Subgroups — #26219645

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 316 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	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic
3	#20246008	C₂	1	2	yes	maximal
4	#20244131	C₂	1	4	yes	maximal, complemented
5	#20244121	C₂	1	4	yes	maximal, complemented
6	#20243108	C₂	1	2	yes	maximal
7	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	#15689161	C₄	1	8	yes	complemented
11	C₂×#13093958	—	2	4	no	—
12	C₂×#13093958	C₂²	1	2	yes	—
13	C₂×#13093958	C₂²	1	2	yes	—
14	C₂×#13093958	C₂²	1	2	yes	—
15	C₂×#13093956	C₂²	1	2	yes	—
16	#15689021	C₂²	1	8	yes	—
17	#15688995	—	2	4	no	—
18	#15688992	—	2	2	no	—
19	#15688988	—	2	2	no	—
20	#15688886	C₂²	1	8	yes	—
21	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#405	—	2	4	no	—
25	C₂×#97×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
27	C₂³×Dic₄₇	—	2	2	no	—
28	#13093983	—	2	4	no	—
29	#13093983	C₂×C₄	1	8	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×Dic₄₇⋊C₄	—	2	8	no	—
36	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	—	2	2	no	—
38	#13093958	—	2	4	no	—
39	#13093958	—	2	4	no	—
40	#13093958	C₂×C₄	1	8	yes	complemented
41	#13093956	C₂×C₄	1	8	yes	complemented
42	C₄₇⋊#95	—	2	8	no	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₄₇	D₄	1	4	yes	—
49	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	#117×C₄₇	—	2	2	no	—
54	#117×C₄₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
59	#97×C₄₇	—	2	4	no	—
60	C₄×C₈×C₄₇	—	2	4	no	—
61	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	4	4	no	—
75	C₂²×Dic₄₇	—	8	4	no	—
76	M₁₆×C₄₇	—	2	8	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
80	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
87	C₄⋊C₄×C₄₇	—	2	2	no	—
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₂²⋊C₄×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₂²⋊Dic₄₇	—	2	8	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	2	4	no	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	—	4	4	no	—
97	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
98	Dic₄₇⋊C₄	—	8	8	no	—
99	C₄×Dic₄₇	—	2	8	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	2	2	no	—
102	C₂³×C₄₇	C₄×D₄	1	2	yes	—
103	C₂³×C₄₇	#121	1	2	yes	—
104	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
105	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
113	C₂×C₄×C₄₇	#121	1	2	yes	—
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
115	C₂×C₄×C₄₇	#125	1	4	yes	—
116	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	8	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	4	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	8	no	—
128	C₂×Dic₄₇	—	8	8	no	—
129	C₈×C₄₇	—	2	4	no	—
130	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
131	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
132	#21156	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	4	no	—
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	#389	1	4	yes	—
138	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
141	C₂²×C₄₇	#576	1	2	yes	—
142	C₂²×C₄₇	#578	1	2	yes	—
143	C₄×C₄₇	—	4	2	no	—
144	C₄×C₄₇	—	4	2	no	—
145	C₄×C₄₇	#389	1	4	yes	—
146	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
149	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
150	Dic₄₇	—	4	8	no	—
151	Dic₄₇	—	8	4	no	—
152	C₂×#427	—	47	1	no	—
153	C₂²×#107	—	47	1	no	—
154	C₂×#405	D₄₇	1	1	yes	characteristic
155	#2317	—	47	2	no	—
156	#1944	—	47	4	no	—
157	#1942	—	47	4	no	—
158	#1736	—	47	2	no	—
159	C₂×C₄₇	—	2	4	no	—
160	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
161	C₂×C₄₇	#2951	1	2	yes	—
162	C₂×C₄₇	#3058	1	1	yes	characteristic
163	C₂×C₄₇	#3059	1	1	yes	characteristic
164	C₂×C₄₇	#3164	1	2	yes	—
165	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
166	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
167	C₂×#117	—	47	2	no	—
168	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
169	C₂²×C₄⋊C₄	—	47	2	no	—
170	#427	—	47	8	no	—
171	C₂×#107	—	47	2	no	—
172	C₂×#107	—	47	2	no	—
173	C₂×#107	—	47	2	no	—
174	C₂×#107	—	94	4	no	—
175	C₂×#106	—	47	2	no	—
176	#405	—	2	4	no	—
177	C₂×#97	C₂×D₄₇	1	2	yes	—
178	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
179	#389	—	47	8	no	—
180	#385	—	94	4	no	—
181	#384	—	94	2	no	—
182	#382	—	94	2	no	—
183	#340	—	47	8	no	—
184	C₄₇	#21156	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	—	94	2	no	—
186	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	Dic₉₄	1	4	yes	—
190	C₂²×C₈	Dic₉₄	1	4	yes	—
