Subgroups — C₂²×#15688907 #26368975

Order 12032.

1174 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	C₂²×#13093964	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688907	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	—	2	3	no	—
6	C₂×#13093964	C₂²	1	6	yes	complemented
7	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
8	C₂²×C₄₇⋊C₁₆	—	2	3	no	—
9	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
10	C₂²×C₄₇⋊C₁₆	C₂²	1	6	yes	complemented
11	#15688907	C₂²	1	16	yes	direct product factor
12	C₂²×#41×C₄₇	C₄	1	1	yes	characteristic
13	C₂×#131×C₄₇	—	2	3	no	—
14	C₂×#131×C₄₇	—	2	3	no	—
15	C₂×#131×C₄₇	C₄	1	6	yes	—
16	C₂×#131×C₄₇	C₂²	1	6	yes	—
17	C₂²×M₁₆×C₄₇	—	2	1	no	—
18	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
19	C₂³×C₈×C₄₇	—	2	1	no	—
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	#13093964	—	2	6	no	—
22	#13093964	C₂³	1	4	yes	complemented
23	C₂×C₄₇⋊C₁₆	—	2	6	no	—
24	C₂×C₄₇⋊C₁₆	—	2	6	no	—
25	C₂×C₄₇⋊C₁₆	—	2	6	no	—
26	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	complemented
27	C₂×#41×C₄₇	—	2	3	no	—
28	C₂×#41×C₄₇	—	2	3	no	—
29	C₂×#41×C₄₇	C₈	1	6	yes	—
30	C₂×#41×C₄₇	C₂×C₄	1	6	yes	—
31	C₂²×Q₈×C₄₇	C₈	1	1	yes	characteristic
32	C₂²×D₄×C₄₇	—	2	1	no	—
33	C₂²×D₄×C₄₇	C₈	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	—	2	1	no	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	#131×C₄₇	—	2	12	no	—
37	#131×C₄₇	—	2	6	no	—
38	#131×C₄₇	—	2	6	no	—
39	#131×C₄₇	C₂×C₄	1	12	yes	—
40	#131×C₄₇	C₂³	1	4	yes	—
41	C₂×M₁₆×C₄₇	—	2	3	no	—
42	C₂×M₁₆×C₄₇	—	2	6	no	—
43	C₂×M₁₆×C₄₇	—	2	6	no	—
44	C₂×M₁₆×C₄₇	C₂×C₄	1	6	yes	—
45	C₂²×C₈×C₄₇	—	2	1	no	—
46	C₂²×C₈×C₄₇	—	2	6	no	—
47	C₂²×C₈×C₄₇	—	2	6	no	—
48	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
49	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	D₄	1	3	yes	—
51	C₂²×C₈×C₄₇	D₄	1	3	yes	—
52	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
54	C₂⁴×C₄₇	—	2	1	no	—
55	C₂⁴×C₄₇	M₁₆	1	1	yes	characteristic
56	#41×C₄₇	—	2	12	no	—
57	#41×C₄₇	—	2	6	no	—
58	#41×C₄₇	—	2	6	no	—
59	#41×C₄₇	C₂×C₈	1	12	yes	—
60	#41×C₄₇	C₂²×C₄	1	4	yes	—
61	C₂×Q₈×C₄₇	—	2	3	no	—
62	C₂×Q₈×C₄₇	C₂×C₈	1	6	yes	—
63	C₂×D₄×C₄₇	—	2	6	no	—
64	C₂×D₄×C₄₇	—	2	6	no	—
65	C₂×D₄×C₄₇	—	2	3	no	—
66	C₂×D₄×C₄₇	C₂×C₈	1	6	yes	—
67	C₂²×C₄×C₄₇	—	2	1	no	—
68	C₂²×C₄×C₄₇	—	2	6	no	—
69	C₂²×C₄×C₄₇	—	2	6	no	—
70	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²⋊C₄	1	3	yes	—
72	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×C₈	1	3	yes	—
74	C₂²×C₄×C₄₇	M₁₆	1	3	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
76	M₁₆×C₄₇	—	2	6	no	—
77	M₁₆×C₄₇	—	2	12	no	—
78	M₁₆×C₄₇	—	2	4	no	—
79	M₁₆×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₄₇	—	2	6	no	—
81	C₂×C₈×C₄₇	—	2	12	no	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	C₂²⋊C₄	1	3	yes	—
84	C₂×C₈×C₄₇	C₂²⋊C₄	1	3	yes	—
85	C₂×C₈×C₄₇	C₂²⋊C₄	1	6	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
89	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
91	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₄₇⋊C₁₆	—	2	4	no	—
93	C₄₇⋊C₁₆	—	2	4	no	—
94	C₂³×C₄₇	—	2	4	no	—
95	C₂³×C₄₇	—	2	6	no	—
96	C₂³×C₄₇	—	4	4	no	—
97	C₂³×C₄₇	C₂²⋊C₈	1	3	yes	—
98	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
99	C₂³×C₄₇	C₂×M₁₆	1	3	yes	—
100	Q₈×C₄₇	—	2	6	no	—
101	Q₈×C₄₇	C₂²×C₈	1	4	yes	—
102	D₄×C₄₇	—	2	4	no	—
103	D₄×C₄₇	—	2	12	no	—
104	D₄×C₄₇	—	2	6	no	—
