Subgroups — #26234578

Order 12032.

1050 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689236	C₂	1	2	yes	maximal
2	C₂×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic
3	#20246299	C₂	1	2	yes	maximal, complemented
4	#20244126	C₂	1	2	yes	maximal, complemented
5	#20243638	C₂	1	4	yes	maximal, complemented
6	#20243552	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243399	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₈₈	C₂²	1	4	yes	—
11	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
12	#15689236	—	2	8	no	—
13	C₄×D₃₇₆	C₄	1	8	yes	complemented
14	C₂×#13093958	C₂²	1	2	yes	—
15	C₂×#13093958	C₂²	1	1	yes	characteristic
16	C₂×#13093956	C₂²	1	2	yes	—
17	C₂×#13093946	C₂²	1	4	yes	—
18	C₂×#13093946	C₂²	1	2	yes	—
19	#15689021	C₂²	1	4	yes	—
20	#15688994	—	2	4	no	—
21	#15688989	C₂²	1	2	yes	—
22	#15688982	—	2	2	no	—
23	#95×D₄₇	—	2	2	no	—
24	#15688886	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#336	C₂²	1	2	yes	—
30	C₂²×D₁₈₈	C₂³	1	2	yes	—
31	C₂²×C₄×D₄₇	—	2	2	no	—
32	C₂²×C₄×D₄₇	—	2	4	no	—
33	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
34	C₂×D₃₇₆	D₄	1	4	yes	—
35	C₂×D₃₇₆	Q₈	1	4	yes	—
36	C₄×D₁₈₈	—	2	4	no	—
37	C₄×D₁₈₈	—	2	8	no	—
38	C₄×D₁₈₈	—	4	4	no	—
39	C₄×D₁₈₈	C₂×C₄	1	8	yes	complemented
40	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×Dic₄₇⋊C₄	—	2	4	no	—
43	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
44	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
45	#13093958	—	4	2	no	—
46	#13093958	C₂×C₄	1	4	yes	complemented
47	#13093956	—	2	4	no	—
48	#13093946	—	2	8	no	—
49	#13093946	C₂×C₄	1	8	yes	complemented
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	4	no	—
52	C₄₇⋊#95	—	2	2	no	—
53	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₇	—	2	2	no	—
60	#97×C₄₇	—	2	4	no	—
61	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
62	#95×C₄₇	D₄	1	2	yes	—
63	C₂³×D₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂×D₁₈₈	—	2	2	no	—
71	C₂×D₁₈₈	—	2	2	no	—
72	C₂×D₁₈₈	—	2	2	no	—
73	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
74	C₂×D₁₈₈	C₂×D₄	1	4	yes	—
75	C₂×D₁₈₈	C₂×Q₈	1	4	yes	—
76	C₂×C₄×D₄₇	—	2	8	no	—
77	C₂×C₄×D₄₇	—	4	8	no	—
78	C₂×C₄×D₄₇	—	4	8	no	—
79	C₂×C₄×D₄₇	—	8	4	no	—
80	C₂×C₄×D₄₇	—	8	4	no	—
81	C₂×C₈×C₄₇	—	2	1	no	—
82	C₂×C₈×C₄₇	—	2	1	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
90	C₄⋊C₄×C₄₇	—	4	2	no	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₄²×C₄₇	—	2	2	no	—
93	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₄₇	#41	1	2	yes	—
95	(C₂×D₄₇)⋊C₄	—	2	8	no	—
96	(C₂×D₄₇)⋊C₄	—	4	8	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
101	Dic₄₇⋊C₄	—	8	4	no	—
102	D₃₇₆	C₄⋊C₄	1	16	yes	complemented
103	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂²×D₄₇	—	2	4	no	—
105	C₂²×D₄₇	—	2	4	no	—
106	C₂²×D₄₇	—	2	4	no	—
107	C₂²×D₄₇	—	4	4	no	—
108	C₂²×D₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×Dic₄₇	—	2	4	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	4	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	D₁₈₈	—	2	4	no	—
128	D₁₈₈	—	2	4	no	—
129	D₁₈₈	—	2	4	no	—
