Subgroups — #26369705

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252305	C₂	1	2	yes	maximal, complemented
2	#20245274	C₂	1	4	yes	maximal, complemented
3	#20245269	C₂	1	4	yes	maximal, complemented
4	C₂×#15688953	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245116	C₂	1	2	yes	maximal, complemented
6	#20244984	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#508	C₂	1	1	yes	maximal, characteristic
8	#15689890	—	2	2	no	—
9	C₂×#13093964	C₂²	1	1	yes	characteristic
10	#15689245	—	2	4	no	—
11	#15689240	C₂²	1	8	yes	complemented
12	C₂×#13093960	C₂²	1	2	yes	—
13	#15689140	—	2	2	no	—
14	#15689078	C₂²	1	1	yes	characteristic
15	#15688953	C₂²	1	2	yes	complemented
16	#15688953	C₂²	1	2	yes	complemented
17	#15688930	C₂²	1	8	yes	complemented
18	#15688898	C₂²	1	2	yes	complemented
19	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
20	C₂×#135×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#508	C₄	1	2	yes	complemented
22	C₄₇×#508	C₄	1	2	yes	complemented
23	C₄₇×#508	C₂²	1	2	yes	—
24	C₄₇×#508	C₂²	1	2	yes	—
25	C₂×Q₃₂×C₄₇	C₄	1	1	yes	characteristic, complemented
26	C₂×SD₃₂×C₄₇	C₄	1	2	yes	complemented
27	C₂×D₁₆×C₄₇	C₄	1	1	yes	characteristic, complemented
28	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
29	#13094031	—	2	8	no	—
30	#13094031	—	4	4	no	—
31	C₂×#10625329	—	2	2	no	—
32	C₂×#10625329	—	4	4	no	—
33	C₂×#10625329	D₄	1	1	yes	characteristic
34	#13093983	D₄	1	1	yes	characteristic
35	#13093964	—	2	1	no	—
36	#13093964	—	2	1	no	—
37	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
38	#13093960	—	2	2	no	—
39	#13093960	C₂³	1	4	yes	—
40	#13093951	—	2	8	no	—
41	#13093934	D₄	1	2	yes	—
42	C₈×Dic₄₇	C₂³	1	2	yes	—
43	C₂×#41×C₄₇	—	2	2	no	—
44	#135×C₄₇	—	2	2	no	—
45	#135×C₄₇	—	2	2	no	—
46	#135×C₄₇	C₂×C₄	1	4	yes	—
47	#135×C₄₇	C₂³	1	4	yes	—
48	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×SD₁₆×C₄₇	—	2	2	no	—
50	C₂×D₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	Q₃₂×C₄₇	C₂×C₄	1	2	yes	—
53	Q₃₂×C₄₇	C₂×C₄	1	2	yes	—
54	SD₃₂×C₄₇	C₂×C₄	1	4	yes	—
55	SD₃₂×C₄₇	C₂×C₄	1	4	yes	—
56	D₁₆×C₄₇	C₂×C₄	1	2	yes	—
57	D₁₆×C₄₇	C₂×C₄	1	2	yes	—
58	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
64	#41×C₄₇	—	2	4	no	—
65	#41×C₄₇	—	2	4	no	—
66	#41×C₄₇	—	4	2	no	—
67	#41×C₄₇	—	4	2	no	—
68	C₂×Q₈×C₄₇	—	2	2	no	—
69	C₂×D₄×C₄₇	—	2	2	no	—
70	C₂×D₄×C₄₇	—	4	2	no	—
71	C₂²×C₄×C₄₇	—	4	2	no	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₄₇	—	2	2	no	—
74	Q₁₆×C₄₇	C₂²×C₄	1	4	yes	—
75	SD₁₆×C₄₇	—	2	4	no	—
76	SD₁₆×C₄₇	—	2	4	no	—
77	D₈×C₄₇	—	2	2	no	—
78	D₈×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
83	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²⋊Dic₄₇	—	2	1	no	—
86	C₄⋊Dic₄₇	—	2	1	no	—
87	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
88	#10625329	—	2	4	no	—
89	#10625329	—	4	2	no	—
90	#10625329	—	4	8	no	—
91	#10625329	D₈	1	2	yes	—
92	#10625329	D₈	1	2	yes	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	4	8	no	—
95	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
96	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
97	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₄₇⋊C₁₆	—	2	2	no	—
