Subgroups — #26371734

Order 12032.

794 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689564	C₂	1	2	yes	maximal, complemented
2	C₂×#15689555	C₂	1	1	yes	maximal, characteristic
3	#20253877	C₂	1	2	yes	maximal, complemented
4	#20245656	C₂	1	4	yes	maximal, complemented
5	#20245650	C₂	1	2	yes	maximal, complemented
6	#20245639	C₂	1	2	yes	maximal, complemented
7	#20244984	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094055	—	2	2	no	—
10	C₂×#13094046	C₂²	1	2	yes	—
11	#15689697	—	2	2	no	—
12	#15689564	—	2	8	no	—
13	#15689555	—	2	4	no	—
14	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
15	C₂×#13093970	C₂²	1	4	yes	—
16	C₂×#13093969	C₂²	1	2	yes	—
17	C₂×#13093967	C₂²	1	4	yes	—
18	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
19	C₂×#13093964	C₂²	1	1	yes	characteristic
20	#15689516	—	2	4	no	—
21	#15689508	C₂²	1	4	yes	—
22	#15689501	C₂²	1	4	yes	—
23	#15689078	C₂²	1	1	yes	characteristic
24	#15688898	C₂²	1	2	yes	—
25	C₁₆×Dic₄₇	C₂²	1	2	yes	—
26	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
27	C₂×Q₃₂×C₄₇	—	2	4	no	—
28	C₂×Q₃₂×C₄₇	C₂²	1	2	yes	—
29	C₂×Q₃₂×C₄₇	C₂²	1	2	yes	—
30	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#10625350	—	2	2	no	—
32	#13094055	—	4	8	no	—
33	#13094046	—	2	4	no	—
34	#13094046	—	2	4	no	—
35	C₂×Dic₁₈₈	C₂³	1	2	yes	—
36	C₂×#10625337	—	2	4	no	—
37	C₂×#10625325	—	2	2	no	—
38	C₂×#10625336	—	2	4	no	—
39	C₂×D₃₇₆	C₂³	1	2	yes	—
40	C₂×#10625329	D₄	1	1	yes	characteristic
41	#13094022	—	2	4	no	—
42	#13094013	—	2	4	no	—
43	#13094007	—	4	4	no	—
44	#13094000	—	4	4	no	—
45	#13093983	D₄	1	1	yes	characteristic
46	Dic₃₇₆	—	2	2	no	—
47	#13093970	—	2	8	no	—
48	#13093969	—	2	4	no	—
49	#13093967	—	2	8	no	—
50	D₇₅₂	—	2	2	no	—
51	#13093964	D₄	1	4	yes	—
52	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
53	#13093934	D₄	1	2	yes	—
54	C₈×Dic₄₇	C₂³	1	2	yes	—
55	C₂²×Q₈×C₄₇	—	2	2	no	—
56	C₂×Q₁₆×C₄₇	—	2	4	no	—
57	C₂×Q₁₆×C₄₇	—	2	4	no	—
58	C₂×Q₁₆×C₄₇	—	2	2	no	—
59	C₂×Q₁₆×C₄₇	C₂³	1	4	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	Q₃₂×C₄₇	—	2	4	no	—
62	Q₃₂×C₄₇	—	2	4	no	—
63	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
64	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₄₇	—	2	4	no	—
67	C₂×Q₈×C₄₇	—	4	2	no	—
68	C₂×Q₈×C₄₇	—	4	4	no	—
69	C₂×Q₈×C₄₇	—	4	4	no	—
70	C₂²×C₄×C₄₇	—	4	2	no	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₂²⋊D₄₇	—	4	2	no	—
73	#10625350	—	4	4	no	—
74	#10625350	—	4	4	no	—
75	C₂×D₁₈₈	—	2	2	no	—
76	C₂×C₄×D₄₇	—	4	2	no	—
77	C₂×Dic₉₄	—	2	2	no	—
78	Q₁₆×C₄₇	—	2	4	no	—
79	Q₁₆×C₄₇	—	2	4	no	—
80	Q₁₆×C₄₇	—	4	4	no	—
81	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²⋊Dic₄₇	D₈	1	2	yes	—
86	#10625337	—	4	8	no	—
87	#10625336	—	4	8	no	—
88	(C₂×D₄₇)⋊C₄	—	4	8	no	—
89	C₄⋊Dic₄₇	D₈	1	2	yes	—
90	Dic₄₇⋊C₄	—	4	8	no	—
91	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
92	#10625329	D₈	1	4	yes	—
93	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
