Subgroups — #26188027

Order 12032.

652 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	#20252514	C₂	1	2	yes	maximal, complemented
2	C₄×#13094046	C₂	1	1	yes	maximal, characteristic
3	#20242296	C₂	1	2	yes	maximal, complemented
4	#20242223	C₂	1	4	yes	maximal, complemented
5	#20242211	C₂	1	2	yes	maximal, complemented
6	#20241825	C₂	1	2	yes	maximal, complemented
7	#20241677	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#1566	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₄	1	1	yes	characteristic
10	C₄×#10625350	C₂²	1	2	yes	—
11	#15689250	C₂²	1	4	yes	—
12	C₄×Dic₁₈₈	C₄	1	1	yes	characteristic
13	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
14	C₄×#10625325	C₄	1	2	yes	—
15	C₄×#10625325	C₂²	1	2	yes	—
16	C₄×D₃₇₆	C₄	1	1	yes	characteristic
17	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
18	#15689192	—	2	4	no	—
19	#15689188	C₂²	1	4	yes	—
20	#15689179	—	2	2	no	—
21	#15689176	C₂²	1	1	yes	characteristic
22	#15689161	C₄	1	1	yes	characteristic
23	C₄⋊C₈×D₄₇	—	2	2	no	—
24	#15689120	C₄	1	2	yes	—
25	C₄×#10625329	C₂²	1	1	yes	characteristic
26	#15688853	C₂²	1	4	yes	—
27	#15688850	C₂²	1	4	yes	—
28	#15688845	C₂²	1	2	yes	—
29	#15688844	C₂²	1	2	yes	—
30	C₄₇×#423	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#334	C₂²	1	2	yes	—
33	C₄₇×#334	C₂²	1	2	yes	—
34	C₂×#10625350	C₂×C₄	1	2	yes	—
35	#13094046	—	2	4	no	—
36	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
37	C₂×#10625325	C₂×C₄	1	2	yes	—
38	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
39	C₂×#10625329	—	2	1	no	—
40	C₂×C₈×D₄₇	—	2	4	no	—
41	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
42	C₄×Dic₉₄	C₂³	1	2	yes	—
43	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
44	C₄×D₁₈₈	C₂³	1	2	yes	—
45	C₄×C₂²⋊D₄₇	—	2	2	no	—
46	C₄×C₂²⋊D₄₇	—	2	2	no	—
47	#13093983	C₂×C₄	1	2	yes	—
48	#13093981	—	2	2	no	—
49	C₄²×D₄₇	—	2	2	no	—
50	#13093958	C₂×C₄	1	1	yes	characteristic
51	#13093958	C₂×C₄	1	1	yes	characteristic
52	#13093956	C₂×C₄	1	2	yes	—
53	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
54	Dic₄₇⋊C₈	—	2	4	no	—
55	#13093949	C₂×C₄	1	2	yes	—
56	#13093949	C₂×C₄	1	2	yes	—
57	#13093946	C₂×C₄	1	2	yes	—
58	#13093946	C₂×C₄	1	2	yes	—
59	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
60	(C₂×D₄₇)⋊C₈	—	2	4	no	—
61	#13093935	—	2	1	no	—
62	#13093934	—	2	4	no	—
63	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
64	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
65	C₂×M₁₆×C₄₇	—	2	2	no	—
66	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₄₇	—	2	2	no	—
70	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
71	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
78	C₂×C₂²⋊D₄₇	—	2	2	no	—
79	#10625350	—	2	4	no	—
80	#10625350	—	4	2	no	—
81	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
82	C₂×C₄×D₄₇	—	2	2	no	—
83	C₂×C₄×D₄₇	—	2	2	no	—
84	C₂×C₄×D₄₇	—	2	2	no	—
85	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
86	M₁₆×C₄₇	—	8	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	4	no	—
89	C₂×C₈×C₄₇	—	4	1	no	—
90	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₄₇	#41	1	1	yes	characteristic
97	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₄₇	—	2	2	no	—
99	(C₂×D₄₇)⋊C₄	—	2	2	no	—
100	(C₂×D₄₇)⋊C₄	—	2	2	no	—
