Subgroups — #26220666

Order 12032.

770 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20252028	C₂	1	2	yes	maximal, complemented
3	#20244277	C₂	1	1	yes	maximal, characteristic
4	#20244136	C₂	1	1	yes	maximal, characteristic, complemented
5	#20244134	C₂	1	1	yes	maximal, characteristic, complemented
6	#20244095	C₂	1	2	yes	maximal, complemented
7	#20244089	C₂	1	2	yes	maximal, complemented
8	#108×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×Dic₄₇	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1974	C₂	1	1	yes	maximal, characteristic
12	#41×Dic₄₇	C₂²	1	2	yes	—
13	Q₁₆×Dic₄₇	C₄	1	2	yes	complemented
14	SD₁₆×Dic₄₇	C₄	1	4	yes	complemented
15	D₈×Dic₄₇	C₄	1	2	yes	complemented
16	#15689181	—	2	4	no	—
17	#15689173	—	2	4	no	—
18	C₂×#13093960	C₂²	1	1	yes	characteristic
19	#15689161	C₂²	1	1	yes	characteristic
20	#15689151	C₂²	1	2	yes	—
21	#15689149	C₂²	1	2	yes	—
22	C₂×#13093951	C₂²	1	2	yes	—
23	#15689119	C₂²	1	2	yes	—
24	#15689090	C₂²	1	2	yes	—
25	M₁₆×Dic₄₇	C₂²	1	2	yes	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
31	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
32	#15688887	C₂²	1	2	yes	—
33	#15688884	C₂²	1	2	yes	—
34	#15688883	C₂²	1	2	yes	—
35	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×#104×C₄₇	C₂²	1	2	yes	—
38	C₄₇×#408	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₂×#10625329	C₂³	1	2	yes	—
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	Q₈×Dic₄₇	C₂×C₄	1	4	yes	—
44	D₄×Dic₄₇	—	2	4	no	—
45	D₄×Dic₄₇	C₂×C₄	1	4	yes	—
46	#13093983	—	2	2	no	—
47	#13093983	C₂³	1	2	yes	—
48	C₂×C₄×Dic₄₇	—	2	2	no	—
49	C₂×C₄×Dic₄₇	—	2	2	no	—
50	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
52	#13093960	—	2	2	no	—
53	#13093958	C₂×C₄	1	2	yes	—
54	#13093956	C₂×C₄	1	2	yes	—
55	Dic₄₇⋊C₈	—	2	4	no	—
56	Dic₄₇⋊C₈	—	2	4	no	—
57	#13093951	—	2	4	no	—
58	#13093948	C₂×C₄	1	4	yes	—
59	#13093945	C₂×C₄	1	4	yes	—
60	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
61	#13093934	—	2	2	no	—
62	C₄×C₄₇⋊C₈	—	2	2	no	—
63	C₈×Dic₄₇	—	2	2	no	—
64	C₈×Dic₄₇	—	2	2	no	—
65	C₈×Dic₄₇	—	2	2	no	—
66	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
67	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
68	C₈×Dic₄₇	D₄	1	2	yes	—
69	C₈×Dic₄₇	D₄	1	2	yes	—
70	C₂×#41×C₄₇	C₂³	1	2	yes	—
71	#135×C₄₇	—	2	2	no	—
72	#135×C₄₇	—	2	2	no	—
73	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
74	C₂×SD₁₆×C₄₇	C₂×C₄	1	2	yes	—
75	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
76	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	#108×C₄₇	—	2	2	no	—
81	#104×C₄₇	—	2	4	no	—
82	C₂²⋊C₈×C₄₇	—	2	4	no	—
83	C₄×C₈×C₄₇	—	2	2	no	—
84	#41×C₄₇	—	2	2	no	—
85	#41×C₄₇	—	2	2	no	—
86	#41×C₄₇	—	2	2	no	—
87	#41×C₄₇	—	2	2	no	—
88	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂×D₄×C₄₇	—	2	2	no	—
