Subgroups — #26215008

Order 12032.

1122 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	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243253	C₂	1	4	yes	maximal, complemented
4	C₈×C₄⋊Dic₄₇	C₂	1	4	yes	maximal, complemented
5	C₂²⋊C₈×Dic₄₇	C₂	1	4	yes	maximal
6	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₄₇	C₄	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
9	#15689176	C₄	1	8	yes	complemented
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
13	C₂×C₈×Dic₄₇	C₂²	1	8	yes	—
14	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
16	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
18	C₂×C₄₇⋊#95	C₄	1	2	yes	—
19	#15688877	C₂²	1	8	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#431	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
25	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	C₂×C₄⋊Dic₄₇	C₈	1	16	yes	complemented
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₄₇	—	4	4	no	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
33	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
34	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
35	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
36	C₈×Dic₄₇	—	4	8	no	—
37	C₄₇⋊#95	C₂×C₄	1	4	yes	—
38	C₄₇⋊#95	C₂×C₄	1	4	yes	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	complemented
50	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
51	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
52	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
57	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂×C₈	1	8	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	4	4	no	—
68	C₂²×Dic₄₇	—	4	4	no	—
69	C₂²×Dic₄₇	C₂×C₈	1	16	yes	—
70	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
74	C₄²×C₄₇	C₄²	1	4	yes	—
75	C₄²×C₄₇	C₄²	1	4	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
80	C₄⋊Dic₄₇	C₂×C₈	1	16	yes	—
81	C₄⋊Dic₄₇	C₂×C₈	1	16	yes	—
82	C₄×Dic₄₇	—	2	8	no	—
83	C₄×Dic₄₇	—	4	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
87	C₂³×C₄₇	—	2	1	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
91	C₂³×C₄₇	C₂²×C₈	1	4	yes	—
92	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	C₄×C₈	1	16	yes	complemented
101	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
107	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
111	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
112	C₂×Dic₄₇	—	2	8	no	—
113	C₂×Dic₄₇	—	2	8	no	—
114	C₂×Dic₄₇	—	4	16	no	—
115	C₂×Dic₄₇	C₂²×C₈	1	16	yes	—
116	C₈×C₄₇	—	2	8	no	—
117	C₄₇⋊C₈	—	2	8	no	—
118	#20533	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	4	no	—
121	C₂²×C₄₇	—	2	4	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	C₂×C₄×C₈	1	4	yes	—
124	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
125	C₂²×C₄₇	#518	1	1	yes	characteristic
126	C₂²×C₄₇	#519	1	1	yes	characteristic
127	C₂²×C₄₇	C₂³×C₈	1	2	yes	—
128	C₂²×C₄₇	#568	1	2	yes	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	C₂×C₄×C₈	1	4	yes	—
133	C₄×C₄₇	C₂×C₄×C₈	1	4	yes	—
134	C₄×C₄₇	#520	1	2	yes	—
135	C₄×C₄₇	#520	1	2	yes	—
136	Dic₄₇	—	2	16	no	—
137	C₂×#431	—	47	1	no	—
138	C₂×#431	D₄₇	1	1	yes	characteristic
139	C₂×C₄×C₄⋊C₄	—	47	1	no	—
140	C₈×C₄⋊C₄	—	47	4	no	—
141	C₈×C₄⋊C₄	—	47	4	no	—
142	C₄×C₂²⋊C₈	—	47	4	no	—
143	C₂×C₄₇	—	2	4	no	—
144	C₂×C₄₇	#2274	1	1	yes	characteristic
145	C₂×C₄₇	C₂²×C₄×C₈	1	1	yes	characteristic
146	C₂×C₄₇	#2873	1	1	yes	characteristic
147	C₂×C₄₇	#2972	1	2	yes	—
148	C₂×C₄₇	#2982	1	2	yes	—
149	C₂²×C₄⋊C₄	—	47	2	no	—
150	C₂²×C₄²	—	47	2	no	—
151	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
152	#431	—	47	8	no	—
153	#431	Dic₄₇	1	8	yes	complemented
154	C₂×C₄⋊C₈	—	47	2	no	—
155	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
156	C₂×C₂²⋊C₈	—	47	2	no	—
157	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
158	C₂×C₄×C₈	—	47	2	no	—
159	C₂×C₄×C₈	—	47	8	no	—
160	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
161	C₄×C₄⋊C₄	—	47	2	no	—
162	C₄×C₄⋊C₄	—	47	2	no	—
163	C₄×C₄⋊C₄	—	47	2	no	—
164	C₄×C₄⋊C₄	—	47	2	no	—
165	C₂×#95	—	47	2	no	—
166	#336	—	47	8	no	—
167	C₄₇	#20533	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	—	47	4	no	—
