Subgroups — #26115342

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251746	C₂	1	1	yes	maximal, characteristic
2	#20243504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688872	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688866	C₂	1	1	yes	maximal, characteristic
6	C₂×#15688866	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₇×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094021	C₂²	1	1	yes	characteristic
9	C₂×#13094013	C₄	1	4	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₄₇	—	2	1	no	—
13	#15689024	—	2	1	no	—
14	#15689024	—	2	1	no	—
15	#15689024	—	2	2	no	—
16	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
17	#15688877	—	2	1	no	—
18	#15688877	—	2	2	no	—
19	#15688872	—	2	2	no	—
20	#15688866	—	2	2	no	—
21	#15688866	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#333	—	2	2	no	—
26	C₂²×D₁₈₈	—	2	1	no	—
27	C₂²×D₁₈₈	—	2	1	no	—
28	C₂²×D₁₈₈	C₂×C₄	1	2	yes	—
29	C₂²×C₄₇⋊C₈	—	2	1	no	—
30	C₂²×C₄₇⋊C₈	—	2	1	no	—
31	#13094021	—	2	1	no	—
32	#13094021	—	2	1	no	—
33	#13094021	—	2	1	no	—
34	#13094021	—	2	1	no	—
35	#13094013	—	4	8	no	—
36	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
37	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
42	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
43	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
46	C₈×Dic₄₇	—	4	2	no	—
47	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₄₇	—	2	1	no	—
49	C₂²×C₈×C₄₇	—	2	1	no	—
50	C₄⋊Q₈×C₄₇	—	2	1	no	—
51	C₄⋊Q₈×C₄₇	—	2	1	no	—
52	C₄⋊Q₈×C₄₇	—	2	1	no	—
53	C₄⋊Q₈×C₄₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₇	—	2	1	no	—
58	C₄⋊C₈×C₄₇	—	2	1	no	—
59	C₄⋊C₈×C₄₇	—	2	1	no	—
60	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
61	C₂³×D₄₇	—	2	2	no	—
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×Q₈×C₄₇	—	2	2	no	—
64	C₂×Q₈×C₄₇	—	4	4	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
71	C₂×D₁₈₈	—	2	2	no	—
72	C₂×D₁₈₈	—	2	2	no	—
73	C₂×D₁₈₈	—	4	4	no	—
74	C₂×D₁₈₈	—	4	1	no	—
75	C₂×D₁₈₈	—	4	1	no	—
76	C₂×D₁₈₈	—	4	1	no	—
77	C₂×D₁₈₈	—	4	1	no	—
78	C₂×D₁₈₈	—	4	1	no	—
79	C₂×D₁₈₈	—	4	1	no	—
80	C₂×D₁₈₈	—	4	2	no	—
81	C₂×D₁₈₈	—	4	2	no	—
82	C₂×D₁₈₈	—	4	1	no	—
83	C₂×D₁₈₈	—	4	1	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	4	1	no	—
86	C₂×C₈×C₄₇	—	4	1	no	—
87	C₂×C₈×C₄₇	—	4	1	no	—
88	C₂×C₈×C₄₇	—	4	2	no	—
89	C₄⋊C₄×C₄₇	—	4	1	no	—
90	C₄⋊C₄×C₄₇	—	4	1	no	—
91	C₄⋊C₄×C₄₇	—	4	1	no	—
92	C₄⋊C₄×C₄₇	—	4	1	no	—
93	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₇	#41	1	1	yes	characteristic
96	C₄²×C₄₇	#41	1	1	yes	characteristic
97	(C₂×D₄₇)⋊C₄	—	4	8	no	—
98	(C₂×D₄₇)⋊C₄	—	4	8	no	—
99	C₄×Dic₄₇	—	2	2	no	—
100	C₄×Dic₄₇	—	2	2	no	—
101	C₄×Dic₄₇	—	2	1	no	—
102	C₄×Dic₄₇	—	2	1	no	—
103	C₄×Dic₄₇	—	2	1	no	—
104	C₄×Dic₄₇	—	2	1	no	—
105	C₄×Dic₄₇	—	2	4	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	—	4	1	no	—
108	C₂×C₄₇⋊C₈	—	4	1	no	—
109	C₂×C₄₇⋊C₈	—	4	1	no	—
110	C₂×C₄₇⋊C₈	—	4	2	no	—
111	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×D₄₇	—	4	2	no	—
113	C₂²×D₄₇	—	4	4	no	—
114	C₂²×D₄₇	—	4	2	no	—
115	C₂²×D₄₇	—	4	4	no	—
116	C₂²×D₄₇	—	4	2	no	—
117	Q₈×C₄₇	—	8	2	no	—
118	Q₈×C₄₇	—	8	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	4	no	—
122	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#135	1	2	yes	—
130	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#135	1	2	yes	—
