Subgroups — #26225766

Order 12032.

1302 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	#20245795	C₂	1	1	yes	maximal, characteristic
3	#20244350	C₂	1	4	yes	maximal, complemented
4	#20244337	C₂	1	4	yes	maximal, complemented
5	#20243478	C₂	1	4	yes	maximal
6	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093983	—	2	1	no	—
8	#15689176	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	#15688993	C₂²	1	2	yes	—
13	#15688989	—	2	1	no	—
14	#15688983	—	2	1	no	—
15	#15688983	C₂²	1	2	yes	—
16	C₄×C₄⋊Dic₄₇	—	2	1	no	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
18	#15688885	C₂²	1	16	yes	—
19	C₂³×Q₈×C₄₇	—	2	1	no	—
20	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
21	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
23	C₂×C₄⋊Q₈×C₄₇	C₄	1	8	yes	complemented
24	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	#13093983	—	4	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
37	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
38	C₄×C₄₇⋊C₈	—	2	4	no	—
39	C₄₇⋊#95	—	2	2	no	—
40	C₄₇⋊#95	—	2	2	no	—
41	C₂²×Q₈×C₄₇	—	2	8	no	—
42	C₂²×Q₈×C₄₇	—	2	2	no	—
43	C₂²×Q₈×C₄₇	—	2	2	no	—
44	C₂²×Q₈×C₄₇	—	4	8	no	—
45	C₂³×C₄×C₄₇	—	2	2	no	—
46	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₄₇	—	2	8	no	—
50	C₄⋊Q₈×C₄₇	—	2	8	no	—
51	C₄⋊Q₈×C₄₇	—	2	4	no	—
52	C₄⋊Q₈×C₄₇	—	2	4	no	—
53	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
54	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
56	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
60	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
61	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
62	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
63	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	D₄	1	4	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×Q₈×C₄₇	—	2	8	no	—
70	C₂×Q₈×C₄₇	—	2	8	no	—
71	C₂×Q₈×C₄₇	—	4	8	no	—
72	C₂×Q₈×C₄₇	—	4	32	no	—
73	C₂×Q₈×C₄₇	—	4	8	no	—
74	C₂²×C₄×C₄₇	—	2	4	no	—
75	C₂²×C₄×C₄₇	—	2	4	no	—
76	C₂²×C₄×C₄₇	—	2	8	no	—
77	C₂²×C₄×C₄₇	—	2	4	no	—
78	C₂²×C₄×C₄₇	—	4	8	no	—
79	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
81	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
82	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₄₇	—	2	4	no	—
92	C₄⋊C₄×C₄₇	—	2	8	no	—
93	C₄⋊C₄×C₄₇	—	2	8	no	—
94	C₄⋊C₄×C₄₇	—	2	8	no	—
95	C₄⋊C₄×C₄₇	—	2	4	no	—
96	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
97	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
98	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
99	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₄₇	C₂×D₄	1	2	yes	—
102	C₄²×C₄₇	C₂×D₄	1	2	yes	—
103	C₂²⋊Dic₄₇	—	4	4	no	—
104	C₄⋊Dic₄₇	—	2	8	no	—
105	C₄⋊Dic₄₇	—	4	4	no	—
106	C₄⋊Dic₄₇	—	4	4	no	—
107	C₄×Dic₄₇	—	4	4	no	—
108	C₂×C₄₇⋊C₈	—	2	4	no	—
109	C₂×C₄₇⋊C₈	—	2	8	no	—
110	C₂³×C₄₇	—	2	2	no	—
111	C₂³×C₄₇	—	2	2	no	—
112	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
114	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
115	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₄₇	—	4	16	no	—
117	Q₈×C₄₇	—	4	16	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	8	no	—
123	C₂×C₄×C₄₇	—	2	8	no	—
124	C₂×C₄×C₄₇	—	4	16	no	—
125	C₂×C₄×C₄₇	—	4	8	no	—
126	C₂×C₄×C₄₇	#103	1	16	yes	—
127	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
135	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
136	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
137	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
140	C₂×Dic₄₇	—	4	4	no	—
141	C₂×Dic₄₇	—	4	4	no	—
142	C₂×Dic₄₇	—	4	4	no	—
143	C₄₇⋊C₈	—	4	4	no	—
144	#21834	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	4	no	—
148	C₂²×C₄₇	—	2	4	no	—
149	C₂²×C₄₇	C₂×#103	1	4	yes	—
150	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄₇	#536	1	2	yes	—
152	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
153	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	4	8	no	—
158	C₄×C₄₇	C₂×#103	1	4	yes	—
159	C₄×C₄₇	C₂×#103	1	4	yes	—
160	C₄×C₄₇	#418	1	4	yes	—
161	C₄×C₄₇	#519	1	2	yes	—
162	C₄×C₄₇	#536	1	2	yes	—
163	Dic₄₇	—	8	4	no	—
164	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
165	C₂×#431	—	47	1	no	—
166	#2285	—	47	1	no	—
167	#1990	—	47	4	no	—
168	#1987	—	47	4	no	—
169	#1824	—	47	4	no	—
170	C₂×C₄₇	—	2	4	no	—
171	C₂×C₄₇	#2310	1	1	yes	characteristic
172	C₂×C₄₇	C₂²×#103	1	1	yes	characteristic
173	C₂×C₄₇	C₂×#418	1	1	yes	characteristic
174	C₂×C₄₇	#3227	1	2	yes	—
175	C₂×C₄₇	#3230	1	2	yes	—
176	C₂³×Q₈	—	2	1	no	—
177	C₂×C₄⋊Q₈	—	2	1	no	—
178	C₂×C₄⋊Q₈	—	2	1	no	—
179	C₂×C₄⋊Q₈	—	2	4	no	—
180	C₂×C₄⋊Q₈	Dic₄₇	1	8	yes	complemented
181	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
182	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
183	C₂×#117	—	94	1	no	—
184	C₂²×C₄⋊C₄	—	2	1	no	—
185	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
