Subgroups — #26121885

Order 12032.

838 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688877	C₂	1	2	yes	maximal
4	C₂×#15688877	C₂	1	2	yes	maximal
5	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	#15688969	—	2	4	no	—
10	C₂×C₄₇⋊#95	C₄	1	4	yes	—
11	#15688878	C₄	1	4	yes	complemented
12	#15688877	—	2	4	no	—
13	#15688877	—	2	4	no	—
14	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	C₄	1	4	yes	complemented
17	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
18	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	2	no	—
28	C₂×C₄×Dic₄₇	—	2	2	no	—
29	C₂×C₄×Dic₄₇	—	2	1	no	—
30	C₂×C₄×Dic₄₇	—	2	1	no	—
31	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
33	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
34	C₄₇⋊#95	—	2	8	no	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	—	2	2	no	—
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	#41	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
64	C₂²×Dic₄₇	—	2	2	no	—
65	C₂²×Dic₄₇	—	2	2	no	—
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	—	2	2	no	—
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	C₂²⋊C₄	1	4	yes	—
73	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	4	2	no	—
78	C₂×C₈×C₄₇	—	4	2	no	—
79	C₂×C₈×C₄₇	—	4	2	no	—
80	C₂×C₈×C₄₇	—	4	2	no	—
81	C₄²×C₄₇	—	2	4	no	—
82	C₄²×C₄₇	—	2	4	no	—
83	C₄×Dic₄₇	—	2	4	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	C₄×Dic₄₇	—	2	4	no	—
86	C₄×Dic₄₇	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
95	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₄₇	#117	1	1	yes	characteristic
98	C₂³×C₄₇	#117	1	1	yes	characteristic
99	C₂³×C₄₇	#117	1	1	yes	characteristic
100	C₂³×C₄₇	#117	1	1	yes	characteristic
101	C₂³×C₄₇	#131	1	4	yes	—
102	C₂³×C₄₇	#131	1	4	yes	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	#117	1	2	yes	—
109	C₂×C₄×C₄₇	#117	1	2	yes	—
110	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#121	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	#123	1	2	yes	—
119	C₂×C₄×C₄₇	#123	1	2	yes	—
120	C₂×C₄×C₄₇	#124	1	2	yes	—
121	C₂×C₄×C₄₇	#125	1	2	yes	—
122	C₂×C₄×C₄₇	#126	1	2	yes	—
123	C₂×C₄×C₄₇	#126	1	2	yes	—
124	C₂×Dic₄₇	—	2	4	no	—
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	2	8	no	—
127	C₂×Dic₄₇	—	2	8	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₈×C₄₇	—	4	4	no	—
135	C₄₇⋊C₈	—	4	4	no	—
136	#10796	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	#376	1	1	yes	characteristic
138	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₄₇	#380	1	1	yes	characteristic
141	C₂²×C₄₇	#381	1	1	yes	characteristic
142	C₂²×C₄₇	#382	1	1	yes	characteristic
143	C₂²×C₄₇	#383	1	1	yes	characteristic
144	C₂²×C₄₇	#405	1	2	yes	—
145	C₂²×C₄₇	#405	1	2	yes	—
146	C₂²×C₄₇	#408	1	4	yes	—
147	C₂²×C₄₇	#424	1	4	yes	—
148	C₂²×C₄₇	#433	1	4	yes	—
149	C₂²×C₄₇	#433	1	2	yes	—
150	C₂²×C₄₇	#433	1	2	yes	—
151	C₂²×C₄₇	#433	1	4	yes	—
152	C₂²×C₄₇	#433	1	2	yes	—
153	C₂²×C₄₇	#433	1	2	yes	—
154	C₄×C₄₇	—	4	4	no	—
155	C₄×C₄₇	#387	1	2	yes	—
156	C₄×C₄₇	#387	1	2	yes	—
157	C₄×C₄₇	#388	1	2	yes	—
158	C₄×C₄₇	#388	1	2	yes	—
159	Dic₄₇	—	4	4	no	—
160	Dic₄₇	—	4	4	no	—
161	C₂×#376	—	47	1	no	—
162	C₂×#336	—	47	1	no	—
163	C₂×#336	—	47	2	no	—
164	C₂×#336	—	47	2	no	—
165	C₂×#336	D₄₇	1	1	yes	characteristic
166	C₂×C₄₇	#1434	1	1	yes	characteristic
167	C₂×C₄₇	#1746	1	1	yes	characteristic
168	C₂×C₄₇	#1746	1	1	yes	characteristic
169	C₂×C₄₇	#1750	1	2	yes	—
170	C₂×C₄₇	#1768	1	2	yes	—
171	C₂×C₄₇	#1798	1	2	yes	—
172	C₂×C₄₇	#1818	1	2	yes	—
173	C₂×C₄₇	#1832	1	2	yes	—
174	C₂×C₄₇	#1840	1	2	yes	—
175	C₂³×C₈	—	47	2	no	—
176	C₂³×C₈	C₂×D₄₇	1	2	yes	—
177	C₂²×C₄²	—	47	1	no	—
178	C₂²×C₄²	—	47	1	no	—
179	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
180	#376	—	94	4	no	—
181	C₂×#95	—	47	4	no	—
182	#336	—	47	4	no	—
183	#336	—	94	4	no	—
184	#336	—	94	4	no	—
185	#336	Dic₄₇	1	4	yes	complemented
186	C₄₇	#10796	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₄	—	47	2	no	—
188	C₂³×C₄	—	47	2	no	—
189	C₂³×C₄	C₄×D₄₇	1	2	yes	—
190	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	47	4	no	—
198	C₂²×C₈	—	94	2	no	—
199	C₂²×C₈	—	94	2	no	—
200	C₂²×C₈	—	94	2	no	—
201	C₂²×C₈	—	94	2	no	—
202	C₂²×C₈	—	94	2	no	—
203	C₂²×C₈	—	94	2	no	—
