Subgroups — #20246264

Order 6016.

548 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#15689022	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689010	C₂	1	2	yes	maximal, complemented
4	#15689004	C₂	1	2	yes	maximal, complemented
5	#15688980	C₂	1	1	yes	maximal, characteristic
6	#15688980	C₂	1	2	yes	maximal
7	C₄⋊C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×(C₂×D₄₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
10	#15688970	C₂	1	2	yes	maximal
11	C₄₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
18	C₄⋊C₄×D₄₇	C₄	1	8	yes	complemented
19	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
20	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
27	C₄₇⋊#95	C₂²	1	1	yes	characteristic
28	C₄₇⋊#95	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	2	yes	—
33	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
35	#95×C₄₇	C₂²	1	2	yes	—
36	C₂³×D₄₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₄₇	C₂³	1	2	yes	—
44	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
45	C₂²×Dic₄₇	C₂³	1	2	yes	—
46	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₄₇	—	2	2	no	—
48	C₂×C₄×D₄₇	—	2	2	no	—
49	C₂×C₄×D₄₇	—	4	2	no	—
50	C₂×C₄×D₄₇	—	4	2	no	—
51	C₂×C₄×D₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄×D₄₇	C₂×C₄	1	8	yes	—
53	C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₄⋊C₄×C₄₇	—	4	2	no	—
55	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
56	C₄²×C₄₇	—	2	2	no	—
57	(C₂×D₄₇)⋊C₄	—	2	4	no	—
58	(C₂×D₄₇)⋊C₄	—	2	2	no	—
59	(C₂×D₄₇)⋊C₄	Q₈	1	4	yes	—
60	C₄⋊Dic₄₇	—	2	2	no	—
61	C₄⋊Dic₄₇	—	4	2	no	—
62	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
63	Dic₄₇⋊C₄	C₂×C₄	1	8	yes	—
64	C₄×Dic₄₇	—	2	2	no	—
65	C₄×Dic₄₇	—	4	2	no	—
66	C₄×Dic₄₇	—	4	2	no	—
67	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×D₄₇	—	2	1	no	—
69	C₂²×D₄₇	—	2	1	no	—
70	C₂²×D₄₇	—	2	1	no	—
71	C₂²×D₄₇	—	2	1	no	—
72	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
73	C₂²×D₄₇	C₂×Q₈	1	2	yes	—
74	C₂²×D₄₇	#41	1	2	yes	—
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	—	2	1	no	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₄₇	#41	1	2	yes	—
89	C₂×C₄×C₄₇	#41	1	2	yes	—
90	C₂×Dic₄₇	—	2	1	no	—
91	C₂×Dic₄₇	—	2	1	no	—
92	C₂×Dic₄₇	—	2	1	no	—
93	C₂×Dic₄₇	—	2	1	no	—
94	C₂×Dic₄₇	—	2	2	no	—
95	C₂×Dic₄₇	—	2	2	no	—
96	C₂×Dic₄₇	—	2	1	no	—
97	C₂×Dic₄₇	—	2	2	no	—
98	C₂×Dic₄₇	—	2	1	no	—
99	C₂×Dic₄₇	—	2	2	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	2	1	no	—
102	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
103	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
104	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
105	C₄×D₄₇	—	2	8	no	—
106	C₄×D₄₇	—	4	8	no	—
107	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
110	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
112	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂×D₄₇	—	2	2	no	—
115	C₂×D₄₇	—	2	2	no	—
116	C₂×D₄₇	—	2	2	no	—
117	C₂×D₄₇	—	2	2	no	—
118	C₂×D₄₇	—	2	2	no	—
119	C₂×D₄₇	—	2	2	no	—
120	C₂×D₄₇	C₄×Q₈	1	4	yes	—
121	C₄×C₄₇	—	2	4	no	—
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	4	1	no	—
124	C₄×C₄₇	—	4	2	no	—
125	C₄×C₄₇	#117	1	4	yes	—
126	Dic₄₇	—	2	2	no	—
127	Dic₄₇	—	2	4	no	—
128	Dic₄₇	—	2	2	no	—
129	Dic₄₇	—	4	1	no	—
130	Dic₄₇	—	4	2	no	—
131	#2345	—	47	1	no	maximal, Sylow
132	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
133	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₂×C₄₇	#518	1	1	yes	characteristic
135	C₂×C₄₇	#553	1	1	yes	characteristic
136	C₂×C₄₇	#553	1	1	yes	characteristic
137	C₂×C₄₇	#554	1	1	yes	characteristic
138	C₂×C₄₇	#554	1	1	yes	characteristic
139	D₄₇	—	2	4	no	—
140	C₂²×C₄⋊C₄	—	47	1	no	—
141	#389	—	47	1	no	—
142	#389	D₄₇	1	1	yes	characteristic, complemented
143	#387	—	47	2	no	—
144	#387	—	47	2	no	—
145	#380	—	47	2	no	—
