Subgroups — D₄₇×#1844 #26216673

Order 12032.

1358 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#106×D₄₇	C₂	1	4	yes	maximal
2	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#384×D₄₇	C₂	1	2	yes	maximal
4	#20243545	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243544	C₂	1	2	yes	maximal, complemented
6	#20243543	C₂	1	4	yes	maximal, complemented
7	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×D₄₇	—	2	4	no	—
10	C₂×C₄⋊C₄×D₄₇	C₂²	1	4	yes	—
11	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
12	C₂×#13093956	C₂²	1	4	yes	—
13	C₂×#13093955	C₂²	1	8	yes	—
14	#106×D₄₇	—	2	16	no	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
17	C₄×C₈×D₄₇	C₄	1	8	yes	complemented
18	#15688993	C₂²	1	2	yes	—
19	#15688992	—	2	4	no	—
20	#15688992	C₂²	1	4	yes	—
21	C₂×#106×C₄₇	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#384	C₂²	1	2	yes	—
24	C₂²×C₄×D₄₇	—	2	4	no	—
25	C₂²×C₄×D₄₇	D₄	1	2	yes	—
26	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
27	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
28	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
29	C₂×C₈×D₄₇	D₄	1	8	yes	—
30	C₂×C₈×D₄₇	Q₈	1	8	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
33	C₂×Dic₄₇⋊C₄	—	2	4	no	—
34	C₂×Dic₄₇⋊C₄	—	2	8	no	—
35	C₄⋊C₄×D₄₇	—	2	16	no	—
36	C₄⋊C₄×D₄₇	—	4	16	no	—
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
39	C₄²×D₄₇	C₂×C₄	1	4	yes	—
40	C₄²×D₄₇	Q₈	1	4	yes	—
41	#13093956	—	2	8	no	—
42	#13093955	—	2	16	no	—
43	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
44	C₈×Dic₄₇	C₂×C₄	1	8	yes	complemented
45	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	#106×C₄₇	—	2	8	no	—
50	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
51	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	2	4	no	—
56	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×D₄₇	—	4	8	no	—
59	C₂×C₄×D₄₇	—	4	16	no	—
60	C₂×C₄×D₄₇	C₄⋊C₄	1	8	yes	—
61	C₂×C₄×D₄₇	SD₁₆	1	16	yes	—
62	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×D₄₇	C₂×D₄	1	4	yes	—
64	C₂×C₄×D₄₇	C₂×Q₈	1	4	yes	—
65	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
66	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
68	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
69	C₄⋊C₄×C₄₇	—	2	8	no	—
70	C₄⋊C₄×C₄₇	—	4	8	no	—
71	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
72	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
73	C₄⋊Dic₄₇	—	2	8	no	—
74	C₄⋊Dic₄₇	—	4	8	no	—
75	Dic₄₇⋊C₄	—	4	8	no	—
76	Dic₄₇⋊C₄	—	4	16	no	—
77	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
78	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
79	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
80	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
81	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
82	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
83	C₈×D₄₇	C₄⋊C₄	1	32	yes	complemented
84	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
85	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂²×D₄₇	#127	1	2	yes	—
87	C₂²×D₄₇	C₄⋊Q₈	1	2	yes	—
88	C₂²×D₄₇	C₂×SD₁₆	1	8	yes	—
