Subgroups — #26216724

Order 12032.

1070 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689163	C₂	1	2	yes	maximal
2	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689078	C₂	1	1	yes	maximal, characteristic
4	#20246012	C₂	1	2	yes	maximal
5	#20243555	C₂	1	4	yes	maximal, complemented
6	#20243551	C₂	1	2	yes	maximal, complemented
7	#20243546	C₂	1	2	yes	maximal, complemented
8	C₂²×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#13093983	C₂²	1	2	yes	—
11	C₂×#13093983	C₂²	1	1	yes	characteristic
12	#15689163	—	2	8	no	—
13	#15689161	—	2	4	no	—
14	C₂×#13093958	C₂²	1	1	yes	characteristic
15	C₂×#13093958	C₂²	1	1	yes	characteristic
16	C₂×#13093957	C₂²	1	4	yes	—
17	C₂×#13093956	C₂²	1	2	yes	—
18	C₂×#13093955	C₂²	1	4	yes	—
19	#15689078	C₄	1	8	yes	complemented
20	C₂×#13093934	C₂²	1	2	yes	—
21	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
22	#15689019	—	2	4	no	—
23	#15688996	—	2	4	no	—
24	#15688992	C₂²	1	4	yes	—
25	#15688988	C₂²	1	4	yes	—
26	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₂×#107×C₄₇	—	2	4	no	—
29	C₂×#107×C₄₇	C₂²	1	2	yes	—
30	C₂×#107×C₄₇	C₂²	1	2	yes	—
31	C₂×#10625329	C₂×C₄	1	4	yes	—
32	C₂×#10625329	D₄	1	4	yes	—
33	C₂×#10625329	Q₈	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	#13093983	—	2	4	no	—
36	#13093983	—	2	4	no	—
37	#13093983	C₂×C₄	1	4	yes	—
38	#13093983	Q₈	1	4	yes	—
39	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×Dic₄₇⋊C₄	—	2	8	no	—
43	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
44	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₇	—	2	2	no	—
46	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
47	#13093958	—	2	2	no	—
48	#13093958	—	2	2	no	—
49	#13093957	—	2	8	no	—
50	#13093956	—	2	4	no	—
51	#13093955	—	2	8	no	—
52	#13093934	C₂×C₄	1	8	yes	complemented
53	C₈×Dic₄₇	C₂×C₄	1	8	yes	complemented
54	C₄₇⋊#95	—	2	8	no	—
55	C₂³×C₄×C₄₇	—	2	2	no	—
56	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
58	C₂²×C₈×C₄₇	D₄	1	4	yes	—
59	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
64	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
66	#107×C₄₇	—	2	4	no	—
67	#107×C₄₇	—	2	4	no	—
68	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	4	4	no	—
71	C₂²×C₄×C₄₇	—	8	4	no	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₇	—	2	2	no	—
81	C₂²×Dic₄₇	—	2	2	no	—
82	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
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	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₄⋊C₄×C₄₇	—	2	4	no	—
92	C₄⋊C₄×C₄₇	—	4	4	no	—
93	C₂²⋊Dic₄₇	—	4	4	no	—
94	C₂²⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
100	Dic₄₇⋊C₄	—	8	8	no	—
101	C₄×Dic₄₇	—	4	4	no	—
102	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
103	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
104	#10625329	C₄⋊C₄	1	16	yes	complemented
105	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
107	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
108	C₂³×C₄₇	—	2	2	no	—
109	C₂³×C₄₇	—	2	2	no	—
110	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂³×C₄₇	#127	1	2	yes	—
112	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
113	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	8	no	—
116	C₂×C₄×C₄₇	—	8	8	no	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₄₇	#127	1	2	yes	—
121	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
122	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	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×Q₈	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	—	2	4	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
137	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
138	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
139	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
140	#20803	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	#384	1	4	yes	—
146	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₄₇	#576	1	2	yes	—
150	C₂²×C₄₇	#578	1	2	yes	—
151	C₄×C₄₇	—	8	4	no	—
152	C₄×C₄₇	#384	1	4	yes	—
153	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	Dic₄₇	—	2	8	no	—
158	Dic₄₇	—	8	2	no	—
159	Dic₄₇	—	8	2	no	—
160	C₂×#428	—	47	2	no	—
161	C₂×#427	—	47	1	no	—
162	C₂²×#107	D₄₇	1	1	yes	characteristic
163	C₂×#405	—	47	1	no	—
164	#2317	—	47	2	no	—
165	#1847	—	47	4	no	—
166	#1846	—	47	2	no	—
167	#1845	—	47	2	no	—
168	C₂×C₄₇	—	2	4	no	—
169	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
170	C₂×C₄₇	#2913	1	2	yes	—
171	C₂×C₄₇	#3155	1	1	yes	characteristic
172	C₂×C₄₇	#3156	1	1	yes	characteristic
173	C₂×C₄₇	#3164	1	2	yes	—
174	C₂²×M₁₆	—	47	1	no	—
175	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
176	C₂×#117	—	47	1	no	—
177	C₂×#117	—	47	2	no	—
178	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
179	#428	—	94	8	no	—
180	#427	—	94	4	no	—
181	C₂×#107	—	2	4	no	—
182	C₂×#107	—	47	1	no	—
183	C₂×#107	—	47	1	no	—