132	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
134	C₂×Dic₄₇	—	2	2	no	—
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	2	2	no	—
137	C₂×Dic₄₇	—	2	2	no	—
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	#100	1	4	yes	—
140	D₁₈₈	—	8	2	no	—
141	D₁₈₈	—	8	2	no	—
142	D₁₈₈	—	8	1	no	—
143	D₁₈₈	—	8	1	no	—
144	D₁₈₈	—	8	1	no	—
145	D₁₈₈	—	8	1	no	—
146	D₁₈₈	—	8	2	no	—
147	D₁₈₈	—	8	1	no	—
148	D₁₈₈	—	8	1	no	—
149	C₈×C₄₇	—	4	2	no	—
150	C₈×C₄₇	—	8	1	no	—
151	C₄₇⋊C₈	—	4	2	no	—
152	C₄₇⋊C₈	—	8	1	no	—
153	#9699	—	47	1	no	maximal, Sylow
154	C₂²×C₄₇	#390	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#100	1	2	yes	—
156	C₂²×C₄₇	#416	1	1	yes	characteristic
157	C₂²×C₄₇	#416	1	2	yes	—
158	C₂²×C₄₇	#417	1	1	yes	characteristic
159	C₂×D₄₇	—	8	4	no	—
160	C₂×D₄₇	—	8	4	no	—
161	C₂×D₄₇	—	8	2	no	—
162	C₂×D₄₇	—	8	2	no	—
163	C₂×D₄₇	—	8	2	no	—
164	C₄×C₄₇	—	2	1	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	8	2	no	—
167	C₄×C₄₇	#463	1	1	yes	characteristic
168	C₄×C₄₇	#463	1	1	yes	characteristic
169	C₄×C₄₇	#464	1	1	yes	characteristic
170	C₄×C₄₇	#464	1	1	yes	characteristic
171	C₄×C₄₇	#488	1	2	yes	—
172	C₄×C₄₇	#495	1	1	yes	characteristic
173	C₄×C₄₇	#495	1	1	yes	characteristic
174	Dic₄₇	—	2	4	no	—
175	Dic₄₇	—	4	2	no	—
176	#2858	—	47	1	no	—
177	#1831	—	47	1	no	—
178	C₄×C₄⋊C₈	—	47	1	no	—
179	C₂×#333	—	47	1	no	—
180	C₂×#333	D₄₇	1	1	yes	characteristic, complemented
181	C₂×#331	—	47	1	no	—
182	C₂×#331	—	47	1	no	—
183	C₂×C₄₇	#1543	1	2	yes	—
184	C₂×C₄₇	#1551	1	2	yes	—
185	C₂×C₄₇	#1958	1	1	yes	characteristic
186	C₂×C₄₇	#1965	1	1	yes	characteristic
187	C₂×C₄₇	#1983	1	1	yes	characteristic
188	D₄₇	—	16	2	no	—
189	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
190	C₂×#127	—	47	1	no	—
191	C₂×#124	—	47	4	no	—
192	C₂×C₄⋊C₈	—	47	1	no	—
193	C₂×C₄⋊C₈	—	47	1	no	—
194	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
195	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
196	C₂×C₄×C₈	—	94	1	no	—
197	#390	—	94	2	no	—
198	#390	—	94	1	no	—
199	#390	—	94	1	no	—
200	C₄³	—	47	1	no	—
201	#336	—	94	2	no	—
202	#336	—	94	1	no	—
203	#333	—	2	2	no	—
204	#333	—	94	2	no	—
205	#331	—	94	2	no	—
206	#331	—	94	2	no	—
207	C₄₇	#9699	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₈	C₄×D₄₇	1	2	yes	—
209	C₂²×D₄	—	47	2	no	—
210	C₂²×D₄	—	94	1	no	—
211	C₂²×D₄	—	94	1	no	—
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	—	94	1	no	—
215	C₂²×C₈	—	94	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	#127	—	94	1	no	—
221	#127	—	94	1	no	—
222	#127	—	94	1	no	—
223	#127	—	94	1	no	—
224	#124	—	188	8	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₂²⋊C₄	—	94	4	no	—
228	C₂×C₂²⋊C₄	—	94	4	no	—
229	C₂×C₄²	—	47	2	no	—
230	C₂×C₄²	—	47	2	no	—
231	C₂×C₄²	—	47	1	no	—
232	C₂×C₄²	—	47	1	no	—
233	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	47	2	no	—
238	C₄⋊C₈	—	94	1	no	—
239	C₄⋊C₈	—	94	1	no	—
240	C₄⋊C₈	—	94	1	no	—
241	C₄⋊C₈	C₂²⋊D₄₇	1	2	yes	—
242	C₄×C₈	—	188	2	no	—
243	C₂⁴	—	94	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	4	no	—
247	C₂×D₄	—	94	2	no	—
248	C₂×D₄	—	94	2	no	—
249	C₂×D₄	—	188	4	no	—
250	C₂×D₄	—	188	1	no	—
251	C₂×D₄	—	188	1	no	—
252	C₂×D₄	—	188	1	no	—
253	C₂×D₄	—	188	1	no	—
254	C₂×D₄	—	188	2	no	—
255	C₂×D₄	—	188	1	no	—
256	C₂×D₄	—	188	2	no	—
257	C₂×D₄	—	188	1	no	—
258	C₂×D₄	—	188	1	no	—
