Subgroups — #26216920

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252019	C₂	1	2	yes	maximal
2	C₂×#15689075	C₂	1	1	yes	maximal, characteristic
3	#20243558	C₂	1	2	yes	maximal, complemented
4	#20243524	C₂	1	2	yes	maximal, complemented
5	#20243429	C₂	1	2	yes	maximal
6	#20243427	C₂	1	2	yes	maximal
7	#20243295	C₂	1	2	yes	maximal, complemented
8	#20243294	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1778	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625324	C₂²	1	1	yes	characteristic
11	C₂²×#10625324	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
13	#15689178	—	2	4	no	—
14	#15689174	—	2	4	no	—
15	C₂×#13093960	C₂²	1	2	yes	—
16	C₂×#13093959	C₂²	1	4	yes	—
17	#15689146	C₂²	1	4	yes	—
18	#15689086	C₂²	1	4	yes	—
19	C₄×#10625329	C₂²	1	2	yes	—
20	#15689075	C₄	1	4	yes	—
21	#15689075	C₄	1	4	yes	—
22	C₂×#13093935	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
25	#15688888	C₂²	1	2	yes	—
26	#15688887	C₂²	1	4	yes	—
27	C₂×#108×C₄₇	C₂²	1	2	yes	—
28	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
29	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#341	C₂²	1	2	yes	—
31	C₂²×C₄×D₄₇	Q₈	1	2	yes	—
32	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
33	C₂×#10625329	C₂³	1	4	yes	—
34	C₂×#10625324	—	2	4	no	—
35	C₂×#10625324	—	2	4	no	—
36	C₂×#10625324	C₂×C₄	1	4	yes	—
37	C₂×#10625324	C₂×C₄	1	4	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
41	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
42	C₄²×D₄₇	C₂×C₄	1	2	yes	—
43	C₄²×D₄₇	C₂×C₄	1	2	yes	—
44	C₄²×D₄₇	D₄	1	4	yes	—
45	#13093960	—	2	4	no	—
46	#13093959	—	2	8	no	—
47	Dic₄₇⋊C₈	—	2	8	no	—
48	(C₂×D₄₇)⋊C₈	—	2	8	no	—
49	#13093935	—	2	2	no	—
50	#13093935	C₂×C₄	1	4	yes	—
51	C₄×C₄₇⋊C₈	—	2	2	no	—
52	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
53	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
54	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
55	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
56	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
57	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	#108×C₄₇	—	2	4	no	—
61	#97×C₄₇	—	2	2	no	—
62	#97×C₄₇	C₂×C₄	1	4	yes	—
63	C₄×C₈×C₄₇	—	2	2	no	—
64	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
68	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
69	C₂×C₄×D₄₇	—	2	2	no	—
70	C₂×C₄×D₄₇	—	2	2	no	—
71	C₂×C₄×D₄₇	—	4	4	no	—
72	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
73	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
74	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
76	M₁₆×C₄₇	—	2	4	no	—
77	M₁₆×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₇	#41	1	4	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₄₇	C₂×D₄	1	2	yes	—
85	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
86	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
87	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
90	#10625329	—	2	4	no	—
91	#10625329	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₄₇⋊C₈	#41	1	4	yes	—
97	#10625324	—	2	8	no	—
98	#10625324	—	2	8	no	—
99	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
100	C₂²×D₄₇	—	2	2	no	—
101	C₂²×D₄₇	—	4	2	no	—
102	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
109	C₂×Dic₄₇	—	2	2	no	—
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₄×D₄₇	—	2	2	no	—
115	C₄×D₄₇	—	2	2	no	—
116	C₄×D₄₇	—	2	4	no	—
117	C₄×D₄₇	—	2	8	no	—
118	C₄×D₄₇	—	4	4	no	—
119	C₈×C₄₇	—	2	2	no	—
120	C₈×C₄₇	—	4	4	no	—
121	C₈×C₄₇	#117	1	4	yes	—
122	C₄₇⋊C₈	—	2	2	no	—
123	C₄₇⋊C₈	—	4	4	no	—
124	C₄₇⋊C₈	#117	1	4	yes	—
125	#20831	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₄₇	#533	1	2	yes	—
129	C₂×D₄₇	—	2	4	no	—
130	C₂×D₄₇	—	2	2	no	—
131	C₂×D₄₇	—	4	4	no	—
132	C₄×C₄₇	—	2	4	no	—
133	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
134	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
135	C₄×C₄₇	#532	1	2	yes	—
136	Dic₄₇	—	2	2	no	—
137	Dic₄₇	—	2	2	no	—
138	Dic₄₇	#430	1	4	yes	—
139	#2920	—	47	2	no	—
140	C₂×C₄×M₁₆	—	47	1	no	—
141	#1848	—	47	2	no	—
142	#1837	—	47	2	no	—
143	#1811	—	47	2	no	—
144	#1811	—	47	2	no	—
145	#1778	—	47	2	no	—
146	#1778	—	47	1	no	—
147	#1778	D₄₇	1	1	yes	characteristic, complemented
148	C₂×C₄₇	—	2	2	no	—
