Subgroups — #26214353

Order 12032.

1126 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689165	C₂	1	2	yes	maximal
2	#20245744	C₂	1	2	yes	maximal
3	#20243285	C₂	1	4	yes	maximal, complemented
4	C₄×#13093958	C₂	1	2	yes	maximal, complemented
5	C₄×#13093956	C₂	1	2	yes	maximal, complemented
6	#20243105	C₂	1	1	yes	maximal, characteristic
7	C₂×#15688886	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	#15689165	C₄	1	16	yes	complemented
12	C₂×#13093958	C₂²	1	4	yes	—
13	C₂×#13093956	C₂²	1	4	yes	—
14	C₄⋊C₄×Dic₄₇	—	2	2	no	—
15	C₄⋊C₄×Dic₄₇	—	2	2	no	—
16	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
18	C₂²⋊C₄×Dic₄₇	—	2	4	no	—
19	#15688886	C₂²	1	1	yes	characteristic
20	#15688886	C₂²	1	2	yes	—
21	#15688886	C₂²	1	1	yes	characteristic
22	#15688886	C₂²	1	2	yes	—
23	#15688886	C₂²	1	2	yes	—
24	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#405	C₄	1	8	yes	complemented
28	C₂×#97×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
30	C₂³×Dic₄₇	—	2	2	no	—
31	#13093983	—	2	4	no	—
32	#13093983	C₂×C₄	1	8	yes	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	4	no	—
35	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	—	2	2	no	—
41	C₂×C₄×Dic₄₇	—	2	4	no	—
42	C₂×C₄×Dic₄₇	—	2	4	no	—
43	#13093958	C₂×C₄	1	16	yes	complemented
44	#13093956	C₂×C₄	1	16	yes	complemented
45	C₄₇⋊#95	—	2	4	no	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
49	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₄₇	—	2	4	no	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
55	#117×C₄₇	C₂×C₄	1	4	yes	—
56	#117×C₄₇	Q₈	1	4	yes	—
57	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
61	#97×C₄₇	C₂×C₄	1	8	yes	complemented
62	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
63	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	2	8	no	—
76	C₂²×Dic₄₇	—	8	2	no	—
77	C₂²×Dic₄₇	—	8	2	no	—
78	M₁₆×C₄₇	C₄²	1	16	yes	complemented
79	C₂×C₈×C₄₇	—	2	8	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₄⋊C₄×C₄₇	C₄⋊C₄	1	4	yes	—
87	C₄⋊C₄×C₄₇	C₄⋊C₄	1	4	yes	—
88	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
91	C₂²⋊Dic₄₇	—	2	8	no	—
92	C₄⋊Dic₄₇	—	2	8	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
97	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
98	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
99	C₄×Dic₄₇	—	2	8	no	—
100	C₄×Dic₄₇	—	8	4	no	—
101	C₄×Dic₄₇	—	8	4	no	—
102	C₂³×C₄₇	—	2	1	no	—
103	C₂³×C₄₇	—	2	2	no	—
104	C₂³×C₄₇	—	2	1	no	—
105	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₄₇	C₄×D₄	1	4	yes	—
107	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₇	#117	1	4	yes	—
116	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
117	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
122	C₂×Dic₄₇	—	2	4	no	—
123	C₂×Dic₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	2	8	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	8	8	no	—
132	C₈×C₄₇	—	2	2	no	—
133	C₈×C₄₇	—	2	2	no	—
134	C₈×C₄₇	C₂×C₄²	1	8	yes	—
135	#20453	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
141	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
143	C₂²×C₄₇	#576	1	1	yes	characteristic
144	C₂²×C₄₇	#577	1	2	yes	—
145	C₂²×C₄₇	#578	1	1	yes	characteristic
146	C₄×C₄₇	—	2	4	no	—
147	C₄×C₄₇	—	2	4	no	—
148	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
149	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
150	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
151	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
152	Dic₄₇	—	4	4	no	—
153	Dic₄₇	—	4	8	no	—
154	Dic₄₇	—	4	4	no	—
155	C₂×#428	—	47	2	no	—
156	C₂×#405	D₄₇	1	1	yes	characteristic
157	#2274	—	47	2	no	—
158	#1776	—	47	4	no	—
159	C₄×#107	—	47	2	no	—
160	C₄×#106	—	47	2	no	—
161	#1735	—	47	1	no	—
162	C₂×#340	—	47	1	no	—
163	C₂×C₄₇	—	2	4	no	—
164	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄₇	#2912	1	1	yes	characteristic
166	C₂×C₄₇	#2913	1	1	yes	characteristic
167	C₂×C₄₇	#2946	1	2	yes	—
168	C₂×C₄₇	#2947	1	2	yes	—
169	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
170	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
171	C₂×#117	—	47	2	no	—
172	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
173	C₂²×C₄⋊C₄	—	47	2	no	—
174	#428	—	47	16	no	—
175	C₂×#107	—	47	4	no	—
176	C₂×#106	—	47	4	no	—
177	#405	Dic₄₇	1	8	yes	complemented
178	C₂×#97	C₂×D₄₇	1	2	yes	—
179	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
180	C₄×C₄⋊C₄	—	47	4	no	—
181	C₄×C₄⋊C₄	—	47	4	no	—
182	C₄×C₄⋊C₄	—	94	2	no	—
183	C₄×C₄⋊C₄	—	94	2	no	—
184	C₄×C₂²⋊C₄	—	94	4	no	—
185	#340	—	47	2	no	—
186	#340	—	47	2	no	—
187	#340	—	47	1	no	—
188	#340	—	47	1	no	—
189	#340	—	47	2	no	—
190	C₄₇	#20453	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	—	94	2	no	—
192	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂×M₁₆	C₄×D₄₇	1	8	yes	—
