Subgroups — #26110924

Order 12032.

586 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252412	C₂	1	1	yes	maximal, characteristic
2	#20241783	C₂	1	2	yes	maximal, complemented
3	#20241709	C₂	1	1	yes	maximal, characteristic
4	#20241221	C₂	1	2	yes	maximal, complemented
5	C₄×#124×C₄₇	C₂	1	1	yes	maximal, characteristic
6	D₄×C₄₇⋊C₈	—	2	4	no	—
7	#15689176	C₂²	1	2	yes	—
8	#15689176	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
10	Q₈⋊(C₄₇⋊C₈)	—	2	4	no	—
11	#15688847	—	2	2	no	—
12	C₂×#124×C₄₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#391	C₄	1	1	yes	characteristic
16	C₄₇×#390	C₄	1	1	yes	characteristic
17	C₄₇×#388	—	2	2	no	—
18	C₄₇×#382	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
20	C₄³×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	4	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₄×C₄₇⋊C₈	—	2	2	no	—
28	C₄×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	D₄	1	2	yes	—
32	#124×C₄₇	—	2	2	no	—
33	#124×C₄₇	C₈	1	4	yes	complemented
34	C₄×Q₈×C₄₇	—	2	2	no	—
35	C₄×Q₈×C₄₇	—	2	2	no	—
36	C₄×D₄×C₄₇	—	2	2	no	—
37	C₄×D₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	—	2	1	no	—
45	C₂×C₄²×C₄₇	—	2	1	no	—
46	C₂×C₄²×C₄₇	—	2	1	no	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	#95×C₄₇	—	2	2	no	—
51	#95×C₄₇	—	2	2	no	—
52	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
53	C₂×Q₈×C₄₇	—	2	1	no	—
54	C₂×Q₈×C₄₇	—	2	2	no	—
55	C₂×Q₈×C₄₇	—	2	2	no	—
56	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	complemented
57	C₂×D₄×C₄₇	—	2	2	no	—
58	C₂×D₄×C₄₇	—	2	2	no	—
59	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₄₇	M₁₆	1	2	yes	—
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	1	no	—
67	C₂²×C₄×C₄₇	—	4	4	no	—
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	—	2	2	no	—
74	C₄⋊C₄×C₄₇	—	4	2	no	—
75	C₂²⋊C₄×C₄₇	—	2	8	no	—
76	C₂²⋊C₄×C₄₇	—	2	4	no	—
77	C₂²⋊C₄×C₄₇	—	4	4	no	—
78	C₄²×C₄₇	—	2	1	no	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	—	4	2	no	—
81	C₄²×C₄₇	—	4	2	no	—
82	C₄²×C₄₇	—	4	1	no	—
83	C₄²×C₄₇	—	4	1	no	—
84	C₄²×C₄₇	C₂×C₈	1	2	yes	complemented
85	C₄²×C₄₇	C₂×D₄	1	2	yes	—
86	C₄²×C₄₇	#41	1	2	yes	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	8	4	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	4	2	no	—
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	C₂²⋊C₈	1	4	yes	—
95	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	Q₈×C₄₇	—	2	4	no	—
97	Q₈×C₄₇	—	4	2	no	—
98	D₄×C₄₇	—	2	4	no	—
99	D₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	4	1	no	—
106	C₂×C₄×C₄₇	—	4	1	no	—
107	C₂×C₄×C₄₇	—	4	1	no	—
108	C₂×C₄×C₄₇	—	4	1	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	1	no	—
116	C₂×C₄×C₄₇	—	8	2	no	—
117	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#121	1	2	yes	—
120	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#131	1	2	yes	—
124	C₄₇⋊C₈	—	4	4	no	—
125	C₄₇⋊C₈	—	8	2	no	—
126	#8957	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	1	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	4	4	no	—
131	C₂²×C₄₇	—	4	2	no	—
132	C₂²×C₄₇	—	4	2	no	—
133	C₂²×C₄₇	—	8	2	no	—
134	C₂²×C₄₇	#386	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₂²×C₄₇	#408	1	1	yes	characteristic
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	—	4	2	no	—
140	C₄×C₄₇	—	4	2	no	—
141	C₄×C₄₇	—	4	2	no	—
142	C₄×C₄₇	—	4	2	no	—
143	C₄×C₄₇	—	8	1	no	—
144	C₄×C₄₇	C₈×D₄	1	2	yes	—
145	C₄×C₄₇	#435	1	2	yes	—
146	C₄×C₄₇	#455	1	2	yes	—
147	#2972	—	47	1	no	—
