Subgroups — #26125166

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688884	C₂	1	2	yes	maximal
3	#20243086	C₂	1	2	yes	maximal
4	C₂×#15688877	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#341	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×#13093983	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	#15689161	—	2	2	no	—
10	#15689161	—	2	2	no	—
11	C₂×#13093958	C₄	1	1	yes	characteristic, complemented
12	C₂×#13093958	C₄	1	1	yes	characteristic, complemented
13	C₂×#13093956	C₄	1	2	yes	complemented
14	#15688884	—	2	4	no	—
15	#15688877	—	2	4	no	—
16	#15688877	—	2	2	no	—
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#341	—	2	2	no	—
20	C₂³×Dic₄₇	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	1	no	—
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
26	#13093983	—	2	2	no	—
27	#13093983	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	#13093983	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	1	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	#13093958	—	2	2	no	—
39	#13093958	—	2	2	no	—
40	#13093956	—	2	4	no	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
48	C₂²×C₈×C₄₇	—	2	2	no	—
49	C₂²×C₈×C₄₇	—	2	2	no	—
50	C₂²×C₈×C₄₇	—	2	2	no	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	D₄	1	2	yes	—
54	C₂²×C₈×C₄₇	D₄	1	2	yes	—
55	C₂⁴×C₄₇	#41	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×Dic₄₇	—	2	4	no	—
67	C₂²×Dic₄₇	—	2	2	no	—
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂²×Dic₄₇	—	4	2	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	M₁₆×C₄₇	—	2	4	no	—
73	M₁₆×C₄₇	—	2	4	no	—
74	M₁₆×C₄₇	—	4	2	no	—
75	M₁₆×C₄₇	—	4	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	4	2	no	—
83	C₂×C₈×C₄₇	—	4	2	no	—
84	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
86	C₂²⋊Dic₄₇	—	4	2	no	—
87	C₂²⋊Dic₄₇	—	4	2	no	—
88	C₄⋊Dic₄₇	—	2	2	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	2	2	no	—
92	C₄×Dic₄₇	—	2	4	no	—
93	C₄×Dic₄₇	—	4	2	no	—
94	C₄×Dic₄₇	—	4	1	no	—
95	C₄×Dic₄₇	—	4	1	no	—
96	C₄×Dic₄₇	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	—	4	1	no	—
100	C₂×C₄₇⋊C₈	—	4	1	no	—
101	C₂×C₄₇⋊C₈	—	4	2	no	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	—	2	2	no	—
104	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
105	C₂³×C₄₇	#121	1	1	yes	characteristic
106	C₂³×C₄₇	#124	1	1	yes	characteristic
107	C₂³×C₄₇	#125	1	2	yes	—
108	C₂³×C₄₇	#126	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#117	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	#123	1	2	yes	—
118	C₂×C₄×C₄₇	#123	1	2	yes	—
119	C₂×C₄×C₄₇	#123	1	2	yes	—
120	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#125	1	2	yes	—
124	C₂×C₄×C₄₇	#126	1	2	yes	—
125	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	4	8	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₈×C₄₇	—	2	2	no	—
136	C₈×C₄₇	—	2	2	no	—
137	C₈×C₄₇	—	4	4	no	—
138	C₄₇⋊C₈	—	4	2	no	—
139	#11376	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	2	4	no	—
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	#387	1	2	yes	—
147	C₂²×C₄₇	#390	1	1	yes	characteristic
148	C₂²×C₄₇	#397	1	1	yes	characteristic
149	C₂²×C₄₇	#398	1	1	yes	characteristic
150	C₂²×C₄₇	#400	1	2	yes	—
151	C₂²×C₄₇	#443	1	2	yes	—
152	C₂²×C₄₇	#444	1	2	yes	—
153	C₄×C₄₇	#381	1	1	yes	characteristic
154	C₄×C₄₇	#387	1	2	yes	—
155	C₄×C₄₇	#391	1	1	yes	characteristic
156	C₄×C₄₇	#397	1	1	yes	characteristic
157	C₄×C₄₇	#398	1	1	yes	characteristic
158	C₄×C₄₇	#399	1	2	yes	—
159	Dic₄₇	—	4	4	no	—
160	Dic₄₇	—	8	2	no	—
161	Dic₄₇	—	8	2	no	—
162	C₂×#427	—	47	1	no	—
163	C₂×#341	D₄₇	1	1	yes	characteristic
164	C₂×#339	—	47	2	no	—
165	#1729	—	47	2	no	—
166	C₂×#336	—	47	1	no	—
167	C₂×C₄₇	—	2	1	no	—
168	C₂×C₄₇	—	2	1	no	—
169	C₂×C₄₇	—	2	2	no	—
170	C₂×C₄₇	#1445	1	1	yes	characteristic
171	C₂×C₄₇	#1922	1	4	yes	—
172	C₂×C₄₇	#1974	1	1	yes	characteristic
173	C₂×C₄₇	#1975	1	1	yes	characteristic
174	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
175	C₂³×C₈	—	47	1	no	—
176	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
177	C₂×#117	—	47	2	no	—
178	C₂²×C₄²	—	47	1	no	—
179	#427	—	94	2	no	—
180	#427	—	94	2	no	—
181	C₂×#107	—	47	1	no	—
182	C₂×#107	—	47	1	no	—
183	C₂×#106	—	47	2	no	—
184	#341	—	2	2	no	—
185	#339	—	94	4	no	—
186	#336	—	94	4	no	—
187	#336	—	94	2	no	—
