Subgroups — #26122105

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689146	C₂	1	2	yes	maximal
2	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688888	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688887	C₂	1	2	yes	maximal
5	C₂×C₄₇×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	#15689146	—	2	4	no	—
10	#15689146	C₄	1	8	yes	—
11	C₂×Dic₄₇⋊C₈	—	2	4	no	—
12	C₂×C₈×Dic₄₇	—	2	4	no	—
13	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
14	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
15	#15688888	—	2	2	no	—
16	#15688887	—	2	4	no	—
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#341	—	2	2	no	—
20	C₂³×Dic₄₇	Q₈	1	2	yes	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
32	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
33	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	Dic₄₇⋊C₈	—	2	16	no	—
39	C₈×Dic₄₇	—	2	4	no	—
40	C₈×Dic₄₇	—	2	4	no	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₄₇	—	2	2	no	—
49	C₂²×C₈×C₄₇	—	2	2	no	—
50	C₂²×C₈×C₄₇	—	2	2	no	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₇	D₄	1	2	yes	—
53	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
54	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×Dic₄₇	—	2	8	no	—
66	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
67	C₂²×Dic₄₇	C₄⋊C₄	1	8	yes	—
68	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
69	M₁₆×C₄₇	—	2	4	no	—
70	M₁₆×C₄₇	—	2	4	no	—
71	M₁₆×C₄₇	—	4	2	no	—
72	M₁₆×C₄₇	—	4	2	no	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	4	2	no	—
79	C₂×C₈×C₄₇	—	4	2	no	—
80	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
81	C₄×Dic₄₇	C₄²	1	2	yes	—
82	C₄×Dic₄₇	C₄²	1	2	yes	—
83	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
84	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
85	C₄×Dic₄₇	C₂²⋊C₄	1	8	yes	—
86	#10625329	—	2	4	no	—
87	#10625329	—	2	4	no	—
88	#10625329	—	4	2	no	—
89	#10625329	—	4	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
98	C₂³×C₄₇	—	2	4	no	—
99	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₄₇	#125	1	2	yes	—
102	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
103	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	#121	1	2	yes	—
116	C₂×C₄×C₄₇	#121	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	#124	1	2	yes	—
120	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×Dic₄₇	—	2	4	no	—
123	C₂×Dic₄₇	—	2	16	no	—
124	C₂×Dic₄₇	#95	1	4	yes	—
125	C₂×Dic₄₇	#95	1	4	yes	—
126	C₂×Dic₄₇	#108	1	8	yes	—
127	C₈×C₄₇	—	2	4	no	—
128	C₈×C₄₇	—	4	4	no	—
129	C₄₇⋊C₈	—	2	4	no	—
130	C₄₇⋊C₈	—	4	4	no	—
131	#10844	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	4	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	4	no	—
136	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
137	C₂²×C₄₇	#382	1	1	yes	characteristic
138	C₂²×C₄₇	#384	1	1	yes	characteristic
139	C₂²×C₄₇	#389	1	2	yes	—
140	C₂²×C₄₇	#389	1	2	yes	—
141	C₂²×C₄₇	C₂×#108	1	4	yes	—
142	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₄₇	#380	1	1	yes	characteristic
145	C₄×C₄₇	#380	1	1	yes	characteristic
146	C₄×C₄₇	#390	1	2	yes	—
147	C₄×C₄₇	#391	1	2	yes	—
148	Dic₄₇	—	2	4	no	—
149	Dic₄₇	#341	1	8	yes	complemented
150	C₂×#423	—	47	2	no	—
151	C₂²×C₄×C₈	—	47	1	no	—
152	C₂×#341	—	47	2	no	—
153	C₂×#341	—	47	1	no	—
154	C₂×#341	D₄₇	1	1	yes	characteristic
155	C₂×C₄₇	—	2	2	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#341	1	2	yes	—
159	C₂×C₄₇	#1837	1	2	yes	—
160	C₂×C₄₇	#1848	1	2	yes	—
161	C₂²×M₁₆	—	47	2	no	—
162	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
163	C₂³×C₈	—	47	1	no	—
164	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
165	C₂²×C₄²	—	47	1	no	—
166	#423	—	47	8	no	—
167	#423	—	94	4	no	—
168	C₂×C₄⋊C₈	—	94	4	no	—
169	C₂×C₄×C₈	—	47	2	no	—
170	C₂×C₄×C₈	—	47	2	no	—
171	C₂×C₄×C₈	—	94	4	no	—
172	#341	—	2	2	no	—
173	#341	—	94	4	no	—
174	#341	—	94	2	no	—
