Subgroups — C₂²×#15689376 #27083399

Order 12032.

3566 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13093998	C₂	1	2	yes	maximal, complemented
3	C₂×#15689376	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13093956	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13093946	C₂	1	2	yes	maximal, complemented
6	C₂³×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₈₈	C₂²	1	2	yes	—
8	C₂³×C₂²⋊D₄₇	—	2	2	no	—
9	C₂²×D₃₇₆	—	2	8	no	—
10	C₂²×D₃₇₆	C₂²	1	12	yes	complemented
11	C₂×#13093998	—	2	12	no	—
12	C₂×#13093998	C₂²	1	24	yes	complemented
13	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂²×(C₂×D₄₇)⋊C₄	—	2	2	no	—
15	#15689376	C₂²	1	64	yes	direct product factor
16	C₂×#13093956	C₂²	1	12	yes	complemented
17	C₂×#13093946	C₂²	1	24	yes	complemented
18	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₄₇	—	2	8	no	—
20	C₂²×M₁₆×C₄₇	C₂²	1	12	yes	complemented
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂⁴×D₄₇	—	2	2	no	—
24	C₂²×D₁₈₈	—	2	8	no	—
25	C₂²×D₁₈₈	—	2	8	no	—
26	C₂²×D₁₈₈	—	2	6	no	—
27	C₂²×D₁₈₈	C₂³	1	12	yes	—
28	C₂²×C₂²⋊D₄₇	—	2	24	no	—
29	C₂²×C₂²⋊D₄₇	—	4	16	no	—
30	C₂³×Dic₄₇	—	2	2	no	—
31	C₂×D₃₇₆	—	2	48	no	—
32	C₂×D₃₇₆	C₂³	1	16	yes	complemented
33	#13093998	—	2	48	no	—
34	#13093998	C₂³	1	32	yes	complemented
35	C₂×C₄⋊Dic₄₇	—	2	6	no	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
37	C₂×(C₂×D₄₇)⋊C₄	—	2	24	no	—
38	#13093956	C₂³	1	16	yes	complemented
39	#13093946	C₂³	1	32	yes	complemented
40	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	—	2	4	no	—
42	C₂³×C₄×C₄₇	—	2	4	no	—
43	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	D₄	1	6	yes	—
45	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₄₇	—	2	48	no	—
48	C₂×M₁₆×C₄₇	C₂³	1	16	yes	complemented
49	C₂²×C₈×C₄₇	—	2	4	no	—
50	C₂²×C₈×C₄₇	D₄	1	8	yes	complemented
51	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
53	C₂⁴×C₄₇	—	2	4	no	—
54	C₂⁴×C₄₇	—	2	8	no	—
55	C₂⁴×C₄₇	C₂×D₄	1	6	yes	—
56	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂³×D₄₇	—	2	12	no	—
58	C₂³×D₄₇	—	4	8	no	—
59	C₂³×D₄₇	—	4	8	no	—
60	C₂³×D₄₇	—	4	8	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	24	no	—
63	C₂²×C₄×C₄₇	—	2	12	no	—
64	C₂²×C₄×C₄₇	—	2	12	no	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₄₇	#41	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
72	C₂×C₂²⋊D₄₇	—	2	32	no	—
73	C₂×C₂²⋊D₄₇	—	4	96	no	—
74	C₂²×Dic₄₇	—	2	12	no	—
75	C₂²×Dic₄₇	—	4	8	no	—
76	C₂×D₁₈₈	—	2	24	no	—
77	C₂×D₁₈₈	—	2	24	no	—
78	C₂×D₁₈₈	—	2	12	no	—
79	C₂×D₁₈₈	—	4	24	no	—
80	C₂×D₁₈₈	C₂⁴	1	8	yes	—
81	M₁₆×C₄₇	—	2	32	no	—
82	C₂×C₈×C₄₇	—	2	12	no	—
83	C₂×C₈×C₄₇	C₂×D₄	1	24	yes	complemented
84	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
85	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
86	(C₂×D₄₇)⋊C₄	—	2	32	no	—
87	C₄⋊Dic₄₇	—	2	12	no	—
88	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
89	D₃₇₆	—	2	32	no	—
90	C₂³×C₄₇	—	2	12	no	—
91	C₂³×C₄₇	—	2	24	no	—
92	C₂³×C₄₇	—	4	16	no	—
93	C₂³×C₄₇	#121	1	4	yes	—
94	C₂³×C₄₇	AGL(1,ℤ/8)	1	8	yes	—
95	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
96	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
97	C₂²×D₄₇	—	2	8	no	—
98	C₂²×D₄₇	—	4	24	no	—
99	C₂²×D₄₇	—	4	24	no	—
100	C₂²×D₄₇	—	4	24	no	—
101	C₂²×D₄₇	—	8	32	no	—
102	C₂×C₄×C₄₇	—	2	12	no	—
103	C₂×C₄×C₄₇	—	2	24	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	#121	1	12	yes	—
107	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
