Subgroups — #26186412

Order 12032.

626 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253415	C₂	1	1	yes	maximal, characteristic
2	#20252468	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252464	C₂	1	1	yes	maximal, characteristic
4	#20242269	C₂	1	2	yes	maximal, complemented
5	#20242234	C₂	1	2	yes	maximal, complemented
6	#20242197	C₂	1	2	yes	maximal, complemented
7	#20242191	C₂	1	2	yes	maximal, complemented
8	#20242166	C₂	1	1	yes	maximal, characteristic, complemented
9	#20241836	C₂	1	1	yes	maximal, characteristic, complemented
10	#20241685	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1484	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
13	#15689439	C₄	1	2	yes	—
14	#15689405	C₄	1	2	yes	—
15	#15689320	C₄	1	2	yes	—
16	#15689261	C₄	1	2	yes	—
17	C₈×Dic₉₄	—	2	2	no	—
18	#15689247	C₂²	1	2	yes	—
19	#15689247	C₂²	1	2	yes	—
20	C₄×#10625335	C₂²	1	2	yes	—
21	C₄×#10625335	C₂²	1	2	yes	—
22	#15689201	C₂²	1	2	yes	—
23	#15689195	—	2	2	no	—
24	D₄×C₄₇⋊C₈	C₂²	1	2	yes	—
25	#15689180	C₂²	1	1	yes	characteristic
26	#15689176	C₂²	1	1	yes	characteristic
27	#15689176	C₂²	1	1	yes	characteristic
28	#15689173	—	2	2	no	—
29	#15689090	—	2	2	no	—
30	#15688875	C₂²	1	2	yes	—
31	#15688873	C₂²	1	2	yes	—
32	#15688855	C₂²	1	2	yes	—
33	#15688855	C₂²	1	2	yes	—
34	#15688845	C₂²	1	1	yes	characteristic
35	#15688845	C₂²	1	1	yes	characteristic
36	#15688844	C₂²	1	1	yes	characteristic
37	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₇×#532	C₂²	1	1	yes	characteristic
39	C₄₇×#431	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₄₇×D₄⋊C₈	C₂²	1	2	yes	—
42	C₄₇×D₄⋊C₈	C₂²	1	2	yes	—
43	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
44	C₂×#10625335	C₂×C₄	1	2	yes	—
45	C₂×#10625335	C₂×C₄	1	2	yes	—
46	C₂×#10625329	—	2	2	no	—
47	C₂²×C₄₇⋊C₈	—	2	2	no	—
48	C₄×Dic₉₄	—	2	1	no	—
49	C₄×Dic₉₄	—	2	1	no	—
50	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
51	C₄×Dic₉₄	C₂³	1	2	yes	—
52	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×Dic₄₇⋊C₄	—	2	1	no	—
54	C₂×C₄×Dic₄₇	—	2	1	no	—
55	#13093955	C₂×C₄	1	2	yes	—
56	#13093955	C₂×C₄	1	2	yes	—
57	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
58	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
59	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
60	Dic₄₇⋊C₈	—	2	2	no	—
61	Dic₄₇⋊C₈	—	2	2	no	—
62	#13093947	C₂×C₄	1	2	yes	—
63	#13093947	C₂×C₄	1	2	yes	—
64	#13093945	C₂×C₄	1	2	yes	—
65	#13093945	C₂×C₄	1	2	yes	—
66	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
67	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
68	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
69	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
70	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
71	#13093935	—	2	1	no	—
72	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×Dic₄₇	—	2	2	no	—
