Subgroups — #26316685

Order 12032.

738 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251749	C₂	1	1	yes	maximal, characteristic
2	#20245955	C₂	1	1	yes	maximal, characteristic
3	#20244764	C₂	1	4	yes	maximal
4	#20244190	C₂	1	1	yes	maximal, characteristic
5	#20244189	C₂	1	1	yes	maximal, characteristic
6	#20243653	C₂	1	4	yes	maximal
7	#20243515	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094024	C₂²	1	1	yes	characteristic
11	C₂×#13094015	—	2	1	no	—
12	C₂×#13093989	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
14	C₂×#13093949	C₂²	1	4	yes	—
15	C₂×#13093947	C₂²	1	4	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#15689028	C₂²	1	2	yes	—
18	#15689022	—	2	1	no	—
19	#15689022	C₂²	1	2	yes	—
20	#15689019	—	2	1	no	—
21	#15689019	—	2	1	no	—
22	#15689019	—	2	1	no	—
23	#15689019	C₂²	1	2	yes	—
24	#15688993	C₂²	1	1	yes	characteristic
25	#15688988	—	2	1	no	—
26	C₄⋊C₄×Dic₄₇	—	2	1	no	—
27	C₄⋊C₄×Dic₄₇	—	2	1	no	—
28	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
29	#15688885	C₂²	1	4	yes	—
30	#15688877	C₂²	1	2	yes	—
31	C₂×#106×C₄₇	C₂²	1	4	yes	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#384	C₂²	1	1	yes	characteristic
34	C₄₇×#384	C₂²	1	1	yes	characteristic
35	C₂²×Dic₉₄	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	#13094024	—	2	2	no	—
38	#13094024	Q₈	1	4	yes	—
39	#13094015	—	4	4	no	—
40	#13093989	—	4	4	no	—
41	C₂×C₄⋊Dic₄₇	—	2	1	no	—
42	C₂×C₄⋊Dic₄₇	—	2	1	no	—
43	C₂×C₄⋊Dic₄₇	—	2	1	no	—
44	C₂×C₄⋊Dic₄₇	—	2	1	no	—
45	C₂×C₄⋊Dic₄₇	—	2	2	no	—
46	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
47	C₂×Dic₄₇⋊C₄	—	2	2	no	—
48	C₂×Dic₄₇⋊C₄	—	2	2	no	—
49	C₂×C₄×Dic₄₇	—	2	2	no	—
50	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
52	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
56	Dic₄₇⋊C₈	—	2	4	no	—
57	#13093949	—	2	8	no	—
58	#13093947	—	4	4	no	—
59	C₄×C₄₇⋊C₈	—	2	2	no	—
60	C₄₇⋊#95	—	2	2	no	—
61	C₄₇⋊#95	—	2	2	no	—
62	C₄₇⋊#95	—	2	2	no	—
63	C₄₇⋊#95	—	2	2	no	—
64	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	#106×C₄₇	—	4	4	no	—
73	C₄×C₈×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	2	2	no	—
80	C₂²×Dic₄₇	—	2	2	no	—
81	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₉₄	—	2	4	no	—
86	C₂×Dic₉₄	—	2	4	no	—
87	C₂×Dic₉₄	—	2	2	no	—
88	C₂×Dic₉₄	C₂×Q₈	1	4	yes	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	#41	1	4	yes	—
92	C₄⋊C₄×C₄₇	—	2	4	no	—
93	C₄⋊C₄×C₄₇	—	2	4	no	—
94	C₄⋊C₄×C₄₇	—	4	2	no	—
95	C₄⋊C₄×C₄₇	—	4	2	no	—
96	C₄⋊C₄×C₄₇	—	4	2	no	—
97	C₄⋊C₄×C₄₇	—	4	2	no	—
98	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
99	C₄²×C₄₇	#41	1	2	yes	—
100	C₄⋊Dic₄₇	—	2	4	no	—
101	C₄⋊Dic₄₇	—	2	4	no	—
102	C₄⋊Dic₄₇	—	2	4	no	—
103	C₄⋊Dic₄₇	—	4	2	no	—
104	C₄⋊Dic₄₇	—	4	2	no	—
105	C₄⋊Dic₄₇	—	4	4	no	—
106	Dic₄₇⋊C₄	—	4	4	no	—
107	Dic₄₇⋊C₄	—	4	4	no	—
108	C₄×Dic₄₇	—	2	2	no	—
109	C₄×Dic₄₇	—	2	2	no	—
110	C₄×Dic₄₇	—	2	2	no	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₄×Dic₄₇	—	2	2	no	—
113	C₄×Dic₄₇	—	4	4	no	—
114	C₄×Dic₄₇	SD₁₆	1	4	yes	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
128	C₂×C₄×C₄₇	#135	1	2	yes	—
129	C₂×C₄×C₄₇	#137	1	2	yes	—
130	C₂×C₄×C₄₇	#137	1	2	yes	—
131	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
134	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
