Subgroups — #26216741

Order 12032.

994 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689170	C₂	1	4	yes	maximal
2	#423×D₄₇	C₂	1	2	yes	maximal
3	C₂×#15689075	C₂	1	1	yes	maximal, characteristic
4	#20243563	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243562	C₂	1	2	yes	maximal, complemented
6	#20243558	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×M₁₆×D₄₇	C₂²	1	4	yes	—
10	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
11	#15689170	—	2	16	no	—
12	C₂×#13093960	C₂²	1	4	yes	—
13	C₂×#13093959	C₂²	1	8	yes	—
14	#15689147	C₂²	1	2	yes	—
15	#15689146	C₂²	1	4	yes	—
16	#15689145	—	2	4	no	—
17	C₄⋊C₈×D₄₇	—	2	4	no	—
18	#15689075	C₄	1	4	yes	—
19	#15689075	C₄	1	4	yes	—
20	C₂×#13093935	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
22	C₂×#108×C₄₇	C₂²	1	4	yes	—
23	C₄₇×#423	C₂²	1	2	yes	—
24	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₄₇	Q₈	1	2	yes	—
26	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
27	C₂×#10625329	C₂³	1	4	yes	—
28	C₂×#10625324	—	2	4	no	—
29	C₂×#10625324	C₂×C₄	1	8	yes	—
30	C₂×#10625324	D₄	1	8	yes	—
31	C₂×#10625324	Q₈	1	8	yes	—
32	M₁₆×D₄₇	—	2	16	no	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	C₂×C₈×D₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
37	C₄²×D₄₇	C₂×C₄	1	2	yes	—
38	C₄²×D₄₇	C₂×C₄	1	2	yes	—
39	C₄²×D₄₇	D₄	1	4	yes	—
40	#13093960	—	2	8	no	—
41	#13093959	—	2	16	no	—
42	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
43	Dic₄₇⋊C₈	—	2	8	no	—
44	#13093935	C₂×C₄	1	4	yes	—
45	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
46	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
47	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	#108×C₄₇	—	2	8	no	—
51	C₄⋊C₈×C₄₇	—	2	4	no	—
52	#97×C₄₇	C₂×C₄	1	4	yes	—
53	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
57	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
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	4	yes	—
63	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×D₄₇	C₂×Q₈	1	4	yes	—
65	M₁₆×C₄₇	—	2	8	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
69	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
70	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
71	C₄²×C₄₇	C₂×D₄	1	2	yes	—
72	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
73	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
74	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
77	#10625329	—	2	8	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
80	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
81	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
82	#10625324	—	4	8	no	—
83	#10625324	C₄⋊C₄	1	32	yes	—
84	C₈×D₄₇	—	4	8	no	—
85	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×D₄₇	—	4	2	no	—
87	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₂²×D₄₇	C₄⋊Q₈	1	4	yes	—
89	C₂×C₄×C₄₇	—	4	2	no	—
90	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
93	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
95	C₂×Dic₄₇	—	2	4	no	—
96	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×Dic₄₇	C₄⋊Q₈	1	4	yes	—
100	C₄×D₄₇	—	2	8	no	—
101	C₄×D₄₇	—	4	4	no	—
102	C₄×D₄₇	C₂×C₄⋊C₄	1	8	yes	—
103	C₄×D₄₇	#127	1	4	yes	—
104	C₄×D₄₇	C₄⋊Q₈	1	4	yes	—
105	C₈×C₄₇	—	4	4	no	—
106	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
107	C₄₇⋊C₈	—	4	4	no	—
108	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
109	#20806	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	2	no	—
111	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
113	C₂×D₄₇	—	2	4	no	—
114	C₂×D₄₇	—	4	4	no	—
115	C₂×D₄₇	#384	1	4	yes	—
116	C₄×C₄₇	—	2	4	no	—
117	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
118	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
119	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	Dic₄₇	—	2	2	no	—
121	Dic₄₇	#384	1	4	yes	—
122	Dic₄₇	#430	1	4	yes	complemented
123	C₂×#430	—	47	4	no	—
124	C₂×#423	—	47	2	no	—
125	C₂×C₄×M₁₆	—	47	1	no	—