101	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
102	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
103	Dic₄₇⋊C₄	—	2	2	no	—
104	Dic₄₇⋊C₄	—	2	2	no	—
105	C₄×Dic₄₇	—	2	2	no	—
106	C₄×Dic₄₇	—	2	2	no	—
107	C₄×Dic₄₇	—	2	2	no	—
108	#10625329	—	4	2	no	—
109	C₂×C₄₇⋊C₈	—	2	2	no	—
110	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
111	Dic₁₈₈	—	2	2	no	—
112	D₃₇₆	M₁₆	1	4	yes	complemented
113	#10625325	—	2	4	no	—
114	C₈×D₄₇	—	8	4	no	—
115	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
116	C₂²×D₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
127	C₂²⋊D₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	2	2	no	—
129	C₂×Dic₄₇	—	2	2	no	—
130	C₂×Dic₄₇	—	2	2	no	—
131	D₁₈₈	—	2	2	no	—
132	D₁₈₈	C₂×M₁₆	1	4	yes	—
133	C₄×D₄₇	—	4	4	no	—
134	C₄×D₄₇	—	4	2	no	—
135	C₄×D₄₇	—	4	2	no	—
136	C₄×D₄₇	—	4	4	no	—
137	Dic₉₄	—	2	2	no	—
138	Dic₉₄	—	2	2	no	—
139	C₈×C₄₇	—	2	1	no	—
140	C₈×C₄₇	—	2	1	no	—
141	C₈×C₄₇	—	2	1	no	—
142	C₈×C₄₇	—	4	4	no	—
143	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
144	C₄₇⋊C₈	—	2	4	no	—
145	C₄₇⋊C₈	—	4	2	no	—
146	#18065	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×D₄₇	—	2	4	no	—
152	C₂×D₄₇	—	4	2	no	—
153	C₂×D₄₇	—	4	2	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	#444	1	2	yes	—
158	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
159	C₄×C₄₇	#568	1	1	yes	characteristic
160	C₄×C₄₇	#577	1	2	yes	—
161	Dic₄₇	—	4	2	no	—
162	Dic₄₇	—	4	2	no	—
163	Dic₄₇	—	4	2	no	—
164	#2990	—	47	2	no	—
165	C₄×#135	—	47	1	no	—
166	#1593	—	47	2	no	—
167	#1582	—	47	4	no	—
168	#1580	—	47	2	no	—
169	#1566	D₄₇	1	1	yes	characteristic, complemented
170	#1489	—	47	2	no	—
171	#1451	—	47	1	no	—
172	C₂×C₄₇	—	4	1	no	—
173	C₂×C₄₇	D₄×M₁₆	1	1	yes	characteristic
174	C₂×C₄₇	#2952	1	1	yes	characteristic
175	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
176	D₄₇	—	4	4	no	—
177	C₂×#135	—	47	1	no	—
178	C₄×#41	—	47	2	no	—
179	#446	—	47	4	no	—
180	C₄×Q₁₆	—	47	1	no	—
181	C₄×Q₁₆	—	47	1	no	—
182	C₄×SD₁₆	—	47	2	no	—
183	C₄×SD₁₆	—	47	2	no	—
184	C₄×D₈	—	47	1	no	—
185	C₄×D₈	—	47	1	no	—
186	#435	—	94	4	no	—
187	C₈×D₄	—	47	4	no	—
188	#432	—	94	2	no	—
189	#431	—	47	1	no	—
190	#427	—	47	1	no	—
191	#423	C₂×D₄₇	1	2	yes	—
192	C₂×C₄⋊C₈	—	94	2	no	—
193	#416	—	47	2	no	—
194	C₄×M₁₆	—	47	1	no	—
195	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
196	#334	C₂×D₄₇	1	2	yes	—
197	#334	C₂×D₄₇	1	2	yes	—
198	Q₈⋊C₈	—	47	4	no	—
199	D₄⋊C₈	—	47	4	no	—
200	#322	—	47	2	no	—
201	#322	—	47	2	no	—
202	C₄₇	#18065	1	1	yes	characteristic, complemented, Sylow
203	C₂×#41	—	47	2	no	—
204	#135	—	94	4	no	—
205	C₂×Q₁₆	—	47	1	no	—
206	C₂×SD₁₆	—	47	2	no	—
207	C₂×D₈	—	47	1	no	—
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	—	94	1	no	—
210	C₂²×C₈	—	94	4	no	—
211	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
213	C₄×Q₈	—	47	2	no	—
214	C₄×Q₈	—	47	2	no	—
215	C₄×D₄	—	47	2	no	—
216	C₄×D₄	—	47	2	no	—
217	C₄×D₄	—	94	2	no	—
218	C₄×D₄	—	94	2	no	—
219	#117	—	47	2	no	—
220	#117	—	94	2	no	—
221	C₂×C₄²	—	94	2	no	—
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	#107	—	47	1	no	—
224	#107	—	47	1	no	—
225	#106	—	47	2	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	47	2	no	—
228	C₄⋊C₈	—	94	4	no	—
229	C₄⋊C₈	C₂²×D₄₇	1	4	yes	—
