Subgroups — #26369264

Order 12032.

706 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₄₇	C₂	1	2	yes	maximal
2	C₂×#15689077	C₂	1	1	yes	maximal, characteristic
3	#20245145	C₂	1	8	yes	maximal, complemented
4	#20245144	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245143	C₂	1	2	yes	maximal, complemented
6	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10625324	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
9	C₂×#13093981	C₄	1	1	yes	characteristic
10	C₂×#13093964	C₂²	1	2	yes	—
11	C₂×#13093963	C₂²	1	4	yes	—
12	M₃₂×D₄₇	—	2	8	no	—
13	C₂×C₁₆×D₄₇	C₂²	1	4	yes	—
14	#15689077	C₄	1	4	yes	—
15	#15689077	C₂²	1	4	yes	—
16	C₂×#13093935	C₄	1	1	yes	characteristic
17	C₂×#13093934	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
20	Dic₄₇⋊C₁₆	C₂²	1	8	yes	complemented
21	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×C₄⋊C₁₆	C₂²	1	4	yes	complemented
24	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×#10625324	—	2	4	no	—
26	C₂×#10625324	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₄₇	D₄	1	4	yes	—
31	C₂×C₈×D₄₇	Q₈	1	4	yes	—
32	C₂×C₈×D₄₇	C₂³	1	2	yes	—
33	#13093981	C₈	1	4	yes	—
34	#13093981	C₂×C₄	1	4	yes	—
35	C₂×Dic₄₇⋊C₄	C₈	1	2	yes	—
36	C₂×(C₂×D₄₇)⋊C₄	C₈	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
38	#13093964	—	2	4	no	—
39	#13093963	—	2	8	no	—
40	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
41	C₁₆×D₄₇	—	2	8	no	—
42	#13093935	C₂×C₄	1	4	yes	—
43	#13093934	C₂×C₄	1	4	yes	—
44	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
45	C₈×Dic₄₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	M₃₂×C₄₇	—	2	4	no	—
50	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
51	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂³×D₄₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
57	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
60	C₂×C₄×D₄₇	M₁₆	1	4	yes	—
61	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₄²×C₄₇	C₂×C₈	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	(C₂×D₄₇)⋊C₄	C₂×C₈	1	8	yes	—
71	Dic₄₇⋊C₄	C₂×C₈	1	8	yes	—
72	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
73	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
74	C₂×C₄₇⋊C₈	—	2	2	no	—
75	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
76	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
78	#10625324	—	2	8	no	—
79	C₈×D₄₇	C₄⋊C₄	1	8	yes	—
80	C₈×D₄₇	C₂×D₄	1	4	yes	—
81	C₈×D₄₇	C₂×Q₈	1	4	yes	—
82	C₁₆×C₄₇	—	2	4	no	—
83	C₄₇⋊C₁₆	—	2	4	no	—
84	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
85	C₂²×D₄₇	—	4	2	no	—
86	C₂²×D₄₇	C₄⋊C₈	1	4	yes	—
87	C₂²×D₄₇	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₄₇	—	4	2	no	—
89	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×Dic₄₇	—	4	2	no	—
95	C₂×Dic₄₇	C₄⋊C₈	1	4	yes	—
96	C₂×Dic₄₇	C₂²×C₈	1	4	yes	—
97	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
98	C₄×D₄₇	C₄⋊C₈	1	8	yes	—
99	C₄×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₄×D₄₇	C₂×M₁₆	1	4	yes	—
101	C₈×C₄₇	—	2	2	no	—
102	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
105	C₄₇⋊C₈	—	2	2	no	—
106	C₄₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
107	#33397	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	—	2	2	no	—
109	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
110	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
111	C₂×D₄₇	—	2	4	no	—
112	C₂×D₄₇	—	4	4	no	—
113	C₂×D₄₇	C₂×C₄⋊C₈	1	4	yes	—
114	C₄×C₄₇	—	2	4	no	—
115	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
116	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
118	Dic₄₇	—	2	4	no	—
119	Dic₄₇	C₂×C₄⋊C₈	1	4	yes	—
120	C₂²×M₃₂	—	47	2	no	—
121	C₂×#405	—	47	1	no	—
122	#2151	—	47	8	no	—
123	#2150	—	47	2	no	—
124	#2150	—	47	1	no	—
125	#2150	D₄₇	1	1	yes	characteristic, complemented
126	C₂×C₄₇	—	2	2	no	—
127	C₂×C₄₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₄₇	#3407	1	2	yes	—
129	D₄₇	—	2	4	no	—
130	C₂²×M₁₆	—	47	1	no	—
131	C₂³×C₈	—	47	1	no	—
