Subgroups — #26214307

Order 12032.

1042 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251978	C₂	1	1	yes	maximal, characteristic
2	#20251806	C₂	1	2	yes	maximal
3	C₂×M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20243275	C₂	1	4	yes	maximal, complemented
5	#20243270	C₂	1	4	yes	maximal, complemented
6	C₂×#15688884	C₂	1	2	yes	maximal
7	C₂×C₄₇×#430	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	1	yes	characteristic
9	C₂×#13093983	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15689165	C₄	1	8	yes	complemented
12	#15689161	C₂²	1	4	yes	—
13	#15689081	C₂²	1	8	yes	—
14	#15689078	—	2	4	no	—
15	M₁₆×Dic₄₇	—	2	4	no	—
16	M₁₆×Dic₄₇	C₄	1	8	yes	complemented
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
19	#15688884	C₂²	1	4	yes	—
20	#15688884	C₂²	1	4	yes	—
21	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
22	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#430	C₄	1	8	yes	complemented
24	C₂×#108×C₄₇	C₂²	1	4	yes	—
25	C₂³×Dic₄₇	—	2	1	no	—
26	C₂×#10625329	C₂×C₄	1	4	yes	—
27	C₂×#10625329	D₄	1	4	yes	—
28	C₂×#10625329	Q₈	1	4	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	#13093983	—	2	2	no	—
31	#13093983	—	2	2	no	—
32	#13093983	C₂×C₄	1	4	yes	—
33	#13093983	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	—	2	1	no	—
38	C₂×C₄×Dic₄₇	—	2	2	no	—
39	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
42	#13093958	C₂×C₄	1	8	yes	complemented
43	#13093956	C₂×C₄	1	8	yes	complemented
44	#13093934	—	2	8	no	—
45	#13093934	C₂×C₄	1	8	yes	complemented
46	C₈×Dic₄₇	—	2	8	no	—
47	C₈×Dic₄₇	C₂×C₄	1	8	yes	complemented
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₄₇	—	2	2	no	—
50	C₂×M₁₆×C₄₇	—	2	4	no	—
51	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
52	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
56	C₂²×C₈×C₄₇	—	2	2	no	—
57	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
58	#108×C₄₇	C₂×C₄	1	16	yes	complemented
59	C₂⁴×C₄₇	#41	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	1	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	1	no	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	4	4	no	—
73	M₁₆×C₄₇	—	2	8	no	—
74	M₁₆×C₄₇	—	2	4	no	—
75	M₁₆×C₄₇	C₄²	1	16	yes	—
76	M₁₆×C₄₇	C₂²×C₄	1	8	yes	—
77	C₂×C₈×C₄₇	—	2	8	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
82	C₂²⋊Dic₄₇	—	2	4	no	—
83	C₂²⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
86	C₄×Dic₄₇	—	2	4	no	—
87	C₄×Dic₄₇	—	2	4	no	—
88	C₄×Dic₄₇	—	2	4	no	—
89	C₄×Dic₄₇	—	2	1	no	—
90	C₄×Dic₄₇	—	2	1	no	—
91	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
92	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
93	#10625329	C₄⋊C₄	1	16	yes	—
94	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
95	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
96	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	#117	1	2	yes	—
99	C₂³×C₄₇	C₄×D₄	1	2	yes	—
100	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
101	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₇	#117	1	2	yes	—
105	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
107	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
108	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₄₇	—	2	4	no	—
116	C₂×Dic₄₇	—	2	2	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	2	8	no	—
119	C₂×Dic₄₇	—	4	8	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	C₂×Dic₄₇	—	8	4	no	—
122	C₈×C₄₇	—	2	8	no	—
123	C₈×C₄₇	C₂×C₄²	1	8	yes	—
124	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
125	#20445	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	4	2	no	—
129	C₂²×C₄₇	—	4	2	no	—
130	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
131	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
135	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
136	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
139	Dic₄₇	—	2	4	no	—
140	Dic₄₇	—	4	2	no	—
141	Dic₄₇	—	4	8	no	—
142	C₂×#430	D₄₇	1	1	yes	characteristic
143	#2911	—	47	1	no	—
144	#2873	—	47	2	no	—
145	C₂×C₄×M₁₆	—	47	1	no	—
146	#1773	—	47	4	no	—
147	#1772	—	47	4	no	—
148	C₂×#339	—	47	2	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	—	2	1	no	—
151	C₂×C₄₇	—	2	4	no	—
152	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄₇	#2956	1	2	yes	—
154	C₂²×M₁₆	—	47	1	no	—
155	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
156	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
157	C₂×#117	—	47	2	no	—
158	C₂²×C₄²	—	47	1	no	—
159	#430	Dic₄₇	1	8	yes	complemented
160	C₂×#108	C₂×D₄₇	1	4	yes	—
161	#428	—	47	8	no	—
162	#427	—	47	4	no	—
163	#405	—	47	8	no	—