175	C₄₇	#10844	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	—	47	2	no	—
177	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	47	4	no	—
184	C₂×M₁₆	—	94	2	no	—
185	C₂×M₁₆	—	94	2	no	—
186	C₂×M₁₆	—	94	2	no	—
187	C₂×M₁₆	—	94	2	no	—
188	C₂×M₁₆	—	94	2	no	—
189	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	47	2	no	—
194	C₂²×C₈	—	47	2	no	—
195	C₂²×C₈	—	47	2	no	—
196	C₂²×C₈	—	94	2	no	—
197	C₂²×C₈	—	94	2	no	—
198	C₂²×C₈	—	94	2	no	—
199	C₂²×C₈	Dic₉₄	1	2	yes	—
200	C₂²×C₈	C₄×D₄₇	1	2	yes	—
201	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
202	C₂×C₄²	—	47	1	no	—
203	C₂×C₄²	—	47	1	no	—
204	C₂×C₄²	—	47	4	no	—
205	C₂×C₄²	—	47	4	no	—
206	C₂×C₄²	—	47	2	no	—
207	C₄⋊C₈	—	94	16	no	—
208	C₄×C₈	—	94	4	no	—
209	C₄×C₈	—	94	4	no	—
210	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
211	C₂²×C₄	—	47	2	no	—
212	C₂²×C₄	—	47	8	no	—
213	C₂²×C₄	—	47	2	no	—
214	C₂²×C₄	—	94	8	no	—
215	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
218	C₂²×C₄	#10625350	1	2	yes	—
219	C₂²×C₄	D₄×D₄₇	1	2	yes	—
220	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
222	C₂²×C₄	#10625352	1	2	yes	—
223	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	94	4	no	—
230	M₁₆	—	94	4	no	—
231	M₁₆	—	188	2	no	—
232	M₁₆	—	188	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	47	4	no	—
241	C₂×C₈	—	94	2	no	—
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	—	94	4	no	—
245	C₂×C₈	—	94	4	no	—
246	C₂×C₈	—	188	2	no	—
247	C₂×C₈	—	188	2	no	—
248	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
249	C₄²	—	47	2	no	—
250	C₄²	—	47	2	no	—
251	C₄²	—	47	2	no	—
252	C₄²	—	47	2	no	—
253	C₄²	—	47	8	no	—
254	C₂³	—	2	4	no	—
255	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
256	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
257	C₂³	#13094014	1	2	yes	—
258	C₂³	#13094022	1	1	yes	characteristic
259	C₂³	#13094023	1	1	yes	characteristic
260	C₂×C₄	—	47	4	no	—
261	C₂×C₄	—	47	8	no	—
262	C₂×C₄	—	47	4	no	—
263	C₂×C₄	—	94	16	no	—
264	C₂×C₄	—	94	4	no	—
265	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
266	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
268	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
270	C₂×C₄	#13093980	1	1	yes	characteristic
271	C₂×C₄	#13093981	1	1	yes	characteristic
272	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
273	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
274	C₂×C₄	#13093989	1	2	yes	—
275	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
276	C₂×C₄	#13093996	1	2	yes	—
277	C₂×C₄	#13093997	1	2	yes	—
278	C₂×C₄	#13094001	1	2	yes	—
279	C₂×C₄	#13094002	1	2	yes	—
280	C₂×C₄	#13094010	1	2	yes	—
281	C₂×C₄	#13094020	1	1	yes	characteristic
282	C₂×C₄	#13094023	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	4	4	no	—
285	C₈	—	94	4	no	—
286	C₈	—	188	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	#95×D₄₇	1	1	yes	characteristic
292	C₂²	#15688987	1	1	yes	characteristic
293	C₂²	#15688992	1	1	yes	characteristic
294	C₂²	#15689018	1	2	yes	—
295	C₂²	#15689020	1	2	yes	—
296	C₂²	#108×D₄₇	1	4	yes	—
297	C₄	—	47	8	no	—
298	C₄	—	94	4	no	—
299	C₄	#15688974	1	1	yes	characteristic
300	C₄	C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
301	C₄	#15688981	1	1	yes	characteristic
302	C₄	#15688982	1	1	yes	characteristic
303	C₄	#15689023	1	2	yes	—
304	C₄	#15689027	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	Dic₄₇⋊#95	1	1	yes	characteristic
308	C₂	#341×D₄₇	1	2	yes	—
309	C₂	#20243524	1	2	yes	—
310	C₂	#20243558	1	2	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	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#341	D₄₇
hypocenter	C₄₇	#10844
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
149. Dic₄₇	#341	semidirect
175. C₄₇	#10844	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689146	2
C₂²×C₈×Dic₄₇	2
C₂×#15688888	2
C₂×#15688887	2
C₂×C₄₇×#341	2
#10844	47