188	1	no	—
232	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	94	1	no	—
240	C₈	—	94	1	no	—
241	C₈	—	94	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	94	1	no	—
246	C₂²	—	94	2	no	—
247	C₂²	—	94	1	no	—
248	C₂²	—	94	2	no	—
249	C₂²	—	94	1	no	—
250	C₂²	—	94	2	no	—
251	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	94	1	no	—
255	C₄	—	94	2	no	—
256	C₄	—	94	1	no	—
257	C₄	—	94	1	no	—
258	C₄	#13094032	1	2	yes	—
259	C₄	#13094069	1	1	yes	characteristic
260	C₄	#13094070	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	188	1	no	—
263	C₂	—	188	2	no	—
264	C₂	#15689595	1	1	yes	characteristic
265	C₂	C₂×#13094032	1	1	yes	characteristic
266	C₂	#15689888	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#436	C₂
hypercenter	#436	D₄₇
hypocenter	C₄₇	#2981
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. #15689202	C₂	semidirect
3. #15689192	C₂	semidirect
4. #15689191	C₂	semidirect
5. D₄×C₄₇⋊C₈	C₂	semidirect
6. #15689188	C₂	semidirect
7. #15689179	C₂	semidirect
8. #15689174	C₂	semidirect
9. #15689091	C₂	semidirect
10. #15689088	C₂	semidirect
11. C₄₇×#436	C₂	semidirect
136. #436	D₄₇	semidirect
145. C₄₇	#2981	semidirect