Subgroups — #26216921

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252020	C₂	1	2	yes	maximal
2	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
3	#20243560	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243525	C₂	1	2	yes	maximal, complemented
5	#20243427	C₂	1	4	yes	maximal
6	#20243296	C₂	1	4	yes	maximal, complemented
7	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625324	C₂²	1	2	yes	—
9	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
10	#15689179	—	2	4	no	—
11	#15689175	—	2	4	no	—
12	C₂×#13093960	C₂²	1	4	yes	—
13	#15689147	C₂²	1	2	yes	—
14	#15689086	C₂²	1	4	yes	—
15	M₁₆×Dic₄₇	C₂²	1	4	yes	—
16	C₄×#10625324	C₄	1	4	yes	—
17	C₄×#10625324	C₄	1	4	yes	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#15688887	C₂²	1	8	yes	—
21	C₂×#108×C₄₇	C₂²	1	4	yes	—
22	C₄₇×#423	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄×D₄₇	Q₈	1	2	yes	—
25	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
26	C₂×#10625329	C₂³	1	4	yes	—
27	C₂×#10625324	—	2	4	no	—
28	C₂×#10625324	—	2	4	no	—
29	C₂×#10625324	C₂×C₄	1	8	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
32	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
33	C₄²×D₄₇	C₂×C₄	1	2	yes	—
34	C₄²×D₄₇	C₂×C₄	1	2	yes	—
35	C₄²×D₄₇	D₄	1	4	yes	—
36	#13093960	—	2	8	no	—
37	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
38	(C₂×D₄₇)⋊C₈	—	2	8	no	—
39	#13093934	—	2	4	no	—
40	#13093934	C₂×C₄	1	8	yes	—
41	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
42	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
43	C₈×Dic₄₇	—	2	4	no	—
44	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	#108×C₄₇	—	2	8	no	—
48	C₄⋊C₈×C₄₇	—	2	4	no	—
49	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
56	C₂×C₄×D₄₇	—	2	2	no	—
57	C₂×C₄×D₄₇	—	2	2	no	—
58	C₂×C₄×D₄₇	—	4	4	no	—
59	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
60	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
61	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
63	M₁₆×C₄₇	—	2	8	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₄₇	#41	1	4	yes	—
67	C₂×C₈×C₄₇	#41	1	4	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×D₄	1	2	yes	—
71	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
72	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
73	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
74	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
75	#10625329	—	2	8	no	—
76	C₂×C₄₇⋊C₈	—	2	2	no	—
77	C₂×C₄₇⋊C₈	—	2	2	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
80	#10625324	—	2	16	no	—
81	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×D₄₇	—	2	2	no	—
83	C₂²×D₄₇	—	4	2	no	—
84	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
92	C₂×Dic₄₇	—	2	2	no	—
93	C₂×Dic₄₇	—	4	2	no	—
94	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₄×D₄₇	—	2	4	no	—
97	C₄×D₄₇	—	2	2	no	—
98	C₄×D₄₇	—	2	2	no	—
99	C₄×D₄₇	—	2	8	no	—
100	C₄×D₄₇	—	4	4	no	—
101	C₈×C₄₇	—	4	4	no	—
102	C₈×C₄₇	#117	1	8	yes	—
103	C₄₇⋊C₈	—	2	4	no	—
104	C₄₇⋊C₈	—	4	4	no	—
105	#20831	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	—	2	2	no	—
107	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₄₇	#533	1	2	yes	—
109	C₂×D₄₇	—	2	4	no	—
110	C₂×D₄₇	—	2	2	no	—
111	C₂×D₄₇	—	4	4	no	—
112	C₄×C₄₇	—	2	2	no	—
113	C₄×C₄₇	#430	1	4	yes	—
114	C₄×C₄₇	C₂×#117	1	2	yes	—
115	C₄×C₄₇	#532	1	1	yes	characteristic
116	C₄×C₄₇	#532	1	1	yes	characteristic
117	Dic₄₇	—	2	4	no	—
118	Dic₄₇	—	2	2	no	—
119	#2920	—	47	2	no	—
120	C₂×C₄×M₁₆	—	47	1	no	—
121	#1848	—	47	1	no	—
122	#1848	D₄₇	1	1	yes	characteristic, complemented
123	#1837	—	47	2	no	—
124	#1811	—	47	4	no	—
125	#1778	—	47	4	no	—
126	C₂×C₄₇	—	2	2	no	—
127	C₂×C₄₇	#2285	1	1	yes	characteristic
128	C₂×C₄₇	C₂×#430	1	2	yes	—
129	D₄₇	—	4	2	no	—
