Subgroups — C₂²×#15689170 #27079969

Order 12032.

2718 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	C₂³×#10625324	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×D₄₇	C₂	1	2	yes	maximal
3	C₂×#15689170	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13093959	C₂	1	2	yes	maximal, complemented
6	C₂²×#108×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₄₇	C₂²	1	1	yes	characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂²×#10625324	—	2	2	no	—
10	C₂²×#10625324	C₄	1	4	yes	—
11	C₂²×#10625324	C₄	1	12	yes	—
12	C₂²×#10625324	C₂²	1	12	yes	—
13	C₂×M₁₆×D₄₇	—	2	12	no	—
14	C₂×M₁₆×D₄₇	C₂²	1	24	yes	—
15	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₈×D₄₇	—	2	2	no	—
17	#15689170	C₂²	1	64	yes	direct product factor
18	C₂×#13093960	C₂²	1	12	yes	complemented
19	C₂×#13093959	C₂²	1	24	yes	complemented
20	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#108×C₄₇	C₂²	1	12	yes	complemented
23	C₂⁴×D₄₇	Q₈	1	1	yes	characteristic
24	C₂³×Dic₄₇	Q₈	1	1	yes	characteristic
25	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×D₄₇	C₂×C₄	1	6	yes	—
27	C₂²×C₄×D₄₇	D₄	1	2	yes	—
28	C₂²×C₄×D₄₇	D₄	1	6	yes	—
29	C₂²×C₄×D₄₇	Q₈	1	6	yes	—
30	C₂²×C₄×D₄₇	C₂³	1	6	yes	—
31	C₂×#10625329	—	2	6	no	—
32	C₂×#10625329	C₂³	1	12	yes	—
33	C₂×#10625324	—	2	24	no	—
34	C₂×#10625324	C₂×C₄	1	48	yes	—
35	C₂×#10625324	C₂×C₄	1	48	yes	—
36	C₂×#10625324	C₂³	1	16	yes	—
37	M₁₆×D₄₇	—	2	48	no	—
38	M₁₆×D₄₇	C₂³	1	32	yes	—
39	C₂²×C₄₇⋊C₈	—	2	2	no	—
40	C₂²×C₄₇⋊C₈	C₂×C₄	1	6	yes	—
41	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
42	C₂²×C₄₇⋊C₈	C₂³	1	6	yes	—
43	C₂×C₈×D₄₇	—	2	24	no	—
44	#13093960	C₂³	1	16	yes	complemented
45	#13093959	C₂³	1	32	yes	complemented
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₄₇	—	2	6	no	—
48	C₂×M₁₆×C₄₇	C₂³	1	12	yes	—
49	C₂²×C₈×C₄₇	—	2	2	no	—
50	C₂²×C₈×C₄₇	C₂×C₄	1	6	yes	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
53	#108×C₄₇	C₂³	1	16	yes	complemented
54	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂³×D₄₇	—	4	4	no	—
56	C₂³×D₄₇	C₄⋊C₄	1	2	yes	—
57	C₂³×D₄₇	C₄⋊C₄	1	6	yes	—
58	C₂³×D₄₇	C₂×Q₈	1	6	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×Q₈	1	3	yes	—
64	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
65	C₂²×Dic₄₇	C₄⋊C₄	1	6	yes	—
66	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
67	C₂²×Dic₄₇	C₂×Q₈	1	6	yes	—
68	C₂×C₄×D₄₇	C₄⋊C₄	1	12	yes	—
69	C₂×C₄×D₄₇	C₄⋊C₄	1	12	yes	—
70	C₂×C₄×D₄₇	C₄⋊C₄	1	24	yes	—
71	C₂×C₄×D₄₇	C₂²×C₄	1	12	yes	—
72	C₂×C₄×D₄₇	C₂²×C₄	1	12	yes	—
73	C₂×C₄×D₄₇	C₂×D₄	1	12	yes	—
74	C₂×C₄×D₄₇	C₂×D₄	1	12	yes	—
75	C₂×C₄×D₄₇	C₂×Q₈	1	12	yes	—
76	C₂×C₄×D₄₇	C₂⁴	1	4	yes	—
77	M₁₆×C₄₇	—	2	12	no	—
78	M₁₆×C₄₇	C₂⁴	1	8	yes	—
79	C₂×C₈×C₄₇	—	2	12	no	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	12	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	12	yes	—
82	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
83	#10625329	—	2	12	no	—
84	#10625329	C₂⁴	1	8	yes	—
85	C₂×C₄₇⋊C₈	—	2	12	no	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	12	yes	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	12	yes	—
88	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
89	#10625324	—	2	32	no	—
90	#10625324	C₂²×C₄	1	64	yes	—
91	C₈×D₄₇	—	2	32	no	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
94	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₄₇	C₂²×Q₈	1	3	yes	—
