Subgroups — #26213623

Order 12032.

1470 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689080	C₂	1	2	yes	maximal
2	C₂×#15689077	C₂	1	2	yes	maximal
3	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
4	C₂×#97×D₄₇	C₂	1	1	yes	maximal, characteristic
5	#20245727	C₂	1	1	yes	maximal, characteristic
6	#20243076	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243075	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243074	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13093934	C₂	1	2	yes	maximal, complemented
10	#97×Dic₄₇	C₂	1	2	yes	maximal, complemented
11	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#10625324	C₂²	1	1	yes	characteristic
13	C₂²×#10625324	C₂²	1	1	yes	characteristic
14	C₂²×C₈×D₄₇	C₂²	1	2	yes	—
15	C₂×#13093981	C₂²	1	2	yes	—
16	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
17	#15689080	C₄	1	16	yes	complemented
18	#15689077	C₄	1	16	yes	complemented
19	C₄×#10625324	C₄	1	8	yes	complemented
20	C₂×#13093935	C₂²	1	1	yes	characteristic
21	C₂×#13093935	C₂²	1	2	yes	—
22	C₂×#13093934	C₂²	1	1	yes	characteristic
23	C₂×#13093934	C₂²	1	2	yes	—
24	C₂×#13093934	C₂²	1	1	yes	characteristic
25	C₂×#13093934	C₂²	1	2	yes	—
26	C₂×#13093934	C₂²	1	2	yes	—
27	#97×D₄₇	C₄	1	8	yes	complemented
28	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
30	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
31	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
32	#15688974	C₄	1	4	yes	—
33	#15688970	C₄	1	4	yes	—
34	#15688969	C₂²	1	1	yes	characteristic
35	C₄²×Dic₄₇	C₂²	1	2	yes	—
36	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×#97×C₄₇	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
40	C₄₇×#376	C₂²	1	1	yes	characteristic
41	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
42	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
43	C₂×#10625324	C₂×C₄	1	4	yes	—
44	C₂×#10625324	C₂×C₄	1	4	yes	—
45	C₂×#10625324	C₂×C₄	1	8	yes	—
46	C₂×#10625324	C₂×C₄	1	8	yes	—
47	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
49	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
51	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
52	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
53	#13093981	C₂×C₄	1	8	yes	—
54	#13093981	C₂×C₄	1	8	yes	—
55	C₂×Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
56	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	4	yes	—
57	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
58	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
59	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
61	C₄²×D₄₇	C₂×C₄	1	4	yes	—
62	C₄²×D₄₇	C₂×C₄	1	4	yes	—
63	#13093935	C₂×C₄	1	4	yes	—
64	#13093935	C₂×C₄	1	8	yes	complemented
65	#13093934	C₂×C₄	1	8	yes	—
66	#13093934	C₂×C₄	1	4	yes	—
67	#13093934	C₂×C₄	1	4	yes	—
68	#13093934	C₂×C₄	1	8	yes	complemented
69	#13093934	C₂×C₄	1	8	yes	complemented
70	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
71	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
72	C₈×Dic₄₇	C₂×C₄	1	8	yes	complemented
73	C₈×Dic₄₇	C₂×C₄	1	8	yes	complemented
74	C₄₇⋊#95	C₂×C₄	1	4	yes	—
75	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
79	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	#97×C₄₇	C₂×C₄	1	4	yes	—
