Subgroups — #26184148

Order 12032.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252473	C₂	1	1	yes	maximal, characteristic
2	#20246353	C₂	1	1	yes	maximal, characteristic
3	#20241952	C₂	1	2	yes	maximal
4	#20241939	C₂	1	4	yes	maximal
5	#20241900	C₂	1	4	yes	maximal
6	#20241887	C₂	1	2	yes	maximal
7	C₄₇×#3502	C₂	1	1	yes	maximal, characteristic
8	#15689254	—	2	4	no	—
9	#15689180	C₂²	1	4	yes	—
10	#15689176	C₂²	1	2	yes	—
11	#15689030	—	2	1	no	—
12	#15689009	—	2	2	no	—
13	#15688991	C₂²	1	2	yes	—
14	#15688989	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₄₇	—	2	1	no	—
16	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
17	#15688969	C₂²	1	2	yes	—
18	#15688860	C₂²	1	4	yes	—
19	#15688860	C₂²	1	8	yes	—
20	#15688860	C₂²	1	4	yes	—
21	Q₈²×C₄₇	C₄	1	4	yes	complemented
22	C₄₇×#557	—	2	4	no	—
23	C₄₇×#557	C₄	1	4	yes	complemented
24	C₄₇×#544	—	2	1	no	—
25	C₄₇×#544	—	2	2	no	—
26	C₄₇×#533	C₂²	1	4	yes	—
27	C₄₇×#533	C₂²	1	2	yes	—
28	C₄₇×#519	—	2	1	no	—
29	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₄₇	—	2	1	no	—
31	C₂×C₄⋊Dic₄₇	—	2	1	no	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
36	C₂²⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
37	#13093935	—	2	4	no	—
38	C₄×C₄₇⋊C₈	—	2	2	no	—
39	C₄₇⋊#95	—	2	2	no	—
40	C₄₇⋊#95	—	2	2	no	—
41	C₄₇⋊#95	—	2	1	no	—
42	C₄₇⋊#95	—	2	2	no	—
43	C₄₇⋊#95	C₂³	1	1	yes	characteristic
44	C₄₇⋊#95	C₂³	1	2	yes	—
45	C₄₇⋊#95	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₄₇	—	2	8	no	—
48	C₄⋊Q₈×C₄₇	—	2	4	no	—
49	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
50	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
52	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
53	#125×C₄₇	—	2	8	no	—
54	#125×C₄₇	—	2	4	no	—
55	#125×C₄₇	—	2	2	no	—
56	#125×C₄₇	—	2	4	no	—
57	C₂²⋊Q₈×C₄₇	—	2	4	no	—
58	C₂²⋊Q₈×C₄₇	—	2	2	no	—
59	C₂²⋊Q₈×C₄₇	C₂³	1	4	yes	—
60	C₄×Q₈×C₄₇	—	2	4	no	—
61	C₄×Q₈×C₄₇	—	2	4	no	—
62	C₄×Q₈×C₄₇	—	2	2	no	—
63	C₄×Q₈×C₄₇	—	2	4	no	—
64	C₄×Q₈×C₄₇	C₂×C₄	1	8	yes	—
65	C₄×Q₈×C₄₇	C₂×C₄	1	4	yes	—
66	C₄×Q₈×C₄₇	C₂×C₄	1	4	yes	—
67	C₄×Q₈×C₄₇	D₄	1	4	yes	—
68	#117×C₄₇	—	2	4	no	—
69	#117×C₄₇	—	2	2	no	—
70	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
72	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
73	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₄₇	—	2	2	no	—
75	C₂×Q₈×C₄₇	—	2	4	no	—
76	C₂×Q₈×C₄₇	—	2	2	no	—
77	C₂×Q₈×C₄₇	—	2	2	no	—
78	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
79	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₇	—	2	1	no	—
86	C₂²×Dic₄₇	—	2	2	no	—
87	C₂²×Dic₄₇	—	2	1	no	—
88	C₄⋊C₄×C₄₇	—	2	4	no	—
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	2	4	no	—
94	C₄⋊C₄×C₄₇	—	2	4	no	—
95	C₄⋊C₄×C₄₇	—	2	8	no	—
96	C₄⋊C₄×C₄₇	—	2	4	no	—
97	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
98	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
99	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
100	C₂²⋊C₄×C₄₇	—	2	4	no	—
101	C₄²×C₄₇	—	2	8	no	—
102	C₄²×C₄₇	—	2	4	no	—
103	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
104	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
