Subgroups — #26224812

Order 12032.

1014 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689230	C₂	1	2	yes	maximal
2	C₂×Q₁₆×Dic₄₇	C₂	1	2	yes	maximal
3	#20246108	C₂	1	2	yes	maximal
4	#20244260	C₂	1	2	yes	maximal
5	#20244250	C₂	1	2	yes	maximal
6	#20244209	C₂	1	2	yes	maximal
7	#20243552	C₂	1	1	yes	maximal, characteristic
8	#20243074	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#494	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	—
11	#15689230	—	2	8	no	—
12	Q₁₆×Dic₄₇	—	2	8	no	—
13	C₂×#13093958	C₂²	1	2	yes	—
14	C₂×#13093956	C₂²	1	2	yes	—
15	C₂×#13093948	C₂²	1	4	yes	—
16	C₂×#13093948	C₂²	1	4	yes	—
17	C₂×#13093935	C₂²	1	1	yes	characteristic
18	C₂×#13093934	C₂²	1	1	yes	characteristic
19	C₂×#13093934	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
22	#15689030	C₂²	1	4	yes	—
23	#15689009	—	2	8	no	—
24	#15688989	C₂²	1	2	yes	—
25	#15688969	C₂²	1	1	yes	characteristic
26	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
27	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
29	C₄₇×#494	C₄	1	8	yes	complemented
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
33	Q₈×Dic₄₇	—	2	8	no	—
34	Q₈×Dic₄₇	—	2	8	no	—
35	C₂×C₄⋊Dic₄₇	—	2	4	no	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
37	C₂×C₄×Dic₄₇	—	2	4	no	—
38	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
40	#13093958	—	2	4	no	—
41	#13093956	—	2	4	no	—
42	#13093948	—	2	8	no	—
43	#13093948	—	2	8	no	—
44	#13093935	—	2	2	no	—
45	#13093934	—	2	2	no	—
46	#13093934	D₄	1	4	yes	—
47	C₈×Dic₄₇	—	2	2	no	—
48	C₈×Dic₄₇	D₄	1	4	yes	—
49	C₄₇⋊#95	—	2	4	no	—
50	C₄₇⋊#95	—	2	8	no	—
51	C₄₇⋊#95	D₄	1	4	yes	—
52	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
53	C₂×Q₁₆×C₄₇	—	2	4	no	—
54	C₂×Q₁₆×C₄₇	—	2	4	no	—
55	C₂×Q₁₆×C₄₇	—	2	4	no	—
56	C₂×Q₁₆×C₄₇	—	2	4	no	—
57	C₂×Q₁₆×C₄₇	C₂×C₄	1	8	yes	—
58	C₂×Q₁₆×C₄₇	C₂×C₄	1	8	yes	—
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₄₇	—	2	4	no	—
62	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
63	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
66	C₂×Q₈×C₄₇	—	2	4	no	—
67	C₂×Q₈×C₄₇	—	2	4	no	—
68	C₂×Q₈×C₄₇	—	2	4	no	—
69	C₂×Q₈×C₄₇	—	2	4	no	—
70	C₂×Q₈×C₄₇	—	2	4	no	—
71	C₂×Q₈×C₄₇	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₄₇	—	2	4	no	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₄₇	—	2	4	no	—
76	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
77	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
78	Q₁₆×C₄₇	—	2	16	no	—
79	Q₁₆×C₄₇	—	2	16	no	—
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	#41	1	2	yes	—
85	C₂×C₈×C₄₇	#41	1	2	yes	—
86	C₄⋊C₄×C₄₇	—	2	16	no	—
87	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₇	#41	1	2	yes	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	4	8	no	—
92	C₄×Dic₄₇	—	2	1	no	—
93	C₄×Dic₄₇	—	2	1	no	—
94	C₄×Dic₄₇	—	4	8	no	—
95	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
96	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
97	C₂×C₄₇⋊C₈	—	2	1	no	—
98	C₂×C₄₇⋊C₈	—	2	1	no	—
99	C₂×C₄₇⋊C₈	—	2	1	no	—
100	C₂×C₄₇⋊C₈	—	2	1	no	—
101	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
103	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₄₇	—	2	8	no	—
105	Q₈×C₄₇	—	2	8	no	—
106	Q₈×C₄₇	—	4	4	no	—
107	Q₈×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	8	no	—
124	C₂×Dic₄₇	#127	1	4	yes	—
125	C₈×C₄₇	C₄×D₄	1	4	yes	—
126	C₈×C₄₇	C₄×D₄	1	4	yes	—
127	C₄₇⋊C₈	—	2	2	no	—
128	C₄₇⋊C₈	—	2	2	no	—
129	#21698	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
132	C₂²×C₄₇	#532	1	1	yes	characteristic
133	C₂²×C₄₇	#578	1	4	yes	—
134	C₄×C₄₇	—	2	4	no	—
135	C₄×C₄₇	—	4	8	no	—
136	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₄×C₄₇	#532	1	1	yes	characteristic
139	C₄×C₄₇	#532	1	1	yes	characteristic
140	Dic₄₇	—	2	4	no	—
141	Dic₄₇	—	4	2	no	—
142	Dic₄₇	—	8	4	no	—
143	C₂×#494	D₄₇	1	1	yes	characteristic
144	C₂×#441	—	47	2	no	—
145	C₂×C₄×Q₁₆	—	47	2	no	—
146	#2327	—	47	2	no	—
147	#1971	—	47	2	no	—
148	#1969	—	47	2	no	—
149	#1960	—	47	2	no	—
150	#1846	—	47	1	no	—
151	#1726	—	47	1	no	—
152	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
153	C₂×C₄₇	#2950	1	2	yes	—
154	C₂×C₄₇	#3156	1	2	yes	—
155	C₂×C₄₇	#3177	1	2	yes	—
156	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