108	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
109	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
110	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
111	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
112	C₂²⋊D₄₇	—	4	64	no	—
113	C₂×Dic₄₇	—	2	8	no	—
114	C₂×Dic₄₇	—	4	24	no	—
115	D₁₈₈	—	2	8	no	—
116	D₁₈₈	—	2	8	no	—
117	D₁₈₈	—	4	24	no	—
118	C₈×C₄₇	—	2	4	no	—
119	C₈×C₄₇	C₂²×D₄	1	8	yes	complemented
120	C₂²×#469	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	4	no	—
122	C₂²×C₄₇	—	2	8	no	—
123	C₂²×C₄₇	—	4	24	no	—
124	C₂²×C₄₇	#469	1	16	yes	complemented
125	C₂²×C₄₇	C₂×#121	1	6	yes	—
126	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	12	yes	—
127	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
128	C₂×D₄₇	—	4	8	no	—
129	C₂×D₄₇	—	4	8	no	—
130	C₂×D₄₇	—	4	8	no	—
131	C₂×D₄₇	—	8	48	no	—
132	C₄×C₄₇	—	2	4	no	—
133	C₄×C₄₇	C₂×#121	1	6	yes	—
134	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
135	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
136	Dic₄₇	—	4	8	no	—
137	C₂³×D₈	—	47	1	no	—
138	C₂³×M₁₆	D₄₇	1	1	yes	characteristic, complemented
139	C₂²×#121	—	47	2	no	—
140	C₂×#469	—	47	24	no	—
141	C₂²×#106	—	47	1	no	—
142	C₂²×#102	—	47	2	no	—
143	C₂×C₄₇	—	4	4	no	—
144	C₂×C₄₇	C₂×#469	1	12	yes	complemented
145	C₂×C₄₇	C₂²×#121	1	1	yes	characteristic
146	C₂×C₄₇	C₂²×AGL(1,ℤ/8)	1	2	yes	—
147	D₄₇	—	8	8	no	—
148	C₂³×D₄	—	47	2	no	—
149	C₂³×D₄	—	94	2	no	—
150	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
151	C₂²×D₈	—	47	12	no	—
152	C₂²×D₈	—	94	8	no	—
153	C₂²×M₁₆	—	2	8	no	—
154	C₂²×M₁₆	C₂×D₄₇	1	12	yes	complemented
155	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
156	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
157	C₂×#121	—	47	24	no	—
158	C₂×#121	—	94	12	no	—
159	C₂²×C₄⋊C₄	—	47	2	no	—
160	C₂²×C₂²⋊C₄	—	94	2	no	—
161	#469	—	47	64	no	—
162	C₂×#106	—	47	12	no	—
163	C₂×#102	—	47	24	no	—
164	C₄₇	C₂²×#469	1	1	yes	characteristic, complemented, Sylow
165	C₂⁵	—	94	2	no	—
166	C₂⁵	D₁₈₈	1	1	yes	characteristic
167	C₂²×D₄	—	47	12	no	—
168	C₂²×D₄	—	94	24	no	—
169	C₂²×D₄	—	94	6	no	—
170	C₂²×D₄	—	94	8	no	—
171	C₂²×D₄	—	94	8	no	—
172	C₂²×D₄	—	188	16	no	—
173	C₂³×C₄	—	2	4	no	—
174	C₂³×C₄	—	2	4	no	—
175	C₂³×C₄	—	94	2	no	—
176	C₂³×C₄	D₁₈₈	1	6	yes	—
177	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
178	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
179	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
180	C₂×D₈	—	47	16	no	—
181	C₂×D₈	—	94	48	no	—
182	C₂×M₁₆	—	2	48	no	—
183	C₂×M₁₆	C₂²×D₄₇	1	16	yes	complemented
184	C₂²×C₈	—	2	4	no	—
185	C₂²×C₈	C₂²⋊D₄₇	1	8	yes	complemented
186	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
187	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
188	#121	—	47	32	no	—
189	#121	—	94	48	no	—
190	C₂×C₄⋊C₄	—	47	12	no	—
191	C₂×C₄⋊C₄	—	94	6	no	—
192	C₂×C₂²⋊C₄	—	94	24	no	—
193	#106	—	47	16	no	—
194	#102	—	47	32	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	—	2	8	no	—
197	C₂⁴	—	94	12	no	—
198	C₂⁴	—	188	8	no	—
199	C₂⁴	—	188	8	no	—
200	C₂⁴	—	188	8	no	—
201	C₂⁴	C₂×D₁₈₈	1	6	yes	—
202	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
203	C₂×D₄	—	47	8	no	—
204	C₂×D₄	—	94	32	no	—
205	C₂×D₄	—	94	24	no	—
206	C₂×D₄	—	94	12	no	—
207	C₂×D₄	—	94	24	no	—
208	C₂×D₄	—	188	96	no	—
209	C₂×D₄	—	188	24	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	24	no	—
212	C₂²×C₄	—	2	12	no	—
213	C₂²×C₄	—	2	12	no	—
214	C₂²×C₄	—	94	12	no	—
215	C₂²×C₄	—	188	8	no	—