194	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
197	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
200	#117	—	47	8	no	—
201	#117	—	94	4	no	—
202	#117	Dic₉₄	1	4	yes	—
203	#117	C₂×Dic₄₇	1	4	yes	—
204	C₂×C₄⋊C₄	—	47	8	no	—
205	C₂×C₄⋊C₄	—	47	4	no	—
206	C₂×C₄⋊C₄	—	47	2	no	—
207	C₂×C₄⋊C₄	—	47	2	no	—
208	C₂×C₄⋊C₄	—	94	2	no	—
209	C₂×C₄⋊C₄	—	94	4	no	—
210	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
211	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	—	94	2	no	—
213	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
214	C₂×C₄²	—	94	2	no	—
215	C₂×C₄²	—	94	4	no	—
216	C₂×C₄²	—	94	4	no	—
217	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
218	#107	—	47	16	no	—
219	#106	—	47	16	no	—
220	#97	C₂×Dic₄₇	1	8	yes	complemented
221	C₄×C₈	C₂×Dic₄₇	1	8	yes	complemented
222	#95	—	94	4	no	—
223	#95	—	94	4	no	—
224	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
225	C₂²×C₄	—	94	8	no	—
226	C₂²×C₄	—	94	2	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	94	4	no	—
229	C₂²×C₄	—	376	2	no	—
230	C₂²×C₄	—	376	2	no	—
231	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
232	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
233	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
234	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
235	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
236	C₂²×C₄	#10625350	1	4	yes	—
237	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
238	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
239	M₁₆	C₄×Dic₄₇	1	16	yes	complemented
240	C₂×C₈	—	2	8	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
244	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
245	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
246	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
247	C₄⋊C₄	—	47	4	no	—
248	C₄⋊C₄	—	47	8	no	—
249	C₄⋊C₄	—	47	4	no	—
250	C₄⋊C₄	—	94	8	no	—
251	C₄⋊C₄	—	94	4	no	—
252	C₄⋊C₄	—	94	2	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₄⋊C₄	C₄⋊Dic₄₇	1	4	yes	—
255	C₄⋊C₄	C₄⋊Dic₄₇	1	4	yes	—
256	C₂²⋊C₄	—	94	8	no	—
257	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
258	C₄²	—	94	8	no	—
259	C₄²	—	376	4	no	—
260	C₄²	—	376	4	no	—
261	C₄²	C₂×Dic₉₄	1	4	yes	—
262	C₄²	C₂²×Dic₄₇	1	4	yes	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	1	no	—
266	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
267	C₂³	C₄×D₁₈₈	1	4	yes	—
268	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	94	4	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	94	8	no	—
275	C₂×C₄	—	376	8	no	—
276	C₂×C₄	—	188	2	no	—
277	C₂×C₄	—	188	4	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	—	188	4	no	—
280	C₂×C₄	—	188	2	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
283	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
284	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
285	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
286	C₂×C₄	#13093983	1	4	yes	—
287	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
288	C₂×C₄	C₄×Dic₉₄	1	8	yes	—
289	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
290	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
292	C₂×C₄	C₂×#10625350	1	2	yes	—
293	C₈	—	2	2	no	—
294	C₈	—	2	2	no	—
295	C₈	C₂×C₄×Dic₄₇	1	8	yes	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	C₄×C₄⋊Dic₄₇	1	4	yes	—
301	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
302	C₂²	C₂×C₄×D₁₈₈	1	2	yes	—
303	C₂²	#15689928	1	1	yes	characteristic
304	C₂²	#15689935	1	2	yes	—
305	C₂²	#15689940	1	1	yes	characteristic
306	C₄	—	2	4	no	—
307	C₄	—	2	4	no	—
308	C₄	—	188	8	no	—
309	C₄	—	188	4	no	—
310	C₄	—	188	4	no	—
311	C₄	C₄×C₄⋊Dic₄₇	1	4	yes	—
312	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
313	C₄	C₂×#13093983	1	1	yes	characteristic
314	C₄	C₂×C₄×Dic₉₄	1	2	yes	—
315	C₂	—	2	4	no	—
316	C₂	C₂×C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
317	C₂	#20251983	1	1	yes	characteristic
318	C₂	#20251988	1	1	yes	characteristic
319	C₂	#20252222	1	2	yes	—
320	C₂	#20252233	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#20453
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #20243285	C₂	semidirect
4. C₄×#13093958	C₂	semidirect
5. C₄×#13093956	C₂	semidirect
11. #15689165	C₄	semidirect
27. C₄₇×#405	C₄	semidirect
43. #13093958	C₂×C₄	semidirect
44. #13093956	C₂×C₄	semidirect
61. #97×C₄₇	C₂×C₄	semidirect
62. C₄×C₈×C₄₇	C₂×C₄	semidirect
78. M₁₆×C₄₇	C₄²	semidirect
177. #405	Dic₄₇	semidirect
190. C₄₇	#20453	semidirect
220. #97	C₂×Dic₄₇	semidirect
221. C₄×C₈	C₂×Dic₄₇	semidirect
239. M₁₆	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689165	2
#20245744	2
#20243285	2
C₄×#13093958	2
C₄×#13093956	2
#20243105	2
C₂×#15688886	2
C₂×C₄₇×#405	2
#20453	47