188	C₄₇	#11376	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₄	—	94	1	no	—
190	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	Dic₉₄	1	4	yes	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	47	2	no	—
201	C₂²×C₈	—	94	1	no	—
202	C₂²×C₈	—	94	2	no	—
203	C₂²×C₈	—	94	2	no	—
204	C₂²×C₈	—	94	1	no	—
205	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
206	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₈	D₁₈₈	1	2	yes	—
208	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
209	#117	—	94	2	no	—
210	#117	—	94	2	no	—
211	#117	—	94	2	no	—
212	#117	—	94	2	no	—
213	C₂×C₄⋊C₄	—	47	2	no	—
214	C₂×C₄⋊C₄	—	47	2	no	—
215	C₂×C₂²⋊C₄	—	94	2	no	—
216	C₂×C₄²	—	47	2	no	—
217	C₂×C₄²	—	94	2	no	—
218	C₂×C₄²	—	94	2	no	—
219	C₂×C₄²	—	94	1	no	—
220	C₂×C₄²	—	94	2	no	—
221	#107	—	94	2	no	—
222	#107	—	94	2	no	—
223	#106	—	94	4	no	—
224	C₂⁴	#10625354	1	1	yes	characteristic
225	C₂²×C₄	—	94	4	no	—
226	C₂²×C₄	—	94	2	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	94	2	no	—
229	C₂²×C₄	—	188	2	no	—
230	C₂²×C₄	—	188	2	no	—
231	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
232	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
234	C₂²×C₄	#10625350	1	2	yes	—
235	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	#10625352	1	2	yes	—
238	C₂²×C₄	#10625352	1	2	yes	—
239	C₂²×C₄	#10625354	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	4	2	no	—
244	M₁₆	—	4	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	94	2	no	—
254	C₂×C₈	—	94	4	no	—
255	C₂×C₈	—	188	1	no	—
256	C₂×C₈	—	188	1	no	—
257	C₂×C₈	—	188	2	no	—
258	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
259	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
260	C₄⋊C₄	—	94	2	no	—
261	C₄⋊C₄	—	94	2	no	—
262	C₄⋊C₄	—	94	2	no	—
263	C₄⋊C₄	—	94	2	no	—
264	C₂²⋊C₄	—	188	2	no	—
265	C₂²⋊C₄	—	188	2	no	—
266	C₄²	—	94	4	no	—
267	C₄²	—	188	2	no	—
268	C₄²	—	188	1	no	—
269	C₄²	—	188	1	no	—
270	C₄²	—	188	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	#13093986	1	1	yes	characteristic
274	C₂³	#13093999	1	1	yes	characteristic
275	C₂³	#13094013	1	1	yes	characteristic
276	C₂³	#13094015	1	2	yes	—
277	C₂³	#13094019	1	2	yes	—
278	C₂×C₄	—	94	4	no	—
279	C₂×C₄	—	188	4	no	—
280	C₂×C₄	—	188	4	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	—	188	2	no	—
283	C₂×C₄	—	188	8	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
287	C₂×C₄	#13093980	1	2	yes	—
288	C₂×C₄	#13093986	1	1	yes	characteristic
289	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
290	C₂×C₄	#13093999	1	1	yes	characteristic
291	C₂×C₄	#13094001	1	2	yes	—
292	C₂×C₄	#13094001	1	2	yes	—
293	C₂×C₄	#13094003	1	2	yes	—
294	C₂×C₄	#13094005	1	2	yes	—
295	C₂×C₄	#13094006	1	2	yes	—
296	C₂×C₄	#13094006	1	2	yes	—
297	C₂×C₄	#13094008	1	1	yes	characteristic
298	C₂×C₄	#13094008	1	1	yes	characteristic
299	C₂×C₄	#13094012	1	1	yes	characteristic
300	C₂×C₄	#13094015	1	2	yes	—
301	C₂×C₄	#13094019	1	2	yes	—
302	C₂×C₄	#13094020	1	1	yes	characteristic
303	C₂×C₄	#13094024	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	4	4	no	—
307	C₈	—	188	2	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	#15689006	1	2	yes	—
315	C₂²	#15689024	1	1	yes	characteristic
316	C₂²	#15689050	1	1	yes	characteristic
317	C₂²	#15689055	1	1	yes	characteristic
318	C₂²	#15689062	1	2	yes	—
319	C₂²	#15689239	1	2	yes	—
320	C₂²	#15689244	1	2	yes	—
321	C₄	—	376	2	no	—
322	C₄	—	376	2	no	—
323	C₄	—	188	4	no	—
324	C₄	#15688984	1	1	yes	characteristic
325	C₄	#15689006	1	2	yes	—
326	C₄	#15689029	1	1	yes	characteristic
327	C₄	#15689050	1	1	yes	characteristic
328	C₄	#15689055	1	1	yes	characteristic
329	C₄	#15689059	1	2	yes	—
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	2	no	—
333	C₂	#20241657	1	1	yes	characteristic
334	C₂	#20243986	1	4	yes	—
335	C₂	#20244274	1	1	yes	characteristic
336	C₂	#20244279	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#341	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#341	D₄₇
hypocenter	C₄₇	#11376
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
11. C₂×#13093958	C₄	semidirect
12. C₂×#13093958	C₄	semidirect
13. C₂×#13093956	C₄	semidirect
188. C₄₇	#11376	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
C₂×#15688884	2
#20243086	2
C₂×#15688877	2
C₂×C₄₇×#341	2
#11376	47