Subgroups — #26109719

Order 12032.

522 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20241702	C₂	1	1	yes	maximal, characteristic
3	#20241220	C₂	1	2	yes	maximal, complemented
4	#20241216	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1458	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
8	#15689176	C₂²	1	2	yes	—
9	C₄⋊C₄×Dic₄₇	—	2	1	no	—
10	C₄⋊C₄×Dic₄₇	—	2	1	no	—
11	C₂²⋊C₄×Dic₄₇	—	2	1	no	—
12	C₂²⋊C₄×Dic₄₇	—	2	1	no	—
13	#15688969	C₄	1	1	yes	characteristic
14	#15688969	C₂²	1	1	yes	characteristic
15	C₄²×Dic₄₇	C₄	1	1	yes	characteristic
16	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
17	#15688849	—	2	1	no	—
18	#15688847	—	2	1	no	—
19	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#431	C₂²	1	2	yes	—
21	C₄₇×#324	—	2	1	no	—
22	C₄₇×#323	—	2	1	no	—
23	C₂³×Dic₄₇	—	2	1	no	—
24	#13093983	—	2	2	no	—
25	#13093983	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	1	no	—
27	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄×Dic₄₇	—	2	1	no	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	1	no	—
31	C₂×C₄×Dic₄₇	—	2	1	no	—
32	C₂×C₄×Dic₄₇	—	2	1	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₂×C₄×Dic₄₇	C₈	1	4	yes	complemented
35	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
39	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
40	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
41	C₄×C₄₇⋊C₈	—	2	2	no	—
42	C₄₇⋊#95	—	2	1	no	—
43	C₄₇⋊#95	—	2	1	no	—
44	C₄₇⋊#95	—	2	1	no	—
45	C₄₇⋊#95	—	2	1	no	—
46	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	#117×C₄₇	—	2	2	no	—
48	#117×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₇	—	2	2	no	—
56	C₂²⋊C₈×C₄₇	—	2	2	no	—
57	C₄×C₈×C₄₇	—	2	2	no	—
58	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	1	no	—
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	—	2	1	no	—
74	C₂²×Dic₄₇	—	2	1	no	—
75	C₂²×Dic₄₇	—	2	1	no	—
76	C₂²×Dic₄₇	—	4	2	no	—
77	C₂²×Dic₄₇	—	4	2	no	—
78	C₂×C₈×C₄₇	—	4	2	no	—
79	C₄⋊C₄×C₄₇	—	2	2	no	—
80	C₄⋊C₄×C₄₇	—	2	2	no	—
81	C₂²⋊C₄×C₄₇	—	2	2	no	—
82	C₂²⋊C₄×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	1	no	—
84	C₄²×C₄₇	—	2	1	no	—
85	C₄²×C₄₇	#41	1	2	yes	—
86	C₄²×C₄₇	#41	1	2	yes	—
87	C₂²⋊Dic₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	—	2	4	no	—
89	C₄×Dic₄₇	—	2	4	no	—
90	C₄×Dic₄₇	—	2	1	no	—
91	C₄×Dic₄₇	—	2	1	no	—
92	C₄×Dic₄₇	—	2	1	no	—
93	C₄×Dic₄₇	—	2	1	no	—
94	C₄×Dic₄₇	—	4	2	no	—
95	C₄×Dic₄₇	—	4	2	no	—
96	C₄×Dic₄₇	—	4	2	no	—
97	C₄×Dic₄₇	—	4	2	no	—
98	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
99	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
100	C₄×Dic₄₇	M₁₆	1	2	yes	—
101	C₄×Dic₄₇	M₁₆	1	2	yes	—
102	C₂×C₄₇⋊C₈	—	4	2	no	—
103	C₂³×C₄₇	—	2	1	no	—
104	C₂³×C₄₇	—	2	1	no	—
105	C₂³×C₄₇	—	4	1	no	—
106	C₂³×C₄₇	—	4	1	no	—
107	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
108	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	4	1	no	—
118	C₂×C₄×C₄₇	—	4	1	no	—
119	C₂×C₄×C₄₇	—	4	1	no	—
120	C₂×C₄×C₄₇	—	4	1	no	—
