Subgroups — #26106289

Order 12032.

678 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	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688991	C₂	1	1	yes	maximal, characteristic
3	#20241266	C₂	1	4	yes	maximal
4	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
5	#15689180	—	2	2	no	—
6	#15689180	C₄	1	4	yes	complemented
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×#13093935	—	2	1	no	—
10	#15688991	—	2	2	no	—
11	#15688991	C₄	1	4	yes	complemented
12	C₂×C₄₇⋊#95	—	2	1	no	—
13	C₂×C₄₇⋊#95	—	2	1	no	—
14	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
15	C₂×#117×C₄₇	—	2	1	no	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#384	C₄	1	2	yes	complemented
18	C₄₇×#382	—	2	1	no	—
19	C₄₇×#380	—	2	1	no	—
20	C₄₇×#380	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
22	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	complemented
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
28	#13093935	—	2	4	no	—
29	C₄₇⋊#95	—	2	4	no	—
30	C₄₇⋊#95	—	2	4	no	—
31	C₄₇⋊#95	—	2	2	no	—
32	C₄₇⋊#95	C₂×C₄	1	4	yes	—
33	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	#117×C₄₇	—	4	4	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
50	#95×C₄₇	—	2	2	no	—
51	#95×C₄₇	—	2	2	no	—
52	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	—	4	4	no	—
74	C₂²×Dic₄₇	—	4	4	no	—
75	C₂²×Dic₄₇	—	4	2	no	—
76	C₂²×Dic₄₇	—	4	2	no	—
77	C₄⋊C₄×C₄₇	—	4	4	no	—
78	C₄⋊C₄×C₄₇	—	4	4	no	—
79	C₄⋊C₄×C₄₇	—	4	4	no	—
80	C₂²⋊C₄×C₄₇	—	4	4	no	—
81	C₄²×C₄₇	—	4	4	no	—
82	C₄²×C₄₇	C₄²	1	2	yes	—
83	C₄²×C₄₇	C₄²	1	2	yes	—
84	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
87	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
88	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
89	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	4	4	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
98	C₂³×C₄₇	#137	1	2	yes	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	—	4	1	no	—
113	C₂×C₄×C₄₇	—	4	1	no	—
114	C₂×C₄×C₄₇	—	4	1	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	#95	1	2	yes	—
119	C₂×C₄×C₄₇	#95	1	2	yes	—
120	C₂×C₄×C₄₇	#95	1	2	yes	—
121	C₂×C₄×C₄₇	#95	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	—	8	4	no	—
130	C₂×Dic₄₇	—	8	4	no	—
131	C₄₇⋊C₈	—	4	4	no	—
132	#8018	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	1	no	—
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	4	no	—
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
140	C₂²×C₄₇	#410	1	2	yes	—
141	C₂²×C₄₇	#417	1	2	yes	—
142	C₂²×C₄₇	#418	1	1	yes	characteristic
143	C₂²×C₄₇	#419	1	1	yes	characteristic
144	C₂²×C₄₇	#421	1	2	yes	—
145	C₂²×C₄₇	#428	1	2	yes	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	8	4	no	—
155	Dic₄₇	—	8	4	no	—
156	C₂×#432	—	47	1	no	—
157	#2285	D₄₇	1	1	yes	characteristic
158	C₂×#383	—	47	1	no	—
159	#1294	—	47	4	no	—
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	#1514	1	2	yes	—
163	C₂×C₄₇	#1515	1	2	yes	—
164	C₂×C₄₇	#1733	1	1	yes	characteristic
165	C₂×C₄₇	#1736	1	1	yes	characteristic
166	C₂×C₄₇	#1740	1	1	yes	characteristic
167	C₂×#117	—	2	1	no	—
168	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
169	#432	—	47	4	no	—
170	#432	—	94	2	no	—
171	C₂×C₄⋊C₈	—	94	1	no	—
172	C₂×C₂²⋊C₈	—	47	2	no	—
173	C₂×#97	—	94	1	no	—
174	#384	Dic₄₇	1	2	yes	complemented
175	#383	—	47	4	no	—