77	C₈×Dic₄₇	—	2	2	no	—
78	C₂×#41×C₄₇	—	2	1	no	—
79	C₂×M₁₆×C₄₇	—	2	1	no	—
80	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
81	#127×C₄₇	C₂×C₄	1	1	yes	characteristic
82	#124×C₄₇	—	2	1	no	—
83	#121×C₄₇	—	2	1	no	—
84	#121×C₄₇	C₂×C₄	1	2	yes	—
85	C₄×D₄×C₄₇	—	2	1	no	—
86	C₄×D₄×C₄₇	C₂³	1	2	yes	—
87	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
89	C₂²⋊C₈×C₄₇	—	2	1	no	—
90	#97×C₄₇	—	2	1	no	—
91	C₄×C₈×C₄₇	—	2	1	no	—
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	#41×C₄₇	—	8	2	no	—
95	C₂×Q₈×C₄₇	—	2	1	no	—
96	C₂×D₄×C₄₇	—	2	2	no	—
97	C₂×D₄×C₄₇	—	2	1	no	—
98	C₂×D₄×C₄₇	—	2	1	no	—
99	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₄₇	—	2	2	no	—
101	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
104	C₂²×Dic₄₇	—	2	1	no	—
105	C₂²×Dic₄₇	—	2	1	no	—
106	C₂×Dic₉₄	—	2	1	no	—
107	C₂×Dic₉₄	—	2	4	no	—
108	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
109	M₁₆×C₄₇	—	4	2	no	—
110	C₂×C₈×C₄₇	—	2	1	no	—
111	C₂×C₈×C₄₇	—	2	2	no	—
112	C₂×C₈×C₄₇	—	2	1	no	—
113	C₄⋊C₄×C₄₇	—	2	1	no	—
114	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
115	C₂²⋊C₄×C₄₇	—	2	2	no	—
116	C₂²⋊C₄×C₄₇	—	2	2	no	—
117	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₇	#41	1	1	yes	characteristic
120	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
121	#10625335	—	2	4	no	—
122	#10625335	—	2	4	no	—
123	C₄⋊Dic₄₇	—	2	1	no	—
124	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
125	Dic₄₇⋊C₄	—	2	2	no	—
126	Dic₄₇⋊C₄	—	2	2	no	—
127	C₄×Dic₄₇	—	2	2	no	—
128	C₄×Dic₄₇	—	2	2	no	—
129	#10625329	—	8	2	no	—
130	C₂×C₄₇⋊C₈	—	2	2	no	—
131	C₂×C₄₇⋊C₈	—	8	2	no	—
132	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
133	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
134	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
135	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
136	C₂³×C₄₇	—	2	2	no	—
137	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₄₇	—	8	1	no	—
139	D₄×C₄₇	—	2	2	no	—
140	D₄×C₄₇	—	2	2	no	—
141	D₄×C₄₇	—	8	1	no	—
142	D₄×C₄₇	—	8	1	no	—
143	D₄×C₄₇	—	8	2	no	—
144	C₂×C₄×C₄₇	—	2	2	no	—
145	C₂×C₄×C₄₇	—	2	2	no	—
146	C₂×C₄×C₄₇	—	2	2	no	—
147	C₂×C₄×C₄₇	—	4	1	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	—	4	2	no	—
150	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
156	C₂×Dic₄₇	—	2	4	no	—
157	C₂×Dic₄₇	—	2	2	no	—
158	C₂×Dic₄₇	—	2	2	no	—
159	C₂×Dic₄₇	—	8	1	no	—
160	Dic₉₄	—	2	4	no	—
161	Dic₉₄	—	2	2	no	—
162	Dic₉₄	#131	1	4	yes	—
163	C₈×C₄₇	—	2	2	no	—
164	C₈×C₄₇	—	4	1	no	—
165	C₈×C₄₇	—	4	2	no	—
166	C₄₇⋊C₈	—	2	2	no	—
167	C₄₇⋊C₈	—	2	2	no	—
168	C₄₇⋊C₈	—	2	2	no	—
169	C₄₇⋊C₈	—	2	2	no	—
170	C₄₇⋊C₈	—	4	2	no	—
171	#17923	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	—	2	1	no	—
173	C₂²×C₄₇	—	2	1	no	—
174	C₂²×C₄₇	—	2	1	no	—
175	C₂²×C₄₇	—	4	2	no	—
176	C₂²×C₄₇	—	4	2	no	—
177	C₂²×C₄₇	—	4	2	no	—
178	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
