Subgroups — #26106403

Order 12032.

954 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245771	C₂	1	2	yes	maximal
2	#20241273	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₇×#407	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093983	—	2	2	no	—
5	#15688986	C₄	1	4	yes	complemented
6	#15688983	—	2	2	no	—
7	#15688983	C₄	1	4	yes	complemented
8	#15688979	—	2	2	no	—
9	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
10	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
11	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₄₇	C₄	1	2	yes	complemented
13	C₄₇×#407	—	2	2	no	—
14	C₄₇×#407	C₄	1	2	yes	complemented
15	C₄₇×#407	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
17	#13093983	—	4	8	no	—
18	C₂×C₄⋊Dic₄₇	—	2	4	no	—
19	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
20	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
21	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
22	C₂×C₄×Dic₄₇	—	2	4	no	—
23	C₄₇⋊#95	—	2	4	no	—
24	C₄₇⋊#95	C₂×C₄	1	8	yes	—
25	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	—	4	4	no	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	D₄	1	2	yes	—
32	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	D₄	1	2	yes	—
34	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	4	4	no	—
39	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂²⋊C₈×C₄₇	—	2	4	no	—
42	C₂²⋊C₈×C₄₇	—	2	4	no	—
43	C₂⁴×C₄₇	—	4	2	no	—
44	C₂⁴×C₄₇	—	4	4	no	—
45	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	2	8	no	—
52	C₂²×C₄×C₄₇	—	4	4	no	—
53	C₂²×C₄×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	—	4	4	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	4	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	8	no	—
79	M₁₆×C₄₇	—	4	4	no	—
80	M₁₆×C₄₇	—	4	4	no	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	4	4	no	—
83	C₂²⋊Dic₄₇	—	4	8	no	—
84	C₂²⋊Dic₄₇	—	4	8	no	—
85	C₂²⋊Dic₄₇	—	4	8	no	—
86	C₄⋊Dic₄₇	—	4	8	no	—
87	C₄×Dic₄₇	—	4	8	no	—
88	C₂³×C₄₇	—	2	4	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	4	no	—
91	C₂³×C₄₇	—	2	4	no	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	—	4	1	no	—
95	C₂³×C₄₇	—	4	1	no	—
96	C₂³×C₄₇	—	4	2	no	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	—	4	2	no	—
99	C₂³×C₄₇	—	4	2	no	—
100	C₂³×C₄₇	—	4	16	no	—
101	C₂³×C₄₇	#95	1	1	yes	characteristic
102	C₂³×C₄₇	#95	1	2	yes	—
103	C₂³×C₄₇	#95	1	1	yes	characteristic
104	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
105	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	1	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	1	no	—
123	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#95	1	2	yes	—
125	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#99	1	4	yes	—
127	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×Dic₄₇	—	4	8	no	—
131	C₂×Dic₄₇	—	4	8	no	—
132	C₂×Dic₄₇	—	4	8	no	—
133	C₈×C₄₇	—	4	4	no	—
134	#8049	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	—	4	4	no	—
146	C₂²×C₄₇	—	4	8	no	—
147	C₂²×C₄₇	—	4	8	no	—
148	C₂²×C₄₇	—	4	4	no	—
149	C₂²×C₄₇	—	4	4	no	—
150	C₂²×C₄₇	#342	1	2	yes	—
151	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
152	C₂²×C₄₇	#421	1	4	yes	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	#342	1	2	yes	—
158	C₄×C₄₇	C₂×#99	1	2	yes	—
159	Dic₄₇	—	8	8	no	—
160	C₂×#407	D₄₇	1	1	yes	characteristic
161	#2280	—	47	2	no	—
162	#1297	—	47	4	no	—
163	C₂×C₄₇	—	2	4	no	—
164	C₂×C₄₇	—	2	2	no	—
165	C₂×C₄₇	—	4	2	no	—
166	C₂×C₄₇	—	4	1	no	—
167	C₂×C₄₇	—	4	1	no	—
168	C₂×C₄₇	#1733	1	2	yes	—
169	C₂×C₄₇	C₂×#342	1	1	yes	characteristic
170	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
171	C₂²×M₁₆	Dic₄₇	1	2	yes	complemented
172	C₂×#117	—	94	2	no	—
173	#407	—	2	2	no	—
174	#407	Dic₄₇	1	2	yes	complemented
175	#407	C₂×D₄₇	1	2	yes	—
176	C₂×C₂²⋊C₈	—	2	2	no	—
177	#381	—	47	4	no	—
178	#380	—	47	4	no	—
179	#380	—	94	2	no	—
180	#379	—	94	2	no	—
181	C₄×C₂²⋊C₄	—	47	4	no	—
182	C₄×C₂²⋊C₄	—	94	2	no	—
