Subgroups — #26216533

Order 12032.

818 conjugacy classes of subgroups.

The subgroup lattice has 362 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	#20243532	C₂	1	4	yes	maximal, complemented
3	#20243530	C₂	1	4	yes	maximal, complemented
4	#20243086	C₂	1	1	yes	maximal, characteristic
5	#20243085	C₂	1	1	yes	maximal, characteristic
6	#20243084	C₂	1	1	yes	maximal, characteristic
7	#20243081	C₂	1	1	yes	maximal, characteristic
8	C₂×#15688877	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1729	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#13093983	C₂²	1	1	yes	characteristic
13	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×Dic₄₇	C₄	1	2	yes	—
15	C₄⋊C₄×Dic₄₇	C₄	1	2	yes	—
16	C₂²⋊C₄×Dic₄₇	C₄	1	4	yes	—
17	#15688969	C₂²	1	2	yes	—
18	C₄²×Dic₄₇	C₂²	1	2	yes	—
19	#15688878	C₂²	1	4	yes	—
20	#15688877	C₂²	1	2	yes	—
21	#15688877	C₂²	1	2	yes	—
22	#15688877	C₂²	1	4	yes	—
23	#15688877	C₂²	1	4	yes	—
24	#15688877	C₂²	1	2	yes	—
25	#15688877	C₂²	1	2	yes	—
26	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#336	C₂²	1	4	yes	—
30	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
31	C₂×#10625329	—	2	2	no	—
32	C₂×#10625329	—	2	2	no	—
33	C₂×#10625329	—	2	2	no	—
34	C₂²×C₄₇⋊C₈	—	2	2	no	—
35	C₂²×C₄₇⋊C₈	—	2	2	no	—
36	C₂²×C₄₇⋊C₈	—	2	2	no	—
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	#13093983	—	2	2	no	—
40	#13093983	—	2	2	no	—
41	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₄₇	—	2	2	no	—
45	C₂×C₄×Dic₄₇	—	2	2	no	—
46	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
51	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
52	C₄₇⋊#95	C₂×C₄	1	4	yes	—
53	C₄₇⋊#95	C₂×C₄	1	4	yes	—
54	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₄₇	—	2	2	no	—
56	C₂×M₁₆×C₄₇	—	2	2	no	—
57	C₂×M₁₆×C₄₇	—	2	2	no	—
58	C₂²×C₈×C₄₇	—	2	2	no	—
59	C₂²×C₈×C₄₇	—	2	2	no	—
60	C₂²×C₈×C₄₇	—	2	2	no	—
61	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
63	#117×C₄₇	—	2	2	no	—
64	#117×C₄₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
68	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	#41	1	2	yes	—
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	#41	1	2	yes	—
74	C₂²×C₄×C₄₇	#41	1	2	yes	—
75	C₂²×C₄×C₄₇	#41	1	2	yes	—
76	C₂²×C₄×C₄₇	#41	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	4	no	—
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂²×Dic₄₇	—	4	2	no	—
81	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
82	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
83	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
85	M₁₆×C₄₇	—	4	4	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	4	4	no	—
93	C₄⋊C₄×C₄₇	—	2	4	no	—
94	C₂²⋊C₄×C₄₇	—	2	4	no	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	—	2	2	no	—
97	C₂²⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	C₄×Dic₄₇	—	2	4	no	—
101	C₄×Dic₄₇	—	2	1	no	—
102	C₄×Dic₄₇	—	2	1	no	—
103	C₄×Dic₄₇	—	2	1	no	—
104	C₄×Dic₄₇	—	2	1	no	—
105	C₄×Dic₄₇	—	2	2	no	—
106	C₄×Dic₄₇	—	2	2	no	—
107	C₄×Dic₄₇	—	2	8	no	—
108	C₄×Dic₄₇	—	4	4	no	—
109	#10625329	—	4	4	no	—
110	C₂×C₄₇⋊C₈	—	2	2	no	—
111	C₂×C₄₇⋊C₈	—	2	2	no	—
112	C₂×C₄₇⋊C₈	—	2	2	no	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	4	4	no	—
117	C₂³×C₄₇	—	2	2	no	—
118	C₂³×C₄₇	—	2	2	no	—
119	C₂³×C₄₇	#117	1	2	yes	—
120	C₂³×C₄₇	#117	1	2	yes	—
121	C₂³×C₄₇	#117	1	2	yes	—
122	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	2	4	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	#117	1	2	yes	—