157	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
158	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
159	C₂×C₄×Q₈	—	47	4	no	—
160	#494	Dic₄₇	1	8	yes	complemented
161	#441	—	94	8	no	—
162	C₄×Q₁₆	—	94	8	no	—
163	C₂×#107	—	47	2	no	—
164	C₂×#106	—	47	2	no	—
165	C₂×#103	—	47	4	no	—
166	C₂×#103	—	47	4	no	—
167	C₂×#97	—	47	1	no	—
168	C₂×#97	—	47	1	no	—
169	C₂×#97	—	47	1	no	—
170	C₂×C₄×C₈	—	47	1	no	—
171	C₂×C₄×C₈	—	47	1	no	—
172	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
173	#391	—	47	4	no	—
174	#387	—	94	8	no	—
175	#382	—	47	2	no	—
176	#376	—	47	1	no	—
177	C₄₇	#21698	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
179	C₂×Q₁₆	—	2	4	no	—
180	C₂×Q₁₆	—	2	4	no	—
181	C₂×Q₁₆	—	2	4	no	—
182	C₂×Q₁₆	—	2	4	no	—
183	C₂×Q₁₆	C₂×Dic₄₇	1	8	yes	—
184	C₂×Q₁₆	C₂×Dic₄₇	1	8	yes	—
185	C₂²×C₈	—	47	1	no	—
186	C₂²×C₈	—	47	1	no	—
187	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
189	C₄⋊Q₈	—	2	4	no	—
190	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
191	C₄×Q₈	—	94	8	no	—
192	C₄×Q₈	—	94	8	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	47	4	no	—
195	C₂×C₄⋊C₄	—	94	4	no	—
196	C₂×C₄²	—	47	1	no	—
197	C₂×C₄²	—	47	1	no	—
198	C₂×C₄²	—	94	4	no	—
199	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
200	#107	—	94	4	no	—
201	#106	—	94	4	no	—
202	#103	—	94	8	no	—
203	#103	—	94	8	no	—
204	#97	—	47	4	no	—
205	#97	—	94	2	no	—
206	#97	—	94	2	no	—
207	C₄×C₈	—	47	4	no	—
208	C₄×C₈	—	94	2	no	—
209	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
210	#95	—	47	4	no	—
211	#95	—	94	8	no	—
212	#95	—	94	4	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	C₂²×Dic₄₇	1	8	yes	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	47	2	no	—
221	C₂²×C₄	—	47	2	no	—
222	C₂²×C₄	—	94	4	no	—
223	C₂²×C₄	D₄×D₄₇	1	2	yes	—
224	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
225	Q₁₆	—	2	16	no	—
226	Q₁₆	—	2	16	no	—
227	C₂×C₈	—	47	2	no	—
228	C₂×C₈	—	47	2	no	—
229	C₂×C₈	—	94	1	no	—
230	C₂×C₈	—	94	1	no	—
231	C₂×C₈	—	94	1	no	—
232	C₂×C₈	—	94	1	no	—
233	C₂×C₈	D₄×D₄₇	1	2	yes	—
234	C₂×C₈	D₄×D₄₇	1	2	yes	—
235	C₂×C₈	#10625352	1	2	yes	—
236	C₂×C₈	#10625352	1	2	yes	—
237	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
238	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
239	C₄⋊C₄	—	2	16	no	—
240	C₄⋊C₄	—	94	4	no	—
241	C₄⋊C₄	—	94	4	no	—
242	C₄⋊C₄	—	188	8	no	—
243	C₄²	—	47	2	no	—
244	C₄²	—	47	2	no	—
245	C₄²	—	94	1	no	—
246	C₄²	—	94	1	no	—
247	C₄²	—	188	8	no	—
248	C₄²	#10625352	1	2	yes	—
249	C₄²	C₂²×Dic₄₇	1	2	yes	—
250	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
251	Q₈	—	2	8	no	—
252	Q₈	—	2	8	no	—
253	Q₈	—	4	4	no	—
254	Q₈	—	4	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	47	4	no	—
260	C₂×C₄	—	94	4	no	—
261	C₂×C₄	—	94	2	no	—
262	C₂×C₄	—	94	4	no	—
263	C₂×C₄	—	188	4	no	—
264	C₂×C₄	—	188	8	no	—
265	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
266	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
270	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
271	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
272	C₈	—	94	2	no	—
273	C₈	—	94	2	no	—
274	C₈	D₄×Dic₄₇	1	4	yes	—
275	C₈	D₄×Dic₄₇	1	4	yes	—
276	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
277	C₂²	#127×D₄₇	1	1	yes	characteristic
278	C₂²	#15689671	1	1	yes	characteristic
279	C₂²	#15689936	1	4	yes	—
280	C₄	—	2	4	no	—
281	C₄	—	4	8	no	—
282	C₄	—	94	4	no	—
283	C₄	—	376	4	no	—
284	C₄	—	188	2	no	—
285	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
286	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
287	C₄	#15689671	1	1	yes	characteristic
288	C₄	#15689671	1	1	yes	characteristic
289	C₂	#127×Dic₄₇	1	1	yes	characteristic
290	C₂	#20252256	1	2	yes	—
291	C₂	#20253876	1	2	yes	—
292	C₂	#20254044	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#494	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#494	D₄₇
hypocenter	C₄₇	#21698
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
29. C₄₇×#494	C₄	semidirect
160. #494	Dic₄₇	semidirect
177. C₄₇	#21698	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689230	2
C₂×Q₁₆×Dic₄₇	2
#20246108	2
#20244260	2
#20244250	2
#20244209	2
#20243552	2
#20243074	2
C₂×C₄₇×#494	2
#21698	47