Subgroups — #26184337

Order 12032.

950 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252021	C₂	1	2	yes	maximal
2	#20242026	C₂	1	4	yes	maximal, complemented
3	#20242015	C₂	1	4	yes	maximal, complemented
4	#20241983	C₂	1	4	yes	maximal, complemented
5	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
6	#15689180	C₂²	1	4	yes	—
7	#15689180	C₂²	1	4	yes	—
8	#15689176	—	2	2	no	—
9	#15689176	—	2	2	no	—
10	#15689147	—	2	2	no	—
11	#15689087	—	2	2	no	—
12	C₄×#10625329	C₂²	1	4	yes	—
13	#15688865	C₂²	1	16	yes	complemented
14	C₄₇×#533	—	2	2	no	—
15	C₄₇×#533	—	2	2	no	—
16	C₄₇×#533	C₄	1	2	yes	—
17	C₄₇×#533	C₄	1	2	yes	—
18	C₄₇×#533	C₂²	1	2	yes	—
19	C₄₇×#533	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
21	C₂×#125×C₄₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
23	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
24	C₂×#117×C₄₇	C₄	1	2	yes	—
25	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#10625329	—	2	4	no	—
27	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
29	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
30	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
31	#13093935	C₂³	1	8	yes	—
32	C₄×C₄₇⋊C₈	—	2	4	no	—
33	C₂²×Q₈×C₄₇	—	2	2	no	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	D₄	1	2	yes	—
36	C₄⋊Q₈×C₄₇	—	2	2	no	—
37	C₄⋊Q₈×C₄₇	—	2	2	no	—
38	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
39	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
40	#125×C₄₇	—	2	4	no	—
41	#125×C₄₇	C₂×C₄	1	4	yes	—
42	#125×C₄₇	C₂³	1	4	yes	—
43	C₂²⋊Q₈×C₄₇	—	2	4	no	—
44	C₂²⋊Q₈×C₄₇	—	2	4	no	—
45	C₂²⋊Q₈×C₄₇	—	2	4	no	—
46	C₂²⋊Q₈×C₄₇	—	2	4	no	—
47	C₄×Q₈×C₄₇	—	2	4	no	—
48	C₄×Q₈×C₄₇	—	2	4	no	—
49	C₄×Q₈×C₄₇	—	2	4	no	—
50	C₄×Q₈×C₄₇	—	2	4	no	—
51	#117×C₄₇	—	2	2	no	—
52	#117×C₄₇	—	2	2	no	—
53	#117×C₄₇	C₂×C₄	1	4	yes	—
54	#117×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
57	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
58	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
59	C₂×C₄²×C₄₇	—	2	2	no	—
60	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	D₄	1	2	yes	—
64	C₂×C₄²×C₄₇	D₄	1	2	yes	—
65	C₂×C₄²×C₄₇	D₄	1	2	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×Q₈×C₄₇	—	2	4	no	—
71	C₂×Q₈×C₄₇	—	2	4	no	—
72	C₂×Q₈×C₄₇	—	4	8	no	—
73	C₂²×C₄×C₄₇	—	2	4	no	—
74	C₂²×C₄×C₄₇	—	2	4	no	—
75	C₂²×C₄×C₄₇	—	4	4	no	—
76	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₄₇	—	2	8	no	—
89	C₄⋊C₄×C₄₇	—	2	4	no	—
90	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₄⋊C₄×C₄₇	—	2	4	no	—
92	C₄⋊C₄×C₄₇	—	2	8	no	—
93	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
94	C₂²⋊C₄×C₄₇	—	2	4	no	—
95	C₂²⋊C₄×C₄₇	—	2	4	no	—
96	C₄²×C₄₇	—	2	4	no	—
97	C₄²×C₄₇	—	2	4	no	—
98	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
99	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
100	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₄₇	C₂×D₄	1	2	yes	—
105	C₄²×C₄₇	C₂×D₄	1	2	yes	—
106	C₄²×C₄₇	C₂×D₄	1	2	yes	—
107	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
108	#10625329	—	4	8	no	—
109	C₂×C₄₇⋊C₈	—	2	8	no	—
110	C₂³×C₄₇	—	2	2	no	—
111	C₂³×C₄₇	—	4	2	no	—
112	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	Q₈×C₄₇	—	4	8	no	—
116	Q₈×C₄₇	—	4	8	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	2	4	no	—
125	C₂×C₄×C₄₇	—	2	8	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	8	no	—
130	C₂×C₄×C₄₇	#101	1	4	yes	—
131	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
136	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
139	C₄₇⋊C₈	—	4	8	no	—
140	#17664	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	4	4	no	—
147	C₂²×C₄₇	C₂×#101	1	2	yes	—
148	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	4	8	no	—
155	C₄×C₄₇	C₂×#101	1	2	yes	—
156	C₄×C₄₇	C₂×#101	1	2	yes	—
157	C₄×C₄₇	#421	1	4	yes	—
158	C₂×#533	D₄₇	1	1	yes	characteristic
159	#2920	—	47	2	no	—
160	#1537	—	47	4	no	—
161	#1534	—	47	4	no	—
162	#1526	—	47	4	no	—
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	—	4	1	no	—
165	C₂×C₄₇	—	4	1	no	—