184	C₂×#107	—	47	4	no	—
185	C₂×#107	C₂×D₄₇	1	2	yes	—
186	C₂×#107	C₂×D₄₇	1	2	yes	—
187	C₂×#106	—	47	2	no	—
188	C₂×#106	—	47	4	no	—
189	#405	—	47	8	no	—
190	C₂×#97	—	47	2	no	—
191	C₂×C₄×C₈	—	47	2	no	—
192	#389	—	94	4	no	—
193	#385	—	94	4	no	—
194	#384	—	47	4	no	—
195	#382	—	47	4	no	—
196	C₄₇	#20803	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₄	—	2	2	no	—
198	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
199	C₂×M₁₆	—	47	4	no	—
200	C₂×M₁₆	—	47	4	no	—
201	C₂×M₁₆	—	47	4	no	—
202	C₂²×C₈	—	47	2	no	—
203	C₂²×C₈	Dic₉₄	1	4	yes	—
204	C₂²×C₈	C₄×D₄₇	1	4	yes	—
205	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
206	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
208	#117	—	47	4	no	—
209	#117	—	47	4	no	—
210	#117	—	94	4	no	—
211	#117	—	94	4	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	47	2	no	—
216	C₂×C₄⋊C₄	—	47	2	no	—
217	C₂×C₄⋊C₄	—	47	2	no	—
218	C₂×C₄⋊C₄	—	94	8	no	—
219	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
220	C₂×C₂²⋊C₄	—	47	2	no	—
221	C₂×C₂²⋊C₄	—	94	2	no	—
222	C₂×C₄²	—	47	2	no	—
223	C₂×C₄²	—	94	2	no	—
224	#107	—	2	4	no	—
225	#107	—	2	4	no	—
226	#107	—	94	8	no	—
227	#107	—	94	2	no	—
228	#107	—	94	2	no	—
229	#106	—	94	8	no	—
230	#106	—	94	4	no	—
231	#97	—	47	8	no	—
232	C₄×C₈	—	47	8	no	—
233	#95	—	94	8	no	—
234	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	8	4	no	—
238	C₂²×C₄	—	47	4	no	—
239	C₂²×C₄	—	94	2	no	—
240	C₂²×C₄	—	94	2	no	—
241	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
242	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
243	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
244	C₂²×C₄	D₄×D₄₇	1	2	yes	—
245	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
246	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
247	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
248	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
249	M₁₆	—	47	16	no	—
250	C₂×C₈	—	47	4	no	—
251	C₂×C₈	—	47	4	no	—
252	C₂×C₈	—	47	4	no	—
253	C₂×C₈	Dic₄₇⋊C₄	1	16	yes	complemented
254	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
255	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
258	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
259	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	47	8	no	—
264	C₄⋊C₄	—	94	2	no	—
265	C₄⋊C₄	—	94	2	no	—
266	C₄⋊C₄	—	94	2	no	—
267	C₄⋊C₄	—	94	2	no	—
268	C₄⋊C₄	—	376	8	no	—
269	C₂²⋊C₄	—	47	8	no	—
270	C₂²⋊C₄	—	188	4	no	—
271	C₄²	—	47	4	no	—
272	C₄²	—	47	4	no	—
273	C₄²	—	188	4	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
277	C₂³	#13094020	1	2	yes	—
278	C₂³	#13094023	1	2	yes	—
279	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	8	no	—
282	C₂×C₄	—	8	8	no	—
283	C₂×C₄	—	47	8	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	94	4	no	—
286	C₂×C₄	—	188	2	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	—	188	4	no	—
289	C₂×C₄	—	188	4	no	—
290	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
291	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
292	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
293	C₂×C₄	#13094020	1	2	yes	—
294	C₂×C₄	#13094022	1	4	yes	—
295	C₂×C₄	#13094023	1	2	yes	—
296	C₂×C₄	#13094023	1	4	yes	—
297	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
302	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
303	C₈	—	47	8	no	—
304	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
305	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	#15688992	1	4	yes	—
311	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
312	C₂²	C₂×#13094020	1	1	yes	characteristic
313	C₂²	C₂×#13094023	1	1	yes	characteristic
314	C₂²	#15689924	1	2	yes	—
315	C₂²	#15689936	1	2	yes	—
316	C₄	—	8	4	no	—
317	C₄	—	94	8	no	—
318	C₄	—	376	2	no	—
319	C₄	—	376	2	no	—
320	C₄	#15688992	1	4	yes	—
321	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
322	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
323	C₄	C₂×#13094022	1	1	yes	characteristic
324	C₄	C₂×#13094023	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	C₂×#15688992	1	1	yes	characteristic
327	C₂	#20251984	1	2	yes	—
328	C₂	#20253869	1	1	yes	characteristic
329	C₂	#20253877	1	1	yes	characteristic
330	C₂	#20253924	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#107×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#107	D₄₇
hypocenter	C₄₇	#20803
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
5. #20243555	C₂	semidirect
6. #20243551	C₂	semidirect
7. #20243546	C₂	semidirect
19. #15689078	C₄	semidirect
52. #13093934	C₂×C₄	semidirect
53. C₈×Dic₄₇	C₂×C₄	semidirect
83. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
104. #10625329	C₄⋊C₄	semidirect
196. C₄₇	#20803	semidirect
253. C₂×C₈	Dic₄₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689163	2
C₂×#15689161	2
C₂×#15689078	2
#20246012	2
#20243555	2
#20243551	2
#20243546	2
C₂²×#107×C₄₇	2
#20803	47