Subgroups — #26115601

Order 12032.

526 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689471	C₂	1	1	yes	maximal, characteristic
2	#20244315	C₂	1	1	yes	maximal, characteristic
3	#20243257	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688873	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688869	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688866	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094021	C₂²	1	1	yes	characteristic
9	#15689471	—	2	2	no	—
10	#15689471	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#13093944	—	2	1	no	—
14	C₂×#13093944	—	2	1	no	—
15	C₂×#13093944	—	2	2	no	—
16	C₂×#13093935	—	2	1	no	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	#15688878	—	2	1	no	—
19	#15688873	—	2	2	no	—
20	#15688869	—	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×#103×C₄₇	—	2	2	no	—
25	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×D₁₈₈	—	2	1	no	—
27	C₂²×D₁₈₈	—	2	1	no	—
28	C₂²×D₁₈₈	—	2	1	no	—
29	C₂²×C₄₇⋊C₈	—	2	1	no	—
30	C₂²×C₄₇⋊C₈	—	2	1	no	—
31	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
32	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
33	#13094021	—	2	1	no	—
34	#13094021	—	2	1	no	—
35	#13094021	—	2	1	no	—
36	#13094021	C₂×C₄	1	2	yes	—
37	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
38	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
39	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
40	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
41	#13093944	—	2	4	no	—
42	#13093944	—	2	4	no	—
43	#13093944	—	4	4	no	—
44	#13093935	—	4	2	no	—
45	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
46	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
47	C₂²×Q₈×C₄₇	—	2	1	no	—
48	C₂²×C₈×C₄₇	—	2	1	no	—
49	C₄⋊Q₈×C₄₇	—	2	1	no	—
50	C₄⋊Q₈×C₄₇	—	2	1	no	—
51	C₄⋊Q₈×C₄₇	—	2	1	no	—
52	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
56	C₂×C₄²×C₄₇	—	2	1	no	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₇	—	2	1	no	—
59	C₄⋊C₈×C₄₇	—	2	1	no	—
60	#103×C₄₇	—	4	4	no	—
61	#95×C₄₇	—	2	1	no	—
62	C₂³×D₄₇	—	4	1	no	—
63	C₂×Q₈×C₄₇	—	2	2	no	—
64	C₂×Q₈×C₄₇	—	4	1	no	—
65	C₂×Q₈×C₄₇	—	4	2	no	—
66	C₂×Q₈×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	4	1	no	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
72	C₂×D₁₈₈	—	2	2	no	—
73	C₂×D₁₈₈	—	2	2	no	—
74	C₂×D₁₈₈	—	2	2	no	—
75	C₂×D₁₈₈	—	4	1	no	—
76	C₂×D₁₈₈	—	4	1	no	—
77	C₂×D₁₈₈	—	4	2	no	—
78	C₂×D₁₈₈	—	4	1	no	—
79	C₂×D₁₈₈	—	4	2	no	—
80	C₂×D₁₈₈	—	4	1	no	—
81	C₂×D₁₈₈	—	4	1	no	—
82	C₂×D₁₈₈	—	4	1	no	—
83	C₂×D₁₈₈	—	4	1	no	—
84	C₂×D₁₈₈	—	8	2	no	—
85	C₂×C₈×C₄₇	—	4	1	no	—
86	C₂×C₈×C₄₇	—	4	2	no	—
87	C₄⋊C₄×C₄₇	—	2	2	no	—
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	—	4	1	no	—
92	C₄⋊C₄×C₄₇	—	4	1	no	—
93	C₄²×C₄₇	—	4	1	no	—
94	C₄²×C₄₇	—	4	1	no	—
95	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₇	#41	1	1	yes	characteristic
99	C₂×C₄₇⋊C₈	—	2	1	no	—
100	C₂×C₄₇⋊C₈	—	2	1	no	—
101	C₂×C₄₇⋊C₈	—	2	1	no	—
102	C₂×C₄₇⋊C₈	—	2	1	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	4	1	no	—
105	C₂×C₄₇⋊C₈	—	4	1	no	—
106	C₂×C₄₇⋊C₈	—	4	1	no	—
107	C₂×C₄₇⋊C₈	—	4	2	no	—
108	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
109	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
110	C₂³×C₄₇	#121	1	1	yes	characteristic