89	C₂×C₄×C₄₇	—	4	4	no	—
90	C₂×C₄×C₄₇	—	4	8	no	—
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₄₇	C₂×SD₁₆	1	8	yes	—
93	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
96	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×Dic₄₇	—	4	4	no	—
98	C₂×Dic₄₇	—	4	8	no	—
99	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
101	C₂×Dic₄₇	#127	1	2	yes	—
102	C₂×Dic₄₇	C₄⋊Q₈	1	2	yes	—
103	C₂×Dic₄₇	C₂×SD₁₆	1	8	yes	—
104	C₄×D₄₇	—	8	16	no	—
105	C₄×D₄₇	#106	1	32	yes	complemented
106	C₄×D₄₇	C₂×C₄⋊C₄	1	8	yes	—
107	C₄×D₄₇	C₄⋊Q₈	1	8	yes	—
108	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	complemented
109	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	complemented
110	C₂×#1844	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
114	C₂²×C₄₇	C₂²×SD₁₆	1	4	yes	—
115	C₂×D₄₇	#384	1	4	yes	—
116	C₂×D₄₇	C₂×#106	1	8	yes	—
117	C₂×D₄₇	#492	1	8	yes	—
118	C₂×D₄₇	#498	1	8	yes	—
119	C₄×C₄₇	—	8	4	no	—
120	C₄×C₄₇	C₂×#106	1	8	yes	complemented
121	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
122	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
123	Dic₄₇	—	8	4	no	—
124	Dic₄₇	#384	1	4	yes	—
125	Dic₄₇	C₂×#106	1	8	yes	complemented
126	C₂²×#106	—	47	4	no	—
127	C₂²×C₄×C₈	—	47	1	no	—
128	C₂×#384	—	47	2	no	—
129	#1844	—	47	4	no	—
130	#1844	—	47	1	no	—
131	#1844	—	47	2	no	—
132	#1844	D₄₇	1	1	yes	characteristic, direct product factor
133	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
134	C₂×C₄₇	C₂²×#106	1	2	yes	—
135	C₂×C₄₇	C₂×#492	1	2	yes	—
136	C₂×C₄₇	C₂×#498	1	2	yes	—
137	D₄₇	#1844	1	8	yes	direct product factor
138	C₂³×C₈	—	47	2	no	—
139	C₂²×C₄⋊C₄	—	47	4	no	—
140	C₂²×C₄⋊C₄	—	94	4	no	—
141	C₂²×C₄²	—	47	1	no	—
142	C₂×#106	—	47	4	no	—
143	C₂×#106	—	47	8	no	—
144	C₂×#106	—	94	16	no	—
145	C₂×#106	C₂×D₄₇	1	4	yes	—
146	C₂×C₄×C₈	—	47	2	no	—
147	C₂×C₄×C₈	—	47	1	no	—
148	C₂×C₄×C₈	—	47	8	no	—
149	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
150	#384	—	47	4	no	—
151	#384	—	47	2	no	—
152	#384	—	94	4	no	—
153	#384	C₂×D₄₇	1	2	yes	—
154	C₄₇	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₄	—	47	2	no	—
156	C₂³×C₄	—	47	1	no	—
157	C₂³×C₄	—	94	4	no	—
158	C₂²×C₈	—	47	2	no	—
159	C₂²×C₈	—	47	8	no	—
160	C₂²×C₈	—	47	8	no	—
161	C₂²×C₈	—	47	8	no	—
162	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	—	47	4	no	—
165	C₂×C₄⋊C₄	—	94	16	no	—
166	C₂×C₄⋊C₄	—	94	4	no	—
167	C₂×C₄⋊C₄	—	94	8	no	—
168	C₂×C₄⋊C₄	—	94	4	no	—
169	C₂×C₄⋊C₄	—	188	16	no	—
170	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
171	C₂×C₄²	—	47	2	no	—
172	C₂×C₄²	—	47	1	no	—
173	C₂×C₄²	—	47	4	no	—
174	C₂×C₄²	—	47	4	no	—
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	#106	—	2	8	no	—
177	#106	—	94	16	no	—
178	#106	—	94	8	no	—
179	C₄×C₈	—	47	8	no	—
180	C₄×C₈	—	47	4	no	—
181	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
182	C₂⁴	—	47	1	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	47	2	no	—
185	C₂²×C₄	—	47	1	no	—
186	C₂²×C₄	—	47	8	no	—
187	C₂²×C₄	—	47	16	no	—
188	C₂²×C₄	—	47	2	no	—
189	C₂²×C₄	—	47	4	no	—
190	C₂²×C₄	—	47	2	no	—
191	C₂²×C₄	—	47	4	no	—
