Subgroups — #20247186

Order 6016.

512 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094012	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689058	C₂	1	2	yes	maximal, complemented
4	#15689052	C₂	1	2	yes	maximal, complemented
5	#15689029	C₂	1	1	yes	maximal, characteristic, 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	#15689013	C₂	1	2	yes	maximal, complemented
10	#15689012	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
13	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
14	#13094012	—	2	4	no	—
15	C₄×Dic₉₄	—	2	4	no	—
16	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
20	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
21	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
23	C₂×(C₂×D₄₇)⋊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×C₄×Dic₄₇	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	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×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	#95×C₄₇	C₂²	1	1	yes	characteristic
36	#95×C₄₇	C₂²	1	1	yes	characteristic
37	C₂³×D₄₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
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×Dic₄₇	C₂³	1	2	yes	—
46	C₂²×Dic₄₇	C₂³	1	2	yes	—
47	C₂²×Dic₄₇	C₂³	1	2	yes	—
48	C₂×D₁₈₈	—	2	2	no	—
49	C₂×D₁₈₈	—	2	2	no	—
50	C₂×D₁₈₈	—	2	2	no	—
51	C₂×Dic₉₄	—	2	2	no	—
52	C₂×Dic₉₄	—	2	2	no	—
53	C₂×Dic₉₄	D₄	1	4	yes	—
54	C₄⋊C₄×C₄₇	—	2	2	no	—
55	C₄⋊C₄×C₄₇	—	2	2	no	—
56	C₄²×C₄₇	—	2	2	no	—
57	C₄²×C₄₇	—	2	2	no	—
58	C₄²×C₄₇	—	2	2	no	—
59	(C₂×D₄₇)⋊C₄	—	2	4	no	—
60	(C₂×D₄₇)⋊C₄	—	2	4	no	—
61	(C₂×D₄₇)⋊C₄	—	2	4	no	—
62	(C₂×D₄₇)⋊C₄	—	2	4	no	—
63	(C₂×D₄₇)⋊C₄	—	4	2	no	—
64	(C₂×D₄₇)⋊C₄	—	4	2	no	—
65	C₄⋊Dic₄₇	—	2	2	no	—
66	C₄⋊Dic₄₇	—	2	2	no	—
67	Dic₄₇⋊C₄	—	2	4	no	—
68	Dic₄₇⋊C₄	—	4	2	no	—
69	C₄×Dic₄₇	—	2	4	no	—
70	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×D₄₇	—	2	2	no	—
72	C₂²×D₄₇	—	2	2	no	—
73	C₂²×D₄₇	—	2	2	no	—
74	C₂²×D₄₇	—	2	2	no	—
75	C₂²×D₄₇	—	2	2	no	—
76	C₂²×D₄₇	—	2	2	no	—
77	C₂²×D₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	1	no	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₄₇	#41	1	2	yes	—
96	C₂×C₄×C₄₇	#41	1	2	yes	—
97	C₂×C₄×C₄₇	#41	1	2	yes	—
98	C₂×C₄×C₄₇	#41	1	2	yes	—
99	C₂×Dic₄₇	—	2	2	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
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	C₂×D₄	1	4	yes	—
108	D₁₈₈	—	4	4	no	—
109	Dic₉₄	—	2	8	no	—
110	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
115	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×D₄₇	—	4	2	no	—
118	C₂×D₄₇	—	4	2	no	—
119	C₂×D₄₇	—	4	2	no	—
120	C₂×D₄₇	—	4	2	no	—
121	C₂×D₄₇	—	4	2	no	—
122	C₂×D₄₇	—	4	2	no	—
123	C₂×D₄₇	—	4	2	no	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	2	2	no	—
126	C₄×C₄₇	—	4	1	no	—
127	C₄×C₄₇	—	4	1	no	—
128	C₄×C₄₇	—	4	1	no	—
129	C₄×C₄₇	—	4	1	no	—
130	C₄×C₄₇	#123	1	4	yes	—
131	Dic₄₇	—	2	4	no	—
132	Dic₄₇	—	4	2	no	—
133	Dic₄₇	—	4	2	no	—
134	#2448	—	47	1	no	maximal, Sylow
135	C₂×C₄₇	C₂×#123	1	1	yes	characteristic
136	C₂×C₄₇	#529	1	1	yes	characteristic
137	C₂×C₄₇	#542	1	1	yes	characteristic
138	C₂×C₄₇	#548	1	1	yes	characteristic
139	C₂×C₄₇	#550	1	1	yes	characteristic
140	C₂×C₄₇	#556	1	1	yes	characteristic
141	C₂×C₄₇	#559	1	1	yes	characteristic
142	D₄₇	—	8	2	no	—
143	C₂×#124	—	47	1	no	—
144	C₂×C₄×Q₈	—	47	1	no	—
145	#399	—	47	2	no	—
146	#397	—	47	2	no	—
147	#391	—	47	1	no	—
148	#390	—	47	1	no	—
149	#390	—	47	1	no	—
