Subgroups — #26216813

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#20245795	C₂	1	1	yes	maximal, characteristic
3	#20243547	C₂	1	2	yes	maximal, complemented
4	#20243515	C₂	1	2	yes	maximal, complemented
5	#20243415	C₂	1	4	yes	maximal
6	C₄×#13093957	C₂	1	4	yes	maximal, complemented
7	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×#13093983	—	2	1	no	—
10	C₂×#13093957	C₂²	1	8	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	#15688993	C₂²	1	2	yes	—
16	#15688989	—	2	1	no	—
17	#15688983	—	2	1	no	—
18	#15688983	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	—	2	1	no	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
21	#15688885	C₂²	1	8	yes	—
22	C₂×#117×C₄₇	—	2	1	no	—
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#384	C₂²	1	2	yes	—
25	C₄₇×#382	—	2	1	no	—
26	C₄₇×#380	—	2	1	no	—
27	C₄₇×#380	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
30	C₂²×C₄₇⋊C₈	—	2	8	no	—
31	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	#13093983	—	4	4	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
38	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	#13093957	—	2	16	no	—
41	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
42	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
43	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
44	C₄₇⋊#95	—	2	2	no	—
45	C₄₇⋊#95	—	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	#117×C₄₇	—	4	4	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
53	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	—	2	2	no	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	#95×C₄₇	—	2	2	no	—
62	#95×C₄₇	—	2	2	no	—
63	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	D₈	1	4	yes	—
70	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₄⋊C₄×C₄₇	—	2	8	no	—
81	C₄⋊C₄×C₄₇	—	4	4	no	—
82	C₄⋊C₄×C₄₇	—	4	4	no	—
83	C₂²⋊C₄×C₄₇	—	4	4	no	—
84	C₄²×C₄₇	—	4	4	no	—
85	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
86	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
91	C₂²⋊Dic₄₇	—	4	4	no	—
92	C₄⋊Dic₄₇	—	2	8	no	—
93	C₄⋊Dic₄₇	—	4	4	no	—
94	C₄⋊Dic₄₇	—	4	4	no	—
95	C₄×Dic₄₇	—	4	4	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	2	16	no	—
98	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₄₇⋊C₈	#41	1	8	yes	—
101	C₂³×C₄₇	—	2	2	no	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₄₇	C₂×D₈	1	2	yes	—
105	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
106	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	8	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	#107	1	16	yes	complemented
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₄₇	#135	1	4	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₄₇⋊C₈	—	2	4	no	—
133	C₄₇⋊C₈	#117	1	8	yes	complemented
134	#20817	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	4	no	—
137	C₂²×C₄₇	—	2	4	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	C₂×#107	1	4	yes	—
140	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₄₇	#532	1	2	yes	—
142	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
143	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×#135	1	2	yes	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	8	4	no	—
148	C₄×C₄₇	C₂×#107	1	4	yes	—
149	C₄×C₄₇	C₂×#107	1	4	yes	—
150	C₄×C₄₇	#427	1	4	yes	—
151	C₄×C₄₇	C₂×#117	1	2	yes	—
152	C₄×C₄₇	#533	1	2	yes	—
153	Dic₄₇	—	8	4	no	—
154	C₂²×C₄×C₈	—	47	1	no	—
155	#2285	—	47	1	no	—
156	#2285	D₄₇	1	1	yes	characteristic
157	#1845	—	47	2	no	—
158	#1834	—	47	2	no	—
159	#1808	—	47	4	no	—
160	C₄×#107	—	47	4	no	—
161	C₂×C₄₇	—	2	4	no	—
162	C₂×C₄₇	#2285	1	1	yes	characteristic
163	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
164	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
165	C₂×C₄₇	#3169	1	2	yes	—
166	C₂×C₄₇	#3170	1	2	yes	—
167	C₂³×C₈	—	47	2	no	—
168	C₂×#117	—	2	1	no	—
169	C₂×#117	—	94	1	no	—
170	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
171	C₂×#107	—	47	8	no	—
172	C₂×C₄×C₈	—	47	8	no	—
173	C₂×C₄×C₈	—	47	1	no	—
174	C₂×C₄×C₈	—	47	1	no	—
175	C₂×C₄×C₈	—	47	2	no	—
176	#384	—	47	2	no	—
177	#384	C₂×D₄₇	1	2	yes	—