111	C₂²×D₄₇	—	4	2	no	—
112	C₂²×D₄₇	—	8	1	no	—
113	C₂²×D₄₇	—	8	1	no	—
114	C₂²×D₄₇	—	8	2	no	—
115	C₂²×D₄₇	—	8	2	no	—
116	Q₈×C₄₇	—	4	4	no	—
117	Q₈×C₄₇	—	8	1	no	—
118	Q₈×C₄₇	—	8	1	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	8	2	no	—
125	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
133	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
137	D₁₈₈	—	4	2	no	—
138	D₁₈₈	—	4	2	no	—
139	D₁₈₈	—	4	4	no	—
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	C₈×C₄₇	—	8	1	no	—
147	C₄₇⋊C₈	—	2	2	no	—
148	C₄₇⋊C₈	—	2	2	no	—
149	C₄₇⋊C₈	—	4	2	no	—
150	C₄₇⋊C₈	—	8	1	no	—
151	#9746	—	47	1	no	maximal, Sylow
152	C₂²×C₄₇	#390	1	1	yes	characteristic
153	C₂²×C₄₇	#463	1	1	yes	characteristic
154	C₂²×C₄₇	#464	1	1	yes	characteristic
155	C₂²×C₄₇	#469	1	2	yes	—
156	C₂²×C₄₇	#471	1	2	yes	—
157	C₂×D₄₇	—	8	2	no	—
158	C₂×D₄₇	—	8	2	no	—
159	C₂×D₄₇	—	8	2	no	—
160	C₂×D₄₇	—	16	2	no	—
161	C₂×D₄₇	—	16	2	no	—
162	C₄×C₄₇	—	2	1	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	8	2	no	—
165	C₄×C₄₇	—	8	2	no	—
166	C₄×C₄₇	#416	1	1	yes	characteristic
167	C₄×C₄₇	#417	1	1	yes	characteristic
168	C₄×C₄₇	#442	1	2	yes	—
169	C₄×C₄₇	#463	1	1	yes	characteristic
170	C₄×C₄₇	#464	1	1	yes	characteristic
171	C₄×C₄₇	#496	1	1	yes	characteristic
172	C₄×C₄₇	#496	1	1	yes	characteristic
173	C₂×#486	—	47	1	no	—
174	#1982	—	47	1	no	—
175	#1981	D₄₇	1	1	yes	characteristic, complemented
176	#1770	—	47	1	no	—
177	C₂×#334	—	47	1	no	—
178	C₂×#332	—	47	1	no	—
179	C₂×#331	—	47	1	no	—
180	C₂×C₄₇	#1701	1	2	yes	—
181	C₂×C₄₇	#1702	1	2	yes	—
182	C₂×C₄₇	#1972	1	1	yes	characteristic
183	C₂×C₄₇	#1977	1	1	yes	characteristic
184	C₂×C₄₇	#1983	1	1	yes	characteristic
185	D₄₇	—	16	2	no	—
186	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
187	C₂×#127	—	47	1	no	—
188	#486	—	47	4	no	—
189	#486	—	94	2	no	—
190	C₂×C₄⋊C₈	—	47	1	no	—
191	C₂×C₄⋊C₈	—	47	1	no	—
192	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
193	C₂×#103	—	2	2	no	—
194	C₂×#102	—	94	1	no	—
195	C₂×#102	—	94	1	no	—
196	C₂×#102	—	94	2	no	—
197	C₂×#97	—	94	1	no	—
198	C₂×C₄×C₈	—	47	1	no	—
199	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
200	#336	—	94	1	no	—
201	#334	—	94	2	no	—
202	#332	—	94	2	no	—
203	#331	—	94	2	no	—
204	C₄₇	#9746	1	1	yes	characteristic, complemented, Sylow
205	C₂²×Q₈	—	2	1	no	—
206	C₂²×D₄	—	94	1	no	—
207	C₂²×D₄	—	94	1	no	—
208	C₂²×D₄	—	94	1	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	47	1	no	—
211	C₂²×C₈	—	47	1	no	—
212	C₂²×C₈	—	94	1	no	—
213	C₂²×C₈	—	94	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	C₄×D₄₇	1	2	yes	—
218	#127	—	47	2	no	—
219	#127	—	94	1	no	—
220	#127	—	94	1	no	—
221	#127	—	94	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
225	C₂×C₄²	—	2	1	no	—
226	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	47	2	no	—
230	C₄⋊C₈	—	94	1	no	—
231	C₄⋊C₈	—	94	1	no	—
232	C₄⋊C₈	—	94	1	no	—
233	#103	—	4	4	no	—
234	#102	—	94	4	no	—
235	#102	—	94	4	no	—
236	#102	—	188	4	no	—
237	#97	—	188	2	no	—
238	C₄×C₈	—	47	2	no	—
239	C₄×C₈	—	47	2	no	—
240	#95	—	2	1	no	—
241	C₂⁴	—	188	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×D₄	—	94	2	no	—
247	C₂×D₄	—	94	2	no	—
248	C₂×D₄	—	94	2	no	—
249	C₂×D₄	—	376	2	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	1	no	—
255	C₂×D₄	—	188	2	no	—
