Subgroups — #26216745

Order 12032.

954 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×#15689171	C₂	1	2	yes	maximal
2	C₂²×#13093960	C₂	1	1	yes	maximal, characteristic
3	#20251951	C₂	1	2	yes	maximal
4	C₂×#15689078	C₂	1	1	yes	maximal, characteristic
5	#20243561	C₂	1	4	yes	maximal, complemented
6	#20243558	C₂	1	4	yes	maximal, complemented
7	C₂²×#108×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#13093983	C₂²	1	1	yes	characteristic
11	#15689171	—	2	8	no	—
12	C₂×#13093960	—	2	4	no	—
13	C₂×#13093960	C₂²	1	2	yes	—
14	C₂×#13093960	C₂²	1	2	yes	—
15	C₂×#13093959	C₂²	1	8	yes	—
16	#15689151	—	2	4	no	—
17	#15689149	—	2	4	no	—
18	#15689146	C₂²	1	8	yes	—
19	#15689078	C₄	1	4	yes	—
20	#15689078	C₄	1	4	yes	—
21	C₂×#13093934	C₂²	1	2	yes	—
22	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
23	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
24	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#108×C₄₇	—	2	4	no	—
26	C₂×#108×C₄₇	C₂²	1	2	yes	—
27	C₂×#108×C₄₇	C₂²	1	2	yes	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	4	no	—
30	C₂×#10625329	—	2	4	no	—
31	C₂×#10625329	C₂×C₄	1	2	yes	—
32	C₂×#10625329	C₂×C₄	1	2	yes	—
33	C₂×#10625329	D₄	1	4	yes	—
34	C₂×#10625329	Q₈	1	4	yes	—
35	C₂×#10625329	C₂³	1	4	yes	—
36	C₂²×C₄₇⋊C₈	—	2	2	no	—
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	#13093983	C₂×C₄	1	2	yes	—
39	#13093983	C₂×C₄	1	2	yes	—
40	#13093983	D₄	1	2	yes	—
41	#13093983	D₄	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	Q₈	1	2	yes	—
43	C₂×C₂²⋊Dic₄₇	Q₈	1	2	yes	—
44	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
45	#13093960	—	2	4	no	—
46	#13093960	—	2	4	no	—
47	#13093959	—	2	16	no	—
48	Dic₄₇⋊C₈	—	2	16	no	—
49	#13093934	C₂×C₄	1	8	yes	—
50	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
51	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₄₇	—	2	2	no	—
53	C₂×M₁₆×C₄₇	—	2	4	no	—
54	C₂×M₁₆×C₄₇	—	2	4	no	—
55	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
56	C₂²×C₈×C₄₇	—	2	2	no	—
57	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₄₇	D₄	1	4	yes	—
60	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	#108×C₄₇	—	2	4	no	—
64	#108×C₄₇	—	2	4	no	—
65	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
77	M₁₆×C₄₇	—	2	4	no	—
78	M₁₆×C₄₇	—	2	4	no	—
79	M₁₆×C₄₇	—	4	4	no	—
80	C₂×C₈×C₄₇	—	2	4	no	—
81	C₂×C₈×C₄₇	—	4	4	no	—
82	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
83	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
90	C₂²⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
91	C₂²⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
92	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
93	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
94	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
95	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
96	#10625329	—	2	4	no	—
97	#10625329	—	2	4	no	—
98	#10625329	—	4	4	no	—
99	#10625329	C₄⋊C₄	1	8	yes	—
100	#10625329	C₄⋊C₄	1	8	yes	—
101	C₂×C₄₇⋊C₈	—	2	4	no	—
102	C₂×C₄₇⋊C₈	—	4	4	no	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
104	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
105	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
106	C₂³×C₄₇	—	4	2	no	—
107	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
110	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
111	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	#127	1	2	yes	—
119	C₂×C₄×C₄₇	#127	1	2	yes	—
120	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
132	C₈×C₄₇	—	4	4	no	—
133	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
134	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
135	C₄₇⋊C₈	—	4	4	no	—
136	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
137	#20807	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	4	2	no	—
142	C₂²×C₄₇	—	4	2	no	—
143	C₂²×C₄₇	#384	1	2	yes	—
144	C₂²×C₄₇	#384	1	2	yes	—
145	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₄×C₄₇	#384	1	2	yes	—
149	C₄×C₄₇	#384	1	2	yes	—
150	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
153	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	Dic₄₇	—	2	8	no	—
155	C₂×#430	—	47	2	no	—
156	C₂²×#108	—	47	1	no	—
157	C₂²×#108	D₄₇	1	1	yes	characteristic
158	#2905	—	47	2	no	—
159	C₂×#405	—	47	1	no	—
160	#1849	—	47	4	no	—
161	#1848	—	47	4	no	—
162	C₂×C₄₇	—	2	1	no	—
163	C₂×C₄₇	—	2	1	no	—
164	C₂×C₄₇	—	2	2	no	—
165	C₂×C₄₇	—	2	2	no	—
166	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
167	C₂×C₄₇	#2916	1	2	yes	—
168	C₂²×M₁₆	—	47	1	no	—
169	C₂²×M₁₆	—	47	2	no	—
170	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
