Subgroups — #26115525

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244316	C₂	1	2	yes	maximal, complemented
2	C₈×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688876	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688867	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#13093945	—	2	4	no	—
9	C₂×C₈×Dic₄₇	—	2	1	no	—
10	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
11	#15688877	—	2	1	no	—
12	#15688876	—	2	2	no	—
13	#15688867	—	2	4	no	—
14	C₂²×D₈×C₄₇	—	2	1	no	—
15	C₂²×D₈×C₄₇	—	2	1	no	—
16	C₂×#127×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#493	—	2	2	no	—
18	C₄₇×#493	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
23	C₂×C₄×Dic₄₇	—	2	1	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
28	#13093945	—	4	8	no	—
29	C₈×Dic₄₇	—	4	2	no	—
30	C₄₇⋊#95	—	2	1	no	—
31	C₂²×D₄×C₄₇	—	2	2	no	—
32	C₂²×D₄×C₄₇	—	2	2	no	—
33	C₂²×D₄×C₄₇	—	2	2	no	—
34	C₂×D₈×C₄₇	—	2	4	no	—
35	C₂×D₈×C₄₇	—	2	4	no	—
36	C₂×D₈×C₄₇	—	4	2	no	—
37	C₂×D₈×C₄₇	—	4	2	no	—
38	C₂×D₈×C₄₇	—	4	2	no	—
39	C₂×D₈×C₄₇	—	4	2	no	—
40	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
42	#127×C₄₇	—	2	2	no	—
43	#127×C₄₇	—	2	2	no	—
44	#127×C₄₇	—	2	2	no	—
45	#127×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₄₇	D₄	1	2	yes	—
49	C₂⁴×C₄₇	—	4	2	no	—
50	C₂×D₄×C₄₇	—	2	4	no	—
51	C₂×D₄×C₄₇	—	2	4	no	—
52	C₂×D₄×C₄₇	—	2	4	no	—
53	C₂×D₄×C₄₇	—	4	2	no	—
54	C₂×D₄×C₄₇	—	4	2	no	—
55	C₂×D₄×C₄₇	—	4	2	no	—
56	C₂×D₄×C₄₇	—	4	2	no	—
57	C₂×D₄×C₄₇	—	4	4	no	—
58	C₂×D₄×C₄₇	—	4	2	no	—
59	C₂×D₄×C₄₇	—	4	4	no	—
60	C₂×D₄×C₄₇	—	4	2	no	—
61	C₂×D₄×C₄₇	—	4	2	no	—
62	C₂×D₄×C₄₇	—	8	4	no	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	2	no	—
67	D₈×C₄₇	—	4	8	no	—
68	D₈×C₄₇	—	4	8	no	—
69	C₂×C₈×C₄₇	—	2	1	no	—
70	C₂×C₈×C₄₇	—	2	1	no	—
71	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₄₇	D₈	1	2	yes	—
74	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₄₇	#41	1	1	yes	characteristic
79	C₄⋊Dic₄₇	—	2	2	no	—
80	C₄⋊Dic₄₇	D₈	1	4	yes	complemented
81	C₄×Dic₄₇	—	4	1	no	—
82	C₄×Dic₄₇	—	4	1	no	—
83	C₂×C₄₇⋊C₈	—	2	2	no	—
84	C₂×C₄₇⋊C₈	—	4	1	no	—
85	C₂×C₄₇⋊C₈	—	4	1	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	4	no	—
88	C₂³×C₄₇	—	4	4	no	—
89	C₂³×C₄₇	—	8	2	no	—
90	C₂³×C₄₇	—	8	2	no	—
91	C₂³×C₄₇	—	8	4	no	—
92	C₂³×C₄₇	—	8	4	no	—
93	C₂³×C₄₇	#121	1	1	yes	characteristic
94	D₄×C₄₇	—	4	4	no	—
95	D₄×C₄₇	—	4	4	no	—
96	D₄×C₄₇	—	4	8	no	—
97	D₄×C₄₇	—	8	4	no	—
98	D₄×C₄₇	—	8	4	no	—
99	D₄×C₄₇	—	8	2	no	—
100	D₄×C₄₇	—	8	2	no	—
101	D₄×C₄₇	—	8	2	no	—
102	D₄×C₄₇	—	8	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#135	1	2	yes	—
114	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×Dic₄₇	—	2	4	no	—
117	C₂×Dic₄₇	—	4	2	no	—
118	C₂×Dic₄₇	—	4	2	no	—
119	C₈×C₄₇	—	2	2	no	—
120	C₈×C₄₇	#102	1	4	yes	complemented
121	C₄₇⋊C₈	—	4	2	no	—
122	C₄₇⋊C₈	—	8	2	no	—
123	#9734	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	8	4	no	—
