Subgroups — #26121997

Order 12032.

734 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	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20243105	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688884	C₂	1	2	yes	maximal
4	#20243086	C₂	1	2	yes	maximal
5	C₂×C₄₇×#341	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×#13093983	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₄₇	—	2	4	no	—
10	C₂×C₈×Dic₄₇	C₄	1	4	yes	complemented
11	#15688886	C₄	1	4	yes	complemented
12	#15688884	—	2	4	no	—
13	#15688877	—	2	4	no	—
14	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#341	C₄	1	4	yes	complemented
17	C₂³×Dic₄₇	—	2	1	no	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	C₂²×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
22	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
23	#13093983	—	2	2	no	—
24	#13093983	—	2	2	no	—
25	#13093983	—	2	2	no	—
26	#13093983	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	1	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
34	C₈×Dic₄₇	—	2	8	no	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₄₇	—	2	2	no	—
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂²×C₈×C₄₇	—	2	2	no	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₂⁴×C₄₇	#41	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	#41	1	2	yes	—
56	C₂²×C₄×C₄₇	#41	1	2	yes	—
57	C₂²×C₄×C₄₇	#41	1	2	yes	—
58	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
59	C₂²×Dic₄₇	—	2	8	no	—
60	C₂²×Dic₄₇	—	2	2	no	—
61	C₂²×Dic₄₇	—	2	4	no	—
62	C₂²×Dic₄₇	—	2	4	no	—
63	M₁₆×C₄₇	—	2	4	no	—
64	M₁₆×C₄₇	—	2	4	no	—
65	M₁₆×C₄₇	—	4	2	no	—
66	M₁₆×C₄₇	—	4	2	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	4	2	no	—
73	C₂×C₈×C₄₇	—	4	2	no	—
74	C₂×C₈×C₄₇	C₄²	1	4	yes	—
75	C₂²⋊Dic₄₇	—	4	2	no	—
76	C₂²⋊Dic₄₇	—	4	2	no	—
77	C₄⋊Dic₄₇	—	2	4	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄×Dic₄₇	—	2	2	no	—
80	C₄×Dic₄₇	—	2	4	no	—
81	C₄×Dic₄₇	—	4	2	no	—
82	C₄×Dic₄₇	—	4	2	no	—
83	C₄×Dic₄₇	C₂²⋊C₄	1	4	yes	—
84	C₂×C₄₇⋊C₈	—	2	2	no	—
85	C₂×C₄₇⋊C₈	—	2	2	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	#117	1	1	yes	characteristic
91	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
92	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
93	C₂³×C₄₇	#123	1	2	yes	—
94	C₂³×C₄₇	#126	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
100	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#117	1	2	yes	—
102	C₂×C₄×C₄₇	#117	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#121	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	#123	1	2	yes	—
110	C₂×C₄×C₄₇	#124	1	2	yes	—
111	C₂×C₄×C₄₇	#125	1	2	yes	—
112	C₂×C₄×C₄₇	#126	1	2	yes	—
113	C₂×Dic₄₇	—	2	16	no	—
114	C₂×Dic₄₇	—	2	8	no	—
115	C₂×Dic₄₇	—	2	4	no	—
116	C₂×Dic₄₇	—	4	4	no	—
117	C₂×Dic₄₇	—	4	8	no	—
118	C₈×C₄₇	—	2	4	no	—
119	C₈×C₄₇	—	4	4	no	—
120	C₄₇⋊C₈	—	2	4	no	—
121	#10818	—	47	1	no	maximal, Sylow
122	C₂²×C₄₇	—	2	4	no	—
123	C₂²×C₄₇	—	2	4	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₄₇	#381	1	1	yes	characteristic
128	C₂²×C₄₇	#383	1	1	yes	characteristic
129	C₂²×C₄₇	#387	1	2	yes	—
130	C₂²×C₄₇	#388	1	2	yes	—
131	C₂²×C₄₇	#443	1	4	yes	—
132	C₄×C₄₇	#376	1	1	yes	characteristic
133	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₄₇	#380	1	1	yes	characteristic
135	C₄×C₄₇	#382	1	1	yes	characteristic
136	C₄×C₄₇	#387	1	2	yes	—
137	C₄×C₄₇	#388	1	2	yes	—
138	Dic₄₇	—	2	8	no	—
139	Dic₄₇	—	8	4	no	—
140	C₂²×C₄×C₈	—	47	1	no	—
141	C₂×#341	D₄₇	1	1	yes	characteristic
142	#1735	—	47	1	no	—
143	C₂×#339	—	47	2	no	—
144	#1729	—	47	2	no	—
145	C₂×C₄₇	—	2	1	no	—
146	C₂×C₄₇	—	2	1	no	—
147	C₂×C₄₇	—	2	2	no	—
148	C₂×C₄₇	#1434	1	1	yes	characteristic
149	C₂×C₄₇	#1772	1	2	yes	—
150	C₂×C₄₇	#1937	1	4	yes	—
151	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
152	C₂³×C₈	—	47	1	no	—
153	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
154	C₂×#117	—	47	2	no	—
155	C₂²×C₄²	—	47	1	no	—
156	C₂×C₄×C₈	—	47	4	no	—