148	C₄×#124	D₄₇	1	1	yes	characteristic
149	#1479	—	47	2	no	—
150	#1460	—	47	1	no	—
151	#1276	—	47	2	no	—
152	C₂×C₄₇	—	2	1	no	—
153	C₂×C₄₇	—	2	1	no	—
154	C₂×C₄₇	—	4	4	no	—
155	C₂×C₄₇	#1869	1	1	yes	characteristic
156	C₂×C₄₇	#1884	1	1	yes	characteristic
157	C₂×C₄₇	#1885	1	1	yes	characteristic
158	C₂×#124	Dic₄₇	1	1	yes	characteristic
159	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
160	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
161	C₈×D₄	—	94	4	no	—
162	#431	—	47	2	no	—
163	#431	—	47	2	no	—
164	C₂×C₂²⋊C₈	—	94	2	no	—
165	#391	Dic₄₇	1	1	yes	characteristic
166	#390	Dic₄₇	1	1	yes	characteristic
167	#388	—	2	2	no	—
168	#382	Dic₄₇	1	1	yes	characteristic
169	C₄×C₂²⋊C₄	—	2	2	no	—
170	C₄³	C₂×D₄₇	1	1	yes	characteristic
171	Q₈⋊C₈	—	94	4	no	—
172	#323	—	94	2	no	—
173	C₄₇	#8957	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
175	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
177	C₂²×C₈	—	94	4	no	—
178	#124	—	2	2	no	—
179	#124	C₄₇⋊C₈	1	4	yes	complemented
180	C₄×Q₈	—	2	2	no	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
196	C₄⋊C₈	—	47	4	no	—
197	C₄⋊C₈	—	94	2	no	—
198	C₂²⋊C₈	—	94	4	no	—
199	C₂²⋊C₈	—	94	2	no	—
200	C₂²⋊C₈	—	94	2	no	—
201	C₄×C₈	—	94	2	no	—
202	C₄×C₈	—	94	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂⁴	C₂²⋊Dic₄₇	1	2	yes	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₂×C₄₇⋊C₈	1	2	yes	complemented
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
213	C₂×D₄	#10625329	1	2	yes	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	—	376	4	no	—
227	C₂×C₈	—	188	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	2	8	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	C₂×C₄₇⋊C₈	1	2	yes	complemented
241	C₄²	D₄×D₄₇	1	2	yes	—
242	C₄²	#10625352	1	2	yes	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
248	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	4	2	no	—
251	D₄	—	2	4	no	—
252	D₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
271	C₂×C₄	#13093992	1	1	yes	characteristic
272	C₂×C₄	#13093996	1	2	yes	—
273	C₂×C₄	#13094005	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
275	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
276	C₂×C₄	#13094031	1	2	yes	—
277	C₈	—	376	2	no	—
278	C₈	—	188	4	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	#15689003	1	1	yes	characteristic
287	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
288	C₂²	#15689090	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	8	1	no	—
296	C₄	D₄×C₄₇⋊C₈	1	2	yes	—
297	C₄	#15689195	1	2	yes	—
298	C₄	#15689300	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	4	4	no	—
302	C₂	#20243681	1	1	yes	characteristic
303	C₂	#20243772	1	1	yes	characteristic
304	C₂	#20243777	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093992
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#124×C₄₇	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#124	D₄₇
hypocenter	C₄₇	#8957
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#124

Complements

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

N	H ≅ G/N	type
2. #20241783	C₂	semidirect
4. #20241221	C₂	semidirect
33. #124×C₄₇	C₈	semidirect
56. C₂×Q₈×C₄₇	C₂×C₈	semidirect
84. C₄²×C₄₇	C₂×C₈	semidirect
173. C₄₇	#8957	semidirect
179. #124	C₄₇⋊C₈	semidirect
209. C₂×Q₈	C₂×C₄₇⋊C₈	semidirect
240. C₄²	C₂×C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20252412	2
#20241783	2
#20241709	2
#20241221	2
C₄×#124×C₄₇	2
#8957	47