Subgroups — #26369363

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689077	C₂	1	1	yes	maximal, characteristic
2	#20245196	C₂	1	4	yes	maximal, complemented
3	#20245193	C₂	1	4	yes	maximal, complemented
4	#20245143	C₂	1	2	yes	maximal, complemented
5	#20244998	C₂	1	2	yes	maximal
6	C₄×#13093964	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625324	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
10	C₂×#13093981	C₄	1	1	yes	characteristic
11	C₂×#13093964	C₂²	1	2	yes	—
12	#15689077	C₄	1	2	yes	—
13	#15689077	C₄	1	2	yes	—
14	#15689077	C₂²	1	2	yes	—
15	#15689077	C₂²	1	2	yes	—
16	C₂×#13093935	C₄	1	1	yes	characteristic
17	C₂×#13093934	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	Dic₄₇⋊C₁₆	C₂²	1	8	yes	complemented
20	(C₂×D₄₇)⋊C₁₆	C₂²	1	8	yes	—
21	#15688899	C₂²	1	2	yes	complemented
22	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
23	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#346	C₂²	1	2	yes	complemented
26	C₄×C₁₆×C₄₇	C₂²	1	2	yes	complemented
27	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×#10625324	—	2	4	no	—
29	C₂×#10625324	C₂×C₄	1	2	yes	—
30	C₂×#10625324	C₂×C₄	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₄₇	—	2	4	no	—
34	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₄₇	C₂³	1	2	yes	—
36	#13093981	C₈	1	4	yes	—
37	#13093981	C₂×C₄	1	2	yes	—
38	#13093981	C₂×C₄	1	2	yes	—
39	C₂×Dic₄₇⋊C₄	C₈	1	2	yes	—
40	C₂×(C₂×D₄₇)⋊C₄	C₈	1	2	yes	—
41	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
42	#13093964	—	2	4	no	—
43	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
44	#13093935	C₂×C₄	1	2	yes	—
45	#13093935	C₂×C₄	1	2	yes	—
46	#13093934	C₂×C₄	1	2	yes	—
47	#13093934	C₂×C₄	1	2	yes	—
48	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
49	C₈×Dic₄₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₄₇	—	2	4	no	—
54	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
55	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂³×D₄₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₄₇	—	2	2	no	—
65	C₂×C₄×D₄₇	—	2	2	no	—
66	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
67	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
69	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	#41	1	4	yes	—
73	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₄²×C₄₇	C₂×C₈	1	2	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	(C₂×D₄₇)⋊C₄	C₂×C₈	1	4	yes	—
78	(C₂×D₄₇)⋊C₄	C₂×C₈	1	4	yes	—
79	Dic₄₇⋊C₄	C₂×C₈	1	4	yes	—
80	Dic₄₇⋊C₄	C₂×C₈	1	4	yes	—
81	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
82	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄₇⋊C₈	—	2	2	no	—
84	C₂×C₄₇⋊C₈	—	2	2	no	—
85	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
87	#10625324	—	2	4	no	—
88	#10625324	—	2	4	no	—
89	C₈×D₄₇	—	2	4	no	—
90	C₈×D₄₇	—	2	4	no	—
91	C₁₆×C₄₇	—	2	4	no	—
92	C₄₇⋊C₁₆	—	2	4	no	—
93	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
94	C₂²×D₄₇	—	2	2	no	—
95	C₂²×D₄₇	—	4	2	no	—
96	C₂²×D₄₇	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	#117	1	2	yes	—
99	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×Dic₄₇	—	2	2	no	—
105	C₂×Dic₄₇	—	4	2	no	—
106	C₂×Dic₄₇	C₂²×C₈	1	4	yes	—
107	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
108	C₄×D₄₇	—	2	2	no	—
109	C₄×D₄₇	—	2	2	no	—
110	C₄×D₄₇	—	2	4	no	—
111	C₈×C₄₇	#117	1	2	yes	—
112	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
113	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
114	C₈×C₄₇	C₂×#41	1	2	yes	—
115	C₄₇⋊C₈	—	2	2	no	—
116	C₄₇⋊C₈	—	2	2	no	—
117	#33419	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	#431	1	2	yes	—
120	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
121	C₂×D₄₇	—	2	4	no	—
122	C₂×D₄₇	—	2	2	no	—
123	C₂×D₄₇	—	4	4	no	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	2	2	no	—
126	C₄×C₄₇	#431	1	2	yes	—
127	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
128	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
129	Dic₄₇	—	2	4	no	—
130	Dic₄₇	—	2	2	no	—
131	C₂×#405	—	47	1	no	—
132	#2164	—	47	4	no	—
133	#2164	—	47	4	no	—
134	#2150	—	47	2	no	—
135	#2112	—	47	2	no	—
136	C₄×M₃₂	—	47	1	no	—
137	C₄×M₃₂	D₄₇	1	1	yes	characteristic, complemented
138	C₂×C₄₇	—	2	2	no	—
139	C₂×C₄₇	C₂×#431	1	1	yes	characteristic
140	C₂×C₄₇	#3407	1	2	yes	—
141	D₄₇	—	4	2	no	—
142	C₂²×M₁₆	—	47	1	no	—
143	C₂³×C₈	—	47	1	no	—