230	#103	—	47	2	no	—
231	#103	—	47	2	no	—
232	#102	—	47	2	no	—
233	#102	—	47	2	no	—
234	C₂²⋊C₈	—	47	2	no	—
235	C₂²⋊C₈	—	94	4	no	—
236	#97	—	94	4	no	—
237	#97	—	94	1	no	—
238	C₄×C₈	—	47	2	no	—
239	C₄×C₈	—	47	2	no	—
240	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
241	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
244	#41	—	94	4	no	—
245	#41	—	188	2	no	—
246	C₂×Q₈	—	47	2	no	—
247	C₂×D₄	—	47	2	no	—
248	C₂×D₄	—	94	2	no	—
249	C₂²×C₄	—	94	2	no	—
250	C₂²×C₄	—	94	2	no	—
251	C₂²×C₄	—	94	2	no	—
252	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
254	C₂²×C₄	#10625354	1	1	yes	characteristic
255	Q₁₆	—	94	2	no	—
256	SD₁₆	—	94	4	no	—
257	D₈	—	47	4	no	—
258	M₁₆	—	8	2	no	—
259	M₁₆	—	188	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	47	4	no	—
264	C₂×C₈	—	94	2	no	—
265	C₂×C₈	—	376	4	no	—
266	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
270	C₄⋊C₄	—	47	2	no	—
271	C₄⋊C₄	—	47	2	no	—
272	C₄⋊C₄	—	94	2	no	—
273	C₄⋊C₄	—	94	2	no	—
274	C₂²⋊C₄	—	94	2	no	—
275	C₂²⋊C₄	—	94	2	no	—
276	C₂²⋊C₄	—	94	2	no	—
277	C₄²	—	94	2	no	—
278	C₄²	—	94	2	no	—
279	C₄²	—	94	2	no	—
280	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
281	C₄²	D₄×D₄₇	1	1	yes	characteristic
282	C₄²	#10625354	1	1	yes	characteristic
283	C₄²	C₂³×D₄₇	1	1	yes	characteristic
284	C₂³	—	94	2	no	—
285	C₂³	#13093986	1	1	yes	characteristic
286	Q₈	—	94	2	no	—
287	Q₈	—	94	2	no	—
288	D₄	—	47	4	no	—
289	D₄	—	94	2	no	—
290	D₄	—	188	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	94	2	no	—
296	C₂×C₄	—	94	2	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	188	4	no	—
299	C₂×C₄	—	188	4	no	—
300	C₂×C₄	—	188	2	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	#13093986	1	1	yes	characteristic
303	C₂×C₄	#13093986	1	1	yes	characteristic
304	C₂×C₄	#13093986	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
307	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	4	no	—
312	C₈	—	94	4	no	—
313	C₈	—	188	2	no	—
314	C₈	M₁₆×D₄₇	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	94	4	no	—
319	C₂²	—	188	2	no	—
320	C₂²	—	188	2	no	—
321	C₂²	C₂×#13093986	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	188	2	no	—
326	C₄	—	188	2	no	—
327	C₄	—	188	2	no	—
328	C₄	#15689244	1	2	yes	—
329	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
330	C₄	#15689889	1	1	yes	characteristic
331	C₄	#15689929	1	2	yes	—
332	C₂	—	4	1	no	—
333	C₂	—	188	4	no	—
334	C₂	#20252123	1	1	yes	characteristic
335	C₂	#20252271	1	1	yes	characteristic
336	C₂	C₂×#15689244	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1566	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#13094046	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1566	D₄₇
hypocenter	C₄₇	#18065
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
1. #20252514	C₂	semidirect
3. #20242296	C₂	semidirect
4. #20242223	C₂	semidirect
5. #20242211	C₂	semidirect
6. #20241825	C₂	semidirect
7. #20241677	C₂	semidirect
8. C₄₇×#1566	C₂	semidirect
112. D₃₇₆	M₁₆	semidirect
169. #1566	D₄₇	semidirect
202. C₄₇	#18065	semidirect

Maximal subgroups

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

subgroup	index
#20252514	2
C₄×#13094046	2
#20242296	2
#20242223	2
#20242211	2
#20241825	2
#20241677	2
C₄₇×#1566	2
#18065	47