Subgroups — #26134536

Order 12032.

574 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689498	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688937	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688936	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688935	C₂	1	1	yes	maximal, characteristic, complemented
5	M₃₂×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243558	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#361	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
9	C₂²×#10625334	—	2	1	no	—
10	C₂×#13094020	—	2	1	no	—
11	C₂×#13093964	C₂²	1	1	yes	characteristic
12	#15689498	—	2	1	no	—
13	#15689498	—	2	1	no	—
14	#15689498	C₄	1	2	yes	—
15	#15689498	C₄	1	2	yes	—
16	C₂×#13093960	C₂²	1	1	yes	characteristic
17	C₂×#13093959	—	2	1	no	—
18	#15689146	—	2	1	no	—
19	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
20	#15688937	—	2	2	no	—
21	#15688936	—	2	2	no	—
22	#15688935	—	2	2	no	—
23	#15688898	—	2	1	no	—
24	C₁₆×Dic₄₇	—	2	1	no	—
25	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#361	—	2	2	no	—
29	C₂²×D₁₈₈	—	2	1	no	—
30	C₂²×C₂²⋊D₄₇	—	4	2	no	—
31	C₂×D₃₇₆	—	2	1	no	—
32	C₂×D₃₇₆	—	2	1	no	—
33	C₂×D₃₇₆	—	2	1	no	—
34	C₂×D₃₇₆	—	2	1	no	—
35	C₂×D₃₇₆	—	2	1	no	—
36	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
37	C₂×#10625334	—	2	2	no	—
38	C₂×#10625334	—	2	2	no	—
39	C₂×#10625334	—	4	2	no	—
40	C₂×#10625334	—	4	2	no	—
41	C₂×#10625329	—	2	1	no	—
42	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
43	#13094020	—	2	2	no	—
44	#13094020	—	2	2	no	—
45	#13094020	—	4	2	no	—
46	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
47	#13093964	—	2	1	no	—
48	#13093964	D₄	1	2	yes	complemented
49	C₂×C₄₇⋊C₁₆	—	2	1	no	—
50	#13093960	—	2	1	no	—
51	#13093960	—	2	1	no	—
52	#13093959	—	4	2	no	—
53	Dic₄₇⋊C₈	—	4	2	no	—
54	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₄₇	D₄	1	1	yes	characteristic
57	C₈×Dic₄₇	D₄	1	1	yes	characteristic
58	C₂²×D₄×C₄₇	—	2	1	no	—
59	C₂×D₈×C₄₇	—	2	1	no	—
60	C₂×D₈×C₄₇	—	2	1	no	—
61	C₂×D₈×C₄₇	—	2	1	no	—
62	C₂×D₈×C₄₇	—	2	1	no	—
63	C₂×D₈×C₄₇	—	2	1	no	—
64	C₂×D₈×C₄₇	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₄₇	—	2	1	no	—
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	M₃₂×C₄₇	—	2	1	no	—
68	M₃₂×C₄₇	D₄	1	2	yes	complemented
69	C₂×C₁₆×C₄₇	—	2	1	no	—
70	#108×C₄₇	—	2	1	no	—
71	#108×C₄₇	—	2	1	no	—
72	C₂⁴×C₄₇	—	4	1	no	—
73	C₂³×D₄₇	—	4	1	no	—
74	C₂×D₄×C₄₇	—	2	2	no	—
75	C₂×D₄×C₄₇	—	4	1	no	—
76	C₂×D₄×C₄₇	—	4	1	no	—
77	C₂×D₄×C₄₇	—	4	1	no	—
78	C₂×D₄×C₄₇	—	4	1	no	—
79	C₂×D₄×C₄₇	—	4	1	no	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₄₇	—	4	4	no	—
82	C₂×C₂²⋊D₄₇	—	4	4	no	—
83	C₂×C₂²⋊D₄₇	—	8	8	no	—
84	C₂²×Dic₄₇	SD₁₆	1	2	yes	—
85	C₂×D₁₈₈	—	2	2	no	—
86	C₂×D₁₈₈	—	4	1	no	—
