Subgroups — #26216893

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 240 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×#15689075	C₂	1	1	yes	maximal, characteristic
3	#20243563	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243528	C₂	1	2	yes	maximal, complemented
5	#20243427	C₂	1	4	yes	maximal
6	#108×Dic₄₇	C₂	1	4	yes	maximal, complemented
7	C₄₇×#1849	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	#15689075	C₄	1	8	yes	—
16	M₁₆×Dic₄₇	C₂²	1	4	yes	—
17	C₂×#13093935	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
19	#15688887	C₂²	1	8	yes	—
20	C₂×#108×C₄₇	C₂²	1	4	yes	—
21	C₄₇×#423	C₂²	1	2	yes	—
22	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₄₇	Q₈	1	2	yes	—
24	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
25	C₂×#10625329	C₂³	1	4	yes	—
26	C₂×#10625324	—	2	4	no	—
27	C₂×#10625324	—	2	4	no	—
28	C₂×#10625324	C₂×C₄	1	8	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
31	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
32	C₄²×D₄₇	C₂×C₄	1	4	yes	—
33	C₄²×D₄₇	D₄	1	2	yes	—
34	C₄²×D₄₇	D₄	1	2	yes	—
35	#13093960	—	2	8	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
37	(C₂×D₄₇)⋊C₈	—	2	8	no	—
38	#13093935	C₂×C₄	1	4	yes	—
39	#13093934	—	2	4	no	—
40	C₈×Dic₄₇	—	2	4	no	—
41	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
42	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	#108×C₄₇	—	2	8	no	—
46	C₄⋊C₈×C₄₇	—	2	4	no	—
47	#97×C₄₇	C₂×C₄	1	4	yes	—
48	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×D₄₇	—	2	2	no	—
54	C₂×C₄×D₄₇	—	2	2	no	—
55	C₂×C₄×D₄₇	—	4	4	no	—
56	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
57	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
58	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
60	M₁₆×C₄₇	—	2	8	no	—
61	C₂×C₈×C₄₇	—	2	4	no	—
62	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₄₇	#41	1	4	yes	—
64	C₂×C₈×C₄₇	#41	1	4	yes	—
65	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
66	C₄²×C₄₇	C₂×D₄	1	2	yes	—
67	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
68	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
69	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
70	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
71	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
72	#10625329	—	2	8	no	—
73	C₂×C₄₇⋊C₈	—	2	2	no	—
74	C₂×C₄₇⋊C₈	—	2	2	no	—
75	C₂×C₄₇⋊C₈	—	2	4	no	—
76	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
77	#10625324	—	2	16	no	—
78	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
79	C₂²×D₄₇	—	2	2	no	—
80	C₂²×D₄₇	—	4	2	no	—
81	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₄₇	—	4	2	no	—
83	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
87	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
88	C₂×Dic₄₇	—	2	2	no	—
89	C₂×Dic₄₇	—	2	4	no	—
90	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₄×D₄₇	—	2	4	no	—
94	C₄×D₄₇	—	2	2	no	—
95	C₄×D₄₇	—	2	2	no	—
96	C₄×D₄₇	—	2	8	no	—
97	C₄×D₄₇	—	4	4	no	—
98	C₈×C₄₇	—	4	4	no	—
99	C₈×C₄₇	#117	1	8	yes	—
100	C₄₇⋊C₈	—	2	4	no	—
101	C₄₇⋊C₈	—	4	4	no	—
102	#20827	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₄₇	#533	1	2	yes	—
106	C₂×D₄₇	—	2	4	no	—
107	C₂×D₄₇	—	2	2	no	—
108	C₂×D₄₇	—	4	4	no	—
109	C₄×C₄₇	—	2	4	no	—
110	C₄×C₄₇	C₂×#117	1	2	yes	—
111	C₄×C₄₇	#532	1	1	yes	characteristic
112	C₄×C₄₇	#532	1	1	yes	characteristic
113	Dic₄₇	—	2	2	no	—
114	Dic₄₇	—	2	2	no	—
115	Dic₄₇	#430	1	4	yes	—
116	#2920	—	47	2	no	—
117	C₂×C₄×M₁₆	—	47	1	no	—
118	#1849	—	47	1	no	—
119	#1849	D₄₇	1	1	yes	characteristic, complemented
120	#1838	—	47	2	no	—
121	#1811	—	47	4	no	—
122	C₄×#108	—	47	4	no	—
123	C₂×C₄₇	—	2	2	no	—
124	C₂×C₄₇	#2285	1	1	yes	characteristic
125	C₂×C₄₇	C₂×#430	1	2	yes	—