183	C₄₇	#8049	1	1	yes	characteristic, complemented, Sylow
184	C₂⁵	D₁₈₈	1	1	yes	characteristic
185	C₂³×C₄	—	4	4	no	—
186	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
187	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
188	C₂³×C₄	C₄×D₄₇	1	2	yes	—
189	C₂³×C₄	C₄×D₄₇	1	2	yes	—
190	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
191	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
192	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
193	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
194	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	4	4	no	—
198	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
199	C₂²×C₈	—	2	2	no	—
200	#117	—	188	8	no	—
201	C₂×C₄⋊C₄	—	94	4	no	—
202	C₂×C₂²⋊C₄	—	94	4	no	—
203	C₂×C₂²⋊C₄	—	94	4	no	—
204	C₂×C₂²⋊C₄	—	94	4	no	—
205	C₂×C₄²	—	94	4	no	—
206	C₂²⋊C₈	—	2	4	no	—
207	C₂²⋊C₈	—	2	4	no	—
208	#95	—	47	8	no	—
209	#95	—	94	4	no	—
210	C₂⁴	—	4	2	no	—
211	C₂⁴	—	4	4	no	—
212	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
213	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
214	C₂⁴	(C₂×D₄₇)⋊C₄	1	2	yes	—
215	C₂⁴	(C₂×D₄₇)⋊C₄	1	2	yes	—
216	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	94	8	no	—
228	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
231	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
232	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
233	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
234	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
235	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
237	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
238	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
239	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
240	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
246	M₁₆	—	4	4	no	—
247	M₁₆	—	4	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	4	4	no	—
250	C₄⋊C₄	—	188	8	no	—
251	C₂²⋊C₄	—	188	8	no	—
252	C₂²⋊C₄	—	188	8	no	—
253	C₂²⋊C₄	—	188	8	no	—
254	C₄²	—	188	8	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	4	1	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	1	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	16	no	—
268	C₂³	C₄₇⋊#95	1	1	yes	characteristic
269	C₂³	C₄₇⋊#95	1	2	yes	—
270	C₂³	C₄₇⋊#95	1	1	yes	characteristic
271	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
272	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	188	8	no	—
291	C₂×C₄	—	188	8	no	—
292	C₂×C₄	—	188	8	no	—
293	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
294	C₂×C₄	C₄₇⋊#95	1	2	yes	—
295	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
296	C₂×C₄	#13093939	1	4	yes	—
297	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
299	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
300	C₈	—	4	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	4	8	no	—
313	C₂²	—	4	8	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	#15688890	1	2	yes	—
317	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
318	C₂²	#15689139	1	4	yes	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	2	no	—
323	C₄	—	376	8	no	—
324	C₄	#15688890	1	2	yes	—
325	C₄	C₂×#13093939	1	2	yes	—
326	C₂	—	2	4	no	—
327	C₂	—	2	2	no	—
328	C₂	—	4	2	no	—
329	C₂	—	4	1	no	—
330	C₂	—	4	1	no	—
331	C₂	#20243099	1	2	yes	—
332	C₂	C₂×#15688890	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#407	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#407	D₄₇
hypocenter	C₄₇	#8049
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂×#407

Complements

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

N	H ≅ G/N	type
2. #20241273	C₂	semidirect
5. #15688986	C₄	semidirect
7. #15688983	C₄	semidirect
12. C₂²×M₁₆×C₄₇	C₄	semidirect
14. C₄₇×#407	C₄	semidirect
171. C₂²×M₁₆	Dic₄₇	semidirect
174. #407	Dic₄₇	semidirect
183. C₄₇	#8049	semidirect

Maximal subgroups

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

subgroup	index
#20245771	2
#20241273	2
C₂×C₄₇×#407	2
#8049	47