169	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
170	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
171	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
172	C₂²×C₈	—	47	4	no	—
173	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
174	C₂×C₄⋊C₄	—	47	16	no	—
175	C₂×C₄⋊C₄	—	47	4	no	—
176	C₂×C₄⋊C₄	—	47	4	no	—
177	C₂×C₄²	—	47	4	no	—
178	C₂×C₄²	—	47	4	no	—
179	C₂×C₄²	—	188	4	no	—
180	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₄₇	1	4	yes	—
183	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
184	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
186	C₄⋊C₈	—	47	8	no	—
187	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	complemented
188	C₂²⋊C₈	—	47	8	no	—
189	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
190	C₄×C₈	—	47	8	no	—
191	C₄×C₈	—	188	8	no	—
192	C₄×C₈	C₂×Dic₄₇	1	8	yes	complemented
193	#95	—	47	4	no	—
194	#95	—	47	4	no	—
195	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	47	8	no	—
200	C₂²×C₄	—	47	16	no	—
201	C₂²×C₄	—	188	4	no	—
202	C₂²×C₄	—	188	4	no	—
203	C₂²×C₄	C₈×D₄₇	1	8	yes	—
204	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
205	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
206	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
210	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
213	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	47	8	no	—
217	C₂×C₈	—	94	4	no	—
218	C₂×C₈	—	94	4	no	—
219	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
220	C₄⋊C₄	—	47	16	no	—
221	C₄⋊C₄	—	47	16	no	—
222	C₄²	—	94	8	no	—
223	C₄²	—	188	4	no	—
224	C₄²	C₄×Dic₄₇	1	4	yes	—
225	C₄²	C₄×Dic₄₇	1	4	yes	—
226	C₄²	C₂×C₄×D₄₇	1	2	yes	—
227	C₄²	C₂×C₄×D₄₇	1	2	yes	—
228	C₄²	C₂²×Dic₄₇	1	2	yes	—
229	C₄²	C₂²×Dic₄₇	1	2	yes	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	2	no	—
233	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
234	C₂³	C₂×C₈×D₄₇	1	4	yes	—
235	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	47	16	no	—
244	C₂×C₄	—	94	8	no	—
245	C₂×C₄	—	94	8	no	—
246	C₂×C₄	—	188	16	no	—
247	C₂×C₄	C₈×Dic₄₇	1	16	yes	complemented
248	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
249	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
250	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
251	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
252	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
253	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
254	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
255	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
257	C₂×C₄	#13094070	1	2	yes	—
258	C₂×C₄	#13094071	1	2	yes	—
259	C₈	—	2	8	no	—
260	C₈	—	94	8	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	C₂×C₈×Dic₄₇	1	4	yes	—
266	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
267	C₂²	#15689597	1	1	yes	characteristic
268	C₂²	#15689600	1	1	yes	characteristic
269	C₂²	C₂²×C₈×D₄₇	1	2	yes	—
270	C₂²	#15689889	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	94	16	no	—
275	C₄	C₂×C₈×Dic₄₇	1	4	yes	—
276	C₄	C₂×C₈×Dic₄₇	1	4	yes	—
277	C₄	#15689603	1	2	yes	—
278	C₄	#15689606	1	2	yes	—
279	C₂	—	2	4	no	—
280	C₂	#20245744	1	1	yes	characteristic
281	C₂	C₂²×C₈×Dic₄₇	1	1	yes	characteristic
282	C₂	#20251806	1	1	yes	characteristic
283	C₂	#20252411	1	2	yes	—
284	C₂	#20252468	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂²×C₄×C₈
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#20533
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20243253	C₂	semidirect
4. C₈×C₄⋊Dic₄₇	C₂	semidirect
9. #15689176	C₄	semidirect
21. C₄₇×#431	C₄	semidirect
27. C₂×C₄⋊Dic₄₇	C₈	semidirect
33. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
35. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
49. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
51. C₄×C₈×C₄₇	C₂×C₄	semidirect
100. C₂×C₄×C₄₇	C₄×C₈	semidirect
153. #431	Dic₄₇	semidirect
167. C₄₇	#20533	semidirect
187. C₄⋊C₈	C₂×Dic₄₇	semidirect
192. C₄×C₈	C₂×Dic₄₇	semidirect
247. C₂×C₄	C₈×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₂×C₄×C₄⋊Dic₄₇	2
#20243253	2
C₈×C₄⋊Dic₄₇	2
C₂²⋊C₈×Dic₄₇	2
C₂×C₄₇×#431	2
#20533	47