135	C₂×Dic₄₇	—	2	1	no	—
136	C₂×Dic₄₇	—	2	1	no	—
137	C₂×Dic₄₇	—	2	1	no	—
138	C₂×Dic₄₇	—	2	1	no	—
139	C₂×Dic₄₇	—	2	4	no	—
140	C₂×Dic₄₇	—	2	1	no	—
141	C₂×Dic₄₇	—	2	1	no	—
142	C₂×Dic₄₇	—	2	1	no	—
143	C₂×Dic₄₇	—	2	1	no	—
144	C₂×Dic₄₇	—	2	1	no	—
145	C₂×Dic₄₇	—	2	1	no	—
146	C₂×Dic₄₇	—	4	2	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	C₂×Dic₄₇	C₂×SD₁₆	1	2	yes	—
152	C₂×Dic₄₇	C₂×SD₁₆	1	2	yes	—
153	Dic₉₄	—	2	8	no	—
154	Dic₉₄	—	4	4	no	—
155	C₈×C₄₇	—	2	4	no	—
156	C₄₇⋊C₈	—	4	2	no	—
157	#29191	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₂²×C₄₇	#532	1	1	yes	characteristic
160	C₂²×C₄₇	#535	1	1	yes	characteristic
161	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
162	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
163	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
164	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
165	C₄×C₄₇	—	8	2	no	—
166	C₄×C₄₇	—	8	2	no	—
167	C₄×C₄₇	#475	1	4	yes	—
168	C₄×C₄₇	#479	1	4	yes	—
169	C₄×C₄₇	#553	1	2	yes	—
170	C₄×C₄₇	#554	1	2	yes	—
171	Dic₄₇	—	2	2	no	—
172	Dic₄₇	—	2	2	no	—
173	Dic₄₇	—	4	4	no	—
174	Dic₄₇	—	4	1	no	—
175	Dic₄₇	—	4	1	no	—
176	Dic₄₇	—	8	2	no	—
177	#2859	—	47	1	no	—
178	C₄×C₄⋊Q₈	—	47	1	no	—
179	#2066	—	47	4	no	—
180	#1956	—	47	1	no	—
181	#1956	—	47	1	no	—
182	#1865	—	47	4	no	—
183	#1844	D₄₇	1	1	yes	characteristic
184	#1834	—	47	1	no	—
185	C₄×C₄⋊C₈	—	47	1	no	—
186	C₂×C₄₇	#2566	1	1	yes	characteristic
187	C₂×C₄₇	C₂×#475	1	1	yes	characteristic
188	C₂×C₄₇	C₂×#479	1	1	yes	characteristic
189	C₂×C₄₇	#3167	1	1	yes	characteristic
190	C₂×C₄₇	#3185	1	1	yes	characteristic
191	C₂×C₄₇	#3206	1	1	yes	characteristic
192	C₂×C₄₇	#3210	1	1	yes	characteristic
193	C₂×C₄⋊Q₈	—	47	1	no	—
194	C₂×#125	—	94	1	no	—
195	C₂×C₄×Q₈	—	47	2	no	—
196	C₂×#106	C₂×D₄₇	1	4	yes	—
197	C₂×C₄⋊C₈	—	47	2	no	—
198	C₂×#103	—	47	4	no	—
199	C₂×#103	—	47	4	no	—
200	C₂×C₄×C₈	—	47	1	no	—
201	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
202	#391	—	47	2	no	—
203	#389	—	47	2	no	—
204	#389	—	47	2	no	—
205	#389	—	94	1	no	—
206	#389	—	94	1	no	—
207	#389	—	94	1	no	—
208	#389	—	94	1	no	—
209	#384	—	47	1	no	—
210	#384	C₂×D₄₇	1	1	yes	characteristic
211	#384	C₂×D₄₇	1	1	yes	characteristic
212	#382	—	94	1	no	—
213	C₄×C₄⋊C₄	—	94	1	no	—
214	C₄×C₄⋊C₄	—	94	1	no	—
215	C₄³	—	47	1	no	—
216	#340	—	47	4	no	—
217	#336	—	47	2	no	—
218	C₄₇	#29191	1	1	yes	characteristic, complemented, Sylow
219	C₂²×Q₈	—	47	2	no	—
220	C₂²×C₈	—	47	2	no	—
221	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
222	C₄⋊Q₈	—	47	4	no	—
223	C₄⋊Q₈	—	94	2	no	—
224	#125	—	188	4	no	—
225	C₄×Q₈	—	188	4	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	47	2	no	—
231	C₂×C₄⋊C₄	—	94	1	no	—
232	C₂×C₄⋊C₄	—	94	1	no	—
233	C₂×C₄⋊C₄	—	94	2	no	—
234	C₂×C₄⋊C₄	—	94	2	no	—
235	C₂×C₄⋊C₄	—	94	2	no	—
236	C₂×C₄⋊C₄	—	94	1	no	—
237	C₂×C₄⋊C₄	—	94	1	no	—
238	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
239	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
240	C₂×C₄²	—	47	1	no	—
241	C₂×C₄²	—	47	1	no	—
242	C₂×C₄²	—	47	1	no	—
243	C₂×C₄²	—	47	1	no	—
244	C₂×C₄²	—	47	1	no	—
245	C₂×C₄²	—	47	1	no	—
246	C₂×C₄²	—	94	2	no	—
247	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
248	#106	—	4	4	no	—
249	C₄⋊C₈	—	94	4	no	—
250	#103	—	94	8	no	—
251	#103	—	188	4	no	—