81	#97×C₄₇	C₂×C₄	1	8	yes	complemented
82	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
83	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
84	#95×C₄₇	C₂×C₄	1	4	yes	—
85	C₂³×D₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
90	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
91	C₂²×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×D₄₇	—	4	4	no	—
93	C₂×C₄×D₄₇	C₄²	1	8	yes	—
94	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
97	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
104	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
105	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
107	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
109	(C₂×D₄₇)⋊C₄	C₄²	1	16	yes	—
110	Dic₄₇⋊C₄	C₄²	1	16	yes	—
111	C₄×Dic₄₇	—	4	8	no	—
112	C₄×Dic₄₇	C₄²	1	8	yes	—
113	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
114	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
115	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
116	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
117	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
118	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
119	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
120	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
121	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
122	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
123	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
124	#10625324	C₄²	1	16	yes	complemented
125	#10625324	C₄²	1	16	yes	complemented
126	C₈×D₄₇	C₄²	1	32	yes	complemented
127	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂²×D₄₇	—	2	4	no	—
129	C₂²×D₄₇	C₂×C₄²	1	4	yes	—
130	C₂²×D₄₇	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₄₇	—	2	8	no	—
132	C₂×C₄×C₄₇	—	2	4	no	—
133	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
134	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
135	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
138	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
139	C₂×Dic₄₇	—	2	4	no	—
140	C₂×Dic₄₇	—	2	8	no	—
141	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
142	C₂×Dic₄₇	C₂×C₄²	1	8	yes	—
143	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
144	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
145	C₄×D₄₇	—	4	8	no	—
146	C₄×D₄₇	C₂×C₄²	1	8	yes	—
147	C₄×D₄₇	C₂×C₄²	1	4	yes	—
148	C₄×D₄₇	C₂×C₄²	1	4	yes	—
149	C₈×C₄₇	C₂×C₄²	1	8	yes	—
150	C₈×C₄₇	C₂×C₄²	1	4	yes	—
151	C₈×C₄₇	C₂×C₄²	1	4	yes	—
152	C₄₇⋊C₈	C₂×C₄²	1	8	yes	—
153	C₄₇⋊C₈	C₂×C₄²	1	4	yes	—
154	C₄₇⋊C₈	C₂×C₄²	1	4	yes	—
155	#20331	—	47	1	no	maximal, Sylow
156	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
157	C₂²×C₄₇	C₂²×C₄²	1	2	yes	—
158	C₂²×C₄₇	#568	1	4	yes	—
159	C₂×D₄₇	—	2	4	no	—
160	C₂×D₄₇	—	2	8	no	—
161	C₂×D₄₇	C₄³	1	4	yes	—
162	C₄×C₄₇	—	2	4	no	—
163	C₄×C₄₇	—	2	8	no	—
164	C₄×C₄₇	C₂²×C₄²	1	2	yes	—
165	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
166	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
167	Dic₄₇	—	2	4	no	—
168	Dic₄₇	—	2	8	no	—
169	Dic₄₇	C₄³	1	4	yes	—
170	C₂×#405	—	47	2	no	—
171	C₂×#405	—	47	2	no	—
172	C₂×C₄×M₁₆	—	47	1	no	—