87	C₂×D₁₈₈	—	4	1	no	—
88	C₂×D₁₈₈	—	4	1	no	—
89	C₂×D₁₈₈	—	4	1	no	—
90	C₂×D₁₈₈	—	4	1	no	—
91	D₈×C₄₇	—	2	2	no	—
92	D₈×C₄₇	—	2	2	no	—
93	D₈×C₄₇	—	4	1	no	—
94	D₈×C₄₇	—	4	1	no	—
95	M₁₆×C₄₇	—	4	1	no	—
96	M₁₆×C₄₇	—	4	1	no	—
97	C₂×C₈×C₄₇	—	4	1	no	—
98	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
103	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
104	#10625334	—	4	4	no	—
105	#10625334	—	4	4	no	—
106	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×Dic₄₇	D₈	1	2	yes	—
109	#10625329	—	4	1	no	—
110	#10625329	—	4	1	no	—
111	C₂×C₄₇⋊C₈	—	4	1	no	—
112	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
115	C₂×C₄₇⋊C₈	SD₁₆	1	2	yes	—
116	D₃₇₆	—	2	2	no	—
117	D₃₇₆	—	2	2	no	—
118	D₃₇₆	—	4	1	no	—
119	D₃₇₆	—	4	1	no	—
120	C₁₆×C₄₇	—	2	2	no	—
121	C₄₇⋊C₁₆	—	2	2	no	—
122	C₂³×C₄₇	—	4	2	no	—
123	C₂³×C₄₇	—	8	4	no	—
124	C₂³×C₄₇	—	8	1	no	—
125	C₂³×C₄₇	—	8	1	no	—
126	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
127	C₂²×D₄₇	—	4	2	no	—
128	C₂²×D₄₇	—	8	4	no	—
129	C₂²×D₄₇	—	8	1	no	—
130	C₂²×D₄₇	—	8	1	no	—
131	D₄×C₄₇	—	4	2	no	—
132	D₄×C₄₇	—	4	2	no	—
133	D₄×C₄₇	—	8	1	no	—
134	D₄×C₄₇	—	8	1	no	—
135	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
141	C₂²⋊D₄₇	—	8	8	no	—
142	C₂²⋊D₄₇	—	8	8	no	—
143	C₂×Dic₄₇	—	2	4	no	—
144	C₂×Dic₄₇	#102	1	2	yes	—
145	C₂×Dic₄₇	#102	1	2	yes	—
146	D₁₈₈	—	4	2	no	—
147	D₁₈₈	—	4	2	no	—
148	D₁₈₈	—	8	1	no	—
149	D₁₈₈	—	8	1	no	—
150	C₈×C₄₇	—	8	1	no	—
151	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
152	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
153	C₈×C₄₇	#121	1	1	yes	characteristic
154	C₈×C₄₇	#121	1	1	yes	characteristic
155	C₄₇⋊C₈	—	8	1	no	—
156	C₄₇⋊C₈	#102	1	2	yes	—
157	C₄₇⋊C₈	#102	1	2	yes	—
158	#13213	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	—	8	4	no	—
161	C₂²×C₄₇	—	8	2	no	—
162	C₂²×C₄₇	—	16	2	no	—
163	C₂²×C₄₇	—	16	2	no	—
164	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
165	C₂²×C₄₇	#486	1	1	yes	characteristic
166	C₂²×C₄₇	#492	1	1	yes	characteristic
167	C₂×D₄₇	—	8	4	no	—
168	C₂×D₄₇	—	8	2	no	—
169	C₂×D₄₇	—	16	2	no	—
170	C₂×D₄₇	—	16	2	no	—
171	C₄×C₄₇	#390	1	1	yes	characteristic
172	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
173	C₄×C₄₇	#486	1	1	yes	characteristic
174	C₄×C₄₇	#493	1	1	yes	characteristic
175	Dic₄₇	—	2	2	no	—
176	Dic₄₇	#361	1	4	yes	complemented
177	C₂×#493	—	47	1	no	—
178	C₂×#361	—	47	1	no	—
179	C₂×#361	—	47	1	no	—
180	C₂×#361	—	47	1	no	—
181	C₂×#361	D₄₇	1	1	yes	characteristic, complemented
182	C₄×M₃₂	—	47	1	no	—
183	#1848	—	47	1	no	—
184	C₂×C₄₇	—	2	1	no	—
185	C₂×C₄₇	—	2	1	no	—
186	C₂×C₄₇	—	16	2	no	—
187	C₂×C₄₇	#1957	1	1	yes	characteristic
188	C₂×C₄₇	C₂×#361	1	2	yes	—
189	D₄₇	—	16	2	no	—
190	C₂²×D₈	—	47	1	no	—
191	C₂²×D₈	—	94	1	no	—