157	C₂×C₄×C₈	—	94	4	no	—
158	#341	Dic₄₇	1	4	yes	complemented
159	#340	—	47	4	no	—
160	#339	—	94	4	no	—
161	#336	—	94	4	no	—
162	C₄₇	#10818	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	—	94	1	no	—
164	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	47	2	no	—
175	C₂²×C₈	—	47	2	no	—
176	C₂²×C₈	—	47	2	no	—
177	C₂²×C₈	—	94	2	no	—
178	C₂²×C₈	—	94	2	no	—
179	C₂²×C₈	C₄×D₄₇	1	2	yes	—
180	C₂²×C₈	C₄×D₄₇	1	2	yes	—
181	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
182	#117	—	94	2	no	—
183	#117	—	94	2	no	—
184	#117	—	94	2	no	—
185	#117	—	94	2	no	—
186	C₂×C₄⋊C₄	—	47	4	no	—
187	C₂×C₂²⋊C₄	—	94	2	no	—
188	C₂×C₄²	—	47	2	no	—
189	C₂×C₄²	—	47	4	no	—
190	C₂×C₄²	—	94	2	no	—
191	C₂×C₄²	—	94	1	no	—
192	C₂×C₄²	—	94	2	no	—
193	C₄×C₈	—	94	8	no	—
194	C₂⁴	#10625354	1	1	yes	characteristic
195	C₂²×C₄	—	94	8	no	—
196	C₂²×C₄	—	94	2	no	—
197	C₂²×C₄	—	94	4	no	—
198	C₂²×C₄	—	94	4	no	—
199	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	#10625350	1	2	yes	—
202	C₂²×C₄	#10625350	1	2	yes	—
203	C₂²×C₄	D₄×D₄₇	1	2	yes	—
204	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	#10625352	1	2	yes	—
206	C₂²×C₄	#10625352	1	1	yes	characteristic
207	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
209	M₁₆	—	2	4	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	47	4	no	—
221	C₂×C₈	—	94	2	no	—
222	C₂×C₈	—	94	2	no	—
223	C₂×C₈	—	94	4	no	—
224	C₂×C₈	—	94	4	no	—
225	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
226	C₄⋊C₄	—	94	4	no	—
227	C₄⋊C₄	—	94	4	no	—
228	C₂²⋊C₄	—	188	2	no	—
229	C₂²⋊C₄	—	188	2	no	—
230	C₄²	—	47	4	no	—
231	C₄²	—	94	2	no	—
232	C₄²	—	94	4	no	—
233	C₄²	—	188	2	no	—
234	C₄²	—	188	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	#13093982	1	1	yes	characteristic
237	C₂³	#13093986	1	1	yes	characteristic
238	C₂³	Q₈×Dic₄₇	1	1	yes	characteristic
239	C₂³	#13094009	1	2	yes	—
240	C₂³	#13094019	1	2	yes	—
241	C₂×C₄	—	94	8	no	—
242	C₂×C₄	—	94	16	no	—
243	C₂×C₄	—	94	4	no	—
244	C₂×C₄	—	188	8	no	—
245	C₂×C₄	—	188	4	no	—
246	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
247	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
248	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
249	C₂×C₄	#13093980	1	1	yes	characteristic
250	C₂×C₄	#13093981	1	1	yes	characteristic
251	C₂×C₄	#13093981	1	1	yes	characteristic
252	C₂×C₄	#13093981	1	2	yes	—
253	C₂×C₄	#13093983	1	2	yes	—
254	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
255	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
256	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
257	C₂×C₄	#13093989	1	1	yes	characteristic
258	C₂×C₄	#13093997	1	2	yes	—
259	C₂×C₄	#13094002	1	2	yes	—
260	C₂×C₄	#13094006	1	2	yes	—
261	C₂×C₄	#13094010	1	2	yes	—
262	C₂×C₄	#13094014	1	2	yes	—
263	C₂×C₄	#13094017	1	2	yes	—
264	C₈	—	2	4	no	—
265	C₈	—	4	4	no	—
266	C₈	—	94	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
272	C₂²	#15688985	1	1	yes	characteristic
273	C₂²	#15688990	1	1	yes	characteristic
274	C₂²	#15689009	1	2	yes	—
275	C₂²	#15689012	1	2	yes	—
276	C₂²	#15689239	1	4	yes	—
277	C₄	—	94	8	no	—
278	C₄	—	376	4	no	—
279	C₄	#15688969	1	1	yes	characteristic
280	C₄	C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
281	C₄	#15688982	1	1	yes	characteristic
282	C₄	#15688987	1	1	yes	characteristic
283	C₄	#15689005	1	2	yes	—
284	C₄	#15689016	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	2	no	—
288	C₂	#20241613	1	1	yes	characteristic
289	C₂	#20243270	1	2	yes	—
290	C₂	#20244090	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#341	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#341	D₄₇
hypocenter	C₄₇	#10818
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
10. C₂×C₈×Dic₄₇	C₄	semidirect
11. #15688886	C₄	semidirect
16. C₄₇×#341	C₄	semidirect
158. #341	Dic₄₇	semidirect
162. C₄₇	#10818	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₄₇	2
#20243105	2
C₂×#15688884	2
#20243086	2
C₂×C₄₇×#341	2
#10818	47