130	C₂²×M₁₆	—	47	2	no	—
131	C₂²×C₄²	—	47	1	no	—
132	#432	—	94	4	no	—
133	#431	—	94	4	no	—
134	C₂×#108	—	47	4	no	—
135	C₂×#108	C₂×D₄₇	1	4	yes	—
136	#423	—	47	2	no	—
137	#423	C₂×D₄₇	1	2	yes	—
138	#407	—	47	4	no	—
139	C₄×M₁₆	—	47	4	no	—
140	C₄×M₁₆	—	47	4	no	—
141	C₄×M₁₆	—	47	4	no	—
142	C₂×#97	—	47	2	no	—
143	C₂×C₄×C₈	—	47	1	no	—
144	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
145	#341	—	47	8	no	—
146	C₄₇	#20831	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	—	47	1	no	—
148	C₂³×C₄	—	47	2	no	—
149	C₂×M₁₆	—	47	8	no	—
150	C₂×M₁₆	—	47	4	no	—
151	C₂×M₁₆	—	94	4	no	—
152	C₂×M₁₆	—	94	4	no	—
153	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
154	C₂²×C₈	—	47	2	no	—
155	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
156	C₂×C₄²	—	47	2	no	—
157	C₂×C₄²	—	47	1	no	—
158	C₂×C₄²	—	47	2	no	—
159	C₂×C₄²	—	47	2	no	—
160	C₂×C₄²	—	47	4	no	—
161	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
162	#108	—	2	8	no	—
163	#108	—	94	8	no	—
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	94	4	no	—
166	C₂²⋊C₈	—	94	8	no	—
167	#97	—	47	8	no	—
168	#97	—	94	4	no	—
169	C₄×C₈	—	47	2	no	—
170	C₄×C₈	—	47	2	no	—
171	C₄×C₈	—	94	4	no	—
172	C₄×C₈	C₄×D₄₇	1	2	yes	—
173	C₄×C₈	C₄×D₄₇	1	2	yes	—
174	C₂⁴	—	47	1	no	—
175	C₂²×C₄	—	47	1	no	—
176	C₂²×C₄	—	47	2	no	—
177	C₂²×C₄	—	47	4	no	—
178	C₂²×C₄	—	47	2	no	—
179	C₂²×C₄	—	47	2	no	—
180	C₂²×C₄	—	47	4	no	—
181	C₂²×C₄	—	94	2	no	—
182	C₂²×C₄	—	94	2	no	—
183	C₂²×C₄	—	188	4	no	—
184	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
185	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
186	M₁₆	—	2	8	no	—
187	M₁₆	—	94	16	no	—
188	M₁₆	—	94	8	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	47	4	no	—
191	C₂×C₈	—	94	2	no	—
192	C₂×C₈	—	94	2	no	—
193	C₂×C₈	—	94	4	no	—
194	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
195	C₂×C₈	#10625352	1	4	yes	—
196	C₂×C₈	#10625354	1	4	yes	—
197	C₄²	—	47	4	no	—
198	C₄²	—	47	4	no	—
199	C₄²	—	47	2	no	—
200	C₄²	—	47	2	no	—
201	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
202	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
203	C₄²	D₄×D₄₇	1	2	yes	—
204	C₂³	—	47	2	no	—
205	C₂³	—	94	2	no	—
206	C₂³	—	188	2	no	—
207	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	47	2	no	—
210	C₂×C₄	—	47	4	no	—
211	C₂×C₄	—	94	2	no	—
212	C₂×C₄	—	94	4	no	—
213	C₂×C₄	—	94	2	no	—
214	C₂×C₄	—	94	2	no	—
215	C₂×C₄	—	94	8	no	—
216	C₂×C₄	—	188	4	no	—
217	C₂×C₄	—	188	2	no	—
218	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
219	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
220	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
222	C₂×C₄	C₂×#10625352	1	2	yes	—
223	C₂×C₄	C₂×#10625354	1	2	yes	—
224	C₈	—	4	4	no	—
225	C₈	—	94	4	no	—
226	C₈	—	188	4	no	—
227	C₈	#13093982	1	8	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	—	94	2	no	—
230	C₂²	—	94	4	no	—
231	C₂²	—	188	4	no	—
232	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
233	C₂²	#15689678	1	2	yes	—
234	C₄	—	2	2	no	—
235	C₄	—	94	2	no	—
236	C₄	—	94	4	no	—
237	C₄	#15689170	1	4	yes	—
238	C₄	C₂×#13093982	1	2	yes	—
239	C₄	#15689671	1	1	yes	characteristic
240	C₄	#15689674	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	188	2	no	—
243	C₂	#20245793	1	1	yes	characteristic
244	C₂	C₂×#15689170	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#20831
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
3. #20243560	C₂	semidirect
4. #20243525	C₂	semidirect
6. #20243296	C₂	semidirect
7. C₄₇×#1848	C₂	semidirect
122. #1848	D₄₇	semidirect
146. C₄₇	#20831	semidirect

Maximal subgroups

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

subgroup	index
#20252020	2
C₂×C₄×#10625324	2
#20243560	2
#20243525	2
#20243427	2
#20243296	2
C₄₇×#1848	2
#20831	47