191	C₂²×C₈	C₄×D₄₇	1	4	yes	—
192	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
193	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
194	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	47	8	no	—
199	#117	—	94	4	no	—
200	C₂×C₄⋊C₄	—	47	4	no	—
201	C₂×C₄⋊C₄	—	47	4	no	—
202	C₂×C₄⋊C₄	—	94	4	no	—
203	C₂×C₄⋊C₄	—	94	4	no	—
204	C₂×C₄⋊C₄	—	94	8	no	—
205	C₂×C₄⋊C₄	—	94	2	no	—
206	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
207	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	—	94	2	no	—
209	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
210	C₂×C₄²	—	94	2	no	—
211	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
212	#107	—	47	8	no	—
213	#107	—	94	4	no	—
214	#107	—	94	4	no	—
215	#106	—	47	8	no	—
216	#97	—	2	4	no	—
217	C₄×C₈	—	2	4	no	—
218	#95	—	94	8	no	—
219	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
220	C₂²×C₄	—	94	4	no	—
221	C₂²×C₄	—	94	4	no	—
222	C₂²×C₄	—	376	4	no	—
223	C₂²×C₄	—	188	4	no	—
224	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
225	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
226	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
227	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
228	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
229	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
230	C₂²×C₄	#10625350	1	2	yes	—
231	C₂²×C₄	#10625350	1	2	yes	—
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
233	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
234	M₁₆	—	2	8	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	Dic₄₇⋊C₄	1	16	yes	complemented
238	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
239	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
240	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
241	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
242	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
243	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
244	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	47	8	no	—
248	C₄⋊C₄	—	94	4	no	—
249	C₄⋊C₄	—	94	4	no	—
250	C₄⋊C₄	—	94	4	no	—
251	C₄⋊C₄	—	376	8	no	—
252	C₄⋊C₄	—	188	4	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	94	8	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	94	8	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₄×D₁₈₈	1	2	yes	—
261	C₂³	#13093998	1	2	yes	—
262	C₂³	#13094024	1	2	yes	—
263	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	94	8	no	—
269	C₂×C₄	—	376	8	no	—
270	C₂×C₄	—	188	4	no	—
271	C₂×C₄	—	188	8	no	—
272	C₂×C₄	—	188	4	no	—
273	C₂×C₄	—	188	4	no	—
274	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
275	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
276	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
277	C₂×C₄	#13093998	1	2	yes	—
278	C₂×C₄	#13094003	1	4	yes	—
279	C₂×C₄	#13094016	1	4	yes	—
280	C₂×C₄	#13094024	1	2	yes	—
281	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
284	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
285	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
286	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
287	C₈	—	2	4	no	—
288	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
289	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	#15689021	1	4	yes	—
295	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
296	C₂²	C₂×#13093998	1	1	yes	characteristic
297	C₂²	C₂×#13094024	1	1	yes	characteristic
298	C₂²	#15689928	1	2	yes	—
299	C₂²	#15689940	1	2	yes	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	376	4	no	—
303	C₄	—	188	8	no	—
304	C₄	#15689021	1	4	yes	—
305	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
306	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
307	C₄	C₂×#13094003	1	1	yes	characteristic
308	C₄	C₂×#13094016	1	1	yes	characteristic
309	C₂	—	2	4	no	—
310	C₂	C₂×#15689021	1	1	yes	characteristic
311	C₂	#20252265	1	2	yes	—
312	C₂	#20253036	1	1	yes	characteristic
313	C₂	#20253043	1	1	yes	characteristic
314	C₂	#20253927	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#21156
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #20244131	C₂	semidirect
5. #20244121	C₂	semidirect
10. #15689161	C₄	semidirect
40. #13093958	C₂×C₄	semidirect
41. #13093956	C₂×C₄	semidirect
79. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
184. C₄₇	#21156	semidirect
237. C₂×C₈	Dic₄₇⋊C₄	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
C₂²×#13093958	2
#20246008	2
#20244131	2
#20244121	2
#20243108	2
C₂×C₄₇×#405	2
#21156	47