130	D₁₈₈	C₂×C₄⋊C₄	1	8	yes	—
131	C₄×D₄₇	—	8	8	no	—
132	C₄×D₄₇	—	8	16	no	—
133	C₈×C₄₇	—	2	2	no	—
134	C₈×C₄₇	—	4	2	no	—
135	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
136	#22672	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₄₇	D₄²	1	1	yes	characteristic
139	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
140	C₂²×C₄₇	#575	1	2	yes	—
141	C₂²×C₄₇	#576	1	2	yes	—
142	C₂×D₄₇	—	2	8	no	—
143	C₂×D₄₇	—	4	8	no	—
144	C₂×D₄₇	—	4	4	no	—
145	C₂×D₄₇	—	4	4	no	—
146	C₂×D₄₇	—	4	8	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	4	4	no	—
150	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₄₇	#520	1	1	yes	characteristic
152	C₄×C₄₇	#547	1	1	yes	characteristic
153	C₄×C₄₇	#554	1	1	yes	characteristic
154	Dic₄₇	—	4	4	no	—
155	Dic₄₇	—	8	4	no	—
156	C₂×#442	—	47	2	no	—
157	C₂×C₄×D₈	—	47	1	no	—
158	#2349	—	47	2	no	—
159	#1943	—	47	2	no	—
160	#1862	—	47	4	no	—
161	#1846	—	47	1	no	—
162	#1804	—	47	2	no	—
163	#1770	D₄₇	1	1	yes	characteristic, complemented
164	C₂×C₄₇	D₄×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄₇	#2943	1	1	yes	characteristic
166	C₂×C₄₇	#2948	1	1	yes	characteristic
167	C₂×C₄₇	#3273	1	2	yes	—
168	C₂×C₄₇	#3389	1	2	yes	—
169	D₄₇	—	4	8	no	—
170	C₂²×D₈	—	47	1	no	—
171	C₂×C₄×D₄	—	47	2	no	—
172	C₂×C₄×D₄	—	47	4	no	—
173	#442	—	94	8	no	—
174	C₄×D₈	—	47	8	no	—
175	C₂×#107	—	47	1	no	—
176	C₂×#107	—	47	2	no	—
177	C₂×#106	—	47	2	no	—
178	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
179	C₂×#102	—	47	2	no	—
180	C₂×#102	—	47	4	no	—
181	C₂×#97	C₂×D₄₇	1	2	yes	—
182	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
183	#389	—	47	4	no	—
184	#385	—	94	4	no	—
185	#382	—	47	2	no	—
186	#380	—	94	2	no	—
187	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
188	C₂×#95	—	94	2	no	—
189	#340	—	47	4	no	—
190	#336	C₂×D₄₇	1	2	yes	—
191	C₄₇	#22672	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₄	—	47	2	no	—
193	C₂³×C₄	—	94	2	no	—
194	C₂³×C₄	—	94	4	no	—
195	C₂×D₈	—	47	4	no	—
196	C₂×D₈	—	47	4	no	—
197	C₂×D₈	—	47	4	no	—
198	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
199	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
201	C₄×D₄	—	47	8	no	—
202	C₄×D₄	—	94	8	no	—
203	C₄×D₄	—	94	4	no	—
204	C₄×D₄	—	188	4	no	—
205	C₂×C₄⋊C₄	—	47	2	no	—
206	C₂×C₄⋊C₄	—	47	4	no	—
207	C₂×C₄⋊C₄	—	94	4	no	—
208	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
209	C₂×C₂²⋊C₄	—	94	2	no	—
210	C₂×C₂²⋊C₄	—	94	4	no	—
211	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
212	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
213	#107	—	47	4	no	—
214	#107	—	188	2	no	—
215	#106	—	94	4	no	—
216	C₄⋊C₈	—	2	2	no	—
217	#102	—	47	8	no	—
218	#102	—	94	8	no	—
219	#97	—	2	4	no	—
220	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
221	#95	—	94	2	no	—
222	#95	—	94	4	no	—
223	#95	—	94	2	no	—
224	#95	D₁₈₈	1	2	yes	—
225	C₂⁴	—	94	2	no	—
226	C₂×D₄	—	47	4	no	—
227	C₂×D₄	—	47	4	no	—
228	C₂×D₄	—	47	4	no	—
229	C₂×D₄	—	94	2	no	—
230	C₂×D₄	—	94	2	no	—
231	C₂×D₄	—	94	2	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	—	94	8	no	—