179	C₄×C₄₇	—	2	1	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	—	2	1	no	—
182	C₄×C₄₇	—	2	1	no	—
183	C₄×C₄₇	—	4	2	no	—
184	C₄×C₄₇	C₂×#131	1	2	yes	—
185	C₄×C₄₇	#577	1	2	yes	—
186	Dic₄₇	—	2	4	no	—
187	Dic₄₇	—	4	2	no	—
188	#3103	—	47	1	no	—
189	#2982	—	47	1	no	—
190	#2981	—	47	1	no	—
191	#1588	—	47	2	no	—
192	#1583	—	47	2	no	—
193	SD₁₆⋊C₈	—	47	2	no	—
194	C₈×SD₁₆	—	47	2	no	—
195	#1572	—	47	1	no	—
196	#1491	—	47	1	no	—
197	#1484	D₄₇	1	1	yes	characteristic
198	#1453	—	47	1	no	—
199	C₂×C₄₇	—	4	1	no	—
200	C₂×C₄₇	—	8	2	no	—
201	C₂×C₄₇	#2929	1	1	yes	characteristic
202	C₂×C₄₇	#2949	1	1	yes	characteristic
203	C₂×C₄₇	#2956	1	1	yes	characteristic
204	#532	C₂×D₄₇	1	1	yes	characteristic
205	C₂×C₄×Q₈	—	47	1	no	—
206	#481	—	47	2	no	—
207	#475	—	47	2	no	—
208	#460	—	47	2	no	—
209	#448	—	47	2	no	—
210	C₈×Q₈	—	47	2	no	—
211	C₈×Q₈	—	47	2	no	—
212	C₈×Q₈	—	94	2	no	—
213	C₄×SD₁₆	—	47	2	no	—
214	C₄×SD₁₆	—	47	2	no	—
215	#436	—	47	2	no	—
216	#435	—	94	2	no	—
217	C₈×D₄	—	47	2	no	—
218	#432	—	47	1	no	—
219	#431	—	47	1	no	—
220	#431	—	47	1	no	—
221	#431	—	94	2	no	—
222	#431	C₂×D₄₇	1	1	yes	characteristic
223	#408	—	94	2	no	—
224	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
225	#335	—	47	2	no	—
226	#334	—	47	2	no	—
227	Q₈⋊C₈	—	47	2	no	—
228	Q₈⋊C₈	—	47	2	no	—
229	D₄⋊C₈	C₂×D₄₇	1	2	yes	—
230	D₄⋊C₈	C₂×D₄₇	1	2	yes	—
231	#322	—	47	1	no	—
232	#322	—	47	1	no	—
233	#322	—	47	1	no	—
234	C₈²	—	47	1	no	—
235	C₄₇	#17923	1	1	yes	characteristic, complemented, Sylow
236	C₂×#41	—	2	1	no	—
237	C₂²×Q₈	—	47	1	no	—
238	C₂×SD₁₆	—	47	2	no	—
239	C₂×SD₁₆	—	47	2	no	—
240	C₂×M₁₆	—	2	1	no	—
241	C₂×M₁₆	—	94	2	no	—
242	C₂²×C₈	—	94	2	no	—
243	C₄⋊Q₈	C₄×D₄₇	1	1	yes	characteristic
244	#127	C₄×D₄₇	1	1	yes	characteristic
245	#124	—	2	1	no	—
246	#121	—	2	1	no	—
247	#121	C₄×D₄₇	1	2	yes	—
248	C₄×Q₈	—	47	2	no	—
249	C₄×Q₈	—	47	2	no	—
250	C₄×Q₈	—	94	1	no	—
251	C₄×Q₈	—	94	1	no	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	C₂²×D₄₇	1	2	yes	—
254	C₂×C₄⋊C₄	—	47	1	no	—
255	C₂×C₄⋊C₄	—	94	1	no	—
256	C₂×C₄²	—	94	1	no	—
257	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
258	#106	—	47	2	no	—
259	#106	—	47	2	no	—
260	C₄⋊C₈	—	47	2	no	—
261	C₄⋊C₈	—	47	2	no	—
262	C₄⋊C₈	—	47	2	no	—
263	C₄⋊C₈	—	94	2	no	—
264	C₄⋊C₈	—	94	2	no	—
265	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
266	#103	—	47	2	no	—
267	#103	—	47	2	no	—
268	#102	—	47	2	no	—
269	#102	—	47	2	no	—
270	C₂²⋊C₈	—	2	1	no	—
271	C₂²⋊C₈	—	47	2	no	—
272	C₂²⋊C₈	—	47	2	no	—
273	C₂²⋊C₈	—	94	2	no	—
274	C₂²⋊C₈	—	94	2	no	—
275	C₂²⋊C₈	—	94	1	no	—
276	#97	—	2	1	no	—
277	#97	—	94	1	no	—
278	C₄×C₈	—	2	1	no	—
279	C₄×C₈	—	47	1	no	—
280	C₄×C₈	—	47	1	no	—
281	C₄×C₈	—	47	1	no	—
282	C₄×C₈	—	47	1	no	—
283	C₄×C₈	—	94	2	no	—
284	C₄×C₈	—	94	2	no	—
285	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
287	#41	—	8	2	no	—