173	C₂²×#97	—	47	1	no	—
174	C₄×#117	—	47	1	no	—
175	#1726	—	47	1	no	—
176	#1726	—	47	1	no	—
177	#1726	—	47	1	no	—
178	#1726	D₄₇	1	1	yes	characteristic, complemented
179	C₄×#97	—	47	2	no	—
180	C₄×#97	—	47	2	no	—
181	C₂×C₄₇	C₂×C₄³	1	1	yes	characteristic
182	C₂×C₄₇	#2873	1	2	yes	—
183	C₂×C₄₇	#2875	1	4	yes	—
184	D₄₇	—	2	4	no	—
185	C₂²×M₁₆	—	47	1	no	—
186	C₂²×M₁₆	—	47	1	no	—
187	C₂³×C₈	—	47	2	no	—
188	C₂×#117	—	47	2	no	—
189	C₂²×C₄²	—	47	1	no	—
190	#405	—	47	16	no	—
191	#405	—	47	16	no	—
192	C₄×M₁₆	—	47	8	no	—
193	C₂×#97	—	47	2	no	—
194	C₂×#97	—	47	1	no	—
195	C₂×#97	—	47	1	no	—
196	C₂×#97	—	47	2	no	—
197	C₂×#97	—	47	2	no	—
198	C₂×#97	—	47	1	no	—
199	C₂×#97	—	47	2	no	—
200	C₂×#97	—	47	8	no	—
201	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
202	C₂×#97	C₂×D₄₇	1	2	yes	—
203	C₂×C₄×C₈	—	47	2	no	—
204	C₂×C₄×C₈	—	47	2	no	—
205	C₂×C₄×C₈	—	47	1	no	—
206	C₂×C₄×C₈	—	47	2	no	—
207	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
208	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
209	C₄×C₄⋊C₄	—	47	4	no	—
210	C₄×C₂²⋊C₄	—	47	4	no	—
211	#376	—	47	1	no	—
212	#376	C₂×D₄₇	1	1	yes	characteristic
213	C₄³	—	47	2	no	—
214	C₄₇	#20331	1	1	yes	characteristic, complemented, Sylow
215	C₂³×C₄	—	47	1	no	—
216	C₂³×C₄	—	47	2	no	—
217	C₂×M₁₆	—	47	4	no	—
218	C₂×M₁₆	—	47	4	no	—
219	C₂×M₁₆	—	47	8	no	—
220	C₂×M₁₆	—	47	8	no	—
221	C₂²×C₈	—	47	2	no	—
222	C₂²×C₈	—	47	1	no	—
223	C₂²×C₈	—	47	1	no	—
224	C₂²×C₈	—	47	8	no	—
225	C₂²×C₈	—	47	8	no	—
226	C₂²×C₈	—	47	8	no	—
227	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
228	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
229	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
230	#117	—	47	8	no	—
231	#117	—	47	8	no	—
232	C₂×C₄⋊C₄	—	47	4	no	—
233	C₂×C₂²⋊C₄	—	47	4	no	—
234	C₂×C₄²	—	47	2	no	—
235	C₂×C₄²	—	47	2	no	—
236	C₂×C₄²	—	47	1	no	—
237	C₂×C₄²	—	47	4	no	—
238	C₂×C₄²	—	47	4	no	—
239	C₂×C₄²	—	47	8	no	—
240	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
241	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
242	#97	—	47	8	no	—
243	#97	—	47	4	no	—
244	#97	—	47	4	no	—
245	#97	—	47	8	no	—
246	#97	—	47	8	no	—
247	#97	—	47	4	no	—
248	#97	—	47	8	no	—
249	#97	C₄×D₄₇	1	4	yes	—
250	#97	C₄×D₄₇	1	8	yes	complemented
251	C₄×C₈	—	47	8	no	—
252	C₄×C₈	—	47	8	no	—
253	C₄×C₈	—	47	4	no	—
254	C₄×C₈	—	47	8	no	—
255	C₄×C₈	C₄×D₄₇	1	4	yes	—
256	C₄×C₈	C₄×D₄₇	1	8	yes	complemented
257	#95	—	47	4	no	—
258	#95	C₄×D₄₇	1	4	yes	—
259	C₂⁴	—	47	1	no	—
260	C₂²×C₄	—	47	1	no	—
261	C₂²×C₄	—	47	2	no	—
262	C₂²×C₄	—	47	2	no	—
263	C₂²×C₄	—	47	4	no	—
264	C₂²×C₄	—	47	4	no	—
265	C₂²×C₄	—	47	4	no	—
266	C₂²×C₄	—	47	2	no	—
267	C₂²×C₄	—	47	8	no	—
268	C₂²×C₄	—	188	4	no	—
269	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
270	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
271	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
272	M₁₆	—	47	16	no	—
273	M₁₆	—	47	16	no	—
274	C₂×C₈	—	47	4	no	—
275	C₂×C₈	—	47	4	no	—
276	C₂×C₈	—	47	2	no	—
277	C₂×C₈	—	47	2	no	—
278	C₂×C₈	—	47	4	no	—
279	C₂×C₈	—	47	4	no	—
280	C₂×C₈	—	47	4	no	—
281	C₂×C₈	—	47	32	no	—
282	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
283	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