178	#382	—	2	1	no	—
179	#382	—	94	1	no	—
180	#380	—	2	1	no	—
181	#380	—	47	2	no	—
182	#380	—	94	1	no	—
183	#380	C₂×D₄₇	1	2	yes	—
184	C₄×C₄⋊C₄	—	2	1	no	—
185	C₄×C₄⋊C₄	—	47	2	no	—
186	C₄×C₄⋊C₄	—	94	1	no	—
187	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
188	#340	—	47	8	no	—
189	C₄₇	#20817	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂²×C₈	—	47	2	no	—
194	C₂²×C₈	—	47	8	no	—
195	C₂²×C₈	—	47	2	no	—
196	C₂²×C₈	—	94	8	no	—
197	#117	—	4	4	no	—
198	#117	—	188	4	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	47	4	no	—
202	C₂×C₄⋊C₄	—	94	2	no	—
203	C₂×C₄⋊C₄	—	94	2	no	—
204	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	—	94	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	94	2	no	—
209	C₂×C₄²	Dic₉₄	1	4	yes	—
210	C₂×C₄²	C₄×D₄₇	1	4	yes	—
211	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
215	#107	—	94	16	no	—
216	C₄×C₈	—	47	4	no	—
217	C₄×C₈	—	47	8	no	—
218	C₄×C₈	—	47	4	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	94	2	no	—
222	#95	—	94	2	no	—
223	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	94	4	no	—
228	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
229	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
230	C₂²×C₄	#10625334	1	4	yes	—
231	C₂²×C₄	#10625337	1	4	yes	—
232	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
233	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
234	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
240	C₂×C₈	—	47	4	no	—
241	C₂×C₈	—	47	8	no	—
242	C₂×C₈	—	47	4	no	—
243	C₂×C₈	—	94	16	no	—
244	C₂×C₈	—	94	4	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	94	8	no	—
249	C₄⋊C₄	—	188	4	no	—
250	C₄⋊C₄	—	188	4	no	—
251	C₂²⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	188	4	no	—
253	C₄²	—	4	4	no	—
254	C₄²	—	188	4	no	—
255	C₄²	Dic₄₇⋊C₄	1	4	yes	—
256	C₄²	Dic₄₇⋊C₄	1	4	yes	—
257	C₄²	C₂×Dic₉₄	1	2	yes	—
258	C₄²	C₂×Dic₉₄	1	2	yes	—
259	C₄²	C₂×C₄×D₄₇	1	2	yes	—
260	C₄²	C₂×C₄×D₄₇	1	2	yes	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
264	C₂³	C₂×#10625334	1	2	yes	—
265	C₂³	C₂×#10625337	1	2	yes	—
266	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	188	4	no	—
276	C₂×C₄	—	188	4	no	—
277	C₂×C₄	—	188	4	no	—
278	C₂×C₄	#13093957	1	16	yes	complemented
279	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
280	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
281	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
282	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
283	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
284	C₂×C₄	C₂×#10625334	1	2	yes	—
285	C₂×C₄	C₂×#10625334	1	2	yes	—
286	C₂×C₄	C₂×#10625337	1	2	yes	—
287	C₂×C₄	C₂×#10625337	1	2	yes	—
288	C₂×C₄	#13094044	1	4	yes	—
289	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
291	C₂×C₄	#41×D₄₇	1	4	yes	—
292	C₈	—	47	8	no	—
293	C₈	—	94	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	C₂×#13093957	1	4	yes	—
299	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
300	C₂²	#15689669	1	2	yes	—
301	C₂²	C₂²×#10625334	1	1	yes	characteristic
302	C₂²	C₂²×#10625337	1	1	yes	characteristic
303	C₂²	C₂×#13094044	1	2	yes	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	8	4	no	—
307	C₄	—	376	4	no	—
308	C₄	C₂×#13093957	1	4	yes	—
309	C₄	C₂×#13093957	1	4	yes	—
310	C₄	#15689159	1	4	yes	—
311	C₄	#117×D₄₇	1	2	yes	—
312	C₄	#15689676	1	2	yes	—
313	C₂	—	2	4	no	—
314	C₂	#20245791	1	1	yes	characteristic
315	C₂	C₂²×#13093957	1	1	yes	characteristic
316	C₂	C₂×#15689159	1	1	yes	characteristic
317	C₂	#20253963	1	2	yes	—
318	C₂	#20253970	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#20817
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #20243547	C₂	semidirect
4. #20243515	C₂	semidirect
6. C₄×#13093957	C₂	semidirect
11. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
115. C₂×C₄×C₄₇	#107	semidirect
133. C₄₇⋊C₈	#117	semidirect
189. C₄₇	#20817	semidirect
278. C₂×C₄	#13093957	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₄₇⋊C₈	2
#20245795	2
#20243547	2
#20243515	2
#20243415	2
C₄×#13093957	2
C₄₇×#2285	2
#20817	47