171	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
172	C₂×#117	—	47	1	no	—
173	#430	—	94	8	no	—
174	C₂×#108	—	2	4	no	—
175	C₂×#108	—	47	8	no	—
176	C₂×#108	—	47	2	no	—
177	C₂×#108	—	47	2	no	—
178	C₂×#108	—	94	4	no	—
179	C₂×#108	C₂×D₄₇	1	2	yes	—
180	C₂×#108	C₂×D₄₇	1	2	yes	—
181	#424	—	94	4	no	—
182	#424	—	94	4	no	—
183	#423	—	47	8	no	—
184	#405	—	47	4	no	—
185	#405	—	47	4	no	—
186	C₂×#97	—	47	2	no	—
187	C₂×C₄×C₈	—	47	2	no	—
188	C₄₇	#20807	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	—	47	2	no	—
194	C₂×M₁₆	—	47	2	no	—
195	C₂×M₁₆	—	47	4	no	—
196	C₂×M₁₆	—	47	4	no	—
197	C₂×M₁₆	—	47	4	no	—
198	C₂×M₁₆	—	94	2	no	—
199	C₂×M₁₆	—	94	4	no	—
200	C₂×M₁₆	—	94	4	no	—
201	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	47	2	no	—
204	C₂²×C₈	—	94	2	no	—
205	C₂²×C₈	Dic₉₄	1	4	yes	—
206	C₂²×C₈	C₄×D₄₇	1	2	yes	—
207	C₂²×C₈	C₄×D₄₇	1	2	yes	—
208	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
209	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
211	#117	—	47	2	no	—
212	#117	—	47	2	no	—
213	#117	—	47	2	no	—
214	#117	—	47	2	no	—
215	C₂×C₄⋊C₄	—	47	2	no	—
216	C₂×C₂²⋊C₄	—	47	2	no	—
217	C₂×C₄²	—	47	2	no	—
218	#108	—	2	4	no	—
219	#108	—	2	4	no	—
220	#108	—	94	16	no	—
221	#108	—	94	4	no	—
222	#108	—	94	4	no	—
223	C₄⋊C₈	—	94	16	no	—
224	#97	—	47	8	no	—
225	C₄×C₈	—	47	8	no	—
226	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	—	47	4	no	—
228	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
229	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₄₇	1	2	yes	—
232	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
235	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
236	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
237	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	4	no	—
241	M₁₆	—	47	8	no	—
242	M₁₆	—	47	8	no	—
243	M₁₆	—	94	4	no	—
244	M₁₆	—	94	4	no	—
245	M₁₆	—	188	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	4	no	—
248	C₂×C₈	—	47	4	no	—
249	C₂×C₈	—	47	4	no	—
250	C₂×C₈	—	47	4	no	—
251	C₂×C₈	—	94	4	no	—
252	C₂×C₈	—	188	4	no	—
253	C₂×C₈	Dic₄₇⋊C₄	1	8	yes	—
254	C₂×C₈	Dic₄₇⋊C₄	1	8	yes	—
255	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
256	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
258	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
259	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
260	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
261	C₄⋊C₄	—	47	4	no	—
262	C₄⋊C₄	—	47	4	no	—
263	C₂²⋊C₄	—	47	4	no	—
264	C₂²⋊C₄	—	47	4	no	—
265	C₄²	—	47	4	no	—
266	C₄²	—	47	4	no	—
267	C₂³	—	4	2	no	—
268	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
269	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
270	C₂³	#13094022	1	2	yes	—
271	C₂³	#13094023	1	2	yes	—
272	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
273	C₂×C₄	—	47	8	no	—
274	C₂×C₄	—	188	4	no	—
275	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
276	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
277	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
278	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
279	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
280	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
281	C₂×C₄	#13094020	1	2	yes	—
282	C₂×C₄	#13094020	1	2	yes	—
283	C₂×C₄	#13094022	1	2	yes	—
284	C₂×C₄	#13094023	1	2	yes	—
285	C₂×C₄	#13094023	1	2	yes	—
286	C₂×C₄	#13094023	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
293	C₈	—	4	4	no	—
294	C₈	—	47	8	no	—
295	C₈	—	188	4	no	—
296	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
297	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#15688992	1	2	yes	—
304	C₂²	#15688992	1	2	yes	—
305	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
306	C₂²	C₂×#13094022	1	1	yes	characteristic
307	C₂²	C₂×#13094023	1	1	yes	characteristic
308	C₄	—	94	8	no	—
309	C₄	#15688992	1	2	yes	—
310	C₄	#15688992	1	2	yes	—
311	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
312	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
313	C₄	C₂×#13094020	1	1	yes	characteristic
314	C₄	C₂×#13094023	1	1	yes	characteristic
315	C₂	—	2	2	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	C₂×#15688992	1	1	yes	characteristic
320	C₂	#20251996	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#108×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#108	D₄₇
hypocenter	C₄₇	#20807
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
5. #20243561	C₂	semidirect
6. #20243558	C₂	semidirect
188. C₄₇	#20807	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689171	2
C₂²×#13093960	2
#20251951	2
C₂×#15689078	2
#20243561	2
#20243558	2
C₂²×#108×C₄₇	2
#20807	47