234	C₂²×C₄	—	94	4	no	—
235	C₂²×C₄	—	376	4	no	—
236	C₂²×C₄	—	376	4	no	—
237	C₂²×C₄	—	188	8	no	—
238	C₂²×C₄	—	188	8	no	—
239	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
240	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
243	D₈	—	47	16	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
251	C₂×C₈	D₄×D₄₇	1	2	yes	—
252	C₂×C₈	Q₈×D₄₇	1	2	yes	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	47	4	no	—
255	C₄⋊C₄	—	94	2	no	—
256	C₄⋊C₄	—	94	4	no	—
257	C₄⋊C₄	—	94	4	no	—
258	C₄⋊C₄	—	376	4	no	—
259	C₂²⋊C₄	—	94	8	no	—
260	C₂²⋊C₄	—	188	8	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	C₂×C₄×D₄₇	1	2	yes	—
264	C₄²	#10625350	1	2	yes	—
265	C₂³	—	94	4	no	—
266	C₂³	—	94	4	no	—
267	C₂³	—	94	4	no	—
268	C₂³	—	188	4	no	—
269	C₂³	—	188	4	no	—
270	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
271	D₄	—	47	8	no	—
272	D₄	—	94	4	no	—
273	D₄	—	94	4	no	—
274	D₄	—	94	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	94	4	no	—
281	C₂×C₄	—	376	8	no	—
282	C₂×C₄	—	376	16	no	—
283	C₂×C₄	—	188	4	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	—	188	4	no	—
287	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
288	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
289	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
293	C₂×C₄	#13094070	1	1	yes	characteristic
294	C₂×C₄	#13094071	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	4	2	no	—
297	C₈	C₄⋊C₄×D₄₇	1	4	yes	—
298	C₂²	—	94	8	no	—
299	C₂²	—	188	8	no	—
300	C₂²	—	188	4	no	—
301	C₂²	—	188	4	no	—
302	C₂²	—	188	8	no	—
303	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
304	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
305	C₂²	Q₈×D₁₈₈	1	1	yes	characteristic
306	C₂²	#15689923	1	2	yes	—
307	C₂²	#15689928	1	2	yes	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	4	4	no	—
311	C₄	—	376	4	no	—
312	C₄	—	188	4	no	—
313	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
314	C₄	#15689603	1	1	yes	characteristic
315	C₄	#15689778	1	1	yes	characteristic
316	C₄	#15689833	1	1	yes	characteristic
317	C₂	—	188	8	no	—
318	C₂	C₄⋊C₄×D₁₈₈	1	1	yes	characteristic
319	C₂	#20252190	1	1	yes	characteristic
320	C₂	#20252241	1	1	yes	characteristic
321	C₂	#20255000	1	2	yes	—
322	C₂	#20256348	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1770	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₃₇₆	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1770	D₄₇
hypocenter	C₄₇	#22672
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
3. #20246299	C₂	semidirect
4. #20244126	C₂	semidirect
5. #20243638	C₂	semidirect
6. #20243552	C₂	semidirect
7. #20243399	C₂	semidirect
8. C₄₇×#1770	C₂	semidirect
13. C₄×D₃₇₆	C₄	semidirect
39. C₄×D₁₈₈	C₂×C₄	semidirect
46. #13093958	C₂×C₄	semidirect
49. #13093946	C₂×C₄	semidirect
61. C₄×C₈×C₄₇	C₂×C₄	semidirect
102. D₃₇₆	C₄⋊C₄	semidirect
163. #1770	D₄₇	semidirect
191. C₄₇	#22672	semidirect
220. C₄×C₈	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689236	2
C₂×C₄×D₃₇₆	2
#20246299	2
#20244126	2
#20243638	2
#20243552	2
#20243399	2
C₄₇×#1770	2
#22672	47