Subgroups — #26216887

Order 12032.

810 conjugacy classes of subgroups.

The subgroup lattice has 268 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×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243560	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243525	C₂	1	2	yes	maximal, complemented
5	#20243423	C₂	1	4	yes	maximal
6	#108×Dic₄₇	C₂	1	4	yes	maximal, complemented
7	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	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	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
18	C₄×C₈×D₄₇	C₄	1	4	yes	—
19	C₄×C₈×D₄₇	C₄	1	4	yes	—
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	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
28	C₂×C₈×D₄₇	—	2	4	no	—
29	C₂×C₈×D₄₇	—	2	4	no	—
30	C₂×C₈×D₄₇	C₂×C₄	1	8	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	2	yes	—
36	C₄²×D₄₇	D₄	1	2	yes	—
37	#13093960	—	2	8	no	—
38	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
39	(C₂×D₄₇)⋊C₈	—	2	8	no	—
40	#13093934	—	2	4	no	—
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	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
45	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	#108×C₄₇	—	2	8	no	—
49	C₄⋊C₈×C₄₇	—	2	4	no	—
50	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
57	C₂×C₄×D₄₇	—	2	2	no	—
58	C₂×C₄×D₄₇	—	2	2	no	—
59	C₂×C₄×D₄₇	—	2	8	no	—
60	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
65	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
66	M₁₆×C₄₇	—	2	8	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₄₇	#41	1	4	yes	—
70	C₂×C₈×C₄₇	#41	1	4	yes	—
71	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₄₇	C₂×D₄	1	2	yes	—
74	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
75	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
76	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
77	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
78	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
79	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
80	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
81	#10625329	—	2	8	no	—
82	C₂×C₄₇⋊C₈	—	2	2	no	—
83	C₂×C₄₇⋊C₈	—	2	2	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
86	C₈×D₄₇	—	2	16	no	—
87	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
88	C₂²×D₄₇	—	2	2	no	—
89	C₂²×D₄₇	—	2	4	no	—
90	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
98	C₂×Dic₄₇	—	2	2	no	—
99	C₂×Dic₄₇	—	2	4	no	—
100	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₄×D₄₇	—	2	4	no	—
104	C₄×D₄₇	—	2	4	no	—
105	C₄×D₄₇	—	2	2	no	—
106	C₄×D₄₇	—	2	2	no	—
107	C₄×D₄₇	—	2	4	no	—
108	C₄×D₄₇	—	2	4	no	—
109	C₄×D₄₇	#108	1	8	yes	complemented
110	C₈×C₄₇	—	4	4	no	—
111	C₈×C₄₇	#117	1	8	yes	—
112	C₄₇⋊C₈	—	2	4	no	—
113	C₄₇⋊C₈	—	4	4	no	—
114	#20826	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₄₇	#533	1	2	yes	—
118	C₂×D₄₇	—	2	8	no	—
119	C₂×D₄₇	—	2	2	no	—
120	C₂×D₄₇	—	2	2	no	—
121	C₂×D₄₇	C₂×#108	1	4	yes	—
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	C₂×#108	1	4	yes	—
124	C₄×C₄₇	C₂×#117	1	2	yes	—
125	C₄×C₄₇	#532	1	1	yes	characteristic
126	C₄×C₄₇	#532	1	1	yes	characteristic
127	Dic₄₇	—	2	2	no	—
128	Dic₄₇	—	2	2	no	—
129	Dic₄₇	C₂×#108	1	4	yes	—
130	#2920	—	47	2	no	—
131	C₂²×C₄×C₈	—	47	1	no	—
132	#1848	—	47	1	no	—
133	#1848	D₄₇	1	1	yes	characteristic, complemented
134	#1837	—	47	2	no	—
135	#1810	—	47	4	no	—
136	C₄×#108	—	47	4	no	—
137	C₂×C₄₇	—	2	2	no	—
138	C₂×C₄₇	#2285	1	1	yes	characteristic
139	C₂×C₄₇	C₂²×#108	1	2	yes	—
140	D₄₇	—	2	4	no	—
141	C₂³×C₈	—	47	2	no	—
142	C₂²×C₄²	—	47	1	no	—
143	#432	—	94	4	no	—
144	#431	—	94	4	no	—
145	C₂×#108	—	47	4	no	—
146	C₂×#108	C₂×D₄₇	1	4	yes	—
147	#423	—	47	2	no	—
148	#423	C₂×D₄₇	1	2	yes	—
149	#407	—	47	4	no	—
