Subgroups — #26105695

Order 12032.

666 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#431×D₄₇	C₂	1	2	yes	maximal
2	C₂×#15689075	C₂	1	1	yes	maximal, characteristic
3	#20241219	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241218	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1275	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10625324	C₄	1	2	yes	—
7	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
8	#15689176	C₂²	1	2	yes	—
9	#15689175	—	2	2	no	—
10	#15689174	C₄	1	4	yes	—
11	#15689173	—	2	2	no	—
12	C₄⋊C₈×D₄₇	—	2	2	no	—
13	C₂²⋊C₈×D₄₇	—	2	2	no	—
14	#15689075	—	2	2	no	—
15	#15689075	—	2	2	no	—
16	C₂×#13093935	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
18	C₄×C₈×D₄₇	C₄	1	4	yes	—
19	C₄₇×#431	C₂²	1	2	yes	—
20	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×D₄₇	Q₈	1	1	yes	characteristic
24	C₂×#10625324	—	2	4	no	—
25	C₂×#10625324	—	2	4	no	—
26	C₂×#10625324	C₈	1	8	yes	complemented
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₈×D₄₇	—	2	4	no	—
29	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₄₇	Q₈	1	1	yes	characteristic
32	C₄²×D₄₇	C₂×C₄	1	2	yes	—
33	C₄²×D₄₇	C₂×C₄	1	2	yes	—
34	C₄²×D₄₇	D₄	1	2	yes	—
35	C₄²×D₄₇	D₄	1	2	yes	—
36	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
37	Dic₄₇⋊C₈	—	2	4	no	—
38	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
39	(C₂×D₄₇)⋊C₈	—	2	4	no	—
40	#13093935	—	2	2	no	—
41	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
42	C₈×Dic₄₇	—	2	4	no	—
43	C₈×Dic₄₇	—	2	2	no	—
44	C₈×Dic₄₇	—	2	2	no	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₄₇	—	2	2	no	—
48	C₂²⋊C₈×C₄₇	—	2	2	no	—
49	#97×C₄₇	—	2	2	no	—
50	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
51	C₂³×D₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×D₄₇	—	4	2	no	—
59	C₂×C₄×D₄₇	—	4	2	no	—
60	C₂×C₄×D₄₇	C₄²	1	2	yes	—
61	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₄₇	C₂×C₈	1	4	yes	—
67	C₂×C₄×D₄₇	M₁₆	1	4	yes	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	complemented
72	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₄₇	C₂×D₄	1	2	yes	—
74	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
75	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
76	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
77	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
80	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
81	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
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	C₂×C₈	1	4	yes	complemented
86	#10625324	—	2	16	no	—
87	C₈×D₄₇	—	4	8	no	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×D₄₇	—	4	1	no	—
90	C₂²×D₄₇	—	4	1	no	—
91	C₂²×D₄₇	#95	1	2	yes	—
92	C₂²×D₄₇	C₄⋊C₈	1	4	yes	—
93	C₂×C₄×C₄₇	—	4	1	no	—
94	C₂×C₄×C₄₇	—	4	1	no	—