284	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
285	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
286	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
287	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
288	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
289	C₄⋊C₄	—	47	16	no	—
290	C₂²⋊C₄	—	47	16	no	—
291	C₄²	—	47	8	no	—
292	C₄²	—	47	4	no	—
293	C₄²	—	47	4	no	—
294	C₄²	—	47	2	no	—
295	C₄²	—	47	2	no	—
296	C₄²	—	188	8	no	—
297	C₄²	C₂×C₄×D₄₇	1	2	yes	—
298	C₄²	C₂×C₄×D₄₇	1	4	yes	—
299	C₄²	C₂×C₄×D₄₇	1	4	yes	—
300	C₄²	C₂×C₄×D₄₇	1	2	yes	—
301	C₂³	—	47	4	no	—
302	C₂³	—	47	2	no	—
303	C₂³	—	94	4	no	—
304	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
305	C₂×C₄	—	2	8	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	47	2	no	—
308	C₂×C₄	—	47	4	no	—
309	C₂×C₄	—	47	8	no	—
310	C₂×C₄	—	47	4	no	—
311	C₂×C₄	—	47	4	no	—
312	C₂×C₄	—	47	4	no	—
313	C₂×C₄	—	47	8	no	—
314	C₂×C₄	—	94	4	no	—
315	C₂×C₄	—	94	8	no	—
316	C₂×C₄	—	188	8	no	—
317	C₂×C₄	C₄²×D₄₇	1	4	yes	—
318	C₂×C₄	C₄²×D₄₇	1	8	yes	—
319	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
321	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
322	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
323	C₈	—	47	8	no	—
324	C₈	—	47	4	no	—
325	C₈	—	47	4	no	—
326	C₈	C₄²×D₄₇	1	8	yes	—
327	C₈	C₄²×D₄₇	1	4	yes	—
328	C₈	C₄²×D₄₇	1	4	yes	—
329	C₂²	—	47	4	no	—
330	C₂²	—	94	4	no	—
331	C₂²	—	94	8	no	—
332	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
333	C₂²	C₂×C₄²×D₄₇	1	2	yes	—
334	C₂²	#15689888	1	4	yes	—
335	C₄	—	2	4	no	—
336	C₄	—	2	8	no	—
337	C₄	—	47	4	no	—
338	C₄	—	94	4	no	—
339	C₄	—	94	8	no	—
340	C₄	C₂×C₄²×D₄₇	1	2	yes	—
341	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
342	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
343	C₂	—	94	4	no	—
344	C₂	C₄³×D₄₇	1	1	yes	characteristic
345	C₂	#20251804	1	2	yes	—
346	C₂	#20251815	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂×C₄³
Fitting	C₄₇×#1726	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂³×D₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#20331
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1726

Complements

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

N	H ≅ G/N	type
6. #20243076	C₂	semidirect
7. #20243075	C₂	semidirect
8. #20243074	C₂	semidirect
9. C₄×#13093934	C₂	semidirect
10. #97×Dic₄₇	C₂	semidirect
11. C₄₇×#1726	C₂	semidirect
17. #15689080	C₄	semidirect
18. #15689077	C₄	semidirect
19. C₄×#10625324	C₄	semidirect
27. #97×D₄₇	C₄	semidirect
64. #13093935	C₂×C₄	semidirect
68. #13093934	C₂×C₄	semidirect
69. #13093934	C₂×C₄	semidirect
71. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
72. C₈×Dic₄₇	C₂×C₄	semidirect
73. C₈×Dic₄₇	C₂×C₄	semidirect
81. #97×C₄₇	C₂×C₄	semidirect
83. C₄×C₈×C₄₇	C₂×C₄	semidirect
124. #10625324	C₄²	semidirect
125. #10625324	C₄²	semidirect
126. C₈×D₄₇	C₄²	semidirect
178. #1726	D₄₇	semidirect
214. C₄₇	#20331	semidirect
250. #97	C₄×D₄₇	semidirect
256. C₄×C₈	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689080	2
C₂×#15689077	2
C₂×C₄×#10625324	2
C₂×#97×D₄₇	2
#20245727	2
#20243076	2
#20243075	2
#20243074	2
C₄×#13093934	2
#97×Dic₄₇	2
C₄₇×#1726	2
#20331	47