101	C₂³×C₄₇	—	4	2	no	—
102	C₂³×C₄₇	#135	1	1	yes	characteristic
103	Q₈×C₄₇	—	2	4	no	—
104	Q₈×C₄₇	—	4	2	no	—
105	D₄×C₄₇	—	2	4	no	—
106	D₄×C₄₇	—	4	4	no	—
107	D₄×C₄₇	—	4	4	no	—
108	D₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×Dic₄₇	—	2	2	no	—
119	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
120	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
121	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
122	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₄₇⋊C₈	—	4	4	no	—
124	C₄₇⋊C₈	C₂×D₈	1	4	yes	—
125	#33465	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	4	1	no	—
127	C₂²×C₄₇	—	4	4	no	—
128	C₂²×C₄₇	—	16	2	no	—
129	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
130	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
132	C₄×C₄₇	—	4	4	no	—
133	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
134	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
135	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
137	Dic₄₇	—	4	2	no	—
138	C₂×#508	D₄₇	1	1	yes	characteristic
139	#2957	—	47	2	no	—
140	#2179	—	47	4	no	—
141	#2178	—	47	4	no	—
142	C₂×#368	—	47	1	no	—
143	#2143	—	47	2	no	—
144	#2108	—	47	1	no	—
145	C₂×C₄₇	—	2	1	no	—
146	C₂×C₄₇	—	2	1	no	—
147	C₂×C₄₇	—	2	1	no	—
148	C₂×C₄₇	—	8	4	no	—
149	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
150	C₂×#135	C₂×D₄₇	1	2	yes	—
151	#568	—	94	2	no	—
152	#508	Dic₄₇	1	2	yes	complemented
153	#508	Dic₄₇	1	2	yes	complemented
154	#508	C₂×D₄₇	1	2	yes	—
155	#508	C₂×D₄₇	1	2	yes	—
156	C₂×Q₃₂	Dic₄₇	1	1	yes	characteristic, complemented
157	C₂×SD₃₂	Dic₄₇	1	2	yes	complemented
158	C₂×D₁₆	Dic₄₇	1	1	yes	characteristic, complemented
159	C₂×M₃₂	—	47	1	no	—
160	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
161	#444	—	94	4	no	—
162	#443	—	47	8	no	—
163	C₂×#108	—	47	2	no	—
164	#421	—	94	2	no	—
165	#405	—	47	1	no	—
166	#368	—	47	2	no	—
167	#368	—	47	2	no	—
168	#359	—	47	8	no	—
169	#346	—	47	2	no	—
170	C₄×C₁₆	—	47	2	no	—
171	C₄₇	#33465	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	2	2	no	—
173	#135	—	2	2	no	—
174	#135	—	2	2	no	—
175	#135	C₂×Dic₄₇	1	4	yes	—
176	#135	C₂²×D₄₇	1	4	yes	—
177	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
178	C₂×SD₁₆	—	2	2	no	—
179	C₂×D₈	C₂×Dic₄₇	1	2	yes	—
180	#131	—	94	8	no	—
181	#131	—	188	4	no	—
182	C₂×M₁₆	—	47	1	no	—
183	C₂×M₁₆	—	94	2	no	—
184	C₂×M₁₆	—	188	4	no	—
185	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
186	#117	—	47	1	no	—
187	Q₃₂	C₂×Dic₄₇	1	2	yes	—
188	Q₃₂	C₂×Dic₄₇	1	2	yes	—
189	SD₃₂	C₂×Dic₄₇	1	4	yes	—
190	SD₃₂	C₂×Dic₄₇	1	4	yes	—
191	D₁₆	C₂×Dic₄₇	1	2	yes	—
192	D₁₆	C₂×Dic₄₇	1	2	yes	—
193	M₃₂	—	94	1	no	—
194	M₃₂	—	94	1	no	—
195	C₂×C₁₆	—	47	2	no	—
196	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
197	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
198	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
199	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
200	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
201	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
202	#108	—	47	4	no	—
203	#108	—	94	2	no	—
204	#104	—	94	8	no	—
205	#97	—	47	2	no	—
