Subgroups — #26181996

Order 12032.

806 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252016	C₂	1	2	yes	maximal
2	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
3	#20241692	C₂	1	4	yes	maximal, complemented
4	#20241684	C₂	1	4	yes	maximal, complemented
5	#20241680	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241672	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625324	C₄	1	2	yes	—
9	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
10	#15689179	C₄	1	4	yes	—
11	#15689177	C₄	1	4	yes	—
12	#15689176	C₂²	1	2	yes	—
13	#15689174	C₂²	1	4	yes	—
14	#15689145	C₄	1	4	yes	—
15	#15689085	C₄	1	4	yes	—
16	C₄×#10625324	C₄	1	2	yes	—
17	C₄×#10625324	C₄	1	2	yes	—
18	C₄×#10625324	C₂²	1	2	yes	—
19	C₄×#10625324	C₂²	1	4	yes	—
20	C₄×#10625324	C₂²	1	2	yes	—
21	C₂×#13093934	C₄	1	2	yes	—
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
23	#15688845	C₂²	1	4	yes	complemented
24	#15688845	C₂²	1	4	yes	complemented
25	C₈×C₄₇⋊C₈	C₂²	1	4	yes	complemented
26	C₄₇×#431	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#322	C₂²	1	4	yes	complemented
29	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
31	C₂×#10625324	—	2	8	no	—
32	C₂×#10625324	C₂×C₄	1	4	yes	—
33	C₂×#10625324	C₂×C₄	1	4	yes	—
34	C₂×#10625324	C₂×C₄	1	8	yes	—
35	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄²×D₄₇	C₂×C₄	1	2	yes	—
41	C₄²×D₄₇	C₂×C₄	1	2	yes	—
42	C₄²×D₄₇	C₂³	1	1	yes	characteristic
43	C₄²×D₄₇	C₂³	1	1	yes	characteristic
44	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
45	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
46	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
47	(C₂×D₄₇)⋊C₈	C₂×C₄	1	8	yes	—
48	#13093934	C₂×C₄	1	4	yes	—
49	#13093934	C₂×C₄	1	4	yes	—
50	#13093934	C₂×C₄	1	8	yes	—
51	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
52	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
53	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
54	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
57	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₄₇	C₂³	1	2	yes	—
60	C₄×C₈×C₄₇	C₂³	1	4	yes	—
61	C₂³×D₄₇	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
65	C₂²×Dic₄₇	C₄²	1	2	yes	—
66	C₂×C₄×D₄₇	—	2	4	no	—
67	C₂×C₄×D₄₇	—	4	4	no	—
68	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
69	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
70	C₂×C₄×D₄₇	C₄²	1	2	yes	—
71	C₂×C₄×D₄₇	C₄²	1	2	yes	—
72	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
73	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₄×Dic₄₇	C₄²	1	2	yes	—
83	C₄×Dic₄₇	C₄²	1	2	yes	—
84	C₄×Dic₄₇	C₄²	1	2	yes	—
85	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
88	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄₇⋊C₈	M₁₆	1	8	yes	complemented
90	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
91	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
92	#10625324	—	2	8	no	—
93	#10625324	—	2	8	no	—
94	#10625324	—	4	8	no	—
95	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