192	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
193	C₂×#127	—	94	1	no	—
194	C₂×M₃₂	—	47	1	no	—
195	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
196	#493	—	47	2	no	—
197	#493	—	47	2	no	—
198	#493	—	94	1	no	—
199	#493	—	94	1	no	—
200	C₂×#108	—	47	1	no	—
201	C₂×#108	—	94	1	no	—
202	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
203	#423	—	94	1	no	—
204	C₂×C₄×C₈	—	47	1	no	—
205	#361	—	2	2	no	—
206	#361	—	94	2	no	—
207	#361	—	94	2	no	—
208	#361	—	94	2	no	—
209	#346	—	94	1	no	—
210	C₄×C₁₆	—	94	1	no	—
211	C₄₇	#13213	1	1	yes	characteristic, complemented, Sylow
212	C₂²×D₄	—	2	1	no	—
213	C₂²×D₄	—	94	1	no	—
214	C₂²×D₄	—	188	2	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	—	2	1	no	—
219	C₂×D₈	—	2	1	no	—
220	C₂×D₈	—	47	2	no	—
221	C₂×D₈	—	94	2	no	—
222	C₂×D₈	—	94	2	no	—
223	C₂×D₈	—	94	1	no	—
224	C₂×D₈	—	94	1	no	—
225	C₂×D₈	—	94	1	no	—
226	C₂×D₈	—	94	1	no	—
227	C₂×D₈	—	94	1	no	—
228	C₂×D₈	—	188	2	no	—
229	C₂×D₈	—	188	2	no	—
230	C₂×D₈	C₄×D₄₇	1	2	yes	—
231	C₂×M₁₆	—	2	1	no	—
232	C₂×M₁₆	—	94	1	no	—
233	C₂²×C₈	—	47	1	no	—
234	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
235	#127	—	94	2	no	—
236	#127	—	94	2	no	—
237	#127	—	188	2	no	—
238	C₂×C₄²	—	47	1	no	—
239	M₃₂	—	2	1	no	—
240	M₃₂	—	47	2	no	—
241	M₃₂	—	94	1	no	—
242	M₃₂	C₂²⋊D₄₇	1	2	yes	complemented
243	C₂×C₁₆	—	2	1	no	—
244	C₂×C₁₆	—	94	1	no	—
245	#108	—	2	1	no	—
246	#108	—	2	1	no	—
247	#108	—	94	1	no	—
248	#108	—	94	1	no	—
249	#108	—	188	2	no	—
250	C₄⋊C₈	—	188	2	no	—
251	C₄×C₈	—	47	1	no	—
252	C₄×C₈	—	47	1	no	—
253	C₄×C₈	—	47	1	no	—
254	C₄×C₈	—	47	1	no	—
255	C₂⁴	—	4	1	no	—
256	C₂⁴	—	188	1	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	94	2	no	—
264	C₂×D₄	—	376	8	no	—
265	C₂×D₄	—	188	4	no	—
266	C₂×D₄	—	188	4	no	—
267	C₂×D₄	—	188	1	no	—
268	C₂×D₄	—	188	1	no	—
269	C₂×D₄	—	188	1	no	—
270	C₂×D₄	—	188	1	no	—
271	C₂×D₄	—	188	1	no	—
272	C₂²×C₄	—	47	2	no	—
273	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
274	D₈	—	2	2	no	—
275	D₈	—	2	2	no	—
276	D₈	—	4	1	no	—
277	D₈	—	4	1	no	—
278	D₈	—	94	2	no	—
279	D₈	—	94	2	no	—
280	D₈	—	188	4	no	—
281	D₈	—	188	4	no	—
282	D₈	—	188	1	no	—
283	D₈	—	188	1	no	—
284	M₁₆	—	4	1	no	—
285	M₁₆	—	4	1	no	—
286	M₁₆	—	188	1	no	—
287	M₁₆	—	188	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	47	1	no	—
290	C₂×C₈	—	47	1	no	—
291	C₂×C₈	—	47	2	no	—
292	C₂×C₈	—	47	2	no	—
293	C₂×C₈	—	188	1	no	—
294	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₈	#10625350	1	1	yes	characteristic
296	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
297	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
298	C₂×C₈	#10625352	1	1	yes	characteristic
299	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₄²	—	47	1	no	—