95	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	#95	1	1	yes	characteristic
104	C₂×Dic₄₇	#95	1	1	yes	characteristic
105	C₂×Dic₄₇	#95	1	2	yes	—
106	C₂×Dic₄₇	#95	1	1	yes	characteristic
107	C₂×Dic₄₇	#95	1	1	yes	characteristic
108	C₂×Dic₄₇	C₄⋊C₈	1	4	yes	—
109	C₄×D₄₇	—	2	4	no	—
110	C₄×D₄₇	—	2	4	no	—
111	C₄×D₄₇	—	4	2	no	—
112	C₄×D₄₇	—	4	2	no	—
113	C₄×D₄₇	C₄×C₈	1	4	yes	—
114	C₄×D₄₇	#97	1	4	yes	—
115	C₄×D₄₇	C₂²⋊C₈	1	8	yes	—
116	C₈×C₄₇	—	2	4	no	—
117	C₈×C₄₇	—	4	4	no	—
118	C₄₇⋊C₈	—	2	4	no	—
119	C₄₇⋊C₈	—	4	4	no	—
120	#7860	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	1	no	—
122	C₂²×C₄₇	—	2	1	no	—
123	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
125	C₂×D₄₇	—	2	2	no	—
126	C₂×D₄₇	—	2	2	no	—
127	C₂×D₄₇	—	4	4	no	—
128	C₂×D₄₇	#336	1	4	yes	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
132	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
133	C₄×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
134	Dic₄₇	—	2	1	no	—
135	Dic₄₇	—	2	1	no	—
136	Dic₄₇	#336	1	4	yes	—
137	Dic₄₇	H(3,ℤ/4)	1	2	yes	—
138	Dic₄₇	#338	1	2	yes	—
139	C₂×#431	—	47	2	no	—
140	C₂×C₄×M₁₆	—	47	1	no	—
141	#1275	—	47	2	no	—
142	#1275	—	47	1	no	—
143	#1275	D₄₇	1	1	yes	characteristic, complemented
144	C₂×C₄₇	—	2	2	no	—
145	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
146	C₂×C₄₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
147	C₂×C₄₇	C₂×#338	1	1	yes	characteristic
148	D₄₇	—	2	4	no	—
149	C₂²×M₁₆	—	47	2	no	—
150	C₂²×C₄²	—	47	1	no	—
151	#431	—	47	2	no	—
152	#431	—	47	4	no	—
153	#431	—	94	2	no	—
154	#431	—	94	2	no	—
155	#431	C₂×D₄₇	1	2	yes	—
156	C₂×C₄⋊C₈	—	94	2	no	—
157	C₂×C₂²⋊C₈	—	94	2	no	—
158	C₄×M₁₆	—	94	2	no	—
159	C₄×M₁₆	—	94	2	no	—
160	C₂×#97	—	47	1	no	—
161	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
162	C₂×C₄×C₈	—	47	4	no	—
163	C₂×C₄×C₈	—	47	2	no	—
164	C₄₇	#7860	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	—	47	1	no	—
166	C₂³×C₄	—	47	1	no	—
167	C₂³×C₄	—	47	1	no	—
168	C₂×M₁₆	—	47	8	no	—
169	C₂×M₁₆	—	94	4	no	—
170	C₂×M₁₆	—	94	4	no	—
171	C₂²×C₈	—	47	2	no	—
172	C₂²×C₈	—	94	4	no	—
173	C₂²×C₈	C₄×D₄₇	1	2	yes	—
174	C₂×C₄²	—	47	1	no	—
175	C₂×C₄²	—	47	1	no	—
176	C₂×C₄²	—	47	1	no	—
177	C₂×C₄²	—	47	2	no	—
178	C₂×C₄²	—	47	2	no	—
179	C₂×C₄²	—	47	2	no	—
180	C₂×C₄²	—	47	2	no	—
181	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	94	4	no	—
184	C₄⋊C₈	—	94	2	no	—
185	C₂²⋊C₈	—	2	2	no	—
186	C₂²⋊C₈	—	94	4	no	—
187	C₂²⋊C₈	—	94	2	no	—
188	#97	—	2	2	no	—
189	#97	—	94	2	no	—
190	C₄×C₈	—	47	4	no	—
191	C₄×C₈	—	94	2	no	—
192	C₄×C₈	—	94	2	no	—
193	C₄×C₈	—	94	4	no	—
194	C₄×C₈	C₄×D₄₇	1	4	yes	—
195	C₂⁴	—	47	1	no	—
196	C₂²×C₄	—	47	1	no	—
197	C₂²×C₄	—	47	2	no	—
198	C₂²×C₄	—	47	2	no	—
199	C₂²×C₄	—	47	4	no	—
200	C₂²×C₄	—	47	4	no	—
201	C₂²×C₄	—	47	1	no	—
202	C₂²×C₄	—	47	1	no	—
203	C₂²×C₄	—	47	2	no	—
204	C₂²×C₄	—	47	2	no	—
205	C₂²×C₄	—	47	2	no	—