126	D₄₇	—	4	2	no	—
127	C₂²×M₁₆	—	47	2	no	—
128	C₂²×C₄²	—	47	1	no	—
129	#432	—	94	4	no	—
130	#431	—	94	4	no	—
131	C₂×#108	—	47	4	no	—
132	C₂×#108	C₂×D₄₇	1	4	yes	—
133	#423	—	47	2	no	—
134	#423	C₂×D₄₇	1	2	yes	—
135	#407	—	47	4	no	—
136	C₄×M₁₆	—	47	8	no	—
137	C₄×M₁₆	—	47	4	no	—
138	C₂×#97	—	47	1	no	—
139	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
140	C₂×C₄×C₈	—	47	2	no	—
141	#341	—	47	8	no	—
142	C₄₇	#20827	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₄	—	47	1	no	—
144	C₂³×C₄	—	47	2	no	—
145	C₂×M₁₆	—	47	8	no	—
146	C₂×M₁₆	—	47	4	no	—
147	C₂×M₁₆	—	94	4	no	—
148	C₂×M₁₆	—	94	4	no	—
149	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
150	C₂²×C₈	—	47	2	no	—
151	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
152	C₂×C₄²	—	47	2	no	—
153	C₂×C₄²	—	47	1	no	—
154	C₂×C₄²	—	47	4	no	—
155	C₂×C₄²	—	47	2	no	—
156	C₂×C₄²	—	47	2	no	—
157	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
158	#108	—	2	8	no	—
159	#108	—	94	8	no	—
160	C₄⋊C₈	—	2	4	no	—
161	C₄⋊C₈	—	94	4	no	—
162	C₂²⋊C₈	—	94	8	no	—
163	#97	—	47	4	no	—
164	#97	—	94	4	no	—
165	#97	C₄×D₄₇	1	4	yes	—
166	C₄×C₈	—	47	8	no	—
167	C₄×C₈	—	94	4	no	—
168	C₂⁴	—	47	1	no	—
169	C₂²×C₄	—	47	2	no	—
170	C₂²×C₄	—	47	1	no	—
171	C₂²×C₄	—	47	4	no	—
172	C₂²×C₄	—	47	2	no	—
173	C₂²×C₄	—	47	2	no	—
174	C₂²×C₄	—	47	4	no	—
175	C₂²×C₄	—	94	2	no	—
176	C₂²×C₄	—	94	2	no	—
177	C₂²×C₄	—	188	4	no	—
178	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
179	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
180	M₁₆	—	2	8	no	—
181	M₁₆	—	94	16	no	—
182	M₁₆	—	94	8	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	47	4	no	—
185	C₂×C₈	—	94	2	no	—
186	C₂×C₈	—	94	2	no	—
187	C₂×C₈	—	94	4	no	—
188	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
189	C₂×C₈	#10625352	1	4	yes	—
190	C₂×C₈	#10625354	1	4	yes	—
191	C₄²	—	47	4	no	—
192	C₄²	—	47	4	no	—
193	C₄²	—	47	2	no	—
194	C₄²	—	47	1	no	—
195	C₄²	—	47	1	no	—
196	C₄²	C₂×C₄×D₄₇	1	2	yes	—
197	C₄²	D₄×D₄₇	1	2	yes	—
198	C₂³	—	47	2	no	—
199	C₂³	—	94	2	no	—
200	C₂³	—	188	2	no	—
201	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	47	2	no	—
204	C₂×C₄	—	47	2	no	—
205	C₂×C₄	—	47	2	no	—
206	C₂×C₄	—	94	8	no	—
207	C₂×C₄	—	94	4	no	—
208	C₂×C₄	—	94	2	no	—
209	C₂×C₄	—	94	4	no	—
210	C₂×C₄	—	94	2	no	—
211	C₂×C₄	—	94	2	no	—
212	C₂×C₄	—	188	4	no	—
213	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
214	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
215	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
216	C₂×C₄	C₂×#10625352	1	2	yes	—
217	C₂×C₄	C₂×#10625354	1	2	yes	—
218	C₈	—	4	4	no	—
219	C₈	—	94	4	no	—
220	C₈	—	188	4	no	—
221	C₈	#13093982	1	8	yes	—
222	C₂²	—	2	2	no	—
223	C₂²	—	94	2	no	—
224	C₂²	—	94	4	no	—
225	C₂²	—	188	4	no	—
226	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
227	C₂²	#15689678	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	47	4	no	—
230	C₄	—	94	2	no	—
231	C₄	—	94	2	no	—
232	C₄	C₂×#13093982	1	2	yes	—
233	C₄	#15689671	1	1	yes	characteristic
234	C₄	#15689674	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	188	2	no	—
237	C₂	#20245793	1	1	yes	characteristic
238	C₂	#430×D₄₇	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×#1849	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1849	D₄₇
hypocenter	C₄₇	#20827
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

Complements

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

N	H ≅ G/N	type
3. #20243563	C₂	semidirect
4. #20243528	C₂	semidirect
6. #108×Dic₄₇	C₂	semidirect
7. C₄₇×#1849	C₂	semidirect
119. #1849	D₄₇	semidirect
142. C₄₇	#20827	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×#15689075	2
#20243563	2
#20243528	2
#20243427	2
#108×Dic₄₇	2
C₄₇×#1849	2
#20827	47