164	#405	—	94	4	no	—
165	C₄×M₁₆	—	47	8	no	—
166	C₄×M₁₆	—	94	4	no	—
167	C₂×#97	—	47	2	no	—
168	C₂×C₄×C₈	—	47	2	no	—
169	#339	—	47	4	no	—
170	#339	—	47	4	no	—
171	C₄₇	#20445	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	—	94	1	no	—
173	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	—	47	4	no	—
177	C₂×M₁₆	—	47	4	no	—
178	C₂×M₁₆	—	47	4	no	—
179	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
180	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
181	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
182	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
183	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	47	2	no	—
186	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
187	#117	—	47	4	no	—
188	#117	—	47	4	no	—
189	#117	—	94	2	no	—
190	#117	—	94	2	no	—
191	C₂×C₄⋊C₄	—	47	4	no	—
192	C₂×C₂²⋊C₄	—	94	2	no	—
193	C₂×C₄²	—	47	4	no	—
194	C₂×C₄²	—	47	1	no	—
195	C₂×C₄²	—	47	1	no	—
196	C₂×C₄²	—	94	2	no	—
197	C₂×C₄²	—	94	2	no	—
198	C₂×C₄²	—	94	1	no	—
199	#108	C₂×Dic₄₇	1	16	yes	complemented
200	#107	—	47	8	no	—
201	#106	—	47	8	no	—
202	#97	—	47	8	no	—
203	#97	—	94	8	no	—
204	C₄×C₈	—	47	8	no	—
205	C₄×C₈	—	94	8	no	—
206	C₂⁴	#10625354	1	1	yes	characteristic
207	C₂²×C₄	—	94	1	no	—
208	C₂²×C₄	—	94	4	no	—
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	—	94	1	no	—
211	C₂²×C₄	—	188	4	no	—
212	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
214	C₂²×C₄	D₄×D₄₇	1	2	yes	—
215	C₂²×C₄	#10625352	1	2	yes	—
216	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
217	C₂²×C₄	#10625354	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
219	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
220	M₁₆	—	2	8	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	47	16	no	—
223	M₁₆	C₄×Dic₄₇	1	16	yes	—
224	M₁₆	C₂²×Dic₄₇	1	8	yes	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	47	4	no	—
228	C₂×C₈	—	47	4	no	—
229	C₂×C₈	—	47	4	no	—
230	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
231	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
232	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
233	C₄⋊C₄	—	47	8	no	—
234	C₄⋊C₄	—	94	4	no	—
235	C₂²⋊C₄	—	94	4	no	—
236	C₂²⋊C₄	—	94	4	no	—
237	C₄²	—	47	2	no	—
238	C₄²	—	47	2	no	—
239	C₄²	—	94	4	no	—
240	C₄²	—	94	4	no	—
241	C₄²	—	94	4	no	—
242	C₄²	—	94	1	no	—
243	C₄²	—	94	1	no	—
244	C₂³	—	4	2	no	—
245	C₂³	#13093982	1	2	yes	—
246	C₂³	#13093986	1	2	yes	—
247	C₂³	Q₈×Dic₄₇	1	2	yes	—
248	C₂³	C₂×#10625354	1	1	yes	characteristic
249	C₂×C₄	—	94	4	no	—
250	C₂×C₄	—	94	2	no	—
251	C₂×C₄	—	94	2	no	—
252	C₂×C₄	—	94	8	no	—
253	C₂×C₄	—	376	4	no	—
254	C₂×C₄	—	188	8	no	—
255	C₂×C₄	—	188	2	no	—
256	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
257	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
258	C₂×C₄	#13093982	1	2	yes	—
259	C₂×C₄	#13093986	1	2	yes	—
260	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
261	C₂×C₄	#13093989	1	4	yes	—
262	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
265	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
268	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
269	C₈	—	2	8	no	—
270	C₈	—	47	8	no	—
271	C₈	C₂×C₄×Dic₄₇	1	8	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₄⋊C₄×Dic₄₇	1	4	yes	—
277	C₂²	C₂×#13093982	1	1	yes	characteristic
278	C₂²	C₂×#13093986	1	1	yes	characteristic
279	C₂²	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
280	C₄	—	94	4	no	—
281	C₄	—	188	2	no	—
282	C₄	—	188	8	no	—
283	C₄	C₄⋊C₄×Dic₄₇	1	4	yes	—
284	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
285	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
286	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
287	C₄	C₂×#13093989	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	4	no	—
290	C₂	—	2	1	no	—
291	C₂	C₂×C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
292	C₂	#20252297	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#430	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#430	D₄₇
hypocenter	C₄₇	#20445
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
4. #20243275	C₂	semidirect
5. #20243270	C₂	semidirect
11. #15689165	C₄	semidirect
16. M₁₆×Dic₄₇	C₄	semidirect
23. C₄₇×#430	C₄	semidirect
42. #13093958	C₂×C₄	semidirect
43. #13093956	C₂×C₄	semidirect
45. #13093934	C₂×C₄	semidirect
47. C₈×Dic₄₇	C₂×C₄	semidirect
58. #108×C₄₇	C₂×C₄	semidirect
159. #430	Dic₄₇	semidirect
171. C₄₇	#20445	semidirect
199. #108	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20251978	2
#20251806	2
C₂×M₁₆×Dic₄₇	2
#20243275	2
#20243270	2
C₂×#15688884	2
C₂×C₄₇×#430	2
#20445	47