204	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	47	2	no	—
208	C₂×C₄²	—	47	4	no	—
209	C₂×C₄²	—	47	2	no	—
210	C₂×C₄²	—	94	2	no	—
211	C₂×C₄²	—	94	2	no	—
212	C₂×C₄²	—	94	2	no	—
213	C₂×C₄²	—	94	1	no	—
214	C₂×C₄²	—	94	1	no	—
215	C₂×C₄²	C₄×D₄₇	1	2	yes	—
216	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
217	#95	—	94	8	no	—
218	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	47	4	no	—
224	C₂²×C₄	—	47	4	no	—
225	C₂²×C₄	—	94	2	no	—
226	C₂²×C₄	—	94	2	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	94	2	no	—
229	C₂²×C₄	—	94	4	no	—
230	C₂²×C₄	—	94	4	no	—
231	C₂²×C₄	—	94	4	no	—
232	C₂²×C₄	—	94	4	no	—
233	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
234	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
235	C₂²×C₄	#10625350	1	2	yes	—
236	C₂²×C₄	#10625350	1	2	yes	—
237	C₂²×C₄	D₄×D₄₇	1	2	yes	—
238	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	#10625352	1	2	yes	—
240	C₂²×C₄	#10625352	1	1	yes	characteristic
241	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	#10625354	1	1	yes	characteristic
243	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	94	4	no	—
252	C₂×C₈	—	94	4	no	—
253	C₂×C₈	—	94	4	no	—
254	C₂×C₈	—	188	2	no	—
255	C₂×C₈	—	188	2	no	—
256	C₂×C₈	—	188	2	no	—
257	C₂×C₈	—	188	2	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	94	4	no	—
261	C₄²	—	94	4	no	—
262	C₄²	—	94	4	no	—
263	C₄²	—	94	4	no	—
264	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
265	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
266	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
267	C₂³	#13093980	1	1	yes	characteristic
268	C₂³	#13093981	1	1	yes	characteristic
269	C₂³	#13093981	1	1	yes	characteristic
270	C₂³	#13093982	1	1	yes	characteristic
271	C₂³	#13094030	1	4	yes	—
272	C₂³	#13094032	1	4	yes	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	94	4	no	—
279	C₂×C₄	—	94	4	no	—
280	C₂×C₄	—	94	8	no	—
281	C₂×C₄	—	94	8	no	—
282	C₂×C₄	—	188	4	no	—
283	C₂×C₄	—	188	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	—	188	2	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	#13093981	1	2	yes	—
289	C₂×C₄	#13093983	1	2	yes	—
290	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
291	C₂×C₄	#13093986	1	1	yes	characteristic
292	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
293	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
294	C₂×C₄	#13093989	1	1	yes	characteristic
295	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
296	C₂×C₄	#13093997	1	2	yes	—
297	C₂×C₄	#13094002	1	2	yes	—
298	C₂×C₄	#13094006	1	2	yes	—
299	C₂×C₄	#13094009	1	2	yes	—
300	C₂×C₄	#13094010	1	2	yes	—
301	C₂×C₄	#13094014	1	2	yes	—
302	C₂×C₄	#13094017	1	2	yes	—
303	C₂×C₄	#13094019	1	2	yes	—
304	C₈	—	4	4	no	—
305	C₈	—	188	4	no	—
306	C₂²	#15688969	1	1	yes	characteristic
307	C₂²	C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
308	C₂²	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
309	C₂²	#15688982	1	1	yes	characteristic
310	C₂²	#15688985	1	1	yes	characteristic
311	C₂²	#15688987	1	1	yes	characteristic
312	C₂²	#15688990	1	1	yes	characteristic
313	C₂²	#15689078	1	2	yes	—
314	C₂²	#15689081	1	2	yes	—
315	C₂²	#15689088	1	4	yes	—
316	C₂²	#15689148	1	4	yes	—
317	C₂²	#15689181	1	4	yes	—
318	C₂²	#15689182	1	2	yes	—
319	C₂²	#15689182	1	2	yes	—
320	C₂²	#15689183	1	4	yes	—
321	C₂²	#15689185	1	2	yes	—
322	C₂²	#15689185	1	2	yes	—
323	C₄	—	4	4	no	—
324	C₄	—	188	4	no	—
325	C₄	—	188	4	no	—
326	C₄	#15689005	1	2	yes	—
327	C₄	#15689009	1	2	yes	—
328	C₄	#15689012	1	2	yes	—
329	C₄	#15689016	1	2	yes	—
330	C₂	#20241613	1	1	yes	characteristic
331	C₂	#20243139	1	1	yes	characteristic
332	C₂	#20243141	1	1	yes	characteristic
333	C₂	#20243156	1	2	yes	—
334	C₂	#20243247	1	2	yes	—
335	C₂	#20243371	1	2	yes	—
336	C₂	#20243458	1	2	yes	—
337	C₂	#20243506	1	2	yes	—
338	C₂	#20243532	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂×C₄²
Fitting	C₂×C₄₇×#336	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	#10796
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
11. #15688878	C₄	semidirect
16. C₄₇×#336	C₄	semidirect
185. #336	Dic₄₇	semidirect
186. C₄₇	#10796	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688969	2
C₂×#15688878	2
C₂×#15688877	2
C₂×C₄₇×#336	2
#10796	47