259	C₂×D₄	—	188	1	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	47	2	no	—
262	C₂²×C₄	—	47	2	no	—
263	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
264	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	—	94	2	no	—
272	C₂×C₈	—	188	1	no	—
273	C₂×C₈	—	188	2	no	—
274	C₂×C₈	—	188	1	no	—
275	C₂×C₈	—	188	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	188	8	no	—
281	C₂²⋊C₄	—	188	8	no	—
282	C₄²	—	94	4	no	—
283	C₄²	—	94	2	no	—
284	C₄²	—	94	1	no	—
285	C₄²	—	94	2	no	—
286	C₄²	—	94	1	no	—
287	C₄²	—	94	1	no	—
288	C₄²	—	94	1	no	—
289	C₄²	#10625350	1	1	yes	characteristic
290	C₄²	D₄×D₄₇	1	1	yes	characteristic
291	C₄²	#10625352	1	1	yes	characteristic
292	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
293	C₂³	—	188	2	no	—
294	C₂³	—	188	4	no	—
295	C₂³	—	188	2	no	—
296	C₂³	—	188	4	no	—
297	C₂³	—	188	2	no	—
298	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
299	Q₈	—	8	2	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	376	2	no	—
302	D₄	—	376	2	no	—
303	D₄	—	376	2	no	—
304	D₄	—	376	1	no	—
305	D₄	—	376	1	no	—
306	D₄	—	376	1	no	—
307	D₄	—	376	1	no	—
308	D₄	—	376	1	no	—
309	D₄	—	376	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	47	4	no	—
314	C₂×C₄	—	94	2	no	—
315	C₂×C₄	—	94	2	no	—
316	C₂×C₄	—	94	2	no	—
317	C₂×C₄	—	94	2	no	—
318	C₂×C₄	—	94	2	no	—
319	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
320	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
321	C₂×C₄	#13093996	1	1	yes	characteristic
322	C₂×C₄	#13093997	1	1	yes	characteristic
323	C₂×C₄	#13094010	1	1	yes	characteristic
324	C₂×C₄	#13094020	1	1	yes	characteristic
325	C₂×C₄	#13094045	1	2	yes	—
326	C₂×C₄	#13094045	1	1	yes	characteristic
327	C₂×C₄	#13094047	1	2	yes	—
328	C₂×C₄	#13094047	1	1	yes	characteristic
329	C₂×C₄	#13094051	1	1	yes	characteristic
330	C₂×C₄	#13094056	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	8	1	no	—
333	C₈	—	376	1	no	—
334	C₈	—	188	2	no	—
335	C₂²	—	376	2	no	—
336	C₂²	—	376	2	no	—
337	C₂²	—	376	4	no	—
338	C₂²	—	376	4	no	—
339	C₂²	—	376	2	no	—
340	C₂²	#15689023	1	1	yes	characteristic
341	C₂²	#100×D₄₇	1	2	yes	—
342	C₂²	#15689118	1	1	yes	characteristic
343	C₂²	#15689118	1	2	yes	—
344	C₂²	#15689123	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	94	4	no	—
349	C₄	—	188	2	no	—
350	C₄	#15689340	1	1	yes	characteristic
351	C₄	#15689341	1	1	yes	characteristic
352	C₄	#15689347	1	1	yes	characteristic
353	C₄	#15689348	1	1	yes	characteristic
354	C₄	#15689479	1	2	yes	—
355	C₄	#15689504	1	1	yes	characteristic
356	C₄	#15689508	1	1	yes	characteristic
357	C₂	—	752	2	no	—
358	C₂	#20242049	1	2	yes	—
359	C₂	#20242082	1	2	yes	—
360	C₂	#20244196	1	1	yes	characteristic
361	C₂	#20244226	1	1	yes	characteristic
362	C₂	#20244317	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#333	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#13094021	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#333	D₄₇
hypocenter	C₄₇	#9699
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
2. #20243504	C₂	semidirect
3. C₄⋊C₈×Dic₄₇	C₂	semidirect
4. C₂×#15688872	C₂	semidirect
7. C₂×C₄₇×#333	C₂	semidirect
180. C₂×#333	D₄₇	semidirect
207. C₄₇	#9699	semidirect

Maximal subgroups

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

subgroup	index
#20251746	2
#20243504	2
C₄⋊C₈×Dic₄₇	2
C₂×#15688872	2
C₂×#15688866	2
C₂×C₄₇×#333	2
#9699	47