90	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₄₇	—	2	1	no	—
92	C₂²×C₄×C₄₇	—	2	2	no	—
93	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₂²×Dic₄₇	—	2	1	no	—
95	C₂²×Dic₄₇	—	2	4	no	—
96	C₂²×Dic₄₇	#41	1	2	yes	—
97	Q₁₆×C₄₇	—	2	2	no	—
98	SD₁₆×C₄₇	—	2	4	no	—
99	D₈×C₄₇	—	2	2	no	—
100	M₁₆×C₄₇	—	2	2	no	—
101	M₁₆×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	1	no	—
103	C₂×C₈×C₄₇	—	2	1	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₄₇	#41	1	2	yes	—
109	C₂×C₈×C₄₇	#41	1	2	yes	—
110	C₄²×C₄₇	—	2	1	no	—
111	C₄²×C₄₇	#41	1	2	yes	—
112	C₂²⋊Dic₄₇	—	2	2	no	—
113	C₂²⋊Dic₄₇	—	2	4	no	—
114	C₄⋊Dic₄₇	—	2	4	no	—
115	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
116	C₄×Dic₄₇	—	2	8	no	—
117	C₄×Dic₄₇	—	2	2	no	—
118	C₄×Dic₄₇	—	2	1	no	—
119	C₄×Dic₄₇	—	2	4	no	—
120	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
121	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
123	C₄×Dic₄₇	#41	1	2	yes	—
124	#10625329	—	2	2	no	—
125	#10625329	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	2	1	no	—
127	C₂×C₄₇⋊C₈	—	2	1	no	—
128	C₂×C₄₇⋊C₈	—	2	1	no	—
129	C₂×C₄₇⋊C₈	—	2	1	no	—
130	C₂×C₄₇⋊C₈	—	2	2	no	—
131	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
132	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
133	C₂³×C₄₇	—	2	2	no	—
134	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
135	Q₈×C₄₇	—	2	2	no	—
136	Q₈×C₄₇	—	2	2	no	—
137	D₄×C₄₇	—	2	2	no	—
138	D₄×C₄₇	—	2	2	no	—
139	D₄×C₄₇	—	4	2	no	—
140	D₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	2	4	no	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	—	2	2	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	—	4	2	no	—
146	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₄₇	—	2	4	no	—
153	C₂×Dic₄₇	—	2	2	no	—
154	C₂×Dic₄₇	—	2	4	no	—
155	C₂×Dic₄₇	—	2	4	no	—
156	C₂×Dic₄₇	—	4	8	no	—
157	C₂×Dic₄₇	C₄×D₄	1	2	yes	—
158	C₂×Dic₄₇	C₄×D₄	1	2	yes	—
159	C₈×C₄₇	—	2	2	no	—
160	C₈×C₄₇	—	4	2	no	—
161	C₈×C₄₇	#117	1	2	yes	—
162	C₈×C₄₇	#117	1	2	yes	—
163	C₄₇⋊C₈	—	2	2	no	—
164	C₄₇⋊C₈	—	2	2	no	—
165	C₄₇⋊C₈	—	4	2	no	—
166	#21292	—	47	1	no	maximal, Sylow
167	C₂²×C₄₇	—	2	2	no	—
168	C₂²×C₄₇	—	4	4	no	—
169	C₂²×C₄₇	—	4	2	no	—
170	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂²×C₄₇	#529	1	1	yes	characteristic
172	C₂²×C₄₇	#529	1	1	yes	characteristic
173	C₄×C₄₇	—	2	4	no	—
174	C₄×C₄₇	—	4	2	no	—
175	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
176	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
177	C₄×C₄₇	#532	1	1	yes	characteristic
178	C₄×C₄₇	#532	1	1	yes	characteristic
179	Dic₄₇	—	2	4	no	—
180	Dic₄₇	—	2	2	no	—
181	Dic₄₇	—	4	4	no	—
182	Dic₄₇	#443	1	4	yes	—
183	C₄×#135	—	47	1	no	—
184	#2922	—	47	2	no	—
185	#1974	—	47	1	no	—
186	#1974	D₄₇	1	1	yes	characteristic