94	Dic₁₈₈	—	2	2	no	—
95	Dic₁₈₈	—	2	2	no	—
96	D₃₇₆	—	2	2	no	—
97	D₃₇₆	—	2	2	no	—
98	#10625325	—	4	4	no	—
99	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
100	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
101	C₄₇⋊C₁₆	C₂×D₄	1	4	yes	—
102	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
103	C₂²×D₄₇	—	4	2	no	—
104	Q₈×C₄₇	—	4	4	no	—
105	Q₈×C₄₇	—	4	4	no	—
106	Q₈×C₄₇	—	8	4	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	8	4	no	—
109	C₂×C₄×C₄₇	#127	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂²⋊D₄₇	—	8	4	no	—
114	C₂×Dic₄₇	—	4	2	no	—
115	C₂×Dic₄₇	C₂×D₈	1	4	yes	—
116	D₁₈₈	—	4	2	no	—
117	D₁₈₈	—	4	2	no	—
118	C₄×D₄₇	—	8	4	no	—
119	Dic₉₄	—	4	2	no	—
120	Dic₉₄	—	4	2	no	—
121	C₈×C₄₇	#127	1	2	yes	—
122	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₄₇⋊C₈	C₂×D₈	1	4	yes	—
125	#33721	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	#493	1	2	yes	—
128	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
129	C₂×D₄₇	—	8	2	no	—
130	C₂×D₄₇	—	8	4	no	—
131	C₄×C₄₇	—	8	4	no	—
132	C₄×C₄₇	#493	1	2	yes	—
133	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
134	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
135	Dic₄₇	—	2	4	no	—
136	Dic₄₇	—	8	2	no	—
137	C₂×#510	—	47	2	no	—
138	C₂×#508	—	47	1	no	—
139	C₂²×Q₃₂	D₄₇	1	1	yes	characteristic, complemented
140	#3156	—	47	2	no	—
141	#2251	—	47	4	no	—
142	#2249	—	47	2	no	—
143	#2247	—	47	2	no	—
144	#2108	—	47	1	no	—
145	C₂×C₄₇	—	2	2	no	—
146	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
147	C₂×C₄₇	#3417	1	2	yes	—
148	D₄₇	—	16	2	no	—
149	C₂×#137	—	94	2	no	—
150	C₂×#135	—	47	2	no	—
151	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
152	#536	—	94	2	no	—
153	#510	—	94	8	no	—
154	#508	—	94	4	no	—
155	C₂×Q₃₂	—	2	4	no	—
156	C₂×Q₃₂	—	47	4	no	—
157	C₂×Q₃₂	—	47	1	no	—
158	C₂×Q₃₂	C₂×D₄₇	1	2	yes	—
159	C₂×Q₃₂	C₂×D₄₇	1	2	yes	—
160	C₂×SD₃₂	—	47	4	no	—
161	C₂×SD₃₂	—	47	2	no	—
162	C₂×D₁₆	—	47	1	no	—
163	C₂×M₃₂	—	47	1	no	—
164	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
165	#497	—	94	4	no	—
166	#495	—	47	4	no	—
167	#494	—	47	4	no	—
168	#405	—	47	1	no	—
169	#346	—	47	2	no	—
170	C₄×C₁₆	—	47	2	no	—
171	C₄₇	#33721	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	94	2	no	—
173	C₂²×Q₈	—	2	2	no	—
174	#137	—	188	8	no	—
175	#135	—	94	4	no	—
176	#135	—	94	4	no	—
177	C₂×Q₁₆	—	2	4	no	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	2	4	no	—
180	C₂×Q₁₆	—	47	2	no	—
181	C₂×Q₁₆	—	94	4	no	—
182	C₂×Q₁₆	C₂²×D₄₇	1	4	yes	—
183	C₂×SD₁₆	—	94	4	no	—
184	C₂×SD₁₆	—	94	2	no	—
185	C₂×D₈	—	47	2	no	—
186	C₂×M₁₆	—	47	1	no	—
187	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
188	C₄⋊Q₈	—	94	4	no	—
189	#124	—	94	4	no	—
190	#123	—	188	4	no	—
191	C₂²⋊Q₈	—	188	4	no	—
192	#117	—	47	1	no	—
193	Q₃₂	—	2	4	no	—
194	Q₃₂	—	2	4	no	—
195	Q₃₂	—	94	8	no	—
196	Q₃₂	—	94	2	no	—
197	SD₃₂	—	94	8	no	—
198	SD₃₂	—	94	4	no	—
199	D₁₆	—	94	2	no	—