206	C₄×C₈	—	47	2	no	—
207	#41	—	2	4	no	—
208	#41	—	2	4	no	—
209	#41	—	4	2	no	—
210	#41	—	4	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	Q₁₆	C₂²×Dic₄₇	1	4	yes	—
218	SD₁₆	—	2	4	no	—
219	SD₁₆	—	2	4	no	—
220	D₈	—	2	2	no	—
221	D₈	C₂²×Dic₄₇	1	4	yes	—
222	M₁₆	—	47	2	no	—
223	M₁₆	—	47	2	no	—
224	M₁₆	—	94	4	no	—
225	M₁₆	—	188	8	no	—
226	M₁₆	—	188	2	no	—
227	C₂×C₈	—	47	2	no	—
228	C₂×C₈	—	188	8	no	—
229	C₂×C₈	—	188	2	no	—
230	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
231	C₂×C₈	#10625352	1	1	yes	characteristic
232	C₂×C₈	#10625352	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
234	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
235	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
236	C₄⋊C₄	—	94	1	no	—
237	C₂²⋊C₄	—	94	1	no	—
238	C₄²	—	47	2	no	—
239	C₁₆	—	94	2	no	—
240	C₁₆	C₂²×Dic₄₇	1	1	yes	characteristic
241	C₁₆	C₂²×Dic₄₇	1	1	yes	characteristic
242	C₁₆	C₂²×Dic₄₇	1	1	yes	characteristic
243	C₁₆	C₂²×Dic₄₇	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	#13094043	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	94	2	no	—
256	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
257	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
258	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
259	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
260	C₂×C₄	#13094043	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
262	C₈	—	47	4	no	—
263	C₈	—	188	4	no	—
264	C₈	D₄×Dic₄₇	1	1	yes	characteristic
265	C₈	D₄×Dic₄₇	1	1	yes	characteristic
266	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
267	C₈	C₂×#10625352	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	16	2	no	—
271	C₂²	D₈×Dic₄₇	1	1	yes	characteristic
272	C₂²	D₈×Dic₄₇	1	1	yes	characteristic
273	C₂²	C₂×#13094043	1	1	yes	characteristic
274	C₄	—	4	4	no	—
275	C₄	—	188	2	no	—
276	C₄	D₈×Dic₄₇	1	1	yes	characteristic
277	C₄	D₈×Dic₄₇	1	1	yes	characteristic
278	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
279	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×D₈×Dic₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₂×C₄₇×#508	C₂
socle	C₂×C₄₇	C₂×C₄×D₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#508	D₄₇
hypocenter	C₄₇	#33465
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #20252305	C₂	semidirect
2. #20245274	C₂	semidirect
3. #20245269	C₂	semidirect
4. C₂×#15688953	C₂	semidirect
5. #20245116	C₂	semidirect
6. #20244984	C₂	semidirect
11. #15689240	C₂²	semidirect
15. #15688953	C₂²	semidirect
16. #15688953	C₂²	semidirect
17. #15688930	C₂²	semidirect
18. #15688898	C₂²	semidirect
19. C₁₆×Dic₄₇	C₂²	semidirect
21. C₄₇×#508	C₄	semidirect
22. C₄₇×#508	C₄	semidirect
25. C₂×Q₃₂×C₄₇	C₄	semidirect
26. C₂×SD₃₂×C₄₇	C₄	semidirect
27. C₂×D₁₆×C₄₇	C₄	semidirect
152. #508	Dic₄₇	semidirect
153. #508	Dic₄₇	semidirect
156. C₂×Q₃₂	Dic₄₇	semidirect
157. C₂×SD₃₂	Dic₄₇	semidirect
158. C₂×D₁₆	Dic₄₇	semidirect
171. C₄₇	#33465	semidirect

Maximal subgroups

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

subgroup	index
#20252305	2
#20245274	2
#20245269	2
C₂×#15688953	2
#20245116	2
#20244984	2
C₂×C₄₇×#508	2
#33465	47