125	C₂²×C₄₇	—	8	4	no	—
126	C₂²×C₄₇	—	8	4	no	—
127	C₂²×C₄₇	—	16	4	no	—
128	C₂²×C₄₇	—	16	4	no	—
129	C₂²×C₄₇	#390	1	1	yes	characteristic
130	C₂²×C₄₇	#459	1	1	yes	characteristic
131	C₂²×C₄₇	#460	1	1	yes	characteristic
132	C₂²×C₄₇	#466	1	2	yes	—
133	C₂²×C₄₇	#472	1	2	yes	—
134	C₄×C₄₇	—	2	1	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
137	C₄×C₄₇	#418	1	1	yes	characteristic
138	C₄×C₄₇	C₄×D₈	1	2	yes	—
139	C₄×C₄₇	#459	1	1	yes	characteristic
140	C₄×C₄₇	#460	1	1	yes	characteristic
141	C₄×C₄₇	#493	1	1	yes	characteristic
142	C₄×C₄₇	#495	1	1	yes	characteristic
143	Dic₄₇	—	4	4	no	—
144	C₂×#493	D₄₇	1	1	yes	characteristic
145	#1982	—	47	2	no	—
146	C₈×C₄⋊C₄	—	47	1	no	—
147	C₂×#335	—	47	1	no	—
148	C₂×#331	—	47	2	no	—
149	C₂×C₄₇	—	16	4	no	—
150	C₂×C₄₇	#1667	1	2	yes	—
151	C₂×C₄₇	#1668	1	2	yes	—
152	C₂×C₄₇	#1970	1	1	yes	characteristic
153	C₂×C₄₇	#1978	1	1	yes	characteristic
154	C₂×C₄₇	#1982	1	1	yes	characteristic
155	C₂²×D₈	—	2	1	no	—
156	C₂²×D₈	—	2	1	no	—
157	C₂×#127	C₂×D₄₇	1	2	yes	—
158	#493	—	2	2	no	—
159	#493	Dic₄₇	1	4	yes	complemented
160	C₂×C₄⋊C₈	—	47	1	no	—
161	C₂×C₄⋊C₈	—	47	2	no	—
162	C₂×#102	—	94	4	no	—
163	C₂×C₄×C₈	—	94	1	no	—
164	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
165	C₄×C₄⋊C₄	—	47	1	no	—
166	#336	—	94	1	no	—
167	#335	—	94	2	no	—
168	#331	—	94	4	no	—
169	C₄₇	#9734	1	1	yes	characteristic, complemented, Sylow
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	2	2	no	—
172	C₂²×D₄	—	2	2	no	—
173	C₂×D₈	—	2	4	no	—
174	C₂×D₈	—	2	4	no	—
175	C₂×D₈	—	4	2	no	—
176	C₂×D₈	—	4	2	no	—
177	C₂×D₈	—	4	2	no	—
178	C₂×D₈	—	4	2	no	—
179	C₂²×C₈	—	94	1	no	—
180	C₂²×C₈	—	94	2	no	—
181	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
183	#127	—	2	2	no	—
184	#127	—	2	2	no	—
185	#127	—	2	2	no	—
186	#127	C₂×Dic₄₇	1	4	yes	—
187	C₂×C₄⋊C₄	—	47	2	no	—
188	C₂×C₄²	—	94	1	no	—
189	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
190	C₄⋊C₈	—	47	2	no	—
191	C₄⋊C₈	—	94	2	no	—
192	C₄⋊C₈	—	94	1	no	—
193	C₄⋊C₈	—	94	2	no	—
194	#102	—	188	8	no	—
195	C₄×C₈	—	188	2	no	—
196	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
197	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
198	#95	—	94	1	no	—
199	C₂⁴	—	4	2	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	8	4	no	—
213	C₂²×C₄	—	94	2	no	—
214	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	#10625352	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	D₈	—	4	8	no	—
218	D₈	—	4	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	94	2	no	—
222	C₂×C₈	—	188	2	no	—
223	C₂×C₈	—	188	4	no	—
224	C₂×C₈	—	188	1	no	—
225	C₂×C₈	—	188	1	no	—
226	C₂×C₈	#10625334	1	2	yes	—
227	C₂×C₈	#10625335	1	2	yes	—
228	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
229	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
230	C₄⋊C₄	—	47	4	no	—
231	C₄⋊C₄	—	94	2	no	—
232	C₄²	—	188	1	no	—