200	M₃₂	—	47	4	no	—
201	C₂×C₁₆	—	47	2	no	—
202	C₂×C₁₆	C₂²⋊D₄₇	1	4	yes	—
203	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
204	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
205	#97	—	47	2	no	—
206	C₄×C₈	—	47	2	no	—
207	#41	—	188	4	no	—
208	#41	—	188	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	—	94	2	no	—
214	C₂×D₄	—	94	2	no	—
215	C₂×D₄	—	188	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	188	2	no	—
218	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
219	Q₁₆	—	2	4	no	—
220	Q₁₆	—	2	4	no	—
221	Q₁₆	—	4	4	no	—
222	Q₁₆	—	94	2	no	—
223	Q₁₆	—	94	2	no	—
224	Q₁₆	—	188	8	no	—
225	SD₁₆	—	188	8	no	—
226	SD₁₆	—	188	4	no	—
227	D₈	—	94	2	no	—
228	D₈	—	94	2	no	—
229	M₁₆	—	47	4	no	—
230	C₂×C₈	—	47	2	no	—
231	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
232	C₂×C₈	D₄×D₄₇	1	2	yes	—
233	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
234	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
235	C₄⋊C₄	—	47	2	no	—
236	C₄⋊C₄	—	188	8	no	—
237	C₂²⋊C₄	—	47	2	no	—
238	C₂²⋊C₄	—	188	8	no	—
239	C₄²	—	47	2	no	—
240	C₁₆	—	47	4	no	—
241	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
242	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
243	C₂³	—	188	2	no	—
244	C₂³	D₈×D₄₇	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	4	4	no	—
247	Q₈	—	8	4	no	—
248	Q₈	—	188	2	no	—
249	Q₈	—	188	2	no	—
250	D₄	—	376	4	no	—
251	D₄	—	188	2	no	—
252	D₄	—	188	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	47	4	no	—
256	C₂×C₄	—	376	4	no	—
257	C₂×C₄	—	188	2	no	—
258	C₂×C₄	#13094020	1	2	yes	—
259	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
260	C₂×C₄	D₈×D₄₇	1	2	yes	—
261	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
262	C₈	—	47	4	no	—
263	C₈	#13094020	1	2	yes	—
264	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
265	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	376	4	no	—
268	C₂²	—	376	2	no	—
269	C₂²	#15689498	1	2	yes	—
270	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
271	C₄	—	8	4	no	—
272	C₄	—	94	4	no	—
273	C₄	—	376	2	no	—
274	C₄	#15689498	1	2	yes	—
275	C₄	C₂×#13094020	1	1	yes	characteristic
276	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	752	2	no	—
279	C₂	C₂×#15689498	1	1	yes	characteristic
280	C₂	#20256537	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂²×Q₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#15689555	C₂
hypercenter	C₂²×Q₃₂	D₄₇
hypocenter	C₄₇	#33721
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#15689564	C₂	semidirect
3. #20253877	C₂	semidirect
4. #20245656	C₂	semidirect
5. #20245650	C₂	semidirect
6. #20245639	C₂	semidirect
7. #20244984	C₂	semidirect
8. C₂²×Q₃₂×C₄₇	C₂	semidirect
139. C₂²×Q₃₂	D₄₇	semidirect
171. C₄₇	#33721	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689564	2
C₂×#15689555	2
#20253877	2
#20245656	2
#20245650	2
#20245639	2
#20244984	2
C₂²×Q₃₂×C₄₇	2
#33721	47