127	C₂×C₄×C₄₇	#117	1	2	yes	—
128	C₂×C₄×C₄₇	#117	1	2	yes	—
129	C₂×C₄×C₄₇	#126	1	4	yes	—
130	C₂×C₄×C₄₇	#126	1	4	yes	—
131	C₂×C₄×C₄₇	#126	1	4	yes	—
132	C₂×C₄×C₄₇	#126	1	4	yes	—
133	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	—	2	2	no	—
144	C₂×Dic₄₇	—	2	4	no	—
145	C₂×Dic₄₇	—	2	2	no	—
146	C₂×Dic₄₇	—	2	4	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	4	8	no	—
149	C₂×Dic₄₇	#131	1	4	yes	—
150	C₈×C₄₇	—	4	2	no	—
151	C₈×C₄₇	—	4	2	no	—
152	C₄₇⋊C₈	—	4	2	no	—
153	C₄₇⋊C₈	—	4	2	no	—
154	#20774	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	—	2	2	no	—
156	C₂²×C₄₇	—	2	2	no	—
157	C₂²×C₄₇	—	2	4	no	—
158	C₂²×C₄₇	—	2	4	no	—
159	C₂²×C₄₇	#383	1	4	yes	—
160	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
161	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
162	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
163	C₂²×C₄₇	C₂×#131	1	2	yes	—
164	C₂²×C₄₇	#568	1	2	yes	—
165	C₄×C₄₇	—	4	4	no	—
166	C₄×C₄₇	#383	1	4	yes	—
167	C₄×C₄₇	C₂×#126	1	1	yes	characteristic
168	C₄×C₄₇	C₂×#126	1	1	yes	characteristic
169	C₄×C₄₇	C₂×#126	1	1	yes	characteristic
170	C₄×C₄₇	C₂×#126	1	1	yes	characteristic
171	Dic₄₇	—	2	4	no	—
172	Dic₄₇	—	4	2	no	—
173	Dic₄₇	—	4	2	no	—
174	Dic₄₇	—	4	2	no	—
175	C₄×#117	—	47	1	no	—
176	#1840	—	47	4	no	—
177	#1839	—	47	4	no	—
178	#1729	—	47	1	no	—
179	#1729	—	47	1	no	—
180	#1729	—	47	1	no	—
181	#1729	D₄₇	1	1	yes	characteristic
182	#1728	—	47	1	no	—
183	C₂×#336	—	47	1	no	—
184	C₂×C₄₇	—	2	4	no	—
185	C₂×C₄₇	C₂×#383	1	1	yes	characteristic
186	C₂×C₄₇	#2922	1	1	yes	characteristic
187	C₂×C₄₇	#2923	1	1	yes	characteristic
188	C₂×C₄₇	#2924	1	2	yes	—
189	C₂×C₄₇	#2925	1	2	yes	—
190	C₂²×M₁₆	—	47	1	no	—
191	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
192	C₂³×C₈	—	47	1	no	—
193	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
194	C₂×#117	—	47	1	no	—
195	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
196	C₂²×C₄²	—	47	1	no	—
197	C₄×C₄⋊C₄	—	47	2	no	—
198	C₄×C₄⋊C₄	—	47	2	no	—
199	C₄×C₂²⋊C₄	—	47	4	no	—
200	#376	—	47	2	no	—
201	C₄³	—	47	2	no	—
202	#336	—	47	2	no	—
203	#336	—	47	2	no	—
204	#336	—	47	4	no	—
205	#336	—	47	4	no	—
206	#336	—	47	2	no	—
207	#336	—	47	2	no	—
208	#336	—	47	4	no	—
209	#336	C₂×D₄₇	1	4	yes	—
210	C₄₇	#20774	1	1	yes	characteristic, complemented, Sylow
211	C₂³×C₄	—	47	2	no	—
212	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	2	2	no	—
215	C₂×M₁₆	—	2	2	no	—
216	C₂×M₁₆	—	94	2	no	—
217	C₂×M₁₆	—	94	2	no	—
218	C₂×M₁₆	—	94	2	no	—
219	C₂²×C₈	—	2	2	no	—
220	C₂²×C₈	—	2	2	no	—
221	C₂²×C₈	—	2	2	no	—
222	C₂²×C₈	—	47	2	no	—
223	C₂²×C₈	—	47	2	no	—
224	C₂²×C₈	—	94	2	no	—
225	C₂²×C₈	—	94	2	no	—
226	C₂²×C₈	—	94	2	no	—
227	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
228	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
229	#117	—	2	2	no	—
230	#117	—	2	2	no	—
231	#117	—	94	2	no	—
232	#117	—	94	2	no	—
233	C₂×C₄⋊C₄	—	47	1	no	—
234	C₂×C₄⋊C₄	—	47	1	no	—
235	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
236	C₂×C₂²⋊C₄	—	47	2	no	—
237	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
238	C₂×C₄²	—	47	4	no	—
239	C₂×C₄²	—	47	2	no	—
240	C₂×C₄²	—	47	2	no	—
241	C₂×C₄²	—	47	1	no	—
242	C₂×C₄²	—	47	1	no	—
243	C₂×C₄²	—	47	4	no	—
244	C₂×C₄²	—	94	2	no	—
245	C₂×C₄²	—	94	2	no	—
246	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
247	#95	—	47	4	no	—
248	#95	—	47	4	no	—
249	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
