Subgroups — #26121517

Order 12032.

1294 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689087	C₂	1	2	yes	maximal
2	C₂×C₄₇⋊#338	C₂	1	4	yes	maximal, complemented
3	C₂³×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	C₂²	1	4	yes	—
5	#15689087	—	2	2	no	—
6	#15689087	—	2	2	no	—
7	#15689087	—	2	4	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	8	yes	—
9	C₄₇⋊#338	—	2	8	no	—
10	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₄₇	—	2	4	no	—
12	C₂²×M₁₆×C₄₇	—	2	8	no	—
13	C₂²×M₁₆×C₄₇	C₄	1	2	yes	—
14	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂³×C₈×C₄₇	C₄	1	2	yes	—
16	C₂×#10625329	—	2	4	no	—
17	C₂×#10625329	—	2	4	no	—
18	C₂×#10625329	—	2	8	no	—
19	C₂×#10625329	—	2	8	no	—
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
23	C₂⁵×C₄₇	Q₈	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	—	2	4	no	—
25	C₂³×C₄×C₄₇	—	2	4	no	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	D₄	1	4	yes	—
32	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₄₇	—	2	32	no	—
35	C₂×M₁₆×C₄₇	—	2	4	no	—
36	C₂×M₁₆×C₄₇	—	2	4	no	—
37	C₂×M₁₆×C₄₇	—	2	4	no	—
38	C₂×M₁₆×C₄₇	—	2	8	no	—
39	C₂×M₁₆×C₄₇	D₄	1	8	yes	—
40	C₂²×C₈×C₄₇	—	2	4	no	—
41	C₂²×C₈×C₄₇	—	2	8	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
43	C₂⁴×C₄₇	—	2	4	no	—
44	C₂⁴×C₄₇	—	4	4	no	—
45	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
47	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	16	no	—
55	C₂²×C₄×C₄₇	—	2	8	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	8	no	—
59	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
60	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₄₇	#41	1	4	yes	—
74	M₁₆×C₄₇	—	2	16	no	—
75	M₁₆×C₄₇	—	2	16	no	—
76	C₂×C₈×C₄₇	—	2	16	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	C₂²⋊C₄	1	8	yes	—
80	#10625329	—	4	4	no	—
81	#10625329	—	4	8	no	—
82	#10625329	—	4	4	no	—
83	C₂×C₄₇⋊C₈	—	2	8	no	—
84	C₂×C₄₇⋊C₈	—	2	16	no	—
85	C₂³×C₄₇	—	2	4	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	2	8	no	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	—	2	8	no	—
90	C₂³×C₄₇	—	4	4	no	—
91	C₂³×C₄₇	—	4	2	no	—
92	C₂³×C₄₇	—	4	2	no	—
93	C₂³×C₄₇	—	4	16	no	—
94	C₂³×C₄₇	#95	1	2	yes	—
95	C₂³×C₄₇	#95	1	2	yes	—
96	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	#95	1	2	yes	—
111	C₂×C₄×C₄₇	#95	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#117	1	2	yes	—
117	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
118	C₂×C₄×C₄₇	#120	1	4	yes	—
119	C₂×C₄×C₄₇	#121	1	4	yes	—
120	C₂×C₄×C₄₇	#123	1	4	yes	—
121	C₈×C₄₇	—	2	8	no	—
122	C₄₇⋊C₈	—	4	8	no	—
123	C₂²⋊#338	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	8	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	4	4	no	—
132	C₂²×C₄₇	—	4	16	no	—
133	C₂²×C₄₇	—	4	4	no	—
134	C₂²×C₄₇	#338	1	4	yes	complemented
135	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
136	C₂²×C₄₇	#380	1	2	yes	—
137	C₂²×C₄₇	#385	1	4	yes	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
141	C₄×C₄₇	#379	1	1	yes	characteristic
142	C₄×C₄₇	#381	1	1	yes	characteristic
143	C₄×C₄₇	#386	1	4	yes	—
144	C₂³×M₁₆	D₄₇	1	1	yes	characteristic
145	C₂×#407	—	47	2	no	—
146	C₂×#338	—	47	4	no	—
147	C₂×C₄₇	—	2	4	no	—
148	C₂×C₄₇	—	2	2	no	—
149	C₂×C₄₇	—	2	4	no	—
150	C₂×C₄₇	—	4	2	no	—
151	C₂×C₄₇	C₂²⋊#95	1	1	yes	characteristic
152	C₂×C₄₇	C₂×#338	1	2	yes	—
153	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
154	C₂²×M₁₆	—	2	4	no	—
155	C₂²×M₁₆	—	2	8	no	—
156	C₂²×M₁₆	—	47	4	no	—
157	C₂²×M₁₆	Dic₄₇	1	2	yes	—
158	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
159	C₂³×C₈	Dic₄₇	1	2	yes	—
160	#407	—	94	2	no	—
161	#407	—	94	2	no	—
162	#407	—	94	4	no	—
163	C₂×C₂²⋊C₈	—	47	8	no	—
164	#338	—	94	8	no	—
165	C₄₇	C₂²⋊#338	1	1	yes	characteristic, complemented, Sylow
166	C₂⁵	Dic₉₄	1	1	yes	characteristic
167	C₂³×C₄	—	2	4	no	—
168	C₂³×C₄	—	2	4	no	—