105	C₄²×C₄₇	C₂×D₄	1	2	yes	—
106	C₄²×C₄₇	C₂×D₄	1	4	yes	—
107	C₄⋊Dic₄₇	—	4	2	no	—
108	C₄⋊Dic₄₇	—	4	2	no	—
109	C₄×Dic₄₇	—	4	4	no	—
110	C₄×Dic₄₇	—	4	2	no	—
111	C₂×C₄₇⋊C₈	—	2	4	no	—
112	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₄₇	—	2	8	no	—
114	Q₈×C₄₇	—	4	4	no	—
115	Q₈×C₄₇	—	8	4	no	—
116	Q₈×C₄₇	—	8	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	1	no	—
123	C₂×C₄×C₄₇	—	4	1	no	—
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
125	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
127	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	#137	1	2	yes	—
129	C₂×C₄×C₄₇	#137	1	2	yes	—
130	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×Dic₄₇	—	2	2	no	—
133	C₂×Dic₄₇	—	4	1	no	—
134	C₂×Dic₄₇	—	4	1	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	1	no	—
137	C₂×Dic₄₇	—	4	1	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	4	1	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₄₇⋊C₈	—	4	4	no	—
142	#17632	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	4	no	—
151	C₄×C₄₇	—	2	4	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	4	4	no	—
154	C₄×C₄₇	—	4	4	no	—
155	C₄×C₄₇	#419	1	4	yes	—
156	Dic₄₇	—	4	2	no	—
157	Dic₄₇	—	8	1	no	—
158	Dic₄₇	—	8	2	no	—
159	#3502	D₄₇	1	1	yes	characteristic
160	#2983	—	47	1	no	—
161	#2356	—	47	1	no	—
162	#1518	—	47	2	no	—
163	#1515	—	47	4	no	—
164	#1506	—	47	4	no	—
165	#1503	—	47	2	no	—
166	C₂×C₄₇	—	2	1	no	—
167	C₂×C₄₇	—	2	1	no	—
168	C₂×C₄₇	#2884	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#419	1	1	yes	characteristic
170	C₂×C₄₇	#2901	1	1	yes	characteristic
171	Q₈²	Dic₄₇	1	4	yes	complemented
172	#557	—	2	4	no	—
173	#557	Dic₄₇	1	4	yes	complemented
174	#544	—	2	1	no	—
175	#544	—	2	2	no	—
176	#533	C₂×D₄₇	1	2	yes	—
177	#533	C₂×D₄₇	1	4	yes	—
178	#519	—	2	1	no	—
179	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
180	#446	—	94	4	no	—
181	#432	—	47	4	no	—
182	#431	—	47	2	no	—
183	#391	—	94	1	no	—
184	#387	—	94	2	no	—
185	#383	—	47	2	no	—
186	#382	—	47	1	no	—
187	C₄×C₄⋊C₄	—	47	1	no	—
188	C₄×C₄⋊C₄	—	94	1	no	—
189	#376	—	47	2	no	—
190	#328	—	47	4	no	—
191	#328	—	47	4	no	—
192	#328	—	47	8	no	—
193	C₄₇	#17632	1	1	yes	characteristic, complemented, Sylow
194	C₂²×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	8	no	—
196	C₄⋊Q₈	—	2	4	no	—
197	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
198	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
199	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
200	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
201	#125	—	2	8	no	—
202	#125	—	2	4	no	—
203	#125	—	2	2	no	—
204	#125	—	2	4	no	—
205	C₂²⋊Q₈	—	2	4	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	C₂²×D₄₇	1	4	yes	—
208	C₄×Q₈	—	2	4	no	—
209	C₄×Q₈	—	2	4	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	C₂×Dic₄₇	1	4	yes	—
213	C₄×Q₈	C₂×Dic₄₇	1	8	yes	—
214	C₄×Q₈	C₂×Dic₄₇	1	4	yes	—
215	C₄×Q₈	C₂²⋊D₄₇	1	4	yes	—
216	#117	—	2	4	no	—
217	#117	—	2	2	no	—
218	C₂×C₄⋊C₄	—	94	1	no	—
219	C₂×C₄⋊C₄	—	94	1	no	—
220	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
222	C₂×C₄²	—	94	2	no	—
223	C₂×C₄²	—	94	1	no	—
