Subgroups — #26185619

Order 12032.

594 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252515	C₂	1	2	yes	maximal
2	#20252354	C₂	1	1	yes	maximal, characteristic
3	#20242174	C₂	1	2	yes	maximal, complemented
4	#20242167	C₂	1	2	yes	maximal, complemented
5	#20242161	C₂	1	2	yes	maximal, complemented
6	#20242152	C₂	1	2	yes	maximal, complemented
7	#20242137	C₂	1	2	yes	maximal, complemented
8	C₂×#15688876	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#10625350	C₂²	1	1	yes	characteristic
11	#15689253	C₄	1	1	yes	characteristic
12	#15689253	C₄	1	1	yes	characteristic
13	#15689251	—	2	2	no	—
14	#15689202	C₄	1	1	yes	characteristic
15	#15689202	C₄	1	1	yes	characteristic
16	#15689196	—	2	2	no	—
17	#15689194	C₄	1	2	yes	—
18	#15689194	C₄	1	2	yes	—
19	#15689191	—	2	2	no	—
20	(C₂²⋊D₄₇)⋊C₈	—	2	2	no	—
21	#15689179	C₂²	1	2	yes	—
22	#15689176	C₂²	1	1	yes	characteristic
23	#15689175	C₂²	1	2	yes	—
24	#15689174	C₂²	1	2	yes	—
25	#15689145	C₂²	1	2	yes	—
26	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
27	C₄⋊C₈×D₄₇	C₂²	1	2	yes	—
28	C₄×#10625324	C₂²	1	2	yes	—
29	#15688876	C₂²	1	1	yes	characteristic
30	#15688876	C₂²	1	1	yes	characteristic
31	#15688876	C₂²	1	2	yes	—
32	#15688875	C₂²	1	2	yes	—
33	#15688875	C₂²	1	2	yes	—
34	#15688873	C₂²	1	2	yes	—
35	#15688873	C₂²	1	2	yes	—
36	C₄₇×#432	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#335	C₂²	1	1	yes	characteristic
39	C₄₇×#335	C₂²	1	1	yes	characteristic
40	C₄₇×#334	C₂²	1	2	yes	—
41	C₂×#10625350	Q₈	1	1	yes	characteristic
42	C₂×#10625324	—	2	2	no	—
43	C₂×#10625324	C₂×C₄	1	2	yes	—
44	C₂×#10625324	C₂×C₄	1	2	yes	—
45	C₂²×C₄₇⋊C₈	—	2	2	no	—
46	C₂×C₈×D₄₇	—	2	2	no	—
47	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₉₄	D₄	1	1	yes	characteristic
49	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
50	C₄×D₁₈₈	D₄	1	1	yes	characteristic
51	C₄×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
52	C₄×C₂²⋊D₄₇	D₄	1	2	yes	—
53	#13093983	Q₈	1	1	yes	characteristic
54	#13093981	Q₈	1	2	yes	—
55	C₄²×D₄₇	C₂³	1	2	yes	—
56	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
57	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
58	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
59	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
60	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
61	Dic₄₇⋊C₈	—	2	2	no	—
62	Dic₄₇⋊C₈	—	2	2	no	—
63	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
64	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
65	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
66	(C₂×D₄₇)⋊C₈	—	2	2	no	—
67	(C₂×D₄₇)⋊C₈	—	2	2	no	—
68	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
69	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
70	#13093934	—	2	2	no	—
71	#13093934	C₂×C₄	1	2	yes	—
72	#13093934	C₂×C₄	1	2	yes	—
73	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
74	C₈×Dic₄₇	—	2	2	no	—
75	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
80	C₂²⋊C₈×C₄₇	—	2	2	no	—
81	#97×C₄₇	—	2	2	no	—
82	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₂²⋊D₄₇	C₄⋊C₄	1	2	yes	—
90	#10625350	—	4	2	no	—
91	C₂×D₁₈₈	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
94	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
95	C₂×Dic₉₄	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	4	1	no	—
100	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
107	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
109	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	2	yes	—
110	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	2	yes	—