126	#1849	—	47	2	no	—
127	#1849	—	47	1	no	—
128	#1849	D₄₇	1	1	yes	characteristic, complemented
129	#1848	—	47	4	no	—
130	C₂×C₄₇	—	2	2	no	—
131	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
132	C₂×C₄₇	C₂×#430	1	2	yes	—
133	D₄₇	—	2	4	no	—
134	C₂²×M₁₆	—	47	2	no	—
135	C₂²×M₁₆	—	47	4	no	—
136	C₂²×C₄²	—	47	1	no	—
137	#430	—	94	16	no	—
138	C₂×#108	—	47	8	no	—
139	C₂×#108	—	47	4	no	—
140	C₂×#108	C₂×D₄₇	1	4	yes	—
141	#423	—	47	4	no	—
142	#423	—	47	2	no	—
143	#423	—	94	4	no	—
144	#423	C₂×D₄₇	1	2	yes	—
145	C₂×C₄⋊C₈	—	94	4	no	—
146	C₄×M₁₆	—	47	4	no	—
147	C₄×M₁₆	—	47	4	no	—
148	C₂×#97	—	47	1	no	—
149	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
150	C₂×C₄×C₈	—	47	2	no	—
151	C₄₇	#20806	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₄	—	47	1	no	—
153	C₂³×C₄	—	47	2	no	—
154	C₂×M₁₆	—	47	8	no	—
155	C₂×M₁₆	—	47	8	no	—
156	C₂×M₁₆	—	47	8	no	—
157	C₂×M₁₆	—	47	4	no	—
158	C₂×M₁₆	—	94	4	no	—
159	C₂×M₁₆	—	94	16	no	—
160	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
161	C₂²×C₈	—	47	2	no	—
162	C₂²×C₈	—	94	4	no	—
163	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
164	C₂×C₄²	—	47	2	no	—
165	C₂×C₄²	—	47	1	no	—
166	C₂×C₄²	—	47	2	no	—
167	C₂×C₄²	—	47	2	no	—
168	C₂×C₄²	—	47	4	no	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	#108	—	2	8	no	—
171	#108	—	94	16	no	—
172	#108	—	94	8	no	—
173	C₄⋊C₈	—	2	4	no	—
174	C₄⋊C₈	—	94	8	no	—
175	C₄⋊C₈	—	94	4	no	—
176	#97	—	47	4	no	—
177	#97	C₄×D₄₇	1	4	yes	—
178	C₄×C₈	—	47	4	no	—
179	C₄×C₈	—	47	4	no	—
180	C₂⁴	—	47	1	no	—
181	C₂²×C₄	—	47	1	no	—
182	C₂²×C₄	—	47	2	no	—
183	C₂²×C₄	—	47	4	no	—
184	C₂²×C₄	—	47	2	no	—
185	C₂²×C₄	—	47	4	no	—
186	C₂²×C₄	—	47	2	no	—
187	C₂²×C₄	—	47	4	no	—
188	C₂²×C₄	—	47	4	no	—
189	C₂²×C₄	—	188	4	no	—
190	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
191	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
192	M₁₆	—	2	8	no	—
193	M₁₆	—	47	32	no	—
194	M₁₆	—	94	8	no	—
195	M₁₆	—	188	8	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	47	4	no	—
198	C₂×C₈	—	47	4	no	—
199	C₂×C₈	—	47	4	no	—
200	C₂×C₈	—	94	4	no	—
201	C₂×C₈	—	188	8	no	—
202	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
203	C₂×C₈	D₄×D₄₇	1	4	yes	—
204	C₂×C₈	Q₈×D₄₇	1	4	yes	—
205	C₄²	—	47	4	no	—
206	C₄²	—	47	4	no	—
207	C₄²	—	47	1	no	—
208	C₄²	—	47	1	no	—
209	C₄²	—	47	2	no	—
210	C₄²	C₂×C₄×D₄₇	1	2	yes	—
211	C₄²	D₄×D₄₇	1	2	yes	—
212	C₂³	—	47	2	no	—
213	C₂³	—	47	4	no	—
214	C₂³	—	188	2	no	—
215	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	47	2	no	—
218	C₂×C₄	—	47	2	no	—
219	C₂×C₄	—	47	4	no	—
220	C₂×C₄	—	47	8	no	—
221	C₂×C₄	—	47	4	no	—
222	C₂×C₄	—	47	4	no	—
223	C₂×C₄	—	47	2	no	—
224	C₂×C₄	—	94	8	no	—
225	C₂×C₄	—	94	4	no	—
226	C₂×C₄	—	188	4	no	—
227	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
230	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
231	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
232	C₈	—	4	4	no	—
233	C₈	—	47	8	no	—
234	C₈	—	188	4	no	—
235	C₈	C₄⋊C₄×D₄₇	1	8	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	47	4	no	—
238	C₂²	—	94	4	no	—
239	C₂²	—	188	4	no	—
240	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
241	C₂²	C₄⋊Q₈×D₄₇	1	2	yes	—
242	C₄	—	2	4	no	—
243	C₄	—	47	4	no	—
244	C₄	—	47	4	no	—
245	C₄	—	94	2	no	—
246	C₄	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
247	C₄	#127×D₄₇	1	1	yes	characteristic
248	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
249	C₂	—	2	2	no	—
250	C₂	—	94	4	no	—
251	C₂	#384×D₄₇	1	1	yes	characteristic
252	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	#20806
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
4. #20243563	C₂	semidirect
5. #20243562	C₂	semidirect
6. #20243558	C₂	semidirect
7. C₄₇×#1849	C₂	semidirect
122. Dic₄₇	#430	semidirect
128. #1849	D₄₇	semidirect
151. C₄₇	#20806	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689170	2
#423×D₄₇	2
C₂×#15689075	2
#20243563	2
#20243562	2
#20243558	2
C₄₇×#1849	2
#20806	47