256	C₂×D₄	—	188	2	no	—
257	C₂×D₄	—	188	1	no	—
258	C₂×D₄	—	188	1	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
262	C₂²×C₄	#10625352	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	47	2	no	—
267	C₂×C₈	—	47	2	no	—
268	C₂×C₈	—	94	1	no	—
269	C₂×C₈	—	94	1	no	—
270	C₂×C₈	—	94	1	no	—
271	C₂×C₈	—	94	1	no	—
272	C₂×C₈	—	94	2	no	—
273	C₂×C₈	—	188	1	no	—
274	C₂×C₈	—	188	2	no	—
275	C₂×C₈	—	188	1	no	—
276	C₂×C₈	—	188	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄²	—	4	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
286	C₄²	#10625350	1	1	yes	characteristic
287	C₄²	D₄×D₄₇	1	1	yes	characteristic
288	C₄²	D₄×D₄₇	1	1	yes	characteristic
289	C₂³	—	376	1	no	—
290	C₂³	—	376	1	no	—
291	C₂³	—	376	2	no	—
292	C₂³	—	376	2	no	—
293	C₂³	—	188	2	no	—
294	C₂³	#13093996	1	1	yes	characteristic
295	Q₈	—	4	4	no	—
296	Q₈	—	8	1	no	—
297	Q₈	—	8	1	no	—
298	D₄	—	376	2	no	—
299	D₄	—	376	2	no	—
300	D₄	—	376	1	no	—
301	D₄	—	376	1	no	—
302	D₄	—	376	1	no	—
303	D₄	—	376	1	no	—
304	D₄	—	188	4	no	—
305	D₄	—	188	2	no	—
306	D₄	—	188	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
314	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
315	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
316	C₂×C₄	#13093997	1	1	yes	characteristic
317	C₂×C₄	#13094010	1	1	yes	characteristic
318	C₂×C₄	#13094020	1	1	yes	characteristic
319	C₂×C₄	#13094045	1	1	yes	characteristic
320	C₂×C₄	#13094047	1	1	yes	characteristic
321	C₂×C₄	#13094049	1	2	yes	—
322	C₂×C₄	#13094050	1	2	yes	—
323	C₂×C₄	#13094051	1	1	yes	characteristic
324	C₂×C₄	#13094056	1	1	yes	characteristic
325	C₈	—	8	1	no	—
326	C₈	—	94	2	no	—
327	C₈	—	94	2	no	—
328	C₈	—	376	1	no	—
329	C₈	—	188	2	no	—
330	C₂²	—	376	2	no	—
331	C₂²	—	376	2	no	—
332	C₂²	—	376	2	no	—
333	C₂²	—	752	2	no	—
334	C₂²	—	752	2	no	—
335	C₂²	#15689023	1	1	yes	characteristic
336	C₂²	#15689340	1	1	yes	characteristic
337	C₂²	#15689347	1	1	yes	characteristic
338	C₂²	#15689374	1	2	yes	—
339	C₂²	#15689388	1	2	yes	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	2	no	—
344	C₄	#15689118	1	1	yes	characteristic
345	C₄	#15689123	1	1	yes	characteristic
346	C₄	#15689233	1	2	yes	—
347	C₄	#15689341	1	1	yes	characteristic
348	C₄	#15689348	1	1	yes	characteristic
349	C₄	#15689510	1	1	yes	characteristic
350	C₄	#15689513	1	1	yes	characteristic
351	C₂	—	752	2	no	—
352	C₂	#20242951	1	2	yes	—
353	C₂	#20242958	1	2	yes	—
354	C₂	#20244263	1	1	yes	characteristic
355	C₂	#20244290	1	1	yes	characteristic
356	C₂	#20244317	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093996
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1981	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂×#13094021	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1981	D₄₇
hypocenter	C₄₇	#9746
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1981

Complements

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

N	H ≅ G/N	type
3. #20243257	C₂	semidirect
4. C₂×#15688873	C₂	semidirect
5. C₂×#15688869	C₂	semidirect
7. C₄₇×#1981	C₂	semidirect
10. #15689471	C₄	semidirect
175. #1981	D₄₇	semidirect
204. C₄₇	#9746	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689471	2
#20244315	2
#20243257	2
C₂×#15688873	2
C₂×#15688869	2
C₂×#15688866	2
C₄₇×#1981	2
#9746	47