111	C₄⋊Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
112	C₄⋊Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
113	Dic₄₇⋊C₄	C₄⋊C₄	1	2	yes	—
114	Dic₄₇⋊C₄	C₄⋊C₄	1	2	yes	—
115	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
116	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
117	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
118	C₂×C₄₇⋊C₈	—	2	2	no	—
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	8	2	no	—
121	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
122	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
123	#10625324	—	2	4	no	—
124	#10625324	—	4	2	no	—
125	#10625324	—	8	2	no	—
126	C₈×D₄₇	—	8	2	no	—
127	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	2	1	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	2	1	no	—
134	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
140	C₂²⋊D₄₇	—	2	4	no	—
141	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
144	D₁₈₈	—	4	1	no	—
145	C₄×D₄₇	—	2	2	no	—
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	—	2	2	no	—
149	C₄×D₄₇	—	4	2	no	—
150	Dic₉₄	—	4	1	no	—
151	C₈×C₄₇	—	2	1	no	—
152	C₈×C₄₇	—	2	1	no	—
153	C₈×C₄₇	—	2	1	no	—
154	C₈×C₄₇	—	4	2	no	—
155	C₈×C₄₇	—	4	2	no	—
156	C₈×C₄₇	#131	1	2	yes	—
157	C₄₇⋊C₈	—	2	2	no	—
158	C₄₇⋊C₈	—	2	2	no	—
159	C₄₇⋊C₈	—	4	2	no	—
160	C₄₇⋊C₈	—	4	2	no	—
161	#17840	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	—	2	1	no	—
164	C₂²×C₄₇	#430	1	2	yes	—
165	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₂×D₄₇	—	2	2	no	—
167	C₂×D₄₇	—	2	2	no	—
168	C₂×D₄₇	—	2	2	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	2	1	no	—
172	C₄×C₄₇	—	2	1	no	—
173	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
174	C₄×C₄₇	#568	1	1	yes	characteristic
175	C₄×C₄₇	#576	1	1	yes	characteristic
176	C₄×C₄₇	#578	1	1	yes	characteristic
177	Dic₄₇	—	2	2	no	—
178	Dic₄₇	—	2	2	no	—
179	Dic₄₇	—	2	2	no	—
180	#2990	—	47	2	no	—
181	#2965	—	47	1	no	—
182	#1573	—	47	2	no	—
183	#1572	—	47	2	no	—
184	#1571	—	47	2	no	—
185	#1569	—	47	2	no	—
186	#1568	D₄₇	1	1	yes	characteristic, complemented
187	#1565	—	47	2	no	—
188	C₂×#335	—	47	1	no	—
189	C₂×C₄₇	—	2	2	no	—
190	C₂×C₄₇	#2906	1	1	yes	characteristic
191	C₂×C₄₇	#2913	1	1	yes	characteristic
192	C₂×C₄₇	C₂×#430	1	1	yes	characteristic
193	D₄₇	—	4	2	no	—
194	C₄×#41	—	47	1	no	—
195	#446	—	47	1	no	—
196	#446	—	47	1	no	—
197	#446	—	94	2	no	—
198	#436	—	47	1	no	—
199	#436	—	47	1	no	—
200	#435	—	47	2	no	—
201	#435	—	47	2	no	—
202	#435	—	94	2	no	—
203	#435	—	94	2	no	—
204	C₈×D₄	—	94	2	no	—
205	#432	—	47	2	no	—
206	#432	C₂×D₄₇	1	2	yes	—
207	#431	—	47	1	no	—
208	#431	—	47	2	no	—
209	#431	—	47	2	no	—
210	#423	—	47	2	no	—
211	C₂×C₄⋊C₈	—	47	2	no	—
212	C₂×C₄⋊C₈	—	47	2	no	—
213	C₄×M₁₆	—	47	2	no	—
214	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
215	#335	—	47	2	no	—
216	#335	—	47	2	no	—
217	#335	—	47	1	no	—
218	#335	—	47	1	no	—
219	#335	—	47	2	no	—
220	#335	C₂×D₄₇	1	1	yes	characteristic
221	#335	C₂×D₄₇	1	1	yes	characteristic
222	#334	—	47	2	no	—
223	#334	—	47	2	no	—
224	#334	C₂×D₄₇	1	2	yes	—
225	C₄₇	#17840	1	1	yes	characteristic, complemented, Sylow
226	C₂×#41	—	47	1	no	—
227	C₂×M₁₆	—	47	2	no	—
228	C₂×M₁₆	—	47	2	no	—
229	C₂×M₁₆	—	94	2	no	—
230	C₂²×C₈	—	94	2	no	—
231	C₂²×C₈	—	94	2	no	—
232	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
233	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
234	C₄×Q₈	—	47	1	no	—
235	C₄×Q₈	—	47	1	no	—
236	C₄×D₄	—	47	2	no	—
237	C₄×D₄	—	47	2	no	—
238	C₄×D₄	—	47	1	no	—
239	C₄×D₄	—	47	1	no	—
240	#117	—	47	1	no	—
241	#117	—	47	2	no	—
242	C₂×C₄²	—	47	2	no	—