186	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
187	#431	—	94	4	no	—
188	C₂×C₄⋊C₈	—	47	2	no	—
189	C₂×C₂²⋊C₈	—	47	2	no	—
190	C₂×C₄×C₈	—	47	2	no	—
191	#384	—	47	2	no	—
192	#382	—	94	1	no	—
193	#380	—	47	2	no	—
194	#380	—	94	1	no	—
195	C₄×C₄⋊C₄	—	47	2	no	—
196	C₄×C₄⋊C₄	—	94	1	no	—
197	#340	—	47	16	no	—
198	C₄₇	#21834	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₈	—	2	8	no	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×Q₈	—	4	8	no	—
203	C₂³×C₄	—	2	2	no	—
204	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²×C₈	—	47	4	no	—
208	C₄⋊Q₈	—	2	8	no	—
209	C₄⋊Q₈	—	2	8	no	—
210	C₄⋊Q₈	—	2	4	no	—
211	C₄⋊Q₈	—	2	4	no	—
212	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
213	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
214	#117	—	188	4	no	—
215	C₂×C₄⋊C₄	—	2	8	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	4	no	—
220	C₂×C₄⋊C₄	—	47	4	no	—
221	C₂×C₄⋊C₄	—	94	2	no	—
222	C₂×C₄⋊C₄	—	94	2	no	—
223	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
224	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
225	C₂×C₂²⋊C₄	—	94	2	no	—
226	C₂×C₄²	—	94	2	no	—
227	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
228	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
231	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
233	C₄⋊C₈	—	94	4	no	—
234	C₂²⋊C₈	—	94	4	no	—
235	C₄×C₈	—	94	4	no	—
236	#95	—	94	2	no	—
237	#95	—	94	2	no	—
238	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂×Q₈	—	2	8	no	—
240	C₂×Q₈	—	2	8	no	—
241	C₂×Q₈	—	4	8	no	—
242	C₂×Q₈	—	4	32	no	—
243	C₂×Q₈	—	4	8	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	8	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	4	8	no	—
249	C₂²×C₄	—	94	4	no	—
250	C₂²×C₄	#10625336	1	4	yes	—
251	C₂²×C₄	#10625337	1	4	yes	—
252	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
253	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
254	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
255	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
258	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
261	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
262	C₂×C₈	—	94	4	no	—
263	C₂×C₈	—	94	8	no	—
264	C₄⋊C₄	—	2	8	no	—
265	C₄⋊C₄	—	2	8	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	94	8	no	—
269	C₄⋊C₄	—	188	4	no	—
270	C₄⋊C₄	—	188	4	no	—
271	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
272	C₂²⋊C₄	—	188	4	no	—
273	C₄²	—	188	4	no	—
274	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
275	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
276	C₄²	C₂²×Dic₄₇	1	2	yes	—
277	C₄²	C₂²×Dic₄₇	1	2	yes	—
278	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
279	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	2	no	—
282	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
283	C₂³	C₂×#10625336	1	2	yes	—
284	C₂³	C₂×#10625337	1	2	yes	—
285	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
286	Q₈	—	4	16	no	—
287	Q₈	—	4	16	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	4	16	no	—
295	C₂×C₄	—	4	8	no	—
296	C₂×C₄	—	188	4	no	—
297	C₂×C₄	—	188	4	no	—
298	C₂×C₄	—	188	4	no	—
299	C₂×C₄	#13093948	1	16	yes	—
300	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
301	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
302	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
303	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
304	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
305	C₂×C₄	C₂×#10625336	1	2	yes	—
306	C₂×C₄	C₂×#10625336	1	2	yes	—
307	C₂×C₄	C₂×#10625337	1	2	yes	—
308	C₂×C₄	C₂×#10625337	1	2	yes	—
309	C₂×C₄	#13094050	1	4	yes	—
310	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
312	C₂×C₄	#13094072	1	4	yes	—
313	C₈	—	188	4	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	C₂×#13093948	1	4	yes	—
319	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
320	C₂²	#15689697	1	2	yes	—
321	C₂²	C₂²×#10625336	1	1	yes	characteristic
322	C₂²	C₂²×#10625337	1	1	yes	characteristic
323	C₂²	C₂×#13094050	1	2	yes	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	8	no	—
327	C₄	—	376	4	no	—
328	C₄	C₂×#13093948	1	4	yes	—
329	C₄	C₂×#13093948	1	4	yes	—
330	C₄	#15689128	1	4	yes	—
331	C₄	#15689601	1	2	yes	—
332	C₄	#15689697	1	2	yes	—
333	C₂	—	2	4	no	—
334	C₂	#20245974	1	1	yes	characteristic
335	C₂	C₂²×#13093948	1	1	yes	characteristic
336	C₂	C₂×#15689128	1	1	yes	characteristic
337	C₂	#20254516	1	2	yes	—
338	C₂	#20254539	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#21834
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #20244350	C₂	semidirect
4. #20244337	C₂	semidirect
23. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
180. C₂×C₄⋊Q₈	Dic₄₇	semidirect
198. C₄₇	#21834	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20245795	2
#20244350	2
#20244337	2
#20243478	2
C₂²×C₄⋊Q₈×C₄₇	2
#21834	47