250	C₂²×C₄	—	47	2	no	—
251	C₂²×C₄	—	47	2	no	—
252	C₂²×C₄	—	47	2	no	—
253	C₂²×C₄	—	47	2	no	—
254	C₂²×C₄	—	94	4	no	—
255	C₂²×C₄	—	94	4	no	—
256	C₂²×C₄	—	188	2	no	—
257	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
258	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
259	C₂²×C₄	#10625350	1	2	yes	—
260	C₂²×C₄	#10625350	1	2	yes	—
261	C₂²×C₄	#10625352	1	2	yes	—
262	C₂²×C₄	#10625352	1	2	yes	—
263	C₂²×C₄	#10625352	1	2	yes	—
264	C₂²×C₄	#10625354	1	2	yes	—
265	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
266	M₁₆	—	4	4	no	—
267	M₁₆	—	188	4	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	—	94	2	no	—
276	C₂×C₈	—	94	2	no	—
277	C₂×C₈	—	94	2	no	—
278	C₂×C₈	—	94	2	no	—
279	C₂×C₈	—	94	2	no	—
280	C₂×C₈	—	94	2	no	—
281	C₂×C₈	—	188	4	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	94	2	no	—
284	C₄⋊C₄	—	94	2	no	—
285	C₂²⋊C₄	—	2	4	no	—
286	C₂²⋊C₄	—	94	4	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	94	8	no	—
290	C₄²	—	94	4	no	—
291	C₄²	—	94	2	no	—
292	C₄²	—	94	1	no	—
293	C₄²	—	94	1	no	—
294	C₄²	—	94	1	no	—
295	C₄²	—	94	1	no	—
296	C₄²	—	94	2	no	—
297	C₄²	—	188	4	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	#13093980	1	2	yes	—
301	C₂³	#13093981	1	2	yes	—
302	C₂³	#13093982	1	2	yes	—
303	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	47	4	no	—
308	C₂×C₄	—	94	2	no	—
309	C₂×C₄	—	94	2	no	—
310	C₂×C₄	—	94	2	no	—
311	C₂×C₄	—	94	2	no	—
312	C₂×C₄	—	94	2	no	—
313	C₂×C₄	—	94	4	no	—
314	C₂×C₄	—	94	2	no	—
315	C₂×C₄	—	94	4	no	—
316	C₂×C₄	—	94	2	no	—
317	C₂×C₄	—	188	8	no	—
318	C₂×C₄	#13093980	1	2	yes	—
319	C₂×C₄	#13093981	1	2	yes	—
320	C₂×C₄	#13093982	1	2	yes	—
321	C₂×C₄	#13094017	1	4	yes	—
322	C₂×C₄	#13094017	1	4	yes	—
323	C₂×C₄	#13094019	1	4	yes	—
324	C₂×C₄	#13094019	1	4	yes	—
325	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
326	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
327	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
328	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
329	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
330	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	4	2	no	—
333	C₈	—	188	2	no	—
334	C₈	—	188	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	4	no	—
338	C₂²	—	2	4	no	—
339	C₂²	#15688990	1	4	yes	—
340	C₂²	C₂×#13093980	1	1	yes	characteristic
341	C₂²	C₂×#13093981	1	1	yes	characteristic
342	C₂²	C₂×#13093982	1	1	yes	characteristic
343	C₂²	#131×D₄₇	1	2	yes	—
344	C₂²	#15689888	1	2	yes	—
345	C₄	—	4	4	no	—
346	C₄	—	94	4	no	—
347	C₄	—	188	2	no	—
348	C₄	—	188	2	no	—
349	C₄	—	188	2	no	—
350	C₄	#15688990	1	4	yes	—
351	C₄	C₂×#13094017	1	1	yes	characteristic
352	C₄	C₂×#13094017	1	1	yes	characteristic
353	C₄	C₂×#13094019	1	1	yes	characteristic
354	C₄	C₂×#13094019	1	1	yes	characteristic
355	C₂	—	2	4	no	—
356	C₂	C₂×#15688990	1	1	yes	characteristic
357	C₂	#20252031	1	1	yes	characteristic
358	C₂	#20252038	1	1	yes	characteristic
359	C₂	#20252043	1	2	yes	—
360	C₂	#20252050	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1729	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1729	D₄₇
hypocenter	C₄₇	#20774
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
2. #20243532	C₂	semidirect
3. #20243530	C₂	semidirect
210. C₄₇	#20774	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
#20243532	2
#20243530	2
#20243086	2
#20243085	2
#20243084	2
#20243081	2
C₂×#15688877	2
C₄₇×#1729	2
#20774	47