150	C₄×M₁₆	—	47	4	no	—
151	C₂×C₄×C₈	—	47	2	no	—
152	C₂×C₄×C₈	—	47	1	no	—
153	C₂×C₄×C₈	—	47	4	no	—
154	C₂×C₄×C₈	—	47	4	no	—
155	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
156	#341	—	47	8	no	—
157	C₄₇	#20826	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	—	47	2	no	—
159	C₂³×C₄	—	47	1	no	—
160	C₂×M₁₆	—	47	4	no	—
161	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
162	C₂²×C₈	—	47	2	no	—
163	C₂²×C₈	—	47	8	no	—
164	C₂²×C₈	—	94	4	no	—
165	C₂²×C₈	—	94	4	no	—
166	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
167	C₂×C₄²	—	47	2	no	—
168	C₂×C₄²	—	47	1	no	—
169	C₂×C₄²	—	47	2	no	—
170	C₂×C₄²	—	47	2	no	—
171	C₂×C₄²	—	47	2	no	—
172	C₂×C₄²	—	47	2	no	—
173	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	#108	—	2	8	no	—
175	#108	—	94	8	no	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	94	4	no	—
178	C₂²⋊C₈	—	94	8	no	—
179	#97	—	94	4	no	—
180	C₄×C₈	—	47	8	no	—
181	C₄×C₈	—	47	2	no	—
182	C₄×C₈	—	47	2	no	—
183	C₄×C₈	—	94	4	no	—
184	C₄×C₈	C₄×D₄₇	1	2	yes	—
185	C₄×C₈	C₄×D₄₇	1	2	yes	—
186	C₂⁴	—	47	1	no	—
187	C₂²×C₄	—	47	2	no	—
188	C₂²×C₄	—	47	2	no	—
189	C₂²×C₄	—	47	1	no	—
190	C₂²×C₄	—	47	2	no	—
191	C₂²×C₄	—	47	2	no	—
192	C₂²×C₄	—	47	2	no	—
193	C₂²×C₄	—	47	2	no	—
194	C₂²×C₄	—	47	2	no	—
195	C₂²×C₄	—	94	8	no	—
196	C₂²×C₄	—	94	2	no	—
197	C₂²×C₄	—	94	2	no	—
198	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
199	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
200	M₁₆	—	2	8	no	—
201	M₁₆	—	94	8	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	47	4	no	—
204	C₂×C₈	—	94	2	no	—
205	C₂×C₈	—	94	2	no	—
206	C₂×C₈	—	94	16	no	—
207	C₂×C₈	—	94	4	no	—
208	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
209	C₂×C₈	#10625352	1	4	yes	—
210	C₂×C₈	#10625354	1	4	yes	—
211	C₄²	—	47	2	no	—
212	C₄²	—	47	2	no	—
213	C₄²	—	47	2	no	—
214	C₄²	—	47	2	no	—
215	C₄²	—	47	2	no	—
216	C₄²	—	47	1	no	—
217	C₄²	—	47	1	no	—
218	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
219	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
220	C₄²	D₄×D₄₇	1	2	yes	—
221	C₂³	—	47	2	no	—
222	C₂³	—	94	4	no	—
223	C₂³	—	94	2	no	—
224	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	47	2	no	—
227	C₂×C₄	—	47	8	no	—
228	C₂×C₄	—	47	2	no	—
229	C₂×C₄	—	47	2	no	—
230	C₂×C₄	—	94	4	no	—
231	C₂×C₄	—	94	4	no	—
232	C₂×C₄	—	94	4	no	—
233	C₂×C₄	—	94	2	no	—
234	C₂×C₄	—	94	4	no	—
235	C₂×C₄	—	94	2	no	—
236	C₂×C₄	—	94	2	no	—
237	C₂×C₄	—	94	4	no	—
238	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
239	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
240	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
242	C₂×C₄	C₂×#10625352	1	2	yes	—
243	C₂×C₄	C₂×#10625354	1	2	yes	—
244	C₈	—	4	4	no	—
245	C₈	—	94	4	no	—
246	C₈	—	188	4	no	—
247	C₈	#13093982	1	8	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	47	4	no	—
250	C₂²	—	94	8	no	—
251	C₂²	—	94	2	no	—
252	C₂²	—	94	2	no	—
253	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
254	C₂²	#15689678	1	2	yes	—
255	C₄	—	2	2	no	—
256	C₄	—	47	4	no	—
257	C₄	—	94	2	no	—
258	C₄	—	94	2	no	—
259	C₄	#108×D₄₇	1	4	yes	—
260	C₄	C₂×#13093982	1	2	yes	—
261	C₄	#15689671	1	1	yes	characteristic
262	C₄	#15689674	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	94	4	no	—
265	C₂	#20245793	1	1	yes	characteristic
266	C₂	C₂×#108×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×#1848	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#20826
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. #108×Dic₄₇	C₂	semidirect
7. C₄₇×#1848	C₂	semidirect
109. C₄×D₄₇	#108	semidirect
133. #1848	D₄₇	semidirect
157. C₄₇	#20826	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×C₈×D₄₇	2
#20243560	2
#20243525	2
#20243423	2
#108×Dic₄₇	2
C₄₇×#1848	2
#20826	47