105	D₄×C₄₇	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₄₇	—	2	6	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	12	no	—
109	C₂×C₄×C₄₇	C₂²⋊C₈	1	3	yes	—
110	C₂×C₄×C₄₇	C₂²⋊C₈	1	3	yes	—
111	C₂×C₄×C₄₇	C₂²⋊C₈	1	6	yes	—
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
114	C₂×C₄×C₄₇	C₂²×C₈	1	3	yes	—
115	C₂×C₄×C₄₇	C₂²×C₈	1	3	yes	—
116	C₂×C₄×C₄₇	C₂×M₁₆	1	3	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₄₇	—	2	4	no	—
119	C₈×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
120	C₈×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
121	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂²×#350	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	6	no	—
125	C₂²×C₄₇	—	2	6	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	4	12	no	—
128	C₂²×C₄₇	#350	1	4	yes	complemented
129	C₂²×C₄₇	C₂×C₂²⋊C₈	1	3	yes	—
130	C₂²×C₄₇	C₂×C₂²⋊C₈	1	3	yes	—
131	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
132	C₄×C₄₇	—	2	4	no	—
133	C₄×C₄₇	C₂×C₂²⋊C₈	1	3	yes	—
134	C₄×C₄₇	C₂×C₂²⋊C₈	1	3	yes	—
135	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
137	C₂²×#131	D₄₇	1	1	yes	characteristic
138	C₂²×M₃₂	—	47	1	no	—
139	C₂³×C₁₆	—	47	1	no	—
140	C₂×#350	—	47	12	no	—
141	C₂×C₄₇	—	2	1	no	—
142	C₂×C₄₇	—	2	3	no	—
143	C₂×C₄₇	—	4	4	no	—
144	C₂×C₄₇	C₂×#350	1	6	yes	complemented
145	C₂×C₄₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
146	C₂²×#41	Dic₄₇	1	1	yes	characteristic
147	C₂×#131	—	2	3	no	—
148	C₂×#131	—	2	3	no	—
149	C₂×#131	Dic₄₇	1	6	yes	—
150	C₂×#131	C₂×D₄₇	1	6	yes	—
151	C₂²×M₁₆	—	2	1	no	—
152	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
153	C₂³×C₈	—	2	1	no	—
154	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
155	C₂×M₃₂	—	47	6	no	—
156	C₂×M₃₂	—	94	3	no	—
157	C₂²×C₁₆	—	47	6	no	—
158	C₂²×C₁₆	—	94	1	no	—
159	C₂²×C₁₆	—	94	3	no	—
160	C₂²×C₁₆	—	94	1	no	—
161	#350	—	47	16	no	—
162	C₄₇	C₂²×#350	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	2	3	no	—
164	C₂×#41	—	2	3	no	—
165	C₂×#41	C₄₇⋊C₈	1	6	yes	—
166	C₂×#41	C₂×Dic₄₇	1	6	yes	—
167	C₂²×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
168	C₂²×D₄	—	2	1	no	—
169	C₂²×D₄	C₄₇⋊C₈	1	1	yes	characteristic
170	C₂³×C₄	—	2	1	no	—
171	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
172	#131	—	2	12	no	—
173	#131	—	2	6	no	—
174	#131	—	2	6	no	—
175	#131	C₂×Dic₄₇	1	12	yes	—
176	#131	C₂²×D₄₇	1	4	yes	—
177	C₂×M₁₆	—	2	3	no	—
178	C₂×M₁₆	—	2	6	no	—
179	C₂×M₁₆	—	2	6	no	—
180	C₂×M₁₆	C₂×Dic₄₇	1	6	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	6	no	—
183	C₂²×C₈	—	2	6	no	—
184	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
185	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
186	C₂²×C₈	C₂²⋊D₄₇	1	3	yes	—
187	C₂²×C₈	C₂²⋊D₄₇	1	3	yes	—
188	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
190	M₃₂	—	47	4	no	—
191	M₃₂	—	94	6	no	—
192	C₂×C₁₆	—	47	4	no	—
193	C₂×C₁₆	—	94	6	no	—
194	C₂×C₁₆	—	94	6	no	—
195	C₂×C₁₆	—	94	6	no	—
196	C₂⁴	—	2	1	no	—
197	C₂⁴	#10625329	1	1	yes	characteristic
198	#41	—	2	12	no	—
199	#41	—	2	6	no	—
200	#41	—	2	6	no	—
201	#41	C₂×C₄₇⋊C₈	1	12	yes	—
202	#41	C₂²×Dic₄₇	1	4	yes	—
203	C₂×Q₈	—	2	3	no	—
204	C₂×Q₈	C₂×C₄₇⋊C₈	1	6	yes	—
205	C₂×D₄	—	2	6	no	—
206	C₂×D₄	—	2	6	no	—
207	C₂×D₄	—	2	3	no	—
208	C₂×D₄	C₂×C₄₇⋊C₈	1	6	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	6	no	—
211	C₂²×C₄	—	2	6	no	—
212	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄₇⋊C₈	1	3	yes	—