96	C₂²×D₄₇	—	2	2	no	—
97	C₂²×D₄₇	—	4	2	no	—
98	C₂²×D₄₇	C₂×C₄²	1	1	yes	characteristic
99	C₂²×D₄₇	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	4	2	no	—
108	C₂×Dic₄₇	C₂×C₄²	1	1	yes	characteristic
109	C₂×Dic₄₇	C₂×C₄²	1	1	yes	characteristic
110	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
111	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
112	C₄×D₄₇	—	2	2	no	—
113	C₄×D₄₇	—	2	2	no	—
114	C₄×D₄₇	—	2	4	no	—
115	C₄×D₄₇	—	2	4	no	—
116	C₄×D₄₇	—	2	2	no	—
117	C₄×D₄₇	—	2	2	no	—
118	C₄×D₄₇	—	4	2	no	—
119	C₄×D₄₇	—	4	2	no	—
120	C₈×C₄₇	—	2	4	no	—
121	C₈×C₄₇	—	2	8	no	—
122	C₄₇⋊C₈	—	2	8	no	—
123	C₄₇⋊C₈	C₂×M₁₆	1	8	yes	—
124	#17334	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
127	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
128	C₂×D₄₇	—	2	1	no	—
129	C₂×D₄₇	—	2	1	no	—
130	C₂×D₄₇	—	2	2	no	—
131	C₂×D₄₇	—	2	2	no	—
132	C₂×D₄₇	—	4	4	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
135	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
136	C₄×C₄₇	#568	1	4	yes	—
137	Dic₄₇	—	2	1	no	—
138	Dic₄₇	—	2	1	no	—
139	Dic₄₇	—	2	2	no	—
140	Dic₄₇	—	2	2	no	—
141	#2920	—	47	2	no	—
142	C₂×C₄×M₁₆	—	47	1	no	—
143	#1455	—	47	4	no	—
144	#1453	—	47	4	no	—
145	#1452	—	47	1	no	—
146	#1452	D₄₇	1	1	yes	characteristic, complemented
147	#1450	—	47	2	no	—
148	C₂×C₄₇	—	2	2	no	—
149	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
150	C₂×C₄₇	#2873	1	2	yes	—
151	D₄₇	—	4	1	no	—
152	D₄₇	—	4	1	no	—
153	C₂²×M₁₆	—	47	2	no	—
154	C₂²×C₄²	—	47	1	no	—
155	#432	—	47	4	no	—
156	#432	—	47	4	no	—
157	#431	—	47	2	no	—
158	#431	—	47	4	no	—
159	#431	C₂×D₄₇	1	2	yes	—
160	#423	—	47	4	no	—
161	#407	—	47	4	no	—
162	C₄×M₁₆	—	47	2	no	—
163	C₄×M₁₆	—	47	2	no	—
164	C₄×M₁₆	—	47	2	no	—
165	C₄×M₁₆	—	47	2	no	—
166	C₄×M₁₆	—	47	4	no	—
167	C₂×#97	—	47	2	no	—
168	C₂×C₄×C₈	—	47	1	no	—
169	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
170	#322	—	47	4	no	—
171	#322	—	47	4	no	—
172	#322	C₂×D₄₇	1	4	yes	complemented
173	C₈²	—	47	4	no	—
174	C₄₇	#17334	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	47	1	no	—
176	C₂³×C₄	—	47	2	no	—
177	C₂×M₁₆	—	47	4	no	—
178	C₂×M₁₆	—	47	4	no	—
179	C₂×M₁₆	—	47	8	no	—
180	C₂×M₁₆	—	94	8	no	—
181	C₂²×C₈	—	47	2	no	—
182	C₂²×C₈	C₄×D₄₇	1	2	yes	—
183	C₂×C₄²	—	47	1	no	—
184	C₂×C₄²	—	47	2	no	—
185	C₂×C₄²	—	47	1	no	—
186	C₂×C₄²	—	47	1	no	—
187	C₂×C₄²	—	47	2	no	—
188	C₂×C₄²	—	47	2	no	—
189	C₂×C₄²	—	47	1	no	—
190	C₂×C₄²	—	47	1	no	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	C₄⋊C₈	—	47	8	no	—
193	C₄⋊C₈	—	47	4	no	—
194	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
195	C₂²⋊C₈	—	47	4	no	—
196	C₂²⋊C₈	—	47	8	no	—
197	C₂²⋊C₈	C₄×D₄₇	1	4	yes	—
198	#97	—	47	4	no	—
199	#97	—	47	4	no	—
200	#97	—	47	8	no	—
201	C₄×C₈	—	47	2	no	—
202	C₄×C₈	—	47	2	no	—
203	C₄×C₈	—	47	4	no	—
204	C₄×C₈	C₄×D₄₇	1	2	yes	—
205	C₄×C₈	C₂²×D₄₇	1	4	yes	—
206	C₄×C₈	C₂²×D₄₇	1	2	yes	—
207	C₂⁴	—	47	1	no	—
208	C₂²×C₄	—	47	1	no	—
209	C₂²×C₄	—	47	2	no	—
210	C₂²×C₄	—	47	1	no	—