150	#389	—	47	1	no	—
151	#388	—	47	2	no	—
152	#388	—	47	2	no	—
153	C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic, complemented
154	C₄₇	#2448	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	47	1	no	—
156	C₂²×D₄	—	47	1	no	—
157	#124	—	94	4	no	—
158	C₄×Q₈	—	94	4	no	—
159	C₂×C₄⋊C₄	—	47	2	no	—
160	C₂×C₄⋊C₄	—	47	1	no	—
161	C₂×C₄⋊C₄	—	47	1	no	—
162	C₂×C₄⋊C₄	—	47	2	no	—
163	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	—	47	2	no	—
166	C₂×C₂²⋊C₄	—	47	2	no	—
167	C₂×C₂²⋊C₄	—	47	2	no	—
168	C₂×C₂²⋊C₄	—	47	2	no	—
169	C₂×C₂²⋊C₄	—	47	2	no	—
170	C₂×C₂²⋊C₄	—	47	2	no	—
171	C₂×C₄²	—	47	2	no	—
172	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
174	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
175	#95	—	47	2	no	—
176	#95	—	47	2	no	—
177	#95	—	47	2	no	—
178	#95	C₂×D₄₇	1	1	yes	characteristic
179	#95	C₂×D₄₇	1	1	yes	characteristic
180	C₂⁴	—	47	2	no	—
181	C₂×Q₈	—	47	4	no	—
182	C₂×Q₈	—	94	2	no	—
183	C₂×Q₈	—	94	2	no	—
184	C₂×D₄	—	94	2	no	—
185	C₂×D₄	—	94	2	no	—
186	C₂×D₄	—	94	2	no	—
187	C₂²×C₄	—	47	2	no	—
188	C₂²×C₄	—	47	2	no	—
189	C₂²×C₄	—	47	2	no	—
190	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	94	4	no	—
200	C₄⋊C₄	—	94	2	no	—
201	C₄⋊C₄	—	94	2	no	—
202	C₄⋊C₄	—	188	2	no	—
203	C₂²⋊C₄	—	94	4	no	—
204	C₂²⋊C₄	—	94	4	no	—
205	C₂²⋊C₄	—	94	4	no	—
206	C₂²⋊C₄	—	94	4	no	—
207	C₂²⋊C₄	—	188	2	no	—
208	C₂²⋊C₄	—	188	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	94	4	no	—
213	C₂³	—	94	2	no	—
214	C₂³	—	94	2	no	—
215	C₂³	—	94	2	no	—
216	C₂³	—	94	2	no	—
217	C₂³	—	94	2	no	—
218	C₂³	—	94	2	no	—
219	C₂³	—	94	2	no	—
220	C₂³	C₂³×D₄₇	1	1	yes	characteristic
221	Q₈	—	94	8	no	—
222	D₄	—	188	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	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	—	47	4	no	—
240	C₂×C₄	—	94	2	no	—
241	C₂×C₄	—	94	2	no	—
242	C₂×C₄	—	94	2	no	—
243	C₂×C₄	—	94	2	no	—
244	C₂×C₄	—	94	2	no	—
245	C₂×C₄	—	94	2	no	—
246	C₂×C₄	—	94	2	no	—
247	C₂×C₄	—	94	2	no	—
248	C₂×C₄	#10625350	1	2	yes	—
249	C₂×C₄	#10625350	1	2	yes	—
250	C₂×C₄	#10625350	1	2	yes	—
251	C₂×C₄	D₄×D₄₇	1	2	yes	—
252	C₂×C₄	#10625354	1	2	yes	—
253	C₂²	—	188	2	no	—
254	C₂²	—	188	2	no	—
255	C₂²	—	188	2	no	—
256	C₂²	—	188	2	no	—
257	C₂²	—	188	2	no	—
258	C₂²	—	188	2	no	—
259	C₂²	—	188	2	no	—
260	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
261	C₂²	C₂×#10625350	1	1	yes	characteristic
262	C₂²	C₂×#10625350	1	1	yes	characteristic
263	C₂²	C₂×#10625350	1	1	yes	characteristic
264	C₂²	C₂×#10625354	1	1	yes	characteristic
265	C₂²	#13094070	1	1	yes	characteristic
266	C₂²	#13094071	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	94	4	no	—
274	C₄	—	188	2	no	—
275	C₄	—	188	2	no	—
276	C₄	#13094009	1	4	yes	—
277	C₂	—	376	2	no	—
278	C₂	C₂×#13094009	1	1	yes	characteristic
279	C₂	#15689657	1	1	yes	characteristic
280	C₂	#15689744	1	1	yes	characteristic
281	C₂	#15689782	1	1	yes	characteristic
282	C₂	#15689794	1	1	yes	characteristic
283	C₂	#15689846	1	1	yes	characteristic
284	C₂	#15689858	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	#2448
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
2. C₂×C₄×Dic₉₄	C₂	semidirect
3. #15689058	C₂	semidirect
4. #15689052	C₂	semidirect
5. #15689029	C₂	semidirect
8. #15689021	C₂	semidirect
9. #15689013	C₂	semidirect
10. #15689012	C₂	semidirect
11. C₄×C₄⋊C₄×C₄₇	C₂	semidirect
153. C₄×C₄⋊C₄	D₄₇	semidirect
154. C₄₇	#2448	semidirect