233	C₄²	—	188	1	no	—
234	C₄²	D₄×D₄₇	1	1	yes	characteristic
235	C₄²	#10625352	1	1	yes	characteristic
236	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
237	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂³	—	4	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	—	8	4	no	—
242	C₂³	—	8	4	no	—
243	C₂³	#13093995	1	1	yes	characteristic
244	D₄	—	4	4	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	8	no	—
247	D₄	—	8	4	no	—
248	D₄	—	8	4	no	—
249	D₄	—	8	2	no	—
250	D₄	—	8	2	no	—
251	D₄	—	8	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	94	4	no	—
255	C₂×C₄	—	188	2	no	—
256	C₂×C₄	—	188	2	no	—
257	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
258	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
259	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
260	C₂×C₄	#13093995	1	1	yes	characteristic
261	C₂×C₄	#13094011	1	1	yes	characteristic
262	C₂×C₄	#13094020	1	1	yes	characteristic
263	C₂×C₄	D₈×D₄₇	1	2	yes	—
264	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
265	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
266	C₂×C₄	#13094043	1	2	yes	—
267	C₂×C₄	#13094050	1	1	yes	characteristic
268	C₂×C₄	#13094050	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	376	2	no	—
271	C₈	—	188	2	no	—
272	C₈	#13093945	1	4	yes	complemented
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	16	4	no	—
277	C₂²	—	16	4	no	—
278	C₂²	#15689026	1	1	yes	characteristic
279	C₂²	#15689315	1	1	yes	characteristic
280	C₂²	#15689320	1	1	yes	characteristic
281	C₂²	#15689356	1	2	yes	—
282	C₂²	#15689390	1	2	yes	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	188	4	no	—
286	C₄	C₂×#13093945	1	1	yes	characteristic
287	C₄	#15689128	1	1	yes	characteristic
288	C₄	D₈×Dic₄₇	1	2	yes	—
289	C₄	#15689315	1	1	yes	characteristic
290	C₄	#15689320	1	1	yes	characteristic
291	C₄	#15689498	1	1	yes	characteristic
292	C₄	#15689506	1	1	yes	characteristic
293	C₂	—	16	4	no	—
294	C₂	#20242740	1	2	yes	—
295	C₂	#20242745	1	2	yes	—
296	C₂	#20244255	1	1	yes	characteristic
297	C₂	#20244299	1	1	yes	characteristic
298	C₂	#20244316	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093995
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting, prime radical	C₂×C₄₇×#493	C₂
socle	C₂³×C₄₇	#121
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#493	D₄₇
hypocenter	C₄₇	#9734
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #20244316	C₂	semidirect
2. C₈×C₄⋊Dic₄₇	C₂	semidirect
3. C₂×#15688876	C₂	semidirect
4. C₂×#15688867	C₂	semidirect
18. C₄₇×#493	C₄	semidirect
80. C₄⋊Dic₄₇	D₈	semidirect
120. C₈×C₄₇	#102	semidirect
159. #493	Dic₄₇	semidirect
169. C₄₇	#9734	semidirect
272. C₈	#13093945	semidirect

Maximal subgroups

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

subgroup	index
#20244316	2
C₈×C₄⋊Dic₄₇	2
C₂×#15688876	2
C₂×#15688867	2
C₂×C₄₇×#493	2
#9734	47