243	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
244	C₄⋊C₈	—	47	2	no	—
245	C₄⋊C₈	—	47	2	no	—
246	C₄⋊C₈	—	47	1	no	—
247	C₄⋊C₈	—	47	1	no	—
248	C₄⋊C₈	—	47	2	no	—
249	C₄⋊C₈	—	47	2	no	—
250	C₄⋊C₈	—	94	2	no	—
251	C₄⋊C₈	—	94	2	no	—
252	C₄⋊C₈	—	94	2	no	—
253	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
254	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	47	2	no	—
257	C₂²⋊C₈	—	47	2	no	—
258	C₂²⋊C₈	—	47	2	no	—
259	C₂²⋊C₈	—	94	2	no	—
260	C₂²⋊C₈	—	94	2	no	—
261	#97	—	2	2	no	—
262	#97	—	47	2	no	—
263	#97	—	47	2	no	—
264	#97	—	94	2	no	—
265	C₄×C₈	—	47	2	no	—
266	C₄×C₈	—	94	2	no	—
267	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
268	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
271	#41	—	188	2	no	—
272	C₂×Q₈	—	47	1	no	—
273	C₂×D₄	—	47	2	no	—
274	C₂×D₄	—	47	1	no	—
275	C₂²×C₄	—	47	2	no	—
276	C₂²×C₄	—	47	2	no	—
277	C₂²×C₄	—	47	2	no	—
278	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
280	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
281	M₁₆	—	94	4	no	—
282	M₁₆	—	376	2	no	—
283	M₁₆	—	188	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	47	2	no	—
289	C₂×C₈	—	47	2	no	—
290	C₂×C₈	—	94	2	no	—
291	C₂×C₈	—	94	2	no	—
292	C₂×C₈	—	376	2	no	—
293	C₂×C₈	—	376	2	no	—
294	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
298	C₄⋊C₄	—	47	2	no	—
299	C₄⋊C₄	—	47	2	no	—
300	C₄⋊C₄	—	47	1	no	—
301	C₄⋊C₄	—	47	1	no	—
302	C₂²⋊C₄	—	47	2	no	—
303	C₂²⋊C₄	—	47	2	no	—
304	C₂²⋊C₄	—	47	2	no	—
305	C₄²	—	47	2	no	—
306	C₄²	—	47	2	no	—
307	C₄²	—	47	2	no	—
308	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
309	C₄²	D₄×D₄₇	1	1	yes	characteristic
310	C₄²	Q₈×D₄₇	1	1	yes	characteristic
311	C₄²	C₂³×D₄₇	1	1	yes	characteristic
312	C₂³	—	47	2	no	—
313	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
314	Q₈	—	188	1	no	—
315	D₄	—	94	4	no	—
316	D₄	—	188	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	47	2	no	—
323	C₂×C₄	—	47	2	no	—
324	C₂×C₄	—	47	2	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
331	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
332	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
335	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	94	2	no	—
342	C₈	—	94	2	no	—
343	C₈	—	188	2	no	—
344	C₈	—	188	2	no	—
345	C₈	#13094032	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	94	2	no	—
349	C₂²	—	94	2	no	—
350	C₂²	—	94	2	no	—
351	C₂²	#15689170	1	2	yes	—
352	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	94	2	no	—
358	C₄	—	94	2	no	—
359	C₄	—	94	2	no	—
360	C₄	C₂×#13094032	1	1	yes	characteristic
361	C₄	#15689889	1	1	yes	characteristic
362	C₄	#15689925	1	1	yes	characteristic
363	C₄	#15689937	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	188	2	no	—
366	C₂	#20251957	1	1	yes	characteristic
367	C₂	#20251985	1	1	yes	characteristic
368	C₂	C₂×#15689170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1568	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂×#10625350	Q₈
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1568	D₄₇
hypocenter	C₄₇	#17840
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #20242174	C₂	semidirect
4. #20242167	C₂	semidirect
5. #20242161	C₂	semidirect
6. #20242152	C₂	semidirect
7. #20242137	C₂	semidirect
8. C₂×#15688876	C₂	semidirect
9. C₄₇×#1568	C₂	semidirect
186. #1568	D₄₇	semidirect
225. C₄₇	#17840	semidirect

Maximal subgroups

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

subgroup	index
#20252515	2
#20252354	2
#20242174	2
#20242167	2
#20242161	2
#20242152	2
#20242137	2
C₂×#15688876	2
C₄₇×#1568	2
#17840	47