121	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
122	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#123	1	2	yes	—
125	C₂×C₄×C₄₇	#123	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
130	C₂×Dic₄₇	—	2	2	no	—
131	C₂×Dic₄₇	—	2	2	no	—
132	C₂×Dic₄₇	—	2	2	no	—
133	C₂×Dic₄₇	—	2	2	no	—
134	C₂×Dic₄₇	—	2	2	no	—
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	4	8	no	—
137	C₂×Dic₄₇	—	4	1	no	—
138	C₂×Dic₄₇	—	4	1	no	—
139	C₂×Dic₄₇	—	4	1	no	—
140	C₂×Dic₄₇	—	4	1	no	—
141	C₂×Dic₄₇	—	4	1	no	—
142	C₂×Dic₄₇	—	4	1	no	—
143	C₈×C₄₇	—	8	2	no	—
144	C₄₇⋊C₈	—	8	2	no	—
145	#8730	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	4	2	no	—
150	C₂²×C₄₇	—	4	2	no	—
151	C₂²×C₄₇	—	4	1	no	—
152	C₂²×C₄₇	—	4	1	no	—
153	C₂²×C₄₇	#381	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
155	C₂²×C₄₇	#407	1	1	yes	characteristic
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	4	1	no	—
160	C₄×C₄₇	—	4	1	no	—
161	C₄×C₄₇	#431	1	1	yes	characteristic
162	C₄×C₄₇	#431	1	1	yes	characteristic
163	C₄×C₄₇	#432	1	1	yes	characteristic
164	C₄×C₄₇	#432	1	1	yes	characteristic
165	Dic₄₇	—	2	4	no	—
166	Dic₄₇	—	4	2	no	—
167	Dic₄₇	—	4	2	no	—
168	Dic₄₇	—	4	2	no	—
169	C₄×#117	—	47	1	no	—
170	#1458	—	47	1	no	—
171	#1458	D₄₇	1	1	yes	characteristic
172	#1276	—	47	2	no	—
173	#1274	—	47	2	no	—
174	C₂×C₄₇	—	2	1	no	—
175	C₂×C₄₇	—	2	1	no	—
176	C₂×C₄₇	—	4	2	no	—
177	C₂×C₄₇	#1817	1	1	yes	characteristic
178	C₂×C₄₇	#1827	1	1	yes	characteristic
179	C₂×C₄₇	#1827	1	1	yes	characteristic
180	C₂×#117	—	47	1	no	—
181	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
182	C₂²×C₄²	—	47	1	no	—
183	#431	—	47	2	no	—
184	#431	C₂×D₄₇	1	2	yes	—
185	C₄×C₄⋊C₄	—	94	1	no	—
186	C₄×C₄⋊C₄	—	94	1	no	—
187	C₄×C₂²⋊C₄	—	94	1	no	—
188	C₄×C₂²⋊C₄	—	94	1	no	—
189	#376	—	47	1	no	—
190	#376	—	47	1	no	—
191	C₄³	—	47	1	no	—
192	C₄³	—	47	1	no	—
193	#324	—	2	1	no	—
194	#324	—	94	1	no	—
195	#323	—	2	1	no	—
196	#323	—	94	1	no	—
197	C₄₇	#8730	1	1	yes	characteristic, complemented, Sylow
198	C₂³×C₄	—	94	1	no	—
199	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	—	94	2	no	—
203	#117	—	94	2	no	—
204	C₂×C₄⋊C₄	—	94	1	no	—
205	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	94	1	no	—
208	C₂×C₂²⋊C₄	D₁₈₈	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
210	C₂×C₄²	—	47	4	no	—
211	C₂×C₄²	—	47	1	no	—
212	C₂×C₄²	—	47	1	no	—
213	C₂×C₄²	—	47	1	no	—
214	C₂×C₄²	—	47	1	no	—
215	C₂×C₄²	—	94	2	no	—
216	C₂×C₄²	—	94	1	no	—
217	C₂×C₄²	—	94	1	no	—
218	C₂×C₄²	—	94	1	no	—
219	C₂×C₄²	—	94	1	no	—
220	C₂×C₄²	—	94	1	no	—
221	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	94	2	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	94	2	no	—
227	C₄×C₈	—	2	2	no	—
228	C₄×C₈	—	94	2	no	—
229	#95	—	94	1	no	—
230	#95	—	94	1	no	—
231	#95	—	94	1	no	—
232	#95	—	94	1	no	—
233	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	94	1	no	—
239	C₂²×C₄	—	94	4	no	—
240	C₂²×C₄	—	94	1	no	—
241	C₂²×C₄	—	94	1	no	—
