Subgroups — #26121473

Order 12032.

926 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689081	C₂	1	2	yes	maximal
2	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243105	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688884	C₂	1	2	yes	maximal
5	C₂×C₄₇×#341	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×#13093983	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	#15689081	—	2	4	no	—
10	#15689081	C₄	1	8	yes	complemented
11	C₂×#13093934	—	2	2	no	—
12	C₂×C₈×Dic₄₇	—	2	4	no	—
13	C₂×C₈×Dic₄₇	—	2	2	no	—
14	C₂×C₈×Dic₄₇	C₄	1	4	yes	complemented
15	#15688886	C₄	1	4	yes	complemented
16	#15688884	C₄	1	8	yes	complemented
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#341	C₄	1	4	yes	complemented
20	C₂³×Dic₄₇	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
23	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
25	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
26	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
27	#13093983	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	#13093983	C₂×C₄	1	4	yes	—
30	#13093983	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	1	no	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
38	#13093934	—	2	8	no	—
39	C₈×Dic₄₇	—	2	8	no	—
40	C₈×Dic₄₇	—	2	8	no	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₄₇	—	2	2	no	—
49	C₂²×C₈×C₄₇	—	2	2	no	—
50	C₂²×C₈×C₄₇	—	2	2	no	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
54	C₂⁴×C₄₇	#41	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
65	C₂²×Dic₄₇	—	2	8	no	—
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	4	no	—
69	M₁₆×C₄₇	—	2	4	no	—
70	M₁₆×C₄₇	—	2	4	no	—
71	M₁₆×C₄₇	—	2	4	no	—
72	M₁₆×C₄₇	C₄²	1	8	yes	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	C₄²	1	4	yes	—
81	C₂²⋊Dic₄₇	—	2	4	no	—
82	C₂²⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	C₄²	1	8	yes	—
85	C₄×Dic₄₇	—	2	2	no	—
86	C₄×Dic₄₇	—	2	4	no	—
87	C₄×Dic₄₇	—	2	4	no	—
88	C₄×Dic₄₇	—	2	4	no	—
89	C₄×Dic₄₇	C₄²	1	4	yes	—
90	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
91	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
92	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
93	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
94	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
95	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
96	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
97	C₂³×C₄₇	—	2	4	no	—
98	C₂³×C₄₇	#117	1	2	yes	—
99	C₂³×C₄₇	#117	1	1	yes	characteristic
100	C₂³×C₄₇	C₄×D₄	1	2	yes	—
101	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
102	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#117	1	2	yes	—
114	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
121	C₂×Dic₄₇	—	2	16	no	—
122	C₂×Dic₄₇	—	2	8	no	—
123	C₂×Dic₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	2	8	no	—
125	C₂×Dic₄₇	—	4	8	no	—
126	C₈×C₄₇	—	2	4	no	—
127	C₈×C₄₇	—	2	8	no	—
128	C₄₇⋊C₈	#95	1	4	yes	—
129	C₄₇⋊C₈	#95	1	4	yes	—
130	#10706	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	—	2	4	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	#376	1	1	yes	characteristic
136	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
138	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
139	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₄₇	#443	1	4	yes	—
141	C₄×C₄₇	C₄³	1	1	yes	characteristic
142	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
143	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
145	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
147	Dic₄₇	—	2	8	no	—
148	Dic₄₇	—	4	8	no	—
149	C₂×#405	—	47	2	no	—
150	C₂²×C₄×C₈	—	47	1	no	—
151	C₂×#341	D₄₇	1	1	yes	characteristic
152	#1735	—	47	1	no	—
153	C₂×#339	—	47	2	no	—
154	C₂×C₄₇	—	2	1	no	—
155	C₂×C₄₇	—	2	1	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
158	C₂×C₄₇	#1764	1	4	yes	—
159	C₂×C₄₇	#1772	1	2	yes	—
160	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
161	C₂³×C₈	—	47	1	no	—
162	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
163	C₂×#117	—	47	2	no	—
164	C₂²×C₄²	—	47	1	no	—
165	#405	—	47	8	no	—
166	#405	—	94	4	no	—
167	C₂×#97	—	94	2	no	—
168	C₂×C₄×C₈	—	47	4	no	—
169	C₂×C₄×C₈	—	94	4	no	—
170	C₂×C₄×C₈	—	94	2	no	—
171	#341	Dic₄₇	1	4	yes	complemented
172	#340	—	47	4	no	—