224	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
225	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
226	C₄⋊C₈	—	94	2	no	—
227	C₄⋊C₈	—	94	4	no	—
228	C₂²⋊C₈	—	47	4	no	—
229	#97	—	94	4	no	—
230	C₄×C₈	—	94	2	no	—
231	#95	—	47	1	no	—
232	#95	—	47	2	no	—
233	#95	—	47	1	no	—
234	#95	—	94	2	no	—
235	#95	—	94	2	no	—
236	#95	—	94	1	no	—
237	#95	—	94	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	C₂²⋊Dic₄₇	1	4	yes	—
243	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
244	C₂²×C₄	—	94	1	no	—
245	C₂²×C₄	—	94	2	no	—
246	C₂²×C₄	—	94	1	no	—
247	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
248	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
252	C₂×C₈	—	94	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	188	2	no	—
263	C₄⋊C₄	—	188	2	no	—
264	C₄⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
265	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
266	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₄²	—	2	8	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	188	4	no	—
271	C₄²	—	188	2	no	—
272	C₄²	C₂²×Dic₄₇	1	4	yes	—
273	C₄²	C₂²×Dic₄₇	1	2	yes	—
274	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
275	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
276	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
277	Q₈	—	2	8	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	8	4	no	—
280	Q₈	—	8	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	188	2	no	—
290	C₂×C₄	—	188	1	no	—
291	C₂×C₄	—	188	2	no	—
292	C₂×C₄	—	188	1	no	—
293	C₂×C₄	—	188	1	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	—	188	1	no	—
296	C₂×C₄	—	188	1	no	—
297	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
298	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
299	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
300	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
301	C₂×C₄	#13094055	1	2	yes	—
302	C₂×C₄	#13094055	1	2	yes	—
303	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
305	C₈	—	188	4	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
309	C₂²	C₂×#13094055	1	1	yes	characteristic
310	C₂²	C₂×#13094055	1	1	yes	characteristic
311	C₄	—	2	4	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	376	2	no	—
319	C₄	—	376	1	no	—
320	C₄	—	188	2	no	—
321	C₄	#15689132	1	4	yes	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	#20251852	1	1	yes	characteristic
325	C₂	C₂×#15689132	1	1	yes	characteristic
326	C₂	#20251935	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#3502	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#3502	D₄₇
hypocenter	C₄₇	#17632
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ #3502

Complements

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

N	H ≅ G/N	type
21. Q₈²×C₄₇	C₄	semidirect
23. C₄₇×#557	C₄	semidirect
171. Q₈²	Dic₄₇	semidirect
173. #557	Dic₄₇	semidirect
193. C₄₇	#17632	semidirect

Maximal subgroups

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

subgroup	index
#20252473	2
#20246353	2
#20241952	2
#20241939	2
#20241900	2
#20241887	2
C₄₇×#3502	2
#17632	47