132	C₂×#117	—	47	1	no	—
133	C₂×M₃₂	—	47	2	no	—
134	C₂×M₃₂	—	47	4	no	—
135	C₂×M₃₂	—	94	8	no	—
136	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
137	C₂²×C₁₆	—	47	4	no	—
138	#405	—	47	4	no	—
139	#405	—	47	4	no	—
140	C₂×#97	—	47	1	no	—
141	C₂×#97	—	47	1	no	—
142	C₂×C₄×C₈	—	47	1	no	—
143	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
144	C₄⋊C₁₆	—	47	8	no	—
145	C₄⋊C₁₆	—	47	4	no	—
146	C₄⋊C₁₆	C₂×D₄₇	1	4	yes	complemented
147	C₄₇	#33397	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	—	47	1	no	—
149	C₂×M₁₆	—	47	4	no	—
150	C₂×M₁₆	—	94	4	no	—
151	C₂²×C₈	—	47	1	no	—
152	C₂²×C₈	—	47	2	no	—
153	C₂²×C₈	—	47	4	no	—
154	C₂²×C₈	—	47	4	no	—
155	C₂²×C₈	—	47	2	no	—
156	C₂²×C₈	—	47	1	no	—
157	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
159	#117	—	47	4	no	—
160	#117	—	47	4	no	—
161	C₂×C₄⋊C₄	—	47	2	no	—
162	C₂×C₂²⋊C₄	—	47	2	no	—
163	C₂×C₄²	—	47	1	no	—
164	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
165	M₃₂	—	2	4	no	—
166	M₃₂	—	94	8	no	—
167	M₃₂	—	94	4	no	—
168	C₂×C₁₆	—	47	4	no	—
169	C₂×C₁₆	—	94	8	no	—
170	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
171	#97	—	47	4	no	—
172	#97	—	47	4	no	—
173	C₄×C₈	—	47	2	no	—
174	C₄×C₈	—	47	2	no	—
175	C₄×C₈	C₄×D₄₇	1	2	yes	—
176	C₄×C₈	C₂²×D₄₇	1	2	yes	—
177	C₂⁴	—	47	1	no	—
178	C₂²×C₄	—	47	1	no	—
179	C₂²×C₄	—	47	4	no	—
180	C₂²×C₄	—	47	2	no	—
181	C₂²×C₄	—	47	4	no	—
182	C₂²×C₄	—	47	2	no	—
183	C₂²×C₄	—	47	2	no	—
184	C₂²×C₄	C₈×D₄₇	1	2	yes	—
185	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
186	M₁₆	—	94	8	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	47	4	no	—
189	C₂×C₈	—	47	2	no	—
190	C₂×C₈	—	47	8	no	—
191	C₂×C₈	—	47	4	no	—
192	C₂×C₈	—	47	4	no	—
193	C₂×C₈	—	47	2	no	—
194	C₂×C₈	—	94	2	no	—
195	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
196	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
197	C₂×C₈	D₄×D₄₇	1	2	yes	—
198	C₂×C₈	Q₈×D₄₇	1	2	yes	—
199	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
200	C₄⋊C₄	—	47	8	no	—
201	C₂²⋊C₄	—	47	8	no	—
202	C₄²	—	47	2	no	—
203	C₄²	—	47	2	no	—
204	C₄²	C₈×D₄₇	1	2	yes	—
205	C₄²	C₂×C₄×D₄₇	1	2	yes	—
206	C₁₆	—	2	4	no	—
207	C₁₆	—	94	4	no	—
208	C₂³	—	47	2	no	—
209	C₂³	—	47	4	no	—
210	C₂³	—	188	2	no	—
211	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	47	2	no	—
214	C₂×C₄	—	47	4	no	—
215	C₂×C₄	—	47	4	no	—
216	C₂×C₄	—	47	4	no	—
217	C₂×C₄	—	47	8	no	—
218	C₂×C₄	—	47	4	no	—
219	C₂×C₄	—	188	2	no	—
220	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
221	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
222	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
223	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
224	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	47	4	no	—
227	C₈	—	94	2	no	—
228	C₈	C₄⋊C₄×D₄₇	1	2	yes	—
229	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
230	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	47	4	no	—
233	C₂²	—	94	4	no	—
234	C₂²	—	188	4	no	—
235	C₂²	C₄⋊C₈×D₄₇	1	2	yes	—
236	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	47	4	no	—
239	C₄	—	94	4	no	—
240	C₄	C₄⋊C₈×D₄₇	1	2	yes	—
241	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
242	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	94	4	no	—
245	C₂	C₂×C₄⋊C₈×D₄₇	1	1	yes	characteristic
246	C₂	#20256490	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄₇×#2150	C₂
prime radical	C₂³×D₄₇	C₂×C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2150	D₄₇
hypocenter	C₄₇	#33397
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #20245145	C₂	semidirect
4. #20245144	C₂	semidirect
5. #20245143	C₂	semidirect
6. C₄₇×#2150	C₂	semidirect
19. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
20. Dic₄₇⋊C₁₆	C₂²	semidirect
23. C₄₇×C₄⋊C₁₆	C₂²	semidirect
125. #2150	D₄₇	semidirect
146. C₄⋊C₁₆	C₂×D₄₇	semidirect
147. C₄₇	#33397	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₄₇	2
C₂×#15689077	2
#20245145	2
#20245144	2
#20245143	2
C₄₇×#2150	2
#33397	47