173	#339	—	47	8	no	—
174	C₄₇	#10706	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	94	1	no	—
176	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
181	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
182	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	—	47	4	no	—
188	C₂²×C₈	—	47	4	no	—
189	C₂²×C₈	—	47	1	no	—
190	C₂²×C₈	—	47	1	no	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	C₄×D₄₇	1	2	yes	—
193	C₂²×C₈	C₄×D₄₇	1	2	yes	—
194	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
195	#117	—	47	4	no	—
196	#117	—	47	4	no	—
197	#117	—	94	2	no	—
198	#117	—	94	2	no	—
199	C₂×C₄⋊C₄	—	47	4	no	—
200	C₂×C₂²⋊C₄	—	94	2	no	—
201	C₂×C₄²	—	47	4	no	—
202	C₂×C₄²	—	47	2	no	—
203	C₂×C₄²	—	94	2	no	—
204	C₂×C₄²	—	94	2	no	—
205	C₂×C₄²	—	94	1	no	—
206	#97	—	94	8	no	—
207	C₄×C₈	—	94	8	no	—
208	C₄×C₈	—	94	8	no	—
209	C₂⁴	#10625354	1	1	yes	characteristic
210	C₂²×C₄	—	94	8	no	—
211	C₂²×C₄	—	94	2	no	—
212	C₂²×C₄	—	94	4	no	—
213	C₂²×C₄	—	94	4	no	—
214	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	D₄×D₄₇	1	2	yes	—
219	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	#10625352	1	1	yes	characteristic
221	C₂²×C₄	#10625352	1	2	yes	—
222	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	C₄×Dic₄₇	1	8	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	47	4	no	—
236	C₂×C₈	—	47	4	no	—
237	C₂×C₈	—	47	4	no	—
238	C₂×C₈	—	47	4	no	—
239	C₂×C₈	—	47	8	no	—
240	C₂×C₈	—	47	2	no	—
241	C₂×C₈	—	47	2	no	—
242	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
243	C₄⋊C₄	—	47	8	no	—
244	C₄⋊C₄	—	94	4	no	—
245	C₂²⋊C₄	—	94	4	no	—
246	C₂²⋊C₄	—	94	4	no	—
247	C₄²	—	47	4	no	—
248	C₄²	—	94	2	no	—
249	C₄²	—	94	4	no	—
250	C₄²	—	94	4	no	—
251	C₄²	—	94	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	#13093982	1	2	yes	—
254	C₂³	#13093982	1	1	yes	characteristic
255	C₂³	#13093986	1	2	yes	—
256	C₂³	#13093986	1	1	yes	characteristic
257	C₂³	Q₈×Dic₄₇	1	1	yes	characteristic
258	C₂×C₄	—	94	16	no	—
259	C₂×C₄	—	94	8	no	—
260	C₂×C₄	—	94	4	no	—
261	C₂×C₄	—	94	8	no	—
262	C₂×C₄	—	188	8	no	—
263	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
264	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
265	C₂×C₄	C₄²×D₄₇	1	2	yes	—
266	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
268	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
270	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
271	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
272	C₂×C₄	#13093980	1	1	yes	characteristic
273	C₂×C₄	#13093980	1	2	yes	—
274	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
275	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
276	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
277	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
278	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
279	C₂×C₄	#13093989	1	2	yes	—
280	C₂×C₄	#13093989	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	2	8	no	—
283	C₈	—	47	4	no	—
284	C₈	—	47	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	#15688968	1	1	yes	characteristic
290	C₂²	#15688970	1	1	yes	characteristic
291	C₂²	#15688970	1	2	yes	—
292	C₂²	C₄⋊C₄×Dic₄₇	1	2	yes	—
293	C₂²	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
294	C₂²	#15689239	1	4	yes	—
295	C₄	—	94	8	no	—
296	C₄	—	188	8	no	—
297	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
298	C₄	#95×D₄₇	1	1	yes	characteristic
299	C₄	C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
300	C₄	C₂²⋊C₄×Dic₄₇	1	2	yes	—
301	C₄	#15688974	1	1	yes	characteristic
302	C₄	#15688974	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	2	no	—
306	C₂	#95×Dic₄₇	1	1	yes	characteristic
307	C₂	#20243224	1	4	yes	—
308	C₂	#20243270	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₄³
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#341	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#341	D₄₇
hypocenter	C₄₇	#10706
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
10. #15689081	C₄	semidirect
14. C₂×C₈×Dic₄₇	C₄	semidirect
15. #15688886	C₄	semidirect
16. #15688884	C₄	semidirect
19. C₄₇×#341	C₄	semidirect
171. #341	Dic₄₇	semidirect
174. C₄₇	#10706	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689081	2
C₂²×C₈×Dic₄₇	2
#20243105	2
C₂×#15688884	2
C₂×C₄₇×#341	2
#10706	47