187	#1945	—	47	1	no	—
188	#1945	—	47	1	no	—
189	#1937	—	47	2	no	—
190	#1937	—	47	2	no	—
191	C₄×#108	—	47	1	no	—
192	C₄×#104	—	47	2	no	—
193	C₄²×C₈	—	47	1	no	—
194	C₂×C₄₇	—	2	1	no	—
195	C₂×C₄₇	—	2	1	no	—
196	C₂×C₄₇	—	8	2	no	—
197	C₂×C₄₇	#2293	1	1	yes	characteristic
198	C₂×C₄₇	C₂×#443	1	2	yes	—
199	C₂×#135	C₂×D₄₇	1	1	yes	characteristic
200	C₄×#41	—	47	2	no	—
201	C₄×Q₁₆	—	47	2	no	—
202	C₄×SD₁₆	—	47	4	no	—
203	C₄×D₈	—	47	2	no	—
204	#433	—	94	4	no	—
205	#431	—	94	4	no	—
206	C₂×#108	—	47	1	no	—
207	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
208	#427	—	47	1	no	—
209	#424	—	47	2	no	—
210	#424	—	47	2	no	—
211	C₂×#104	—	47	2	no	—
212	C₂×#104	C₂×D₄₇	1	2	yes	—
213	#416	—	47	2	no	—
214	#408	—	47	2	no	—
215	#408	C₂×D₄₇	1	2	yes	—
216	C₄×M₁₆	—	47	2	no	—
217	C₂×C₄×C₈	—	47	1	no	—
218	C₂×C₄×C₈	—	47	1	no	—
219	C₂×C₄×C₈	—	47	1	no	—
220	C₂×C₄×C₈	—	47	1	no	—
221	C₂×C₄×C₈	—	47	1	no	—
222	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
223	C₄³	—	47	1	no	—
224	#341	—	47	2	no	—
225	#339	—	47	2	no	—
226	#339	—	47	2	no	—
227	C₄₇	#21292	1	1	yes	characteristic, complemented, Sylow
228	C₂×#41	C₂²×D₄₇	1	2	yes	—
229	#135	—	2	2	no	—
230	#135	—	2	2	no	—
231	C₂×Q₁₆	C₄×D₄₇	1	1	yes	characteristic
232	C₂×SD₁₆	C₄×D₄₇	1	2	yes	—
233	C₂×D₈	C₄×D₄₇	1	1	yes	characteristic
234	C₂×M₁₆	—	47	2	no	—
235	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
236	C₂²×C₈	—	47	1	no	—
237	C₂²×C₈	—	47	1	no	—
238	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
239	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
240	C₄×Q₈	—	47	4	no	—
241	C₄×D₄	—	47	4	no	—
242	C₄×D₄	—	94	4	no	—
243	#117	—	47	2	no	—
244	#117	—	94	2	no	—
245	C₂×C₄²	—	47	1	no	—
246	C₂×C₄²	—	47	1	no	—
247	C₂×C₄²	—	94	2	no	—
248	C₂×C₄²	—	94	2	no	—
249	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
250	#108	—	2	2	no	—
251	#108	—	94	2	no	—
252	#107	—	47	2	no	—
253	#106	—	47	2	no	—
254	C₄⋊C₈	—	94	4	no	—
255	C₄⋊C₈	—	94	4	no	—
256	#104	—	2	4	no	—
257	#104	—	94	4	no	—
258	#103	—	47	4	no	—
259	#102	—	47	4	no	—
260	C₂²⋊C₈	—	2	4	no	—
261	C₂²⋊C₈	—	94	4	no	—
262	#97	—	94	2	no	—
263	C₄×C₈	—	2	2	no	—
264	C₄×C₈	—	47	2	no	—
265	C₄×C₈	—	47	2	no	—
266	C₄×C₈	—	47	2	no	—
267	C₄×C₈	—	47	2	no	—
268	C₄×C₈	—	94	2	no	—
269	C₄×C₈	—	94	2	no	—
270	C₄×C₈	—	94	2	no	—
271	C₄×C₈	—	94	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	#41	—	2	2	no	—
276	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	47	2	no	—
282	C₂²×C₄	—	94	4	no	—
283	C₂²×C₄	—	94	1	no	—
284	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
285	Q₁₆	—	2	2	no	—
286	SD₁₆	—	2	4	no	—
287	D₈	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	94	2	no	—
291	M₁₆	—	94	2	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	47	2	no	—
296	C₂×C₈	—	47	2	no	—
297	C₂×C₈	—	94	1	no	—