149	C₂×C₄₇	#2285	1	1	yes	characteristic
150	C₂×C₄₇	C₂×#430	1	2	yes	—
151	D₄₇	—	4	2	no	—
152	C₂²×M₁₆	—	47	1	no	—
153	C₂²×M₁₆	—	47	1	no	—
154	C₂²×C₄²	—	47	1	no	—
155	#432	—	94	4	no	—
156	#431	—	94	4	no	—
157	C₂×#108	—	47	4	no	—
158	C₂×#108	—	47	2	no	—
159	C₂×#108	C₂×D₄₇	1	2	yes	—
160	#423	—	47	4	no	—
161	#407	—	47	4	no	—
162	C₄×M₁₆	—	47	4	no	—
163	C₄×M₁₆	—	47	4	no	—
164	C₄×M₁₆	—	47	2	no	—
165	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
166	C₂×#97	—	47	1	no	—
167	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
168	C₂×C₄×C₈	—	47	1	no	—
169	C₂×C₄×C₈	—	47	1	no	—
170	#341	—	47	4	no	—
171	#341	—	47	2	no	—
172	#341	C₂×D₄₇	1	2	yes	—
173	C₄₇	#20831	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	—	47	1	no	—
175	C₂³×C₄	—	47	2	no	—
176	C₂×M₁₆	—	47	4	no	—
177	C₂×M₁₆	—	47	4	no	—
178	C₂×M₁₆	—	47	4	no	—
179	C₂×M₁₆	—	94	4	no	—
180	C₂×M₁₆	—	94	4	no	—
181	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
182	C₂²×C₈	—	47	1	no	—
183	C₂²×C₈	—	47	1	no	—
184	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×C₄²	—	47	2	no	—
187	C₂×C₄²	—	47	1	no	—
188	C₂×C₄²	—	47	2	no	—
189	C₂×C₄²	—	47	2	no	—
190	C₂×C₄²	—	47	4	no	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	#108	—	2	4	no	—
193	#108	—	94	8	no	—
194	#108	—	94	4	no	—
195	C₄⋊C₈	—	94	8	no	—
196	C₂²⋊C₈	—	94	8	no	—
197	#97	—	2	2	no	—
198	#97	—	47	4	no	—
199	#97	—	94	2	no	—
200	#97	C₄×D₄₇	1	4	yes	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	—	47	2	no	—
203	C₄×C₈	—	47	2	no	—
204	C₄×C₈	—	47	2	no	—
205	C₄×C₈	—	47	2	no	—
206	C₄×C₈	—	94	2	no	—
207	C₂⁴	—	47	1	no	—
208	C₂²×C₄	—	47	2	no	—
209	C₂²×C₄	—	47	1	no	—
210	C₂²×C₄	—	47	4	no	—
211	C₂²×C₄	—	47	2	no	—
212	C₂²×C₄	—	47	2	no	—
213	C₂²×C₄	—	47	4	no	—
214	C₂²×C₄	—	94	2	no	—
215	C₂²×C₄	—	94	2	no	—
216	C₂²×C₄	—	188	4	no	—
217	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
218	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	94	8	no	—
222	M₁₆	—	94	8	no	—
223	M₁₆	—	94	4	no	—
224	M₁₆	—	94	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	47	2	no	—
228	C₂×C₈	—	47	4	no	—
229	C₂×C₈	—	47	2	no	—
230	C₂×C₈	—	94	2	no	—
231	C₂×C₈	—	94	4	no	—
232	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
234	C₂×C₈	#10625350	1	4	yes	—
235	C₄²	—	47	4	no	—
236	C₄²	—	47	4	no	—
237	C₄²	—	47	1	no	—
238	C₄²	—	47	1	no	—
239	C₄²	—	47	2	no	—
240	C₄²	C₂×C₄×D₄₇	1	2	yes	—
241	C₄²	D₄×D₄₇	1	2	yes	—
242	C₂³	—	47	2	no	—
243	C₂³	—	94	2	no	—
244	C₂³	—	188	2	no	—
245	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	47	2	no	—
248	C₂×C₄	—	47	2	no	—
249	C₂×C₄	—	47	2	no	—
250	C₂×C₄	—	94	8	no	—
251	C₂×C₄	—	94	4	no	—
252	C₂×C₄	—	94	2	no	—
253	C₂×C₄	—	94	2	no	—
254	C₂×C₄	—	94	2	no	—
255	C₂×C₄	—	94	4	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
258	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
259	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
260	C₂×C₄	#41×D₄₇	1	2	yes	—
261	C₂×C₄	C₂×#10625350	1	2	yes	—
262	C₈	—	2	2	no	—
263	C₈	—	4	4	no	—
264	C₈	—	47	4	no	—
265	C₈	—	94	2	no	—
266	C₈	—	188	4	no	—
267	C₈	#13093981	1	4	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	94	2	no	—
270	C₂²	—	94	4	no	—
271	C₂²	—	188	4	no	—
272	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
273	C₂²	#15689677	1	2	yes	—
274	C₄	—	2	4	no	—
275	C₄	—	47	4	no	—
276	C₄	—	94	2	no	—
277	C₄	—	94	2	no	—
278	C₄	#117×D₄₇	1	1	yes	characteristic
279	C₄	C₂×#13093981	1	1	yes	characteristic
280	C₄	#15689670	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	188	2	no	—
283	C₂	#20245792	1	1	yes	characteristic
284	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×#1778	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1778	D₄₇
hypocenter	C₄₇	#20831
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

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

N	H ≅ G/N	type
3. #20243558	C₂	semidirect
4. #20243524	C₂	semidirect
7. #20243295	C₂	semidirect
8. #20243294	C₂	semidirect
9. C₄₇×#1778	C₂	semidirect
147. #1778	D₄₇	semidirect
173. C₄₇	#20831	semidirect

Maximal subgroups

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

subgroup	index
#20252019	2
C₂×#15689075	2
#20243558	2
#20243524	2
#20243429	2
#20243427	2
#20243295	2
#20243294	2
C₄₇×#1778	2
#20831	47