206	C₂²×C₄	—	47	2	no	—
207	C₂²×C₄	—	188	2	no	—
208	C₂²×C₄	—	188	2	no	—
209	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
211	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
212	M₁₆	—	94	16	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	47	4	no	—
217	C₂×C₈	—	94	2	no	—
218	C₂×C₈	—	94	2	no	—
219	C₂×C₈	—	94	4	no	—
220	C₂×C₈	—	188	8	no	—
221	C₂×C₈	C₈×D₄₇	1	4	yes	complemented
222	C₄²	—	47	1	no	—
223	C₄²	—	47	1	no	—
224	C₄²	—	47	2	no	—
225	C₄²	—	47	1	no	—
226	C₄²	—	47	1	no	—
227	C₄²	—	47	2	no	—
228	C₄²	—	47	2	no	—
229	C₄²	—	47	2	no	—
230	C₄²	C₂×C₄×D₄₇	1	2	yes	—
231	C₄²	D₄×D₄₇	1	2	yes	—
232	C₂³	—	47	2	no	—
233	C₂³	—	47	4	no	—
234	C₂³	—	188	1	no	—
235	C₂³	—	188	1	no	—
236	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	47	2	no	—
240	C₂×C₄	—	47	1	no	—
241	C₂×C₄	—	47	1	no	—
242	C₂×C₄	—	47	4	no	—
243	C₂×C₄	—	47	4	no	—
244	C₂×C₄	—	47	4	no	—
245	C₂×C₄	—	47	8	no	—
246	C₂×C₄	—	47	1	no	—
247	C₂×C₄	—	47	1	no	—
248	C₂×C₄	—	94	4	no	—
249	C₂×C₄	—	94	4	no	—
250	C₂×C₄	—	94	2	no	—
251	C₂×C₄	—	94	2	no	—
252	C₂×C₄	—	188	2	no	—
253	C₂×C₄	—	188	2	no	—
254	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
255	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
256	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
257	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
259	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
260	C₈	—	2	4	no	—
261	C₈	—	4	4	no	—
262	C₈	—	94	4	no	—
263	C₈	—	188	4	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	47	4	no	—
267	C₂²	—	94	2	no	—
268	C₂²	—	94	2	no	—
269	C₂²	—	188	4	no	—
270	C₂²	#95×D₄₇	1	1	yes	characteristic
271	C₂²	C₄⋊C₈×D₄₇	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	47	4	no	—
275	C₄	—	47	2	no	—
276	C₄	—	47	2	no	—
277	C₄	—	94	1	no	—
278	C₄	—	94	1	no	—
279	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
280	C₄	#97×D₄₇	1	1	yes	characteristic
281	C₄	C₂²⋊C₈×D₄₇	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	94	4	no	—
284	C₂	#336×D₄₇	1	1	yes	characteristic
285	C₂	H(3,ℤ/4)×D₄₇	1	1	yes	characteristic
286	C₂	#338×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1275	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂³×D₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1275	D₄₇
hypocenter	C₄₇	#7860
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #1275

Complements

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

N	H ≅ G/N	type
3. #20241219	C₂	semidirect
4. #20241218	C₂	semidirect
5. C₄₇×#1275	C₂	semidirect
26. C₂×#10625324	C₈	semidirect
71. C₂×C₈×C₄₇	C₂×C₈	semidirect
85. C₂×C₄₇⋊C₈	C₂×C₈	semidirect
143. #1275	D₄₇	semidirect
164. C₄₇	#7860	semidirect
221. C₂×C₈	C₈×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#431×D₄₇	2
C₂×#15689075	2
#20241219	2
#20241218	2
C₄₇×#1275	2
#7860	47