252	C₄×C₈	—	2	2	no	—
253	C₄×C₈	—	94	2	no	—
254	#95	—	94	2	no	—
255	#95	—	94	2	no	—
256	#95	—	94	2	no	—
257	#95	—	94	2	no	—
258	C₂×Q₈	—	47	4	no	—
259	C₂×Q₈	—	94	2	no	—
260	C₂×Q₈	—	94	4	no	—
261	C₂×Q₈	—	94	4	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	47	1	no	—
265	C₂²×C₄	—	47	1	no	—
266	C₂²×C₄	—	47	1	no	—
267	C₂²×C₄	—	47	1	no	—
268	C₂²×C₄	—	94	2	no	—
269	C₂²×C₄	—	94	2	no	—
270	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	94	2	no	—
276	C₂×C₈	—	94	2	no	—
277	C₂×C₈	—	94	2	no	—
278	C₂×C₈	#10625354	1	4	yes	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	94	4	no	—
286	C₄⋊C₄	—	94	4	no	—
287	C₄⋊C₄	—	94	4	no	—
288	C₄⋊C₄	—	188	4	no	—
289	C₄⋊C₄	—	188	4	no	—
290	C₄⋊C₄	—	188	4	no	—
291	C₄⋊C₄	—	188	2	no	—
292	C₄⋊C₄	—	188	2	no	—
293	C₄²	—	47	4	no	—
294	C₄²	—	94	2	no	—
295	C₄²	—	94	2	no	—
296	C₄²	—	94	2	no	—
297	C₄²	—	94	2	no	—
298	C₄²	—	94	2	no	—
299	C₄²	—	188	4	no	—
300	C₄²	#10625350	1	2	yes	—
301	C₄²	Q₈×D₄₇	1	2	yes	—
302	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
303	Q₈	—	94	8	no	—
304	Q₈	—	188	4	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	47	2	no	—
312	C₂×C₄	—	47	2	no	—
313	C₂×C₄	—	94	1	no	—
314	C₂×C₄	—	94	1	no	—
315	C₂×C₄	—	94	1	no	—
316	C₂×C₄	—	94	1	no	—
317	C₂×C₄	—	94	4	no	—
318	C₂×C₄	—	94	1	no	—
319	C₂×C₄	—	94	1	no	—
320	C₂×C₄	—	94	1	no	—
321	C₂×C₄	—	94	1	no	—
322	C₂×C₄	—	94	1	no	—
323	C₂×C₄	—	94	1	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	—	188	2	no	—
327	C₂×C₄	—	188	2	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	#13094002	1	2	yes	—
330	C₂×C₄	#13094002	1	2	yes	—
331	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
332	C₂×C₄	#13094045	1	2	yes	—
333	C₂×C₄	#13094053	1	2	yes	—
334	C₂×C₄	#13094053	1	2	yes	—
335	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
336	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
337	C₂×C₄	C₂×#10625354	1	2	yes	—
338	C₂×C₄	#13094069	1	2	yes	—
339	C₈	—	2	4	no	—
340	C₈	—	188	2	no	—
341	C₂²	C₂×#13094002	1	1	yes	characteristic
342	C₂²	#15689674	1	1	yes	characteristic
343	C₂²	#15689686	1	1	yes	characteristic
344	C₂²	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
345	C₂²	C₂×#13094045	1	1	yes	characteristic
346	C₂²	C₂×#13094053	1	1	yes	characteristic
347	C₂²	C₂×#13094053	1	1	yes	characteristic
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	94	2	no	—
351	C₄	—	94	2	no	—
352	C₄	—	376	2	no	—
353	C₄	—	188	4	no	—
354	C₄	—	188	1	no	—
355	C₄	—	188	1	no	—
356	C₄	#15689407	1	4	yes	—
357	C₄	#15689433	1	4	yes	—
358	C₄	#15689821	1	2	yes	—
359	C₄	#15689834	1	2	yes	—
360	C₂	#20248585	1	1	yes	characteristic
361	C₂	C₂×#15689407	1	1	yes	characteristic
362	C₂	C₂×#15689433	1	1	yes	characteristic
363	C₂	#20253949	1	1	yes	characteristic
364	C₂	#20254134	1	1	yes	characteristic
365	C₂	#20254309	1	1	yes	characteristic
366	C₂	#20254339	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1844	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#29191
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
218. C₄₇	#29191	semidirect

Maximal subgroups

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

subgroup	index
#20251749	2
#20245955	2
#20244764	2
#20244190	2
#20244189	2
#20243653	2
#20243515	2
C₄×Dic₄₇⋊C₈	2
C₄₇×#1844	2
#29191	47