211	C₂²×C₄	—	47	1	no	—
212	C₂²×C₄	—	47	2	no	—
213	C₂²×C₄	—	47	1	no	—
214	C₂²×C₄	—	47	1	no	—
215	C₂²×C₄	—	47	2	no	—
216	C₂²×C₄	—	47	2	no	—
217	C₂²×C₄	—	47	2	no	—
218	C₂²×C₄	—	94	4	no	—
219	C₂²×C₄	—	188	4	no	—
220	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
222	M₁₆	—	94	8	no	—
223	M₁₆	—	94	8	no	—
224	M₁₆	—	188	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	47	8	no	—
227	C₂×C₈	—	47	4	no	—
228	C₂×C₈	—	47	8	no	—
229	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
230	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
231	C₄²	—	47	2	no	—
232	C₄²	—	47	2	no	—
233	C₄²	—	47	2	no	—
234	C₄²	—	47	1	no	—
235	C₄²	—	47	1	no	—
236	C₄²	—	47	2	no	—
237	C₄²	—	47	2	no	—
238	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₄²	C₂×C₄×D₄₇	1	2	yes	—
240	C₄²	C₂³×D₄₇	1	1	yes	characteristic
241	C₂³	—	47	1	no	—
242	C₂³	—	47	1	no	—
243	C₂³	—	94	2	no	—
244	C₂³	—	188	2	no	—
245	C₂³	C₄²×D₄₇	1	1	yes	characteristic
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	47	1	no	—
248	C₂×C₄	—	47	1	no	—
249	C₂×C₄	—	47	2	no	—
250	C₂×C₄	—	47	2	no	—
251	C₂×C₄	—	94	2	no	—
252	C₂×C₄	—	94	2	no	—
253	C₂×C₄	—	94	2	no	—
254	C₂×C₄	—	94	2	no	—
255	C₂×C₄	—	94	2	no	—
256	C₂×C₄	—	94	4	no	—
257	C₂×C₄	—	94	4	no	—
258	C₂×C₄	—	188	2	no	—
259	C₂×C₄	—	188	2	no	—
260	C₂×C₄	—	188	2	no	—
261	C₂×C₄	C₄²×D₄₇	1	2	yes	—
262	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
263	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
264	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
266	C₈	—	2	4	no	—
267	C₈	—	2	8	no	—
268	C₈	—	47	8	no	—
269	C₈	—	94	8	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	94	1	no	—
272	C₂²	—	94	1	no	—
273	C₂²	—	94	2	no	—
274	C₂²	—	94	2	no	—
275	C₂²	—	188	4	no	—
276	C₂²	#15689075	1	2	yes	—
277	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	94	1	no	—
280	C₄	—	94	1	no	—
281	C₄	—	94	2	no	—
282	C₄	—	94	2	no	—
283	C₄	#15689075	1	2	yes	—
284	C₄	C₂×M₁₆×D₄₇	1	2	yes	—
285	C₄	#15689888	1	4	yes	—
286	C₂	—	2	2	no	—
287	C₂	—	188	1	no	—
288	C₂	—	188	1	no	—
289	C₂	C₂×#15689075	1	1	yes	characteristic
290	C₂	#20251804	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#1452	C₂
prime radical	C₂³×D₄₇	C₄²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1452	D₄₇
hypocenter	C₄₇	#17334
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ #1452

Complements

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

N	H ≅ G/N	type
3. #20241692	C₂	semidirect
4. #20241684	C₂	semidirect
5. #20241680	C₂	semidirect
6. #20241672	C₂	semidirect
7. C₄₇×#1452	C₂	semidirect
23. #15688845	C₂²	semidirect
24. #15688845	C₂²	semidirect
25. C₈×C₄₇⋊C₈	C₂²	semidirect
28. C₄₇×#322	C₂²	semidirect
89. C₂×C₄₇⋊C₈	M₁₆	semidirect
146. #1452	D₄₇	semidirect
172. #322	C₂×D₄₇	semidirect
174. C₄₇	#17334	semidirect

Maximal subgroups

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

subgroup	index
#20252016	2
C₂×C₄×#10625324	2
#20241692	2
#20241684	2
#20241680	2
#20241672	2
C₄₇×#1452	2
#17334	47