176	#383	—	94	2	no	—
177	#382	—	2	1	no	—
178	#380	—	2	1	no	—
179	#380	C₂×D₄₇	1	2	yes	—
180	C₄×C₄⋊C₄	—	2	1	no	—
181	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	complemented
182	C₂×#95	—	47	2	no	—
183	C₂×#95	—	94	1	no	—
184	C₂×#95	—	94	1	no	—
185	C₄₇	#8018	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₄	—	94	2	no	—
187	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
189	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
190	C₂²×C₈	—	94	2	no	—
191	#117	—	4	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	Dic₉₄	1	2	yes	—
198	C₂×C₄²	C₄×D₄₇	1	2	yes	—
199	C₂×C₄²	C₄×D₄₇	1	2	yes	—
200	C₂×C₄²	D₁₈₈	1	2	yes	—
201	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
202	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₄⋊C₈	—	94	4	no	—
206	C₂²⋊C₈	—	47	4	no	—
207	C₂²⋊C₈	—	94	2	no	—
208	#97	—	94	4	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	#95	—	47	4	no	—
212	#95	—	94	4	no	—
213	#95	—	94	4	no	—
214	#95	—	94	2	no	—
215	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	94	4	no	—
222	C₂²×C₄	—	188	4	no	—
223	C₂²×C₄	—	188	4	no	—
224	C₂²×C₄	—	188	2	no	—
225	C₂²×C₄	—	188	2	no	—
226	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
227	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
228	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
229	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
230	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
240	C₂×C₈	—	94	4	no	—
241	C₂×C₈	—	188	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₄²	—	4	4	no	—
247	C₄²	C₄×Dic₄₇	1	2	yes	—
248	C₄²	C₄×Dic₄₇	1	2	yes	—
249	C₄²	Dic₄₇⋊C₄	1	2	yes	—
250	C₄²	Dic₄₇⋊C₄	1	2	yes	—
251	C₄²	C₄⋊Dic₄₇	1	2	yes	—
252	C₄²	C₄⋊Dic₄₇	1	2	yes	—
253	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
254	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
258	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
259	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
260	C₂³	#13094050	1	2	yes	—
261	C₂³	#13094055	1	2	yes	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	376	4	no	—
282	C₂×C₄	—	376	4	no	—
283	C₂×C₄	—	188	4	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	C₄₇⋊#95	1	2	yes	—
287	C₂×C₄	C₄₇⋊#95	1	2	yes	—
288	C₂×C₄	C₄₇⋊#95	1	2	yes	—
289	C₂×C₄	C₄₇⋊#95	1	2	yes	—
290	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
294	C₈	—	188	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
302	C₂²	#15689097	1	2	yes	—
303	C₂²	#15689122	1	2	yes	—
304	C₂²	#15689128	1	1	yes	characteristic
305	C₂²	#15689132	1	1	yes	characteristic
306	C₂²	#15689140	1	2	yes	—
307	C₂²	#15689163	1	2	yes	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	8	4	no	—
317	C₄	—	376	4	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	#20241935	1	2	yes	—
321	C₂	#20241939	1	2	yes	—
322	C₂	#20243097	1	1	yes	characteristic
323	C₂	#20243107	1	1	yes	characteristic
324	C₂	#20243121	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#8018
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
6. #15689180	C₄	semidirect
11. #15688991	C₄	semidirect
17. C₄₇×#384	C₄	semidirect
22. C₄×C₄⋊C₄×C₄₇	C₄	semidirect
174. #384	Dic₄₇	semidirect
181. C₄×C₄⋊C₄	Dic₄₇	semidirect
185. C₄₇	#8018	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
C₂×#15688991	2
#20241266	2
C₄₇×#2285	2
#8018	47