216	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
217	C₂²×C₄	C₂×D₁₈₈	1	12	yes	—
218	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
219	C₂²×C₄	#10625350	1	4	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
221	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
222	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
223	D₈	—	94	32	no	—
224	M₁₆	—	2	32	no	—
225	C₂×C₈	—	2	12	no	—
226	C₂×C₈	C₂×C₂²⋊D₄₇	1	24	yes	complemented
227	C₂×C₈	C₂³×D₄₇	1	4	yes	—
228	C₂×C₈	C₂³×D₄₇	1	4	yes	—
229	C₄⋊C₄	—	47	8	no	—
230	C₄⋊C₄	—	94	12	no	—
231	C₂²⋊C₄	—	94	32	no	—
232	C₂³	—	2	12	no	—
233	C₂³	—	2	24	no	—
234	C₂³	—	4	16	no	—
235	C₂³	—	94	8	no	—
236	C₂³	—	376	32	no	—
237	C₂³	—	188	24	no	—
238	C₂³	—	188	24	no	—
239	C₂³	—	188	24	no	—
240	C₂³	#13093998	1	4	yes	—
241	C₂³	#13094052	1	8	yes	—
242	C₂³	C₂²×D₁₈₈	1	4	yes	—
243	C₂³	C₂²×D₁₈₈	1	3	yes	—
244	D₄	—	94	8	no	—
245	D₄	—	94	8	no	—
246	D₄	—	188	64	no	—
247	D₄	—	188	24	no	—
248	C₂×C₄	—	2	12	no	—
249	C₂×C₄	—	2	24	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	94	8	no	—
253	C₂×C₄	—	188	24	no	—
254	C₂×C₄	#13093998	1	12	yes	—
255	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
256	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
257	C₂×C₄	C₂²×D₁₈₈	1	3	yes	—
258	C₂×C₄	C₂×#10625350	1	6	yes	—
259	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	C₂²×C₂²⋊D₄₇	1	8	yes	complemented
262	C₂²	—	2	8	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	24	no	—
265	C₂²	—	376	48	no	—
266	C₂²	—	188	8	no	—
267	C₂²	—	188	8	no	—
268	C₂²	—	188	8	no	—
269	C₂²	#15689376	1	16	yes	direct product factor
270	C₂²	C₂×#13093998	1	6	yes	—
271	C₂²	C₂×#13094052	1	12	yes	—
272	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	188	8	no	—
275	C₄	C₂×#13093998	1	6	yes	—
276	C₄	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
277	C₄	C₂²×#10625350	1	1	yes	characteristic
278	C₂	—	4	4	no	—
279	C₂	—	376	8	no	—
280	C₂	C₂×#15689376	1	12	yes	direct product factor
281	C₂	C₂²×#13093998	1	1	yes	characteristic
282	C₂	C₂²×#13094052	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂³×M₁₆×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂³×M₁₆	D₄₇
hypocenter	C₄₇	C₂²×#469
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂³×D₃₇₆	C₂	semidirect
2. C₂²×#13093998	C₂	semidirect
3. C₂×#15689376	C₂	direct (N × H)
4. C₂²×#13093956	C₂	semidirect
5. C₂²×#13093946	C₂	semidirect
6. C₂³×M₁₆×C₄₇	C₂	semidirect
10. C₂²×D₃₇₆	C₂²	semidirect
12. C₂×#13093998	C₂²	semidirect
15. #15689376	C₂²	direct (N × H)
16. C₂×#13093956	C₂²	semidirect
17. C₂×#13093946	C₂²	semidirect
20. C₂²×M₁₆×C₄₇	C₂²	semidirect
32. C₂×D₃₇₆	C₂³	semidirect
34. #13093998	C₂³	semidirect
38. #13093956	C₂³	semidirect
39. #13093946	C₂³	semidirect
48. C₂×M₁₆×C₄₇	C₂³	semidirect
50. C₂²×C₈×C₄₇	D₄	semidirect
83. C₂×C₈×C₄₇	C₂×D₄	semidirect
119. C₈×C₄₇	C₂²×D₄	semidirect
124. C₂²×C₄₇	#469	semidirect
138. C₂³×M₁₆	D₄₇	semidirect
144. C₂×C₄₇	C₂×#469	semidirect
154. C₂²×M₁₆	C₂×D₄₇	semidirect
164. C₄₇	C₂²×#469	semidirect
183. C₂×M₁₆	C₂²×D₄₇	semidirect
185. C₂²×C₈	C₂²⋊D₄₇	semidirect
226. C₂×C₈	C₂×C₂²⋊D₄₇	semidirect
261. C₈	C₂²×C₂²⋊D₄₇	semidirect
269. C₂²	#15689376	direct (N × H)
280. C₂	C₂×#15689376	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₃₇₆	2
C₂²×#13093998	2
C₂×#15689376	2
C₂²×#13093956	2
C₂²×#13093946	2
C₂³×M₁₆×C₄₇	2
C₂²×#469	47

Subgroups — C22×#15689376 #27083399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689376 #27083399