301	C₄²	—	47	1	no	—
302	C₄²	—	47	2	no	—
303	C₁₆	—	2	2	no	—
304	C₁₆	—	94	2	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	8	4	no	—
307	C₂³	—	8	1	no	—
308	C₂³	—	8	1	no	—
309	C₂³	—	376	4	no	—
310	C₂³	—	376	1	no	—
311	C₂³	—	376	1	no	—
312	C₂³	—	188	2	no	—
313	C₂³	SD₁₆×D₄₇	1	1	yes	characteristic
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	8	1	no	—
317	D₄	—	8	1	no	—
318	D₄	—	376	8	no	—
319	D₄	—	376	8	no	—
320	D₄	—	376	1	no	—
321	D₄	—	376	1	no	—
322	D₄	—	188	2	no	—
323	D₄	—	188	2	no	—
324	C₂×C₄	—	47	2	no	—
325	C₂×C₄	—	47	2	no	—
326	C₂×C₄	—	94	4	no	—
327	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
328	C₂×C₄	#13094010	1	1	yes	characteristic
329	C₂×C₄	#13094020	1	1	yes	characteristic
330	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
331	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
332	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
333	C₈	—	8	1	no	—
334	C₈	—	47	2	no	—
335	C₈	—	47	2	no	—
336	C₈	—	376	1	no	—
337	C₈	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
338	C₈	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
339	C₈	#13093996	1	1	yes	characteristic
340	C₈	#13093997	1	1	yes	characteristic
341	C₂²	—	2	2	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	16	2	no	—
345	C₂²	—	16	2	no	—
346	C₂²	—	376	4	no	—
347	C₂²	—	376	2	no	—
348	C₂²	—	752	2	no	—
349	C₂²	—	752	2	no	—
350	C₂²	#102×D₄₇	1	1	yes	characteristic
351	C₂²	#15689468	1	1	yes	characteristic
352	C₂²	#15689494	1	1	yes	characteristic
353	C₄	—	47	4	no	—
354	C₄	—	94	2	no	—
355	C₄	#15689023	1	1	yes	characteristic
356	C₄	#102×D₄₇	1	1	yes	characteristic
357	C₄	#15689468	1	1	yes	characteristic
358	C₄	#15689498	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	16	2	no	—
362	C₂	—	752	2	no	—
363	C₂	#20244192	1	1	yes	characteristic
364	C₂	#361×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂²	#102×D₄₇
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#361	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂×#15689498	C₂
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#361	D₄₇
hypocenter	C₄₇	#13213
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

Complements

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

N	H ≅ G/N	type
2. C₂×#15688937	C₂	semidirect
3. C₂×#15688936	C₂	semidirect
4. C₂×#15688935	C₂	semidirect
5. M₃₂×Dic₄₇	C₂	semidirect
6. #20243558	C₂	semidirect
7. C₂×C₄₇×#361	C₂	semidirect
48. #13093964	D₄	semidirect
68. M₃₂×C₄₇	D₄	semidirect
176. Dic₄₇	#361	semidirect
181. C₂×#361	D₄₇	semidirect
211. C₄₇	#13213	semidirect
242. M₃₂	C₂²⋊D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689498	2
C₂×#15688937	2
C₂×#15688936	2
C₂×#15688935	2
M₃₂×Dic₄₇	2
#20243558	2
C₂×C₄₇×#361	2
#13213	47