146	#380	—	47	1	no	—
147	C₄×C₄⋊C₄	—	47	1	no	—
148	C₄×C₄⋊C₄	—	47	1	no	—
149	C₄×C₂²⋊C₄	—	47	1	no	—
150	C₄×C₂²⋊C₄	—	47	2	no	—
151	C₄₇	#2345	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₄	—	47	1	no	—
153	C₂³×C₄	—	47	2	no	—
154	C₂×C₄⋊C₄	—	47	2	no	—
155	C₂×C₄⋊C₄	—	47	1	no	—
156	C₂×C₄⋊C₄	—	47	8	no	—
157	C₂×C₄⋊C₄	—	47	1	no	—
158	C₂×C₄⋊C₄	—	47	2	no	—
159	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
161	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	—	47	1	no	—
163	C₂×C₂²⋊C₄	—	47	1	no	—
164	C₂×C₂²⋊C₄	—	47	2	no	—
165	C₂×C₄²	—	47	1	no	—
166	C₂×C₄²	—	47	2	no	—
167	C₂×C₄²	—	47	2	no	—
168	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
169	#95	—	47	1	no	—
170	#95	—	47	1	no	—
171	#95	—	47	2	no	—
172	#95	—	47	2	no	—
173	#95	—	47	2	no	—
174	#95	—	47	2	no	—
175	#95	C₂×D₄₇	1	2	yes	—
176	C₂⁴	—	47	1	no	—
177	C₂²×C₄	—	47	1	no	—
178	C₂²×C₄	—	47	1	no	—
179	C₂²×C₄	—	47	2	no	—
180	C₂²×C₄	—	47	4	no	—
181	C₂²×C₄	—	47	8	no	—
182	C₂²×C₄	—	47	1	no	—
183	C₂²×C₄	—	47	2	no	—
184	C₂²×C₄	—	94	2	no	—
185	C₂²×C₄	—	94	2	no	—
186	C₂²×C₄	—	188	2	no	—
187	C₂²×C₄	—	188	2	no	—
188	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
190	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
191	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	47	8	no	—
196	C₄⋊C₄	—	47	4	no	—
197	C₄⋊C₄	—	94	2	no	—
198	C₄⋊C₄	—	188	2	no	—
199	C₄⋊C₄	C₄×D₄₇	1	4	yes	—
200	C₂²⋊C₄	—	47	4	no	—
201	C₂²⋊C₄	—	94	2	no	—
202	C₂²⋊C₄	—	94	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	94	2	no	—
205	C₄²	—	188	2	no	—
206	C₄²	—	188	2	no	—
207	C₂³	—	47	2	no	—
208	C₂³	—	47	2	no	—
209	C₂³	—	47	2	no	—
210	C₂³	—	94	1	no	—
211	C₂³	—	94	1	no	—
212	C₂³	—	94	1	no	—
213	C₂³	—	94	1	no	—
214	C₂³	C₂³×D₄₇	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	47	2	no	—
226	C₂×C₄	—	47	4	no	—
227	C₂×C₄	—	47	2	no	—
228	C₂×C₄	—	94	1	no	—
229	C₂×C₄	—	94	1	no	—
230	C₂×C₄	—	94	1	no	—
231	C₂×C₄	—	94	1	no	—
232	C₂×C₄	—	94	2	no	—
233	C₂×C₄	—	94	2	no	—
234	C₂×C₄	—	94	8	no	—
235	C₂×C₄	—	94	1	no	—
236	C₂×C₄	—	94	2	no	—
237	C₂×C₄	—	94	1	no	—
238	C₂×C₄	—	94	2	no	—
239	C₂×C₄	—	94	2	no	—
240	C₂×C₄	—	94	1	no	—
241	C₂×C₄	—	188	8	no	—
242	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
243	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
244	C₂×C₄	#10625352	1	2	yes	—
245	C₂×C₄	Q₈×D₄₇	1	2	yes	—
246	C₂×C₄	#10625354	1	2	yes	—
247	C₂²	—	47	4	no	—
248	C₂²	—	94	2	no	—
249	C₂²	—	94	2	no	—
250	C₂²	—	94	2	no	—
251	C₂²	—	94	2	no	—
252	C₂²	—	94	2	no	—
253	C₂²	—	94	2	no	—
254	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
255	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
256	C₂²	#41×D₄₇	1	1	yes	characteristic
257	C₂²	C₂×#10625352	1	1	yes	characteristic
258	C₂²	C₂×#10625354	1	1	yes	characteristic
259	C₂²	#13094069	1	1	yes	characteristic
260	C₂²	#13094070	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	4	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	2	no	—
265	C₄	—	94	2	no	—
266	C₄	—	94	2	no	—
267	C₄	—	94	4	no	—
268	C₄	—	188	1	no	—
269	C₄	—	188	2	no	—
270	C₄	#13093982	1	4	yes	—
271	C₂	—	94	4	no	—
272	C₂	C₂×#13093982	1	1	yes	characteristic
273	C₂	C₄×Q₈×D₄₇	1	1	yes	characteristic
274	C₂	#15689595	1	1	yes	characteristic
275	C₂	#15689820	1	1	yes	characteristic
276	C₂	#15689826	1	1	yes	characteristic
277	C₂	#15689831	1	1	yes	characteristic
278	C₂	#15689834	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#389	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	#389	D₄₇
hypocenter	C₄₇	#2345
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
2. #15689022	C₂	semidirect
3. #15689010	C₂	semidirect
4. #15689004	C₂	semidirect
7. C₄⋊C₄×Dic₄₇	C₂	semidirect
8. C₄×C₄⋊Dic₄₇	C₂	semidirect
11. C₄₇×#389	C₂	semidirect
18. C₄⋊C₄×D₄₇	C₄	semidirect
142. #389	D₄₇	semidirect
151. C₄₇	#2345	semidirect