144	C₂×#117	—	47	1	no	—
145	C₂×M₃₂	—	47	2	no	—
146	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
147	#405	—	47	2	no	—
148	#405	—	47	2	no	—
149	#405	—	47	2	no	—
150	#405	—	47	2	no	—
151	C₂×#97	—	47	1	no	—
152	C₂×#97	—	47	1	no	—
153	C₂×C₄×C₈	—	47	1	no	—
154	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
155	C₄⋊C₁₆	—	47	8	no	—
156	C₂²⋊C₁₆	—	47	8	no	—
157	#346	—	47	2	no	—
158	#346	C₂×D₄₇	1	2	yes	complemented
159	C₄×C₁₆	—	47	2	no	—
160	C₄×C₁₆	C₂×D₄₇	1	2	yes	complemented
161	C₄₇	#33419	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	—	47	1	no	—
163	C₂×M₁₆	—	47	2	no	—
164	C₂×M₁₆	—	47	2	no	—
165	C₂×M₁₆	—	94	4	no	—
166	C₂²×C₈	—	47	1	no	—
167	C₂²×C₈	—	47	1	no	—
168	C₂²×C₈	—	47	2	no	—
169	C₂²×C₈	—	47	2	no	—
170	C₂²×C₈	—	94	4	no	—
171	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
173	#117	—	47	4	no	—
174	#117	—	47	2	no	—
175	#117	—	47	2	no	—
176	C₂×C₄⋊C₄	—	47	2	no	—
177	C₂×C₂²⋊C₄	—	47	2	no	—
178	C₂×C₄²	—	47	1	no	—
179	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
180	M₃₂	—	2	4	no	—
181	M₃₂	—	94	4	no	—
182	C₂×C₁₆	—	47	4	no	—
183	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
184	#97	—	47	2	no	—
185	#97	—	47	2	no	—
186	#97	—	47	2	no	—
187	#97	—	47	2	no	—
188	C₄×C₈	—	47	2	no	—
189	C₄×C₈	—	47	2	no	—
190	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
191	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
192	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
194	C₂⁴	—	47	1	no	—
195	C₂²×C₄	—	47	1	no	—
196	C₂²×C₄	—	47	2	no	—
197	C₂²×C₄	—	47	2	no	—
198	C₂²×C₄	—	47	2	no	—
199	C₂²×C₄	—	94	2	no	—
200	C₂²×C₄	—	94	2	no	—
201	C₂²×C₄	C₈×D₄₇	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
203	M₁₆	—	94	4	no	—
204	M₁₆	—	94	4	no	—
205	C₂×C₈	—	47	2	no	—
206	C₂×C₈	—	47	2	no	—
207	C₂×C₈	—	94	2	no	—
208	C₂×C₈	—	94	2	no	—
209	C₂×C₈	—	94	4	no	—
210	C₂×C₈	—	94	4	no	—
211	C₂×C₈	#10625324	1	4	yes	—
212	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₂×C₈	#10625350	1	4	yes	—
216	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
217	C₄⋊C₄	—	47	4	no	—
218	C₄⋊C₄	—	47	4	no	—
219	C₂²⋊C₄	—	47	4	no	—
220	C₂²⋊C₄	—	47	4	no	—
221	C₄²	—	47	2	no	—
222	C₄²	—	47	2	no	—
223	C₄²	C₈×D₄₇	1	2	yes	—
224	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
225	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
226	C₁₆	—	2	4	no	—
227	C₁₆	—	94	4	no	—
228	C₂³	—	47	2	no	—
229	C₂³	—	94	2	no	—
230	C₂³	—	188	2	no	—
231	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	47	4	no	—
234	C₂×C₄	—	47	2	no	—
235	C₂×C₄	—	94	2	no	—
236	C₂×C₄	—	94	2	no	—
237	C₂×C₄	—	94	2	no	—
238	C₂×C₄	—	94	4	no	—
239	C₂×C₄	—	188	2	no	—
240	C₂×C₄	#13093981	1	2	yes	—
241	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
242	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
243	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
244	C₂×C₄	C₂×#10625324	1	2	yes	—
245	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
246	C₈	—	94	2	no	—
247	C₈	—	94	2	no	—
248	C₈	#13093981	1	2	yes	—
249	C₈	C₂×#10625324	1	2	yes	—
250	C₈	C₂×#10625324	1	2	yes	—
251	C₈	C₂×#10625350	1	2	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	94	2	no	—
254	C₂²	—	94	4	no	—
255	C₂²	—	188	4	no	—
256	C₂²	#15689174	1	2	yes	—
257	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	94	2	no	—
261	C₄	—	94	4	no	—
262	C₄	#15689174	1	2	yes	—
263	C₄	C₂×#13093981	1	1	yes	characteristic
264	C₄	C₂²×#10625324	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	188	2	no	—
267	C₂	C₂×#15689174	1	1	yes	characteristic
268	C₂	#20256490	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄×M₃₂×C₄₇	C₂
prime radical	C₂³×D₄₇	C₂×C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₄×M₃₂	D₄₇
hypocenter	C₄₇	#33419
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20245196	C₂	semidirect
3. #20245193	C₂	semidirect
4. #20245143	C₂	semidirect
6. C₄×#13093964	C₂	semidirect
7. C₄×M₃₂×C₄₇	C₂	semidirect
19. Dic₄₇⋊C₁₆	C₂²	semidirect
21. #15688899	C₂²	semidirect
22. C₄×C₄₇⋊C₁₆	C₂²	semidirect
25. C₄₇×#346	C₂²	semidirect
26. C₄×C₁₆×C₄₇	C₂²	semidirect
137. C₄×M₃₂	D₄₇	semidirect
158. #346	C₂×D₄₇	semidirect
160. C₄×C₁₆	C₂×D₄₇	semidirect
161. C₄₇	#33419	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689077	2
#20245196	2
#20245193	2
#20245143	2
#20244998	2
C₄×#13093964	2
C₄×M₃₂×C₄₇	2
#33419	47