298	C₂×C₈	—	94	1	no	—
299	C₂×C₈	—	94	1	no	—
300	C₂×C₈	—	94	1	no	—
301	C₂×C₈	—	94	2	no	—
302	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
304	C₂×C₈	D₄×D₄₇	1	2	yes	—
305	C₂×C₈	#10625352	1	2	yes	—
306	C₂×C₈	#10625352	1	2	yes	—
307	C₄⋊C₄	—	47	4	no	—
308	C₄⋊C₄	—	94	4	no	—
309	C₂²⋊C₄	—	94	4	no	—
310	C₂²⋊C₄	—	94	2	no	—
311	C₄²	—	2	1	no	—
312	C₄²	—	47	2	no	—
313	C₄²	—	47	1	no	—
314	C₄²	—	47	1	no	—
315	C₄²	—	47	2	no	—
316	C₄²	—	94	4	no	—
317	C₄²	—	94	8	no	—
318	C₄²	—	94	2	no	—
319	C₄²	—	94	1	no	—
320	C₄²	#10625352	1	2	yes	—
321	C₂³	—	2	2	no	—
322	C₂³	#41×D₄₇	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	4	2	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	47	2	no	—
335	C₂×C₄	—	47	2	no	—
336	C₂×C₄	—	94	4	no	—
337	C₂×C₄	—	94	4	no	—
338	C₂×C₄	—	94	2	no	—
339	C₂×C₄	—	94	4	no	—
340	C₂×C₄	—	188	8	no	—
341	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
343	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
344	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
345	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
346	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	94	2	no	—
350	C₈	—	94	2	no	—
351	C₈	—	188	2	no	—
352	C₈	#13093980	1	2	yes	—
353	C₈	#13093980	1	2	yes	—
354	C₂²	—	2	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	4	no	—
357	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
358	C₂²	#15689652	1	1	yes	characteristic
359	C₂²	#15689654	1	1	yes	characteristic
360	C₄	—	2	4	no	—
361	C₄	—	4	2	no	—
362	C₄	—	47	4	no	—
363	C₄	—	94	4	no	—
364	C₄	—	94	2	no	—
365	C₄	—	188	4	no	—
366	C₄	C₂×#13093980	1	1	yes	characteristic
367	C₄	C₄×#10625352	1	1	yes	characteristic
368	C₄	#15689668	1	1	yes	characteristic
369	C₄	#15689671	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	8	2	no	—
373	C₂	#20245850	1	1	yes	characteristic
374	C₂	#443×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1974	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#135×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1974	D₄₇
hypocenter	C₄₇	#21292
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1974

Complements

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

N	H ≅ G/N	type
2. #20252028	C₂	semidirect
4. #20244136	C₂	semidirect
5. #20244134	C₂	semidirect
6. #20244095	C₂	semidirect
7. #20244089	C₂	semidirect
8. #108×Dic₄₇	C₂	semidirect
9. #104×Dic₄₇	C₂	semidirect
10. C₄×C₈×Dic₄₇	C₂	semidirect
13. Q₁₆×Dic₄₇	C₄	semidirect
14. SD₁₆×Dic₄₇	C₄	semidirect
15. D₈×Dic₄₇	C₄	semidirect
227. C₄₇	#21292	semidirect

Maximal subgroups

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

subgroup	index
#135×Dic₄₇	2
#20252028	2
#20244277	2
#20244136	2
#20244134	2
#20244095	2
#20244089	2
#108×Dic₄₇	2
#104×Dic₄₇	2
C₄×C₈×Dic₄₇	2
C₄₇×#1974	2
#21292	47