214	C₂²×C₄	#10625329	1	3	yes	—
215	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Dic₄₇	1	3	yes	—
217	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
218	M₁₆	—	2	6	no	—
219	M₁₆	—	2	12	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	C₂²×Dic₄₇	1	4	yes	—
222	C₂×C₈	—	2	6	no	—
223	C₂×C₈	—	2	12	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂²⋊Dic₄₇	1	3	yes	—
226	C₂×C₈	C₂²⋊Dic₄₇	1	3	yes	—
227	C₂×C₈	C₂²⋊Dic₄₇	1	6	yes	—
228	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
229	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
230	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
231	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
232	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
233	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
234	C₁₆	—	94	4	no	—
235	C₁₆	—	94	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	6	no	—
238	C₂³	—	4	4	no	—
239	C₂³	C₂²⋊(C₄₇⋊C₈)	1	3	yes	—
240	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
241	C₂³	C₂×#10625329	1	3	yes	—
242	Q₈	—	2	6	no	—
243	Q₈	C₂²×C₄₇⋊C₈	1	4	yes	—
244	D₄	—	2	12	no	—
245	D₄	—	2	4	no	—
246	D₄	—	2	6	no	—
247	D₄	C₂²×C₄₇⋊C₈	1	4	yes	—
248	C₂×C₄	—	2	6	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	3	yes	—
252	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	3	yes	—
253	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	6	yes	—
254	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
255	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
256	C₂×C₄	C₂²×C₄₇⋊C₈	1	3	yes	—
257	C₂×C₄	C₂²×C₄₇⋊C₈	1	3	yes	—
258	C₂×C₄	C₂×#10625329	1	3	yes	—
259	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	C₂×C₂²⋊Dic₄₇	1	3	yes	—
262	C₈	C₂×C₂²⋊Dic₄₇	1	3	yes	—
263	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
264	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
265	C₂²	—	2	6	no	—
266	C₂²	—	2	6	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	12	no	—
269	C₂²	#15688907	1	4	yes	direct product factor
270	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	3	yes	—
271	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	3	yes	—
272	C₂²	C₂²×#10625329	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	3	yes	—
275	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	3	yes	—
276	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
277	C₄	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
278	C₂	—	2	3	no	—
279	C₂	—	2	1	no	—
280	C₂	—	4	4	no	—
281	C₂	C₂×#15688907	1	6	yes	direct product factor
282	C₂	C₂²×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂²×#131×C₄₇	C₂
socle	C₂³×C₄₇	C₂²⋊C₈
prime radical	C₂²×#41×C₄₇	C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	C₂²×#350
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093964	C₂	semidirect
2. C₂³×C₄₇⋊C₁₆	C₂	semidirect
3. C₂×#15688907	C₂	direct (N × H)
6. C₂×#13093964	C₂²	semidirect
10. C₂²×C₄₇⋊C₁₆	C₂²	semidirect
11. #15688907	C₂²	direct (N × H)
22. #13093964	C₂³	semidirect
26. C₂×C₄₇⋊C₁₆	C₂³	semidirect
128. C₂²×C₄₇	#350	semidirect
144. C₂×C₄₇	C₂×#350	semidirect
162. C₄₇	C₂²×#350	semidirect
269. C₂²	#15688907	direct (N × H)
281. C₂	C₂×#15688907	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13093964	2
C₂³×C₄₇⋊C₁₆	2
C₂×#15688907	2
C₂²×#131×C₄₇	2
C₂²×#350	47

Subgroups — C22×#15688907 #26368975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688907 #26368975