242	C₂²×C₄	—	94	1	no	—
243	C₂²×C₄	—	188	2	no	—
244	C₂²×C₄	—	188	2	no	—
245	C₂²×C₄	C₈×D₄₇	1	2	yes	—
246	C₂²×C₄	#10625324	1	2	yes	—
247	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
248	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	188	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	94	4	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	94	4	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	47	2	no	—
264	C₄²	—	47	2	no	—
265	C₄²	—	47	2	no	—
266	C₄²	—	47	2	no	—
267	C₄²	—	94	4	no	—
268	C₄²	—	94	1	no	—
269	C₄²	—	94	1	no	—
270	C₄²	—	94	1	no	—
271	C₄²	—	94	1	no	—
272	C₄²	—	188	2	no	—
273	C₄²	—	188	2	no	—
274	C₄²	—	188	2	no	—
275	C₄²	—	188	2	no	—
276	C₄²	#10625352	1	2	yes	—
277	C₄²	#10625352	1	2	yes	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	4	1	no	—
281	C₂³	—	4	1	no	—
282	C₂³	(C₂×D₄₇)⋊C₈	1	2	yes	—
283	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	94	2	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	94	2	no	—
299	C₂×C₄	—	94	2	no	—
300	C₂×C₄	—	94	2	no	—
301	C₂×C₄	—	94	2	no	—
302	C₂×C₄	—	188	8	no	—
303	C₂×C₄	—	188	1	no	—
304	C₂×C₄	—	188	1	no	—
305	C₂×C₄	—	188	1	no	—
306	C₂×C₄	—	188	1	no	—
307	C₂×C₄	—	188	1	no	—
308	C₂×C₄	—	188	1	no	—
309	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
310	C₂×C₄	#13093980	1	1	yes	characteristic
311	C₂×C₄	#13093980	1	1	yes	characteristic
312	C₂×C₄	#13094005	1	2	yes	—
313	C₂×C₄	#13094008	1	2	yes	—
314	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
315	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
316	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
317	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
318	C₈	—	8	2	no	—
319	C₈	—	376	2	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	#15688984	1	1	yes	characteristic
328	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
329	C₂²	#15689086	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	94	4	no	—
336	C₄	—	188	2	no	—
337	C₄	—	188	2	no	—
338	C₄	—	188	2	no	—
339	C₄	#15689173	1	1	yes	characteristic
340	C₄	#15689173	1	1	yes	characteristic
341	C₄	#15689177	1	1	yes	characteristic
342	C₄	#15689177	1	1	yes	characteristic
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	#20243454	1	1	yes	characteristic
347	C₂	#20243489	1	1	yes	characteristic
348	C₂	#20243493	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093980
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1458	C₂
socle	C₂²×C₄₇	#381
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1458	D₄₇
hypocenter	C₄₇	#8730
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

Complements

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

N	H ≅ G/N	type
3. #20241220	C₂	semidirect
4. #20241216	C₂	semidirect
34. C₂×C₄×Dic₄₇	C₈	semidirect
197. C₄₇	#8730	semidirect

Maximal subgroups

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

subgroup	index
#117×Dic₄₇	2
#20241702	2
#20241220	2
#20241216	2
C₄₇×#1458	2
#8730	47