288	C₂×Q₈	—	2	1	no	—
289	C₂×Q₈	—	47	2	no	—
290	C₂×Q₈	—	94	4	no	—
291	C₂×Q₈	—	94	1	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	94	1	no	—
298	C₂²×C₄	—	94	1	no	—
299	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
300	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
301	C₂²×C₄	#10625352	1	1	yes	characteristic
302	SD₁₆	—	94	4	no	—
303	SD₁₆	—	94	4	no	—
304	M₁₆	—	4	2	no	—
305	M₁₆	—	376	2	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	47	2	no	—
310	C₂×C₈	—	47	2	no	—
311	C₂×C₈	—	47	2	no	—
312	C₂×C₈	—	47	2	no	—
313	C₂×C₈	—	94	2	no	—
314	C₂×C₈	—	376	2	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	47	2	no	—
317	C₄⋊C₄	—	94	2	no	—
318	C₄⋊C₄	—	94	1	no	—
319	C₄⋊C₄	—	94	2	no	—
320	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₄²	—	94	2	no	—
324	C₄²	—	94	2	no	—
325	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
326	C₄²	D₄×D₄₇	1	1	yes	characteristic
327	C₄²	#10625352	1	1	yes	characteristic
328	C₄²	C₂³×D₄₇	1	1	yes	characteristic
329	C₂³	—	2	2	no	—
330	C₂³	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
331	Q₈	—	8	1	no	—
332	Q₈	—	47	4	no	—
333	Q₈	—	94	4	no	—
334	Q₈	—	94	2	no	—
335	D₄	—	2	2	no	—
336	D₄	—	2	2	no	—
337	D₄	—	8	1	no	—
338	D₄	—	8	1	no	—
339	D₄	—	8	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	94	4	no	—
347	C₂×C₄	—	94	2	no	—
348	C₂×C₄	—	94	2	no	—
349	C₂×C₄	—	376	1	no	—
350	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
351	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
352	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
355	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	4	1	no	—
358	C₈	—	4	2	no	—
359	C₈	—	94	2	no	—
360	C₈	—	94	2	no	—
361	C₈	—	94	2	no	—
362	C₈	—	94	2	no	—
363	C₈	—	188	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	94	4	no	—
377	C₄	—	188	2	no	—
378	C₄	#131×D₄₇	1	2	yes	—
379	C₄	#15689933	1	2	yes	—
380	C₂	—	4	1	no	—
381	C₂	—	8	2	no	—
382	C₂	#20252085	1	1	yes	characteristic
383	C₂	#20252250	1	1	yes	characteristic
384	C₂	#20252294	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1484	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₄₇×#532	C₂²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1484	D₄₇
hypocenter	C₄₇	#17923
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1484

Complements

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

N	H ≅ G/N	type
2. #20252468	C₂	semidirect
4. #20242269	C₂	semidirect
5. #20242234	C₂	semidirect
6. #20242197	C₂	semidirect
7. #20242191	C₂	semidirect
8. #20242166	C₂	semidirect
9. #20241836	C₂	semidirect
10. #20241685	C₂	semidirect
235. C₄₇	#17923	semidirect

Maximal subgroups

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

subgroup	index
#20253415	2
#20252468	2
#20252464	2
#20242269	2
#20242234	2
#20242197	2
#20242191	2
#20242166	2
#20241836	2
#20241685	2
C₄₇×#1484	2
#17923	47