192	C₂²×C₄	—	94	4	no	—
193	C₂²×C₄	—	188	8	no	—
194	C₂²×C₄	—	188	16	no	—
195	C₂²×C₄	D₄×D₄₇	1	2	yes	—
196	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
197	C₂×C₈	—	47	4	no	—
198	C₂×C₈	—	47	4	no	—
199	C₂×C₈	—	47	4	no	—
200	C₂×C₈	—	47	32	no	—
201	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
202	C₂×C₈	D₄×D₄₇	1	4	yes	—
203	C₂×C₈	Q₈×D₄₇	1	4	yes	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄⋊C₄	—	4	8	no	—
206	C₄⋊C₄	—	94	8	no	—
207	C₄⋊C₄	—	188	8	no	—
208	C₄⋊C₄	—	188	16	no	—
209	C₄⋊C₄	—	188	8	no	—
210	C₄²	—	47	8	no	—
211	C₄²	—	47	2	no	—
212	C₄²	—	47	2	no	—
213	C₄²	C₂×C₄×D₄₇	1	2	yes	—
214	C₄²	Q₈×D₄₇	1	2	yes	—
215	C₂³	—	47	2	no	—
216	C₂³	—	47	8	no	—
217	C₂³	—	47	2	no	—
218	C₂³	—	47	2	no	—
219	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	47	2	no	—
223	C₂×C₄	—	47	4	no	—
224	C₂×C₄	—	47	8	no	—
225	C₂×C₄	—	47	2	no	—
226	C₂×C₄	—	47	2	no	—
227	C₂×C₄	—	47	8	no	—
228	C₂×C₄	—	47	32	no	—
229	C₂×C₄	—	47	8	no	—
230	C₂×C₄	—	376	16	no	—
231	C₂×C₄	—	188	8	no	—
232	C₂×C₄	—	188	4	no	—
233	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
234	C₂×C₄	SD₁₆×D₄₇	1	8	yes	—
235	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
237	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
238	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
239	C₈	—	47	8	no	—
240	C₈	C₄⋊C₄×D₄₇	1	8	yes	complemented
241	C₂²	—	47	4	no	—
242	C₂²	—	47	8	no	—
243	C₂²	—	47	8	no	—
244	C₂²	—	47	8	no	—
245	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
246	C₂²	#127×D₄₇	1	1	yes	characteristic
247	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
248	C₂²	C₂×SD₁₆×D₄₇	1	4	yes	—
249	C₄	—	8	4	no	—
250	C₄	—	47	4	no	—
251	C₄	—	47	8	no	—
252	C₄	—	376	4	no	—
253	C₄	#106×D₄₇	1	8	yes	complemented
254	C₄	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
255	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
256	C₂	—	47	8	no	—
257	C₂	#384×D₄₇	1	1	yes	characteristic
258	C₂	C₂×#106×D₄₇	1	2	yes	—
259	C₂	#492×D₄₇	1	2	yes	—
260	C₂	#498×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1844	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1844	D₄₇
hypocenter	C₄₇	C₂×#1844
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
4. #20243545	C₂	semidirect
5. #20243544	C₂	semidirect
6. #20243543	C₂	semidirect
7. C₄₇×#1844	C₂	semidirect
17. C₄×C₈×D₄₇	C₄	semidirect
43. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
44. C₈×Dic₄₇	C₂×C₄	semidirect
50. C₄×C₈×C₄₇	C₂×C₄	semidirect
83. C₈×D₄₇	C₄⋊C₄	semidirect
105. C₄×D₄₇	#106	semidirect
108. C₈×C₄₇	C₂×C₄⋊C₄	semidirect
109. C₄₇⋊C₈	C₂×C₄⋊C₄	semidirect
120. C₄×C₄₇	C₂×#106	semidirect
125. Dic₄₇	C₂×#106	semidirect
132. #1844	D₄₇	direct (N × H)
137. D₄₇	#1844	direct (N × H)
154. C₄₇	C₂×#1844	semidirect
181. C₄×C₈	C₄×D₄₇	semidirect
240. C₈	C₄⋊C₄×D₄₇	semidirect
253. C₄	#106×D₄₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#106×D₄₇	2
C₂×C₄×C₈×D₄₇	2
#384×D₄₇	2
#20243545	2
#20243544	2
#20243543	2
C₄₇×#1844	2
C₂×#1844	47

Subgroups — D47×#1844 #26216673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₇×#1844 #26216673