166	C₂×C₄₇	C₂²×#101	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#421	1	2	yes	—
168	#533	—	2	2	no	—
169	#533	—	2	2	no	—
170	#533	Dic₄₇	1	2	yes	—
171	#533	Dic₄₇	1	2	yes	—
172	#533	C₂×D₄₇	1	2	yes	—
173	#533	C₂×D₄₇	1	2	yes	—
174	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	—
175	C₂×#125	C₂×D₄₇	1	2	yes	—
176	C₂×C₂²⋊Q₈	—	2	2	no	—
177	C₂×C₄×Q₈	—	2	2	no	—
178	C₂×#117	Dic₄₇	1	2	yes	—
179	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
180	#432	—	47	4	no	—
181	#432	—	47	4	no	—
182	#431	—	94	2	no	—
183	#431	—	94	2	no	—
184	#423	—	94	2	no	—
185	#407	—	94	2	no	—
186	C₄×M₁₆	—	47	4	no	—
187	#330	—	47	16	no	—
188	C₄₇	#17664	1	1	yes	characteristic, complemented, Sylow
189	C₂²×Q₈	—	2	2	no	—
190	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
192	C₂×M₁₆	—	94	4	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
196	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
197	#125	—	2	4	no	—
198	#125	C₂×Dic₄₇	1	4	yes	—
199	#125	C₂²×D₄₇	1	4	yes	—
200	C₂²⋊Q₈	—	2	4	no	—
201	C₂²⋊Q₈	—	2	4	no	—
202	C₂²⋊Q₈	—	2	4	no	—
203	C₂²⋊Q₈	—	2	4	no	—
204	C₄×Q₈	—	2	4	no	—
205	C₄×Q₈	—	2	4	no	—
206	C₄×Q₈	—	2	4	no	—
207	C₄×Q₈	—	2	4	no	—
208	#117	—	2	2	no	—
209	#117	—	2	2	no	—
210	#117	C₂×Dic₄₇	1	4	yes	—
211	#117	C₂×Dic₄₇	1	4	yes	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
219	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
221	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
222	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
223	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
225	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
226	C₄⋊C₈	—	94	4	no	—
227	C₄⋊C₈	—	94	4	no	—
228	C₂²⋊C₈	—	94	4	no	—
229	C₂²⋊C₈	—	94	4	no	—
230	#97	—	47	8	no	—
231	C₄×C₈	—	94	4	no	—
232	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	4	8	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
240	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
241	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
244	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
250	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
251	M₁₆	—	188	8	no	—
252	C₂×C₈	—	94	8	no	—
253	C₄⋊C₄	—	2	8	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
264	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
265	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
266	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
267	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
268	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
270	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
271	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
272	C₄²	C₂³×D₄₇	1	1	yes	characteristic
273	C₂³	—	2	2	no	—
274	C₂³	—	4	2	no	—
275	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
276	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
277	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
278	Q₈	—	4	8	no	—
279	Q₈	—	4	8	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	8	no	—
293	C₂×C₄	#13093943	1	4	yes	—
294	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
295	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
296	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
299	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
302	C₈	—	188	8	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	4	4	no	—
309	C₂²	C₂×#13093943	1	2	yes	—
310	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	8	no	—
317	C₄	C₂×#13093943	1	2	yes	—
318	C₄	C₂×#13093943	1	2	yes	—
319	C₄	#15689140	1	4	yes	—
320	C₂	—	2	2	no	—
321	C₂	—	4	1	no	—
322	C₂	—	4	1	no	—
323	C₂	C₂²×#13093943	1	1	yes	characteristic
324	C₂	C₂×#15689140	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	#17664
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
2. #20242026	C₂	semidirect
3. #20242015	C₂	semidirect
4. #20241983	C₂	semidirect
13. #15688865	C₂²	semidirect
188. C₄₇	#17664	semidirect

Maximal subgroups

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

subgroup	index
#20252021	2
#20242026	2
#20242015	2
#20241983	2
C₂×C₄₇×#533	2
#17664	47