169	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
170	C₂³×C₄	C₄×D₄₇	1	4	yes	—
171	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
172	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
173	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
174	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
175	C₂³×C₄	C₂²⋊D₄₇	1	4	yes	—
176	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
177	C₂×M₁₆	—	2	32	no	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	8	no	—
182	C₂×M₁₆	—	94	4	no	—
183	C₂×M₁₆	—	94	8	no	—
184	C₂×M₁₆	—	94	8	no	—
185	C₂×M₁₆	—	94	4	no	—
186	C₂×M₁₆	C₂²⋊D₄₇	1	8	yes	—
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	—	2	8	no	—
189	C₂²×C₈	—	47	8	no	—
190	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
191	C₂²⋊C₈	—	94	8	no	—
192	C₂²⋊C₈	—	94	8	no	—
193	C₂⁴	—	2	4	no	—
194	C₂⁴	—	4	4	no	—
195	C₂⁴	Dic₄₇⋊C₄	1	4	yes	—
196	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
197	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
198	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	16	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
210	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
211	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
212	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
214	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
215	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
216	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
217	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
219	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
220	C₂²×C₄	#10625350	1	4	yes	—
221	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
224	M₁₆	—	2	16	no	—
225	M₁₆	—	2	16	no	—
226	M₁₆	—	188	4	no	—
227	M₁₆	—	188	8	no	—
228	M₁₆	—	188	4	no	—
229	C₂×C₈	—	2	16	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	94	8	no	—
233	C₂×C₈	—	94	16	no	—
234	C₂×C₈	C₂²⋊Dic₄₇	1	8	yes	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	8	no	—
239	C₂³	—	2	8	no	—
240	C₂³	—	4	4	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	16	no	—
244	C₂³	C₄₇⋊#95	1	2	yes	—
245	C₂³	C₄₇⋊#95	1	2	yes	—
246	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
247	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
248	C₂³	#13094003	1	4	yes	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	C₄₇⋊#95	1	2	yes	—
261	C₂×C₄	C₄₇⋊#95	1	2	yes	—
262	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
263	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
265	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
266	C₂×C₄	#13093983	1	2	yes	—
267	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
268	C₂×C₄	#13093993	1	4	yes	—
269	C₂×C₄	#13093998	1	4	yes	—
270	C₂×C₄	#13094007	1	4	yes	—
271	C₈	—	2	8	no	—
272	C₈	—	188	8	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	16	no	—
282	C₂²	—	4	4	no	—
283	C₂²	C₄₇⋊#338	1	4	yes	complemented
284	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
285	C₂²	#15688983	1	2	yes	—
286	C₂²	#15688996	1	4	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	C₄×C₂²⋊Dic₄₇	1	2	yes	—
290	C₄	#15688979	1	1	yes	characteristic
291	C₄	#15688986	1	1	yes	characteristic
292	C₄	#15689002	1	4	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	2	4	no	—
295	C₂	—	2	4	no	—
296	C₂	—	4	2	no	—
297	C₂	#20241611	1	1	yes	characteristic
298	C₂	C₂×C₄₇⋊#338	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂³×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁵×C₄₇	Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂³×M₁₆	D₄₇
hypocenter	C₄₇	C₂²⋊#338
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇⋊#338	C₂	semidirect
134. C₂²×C₄₇	#338	semidirect
165. C₄₇	C₂²⋊#338	semidirect
283. C₂²	C₄₇⋊#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689087	2
C₂×C₄₇⋊#338	2
C₂³×M₁₆×C₄₇	2
C₂²⋊#338	47