96	C₂²×D₄₇	—	2	8	no	—
97	C₂²×D₄₇	—	4	12	no	—
98	C₂²×D₄₇	—	4	12	no	—
99	C₂²×D₄₇	C₂×C₄⋊C₄	1	12	yes	—
100	C₂²×D₄₇	C₂×C₄⋊C₄	1	12	yes	—
101	C₂²×D₄₇	C₂²×Q₈	1	4	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	3	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	3	yes	—
107	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
108	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
109	C₂×C₄×C₄₇	C₂²×Q₈	1	3	yes	—
110	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
111	C₂×Dic₄₇	C₂×C₄⋊C₄	1	12	yes	—
112	C₂×Dic₄₇	C₂×C₄⋊C₄	1	12	yes	—
113	C₂×Dic₄₇	C₂²×Q₈	1	4	yes	—
114	C₄×D₄₇	C₂×C₄⋊C₄	1	24	yes	—
115	C₄×D₄₇	C₂×C₄⋊C₄	1	24	yes	—
116	C₄×D₄₇	C₂³×C₄	1	8	yes	—
117	C₄×D₄₇	C₂²×D₄	1	8	yes	—
118	C₈×C₄₇	—	2	8	no	—
119	C₈×C₄₇	C₂³×C₄	1	8	yes	—
120	C₄₇⋊C₈	—	2	8	no	—
121	C₄₇⋊C₈	C₂³×C₄	1	8	yes	—
122	C₂²×#430	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	6	no	—
124	C₂²×C₄₇	—	2	6	no	—
125	C₂²×C₄₇	#430	1	4	yes	complemented
126	C₂²×C₄₇	C₂²×C₄⋊C₄	1	3	yes	—
127	C₂²×C₄₇	C₂²×C₄⋊C₄	1	3	yes	—
128	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
129	C₂×D₄₇	—	2	24	no	—
130	C₂×D₄₇	—	4	12	no	—
131	C₂×D₄₇	—	4	4	no	—
132	C₂×D₄₇	C₂²×C₄⋊C₄	1	8	yes	—
133	C₄×C₄₇	C₂²×C₄⋊C₄	1	3	yes	—
134	C₄×C₄₇	C₂²×C₄⋊C₄	1	3	yes	—
135	C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
136	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
137	Dic₄₇	C₂²×C₄⋊C₄	1	8	yes	—
138	C₂³×M₁₆	—	47	1	no	—
139	C₂³×M₁₆	—	47	2	no	—
140	C₂×#430	—	47	24	no	—
141	C₂²×#108	—	47	2	no	—
142	C₂²×#108	—	47	1	no	—
143	C₂²×#108	D₄₇	1	1	yes	characteristic, complemented
144	C₂×C₄₇	—	2	3	no	—
145	C₂×C₄₇	—	2	1	no	—
146	C₂×C₄₇	C₂×#430	1	6	yes	complemented
147	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
148	D₄₇	—	2	8	no	—
149	C₂⁴×C₄	—	47	1	no	—
150	C₂²×M₁₆	—	47	4	no	—
151	C₂²×M₁₆	—	47	12	no	—
152	C₂²×M₁₆	—	47	12	no	—
153	C₂²×M₁₆	—	47	2	no	—
154	C₂²×M₁₆	—	47	24	no	—
155	C₂²×M₁₆	—	94	2	no	—
156	C₂²×M₁₆	—	94	12	no	—
157	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
158	C₂³×C₈	—	47	1	no	—
159	C₂³×C₈	—	94	2	no	—
160	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
161	#430	—	47	64	no	—
162	C₂×#108	—	47	24	no	—
163	C₂×#108	—	47	12	no	—
164	C₂×#108	C₂×D₄₇	1	12	yes	complemented
165	C₄₇	C₂²×#430	1	1	yes	characteristic, complemented, Sylow
166	C₂⁵	—	47	1	no	—
167	C₂³×C₄	—	47	1	no	—
168	C₂³×C₄	—	47	2	no	—
169	C₂³×C₄	—	47	6	no	—
170	C₂³×C₄	—	47	6	no	—
171	C₂³×C₄	—	47	2	no	—
172	C₂³×C₄	—	47	6	no	—
173	C₂³×C₄	—	47	6	no	—
174	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
175	C₂×M₁₆	—	2	6	no	—
176	C₂×M₁₆	—	47	48	no	—
177	C₂×M₁₆	—	47	48	no	—
178	C₂×M₁₆	—	47	16	no	—
179	C₂×M₁₆	—	47	12	no	—
180	C₂×M₁₆	—	47	32	no	—
181	C₂×M₁₆	—	94	24	no	—
182	C₂×M₁₆	—	94	48	no	—
183	C₂×M₁₆	—	94	6	no	—
184	C₂×M₁₆	C₂²×D₄₇	1	12	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	—	47	6	no	—
188	C₂²×C₈	—	47	6	no	—
189	C₂²×C₈	—	94	24	no	—
190	C₂²×C₈	—	94	2	no	—
191	C₂²×C₈	C₄×D₄₇	1	2	yes	—
192	C₂²×C₈	C₄×D₄₇	1	6	yes	—
193	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
194	#108	—	47	32	no	—
195	#108	—	47	16	no	—
196	#108	C₂²×D₄₇	1	16	yes	complemented
197	C₂⁴	—	47	2	no	—
198	C₂⁴	—	47	6	no	—
199	C₂⁴	—	47	6	no	—
200	C₂⁴	—	188	4	no	—
201	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	—	47	6	no	—
203	C₂²×C₄	—	47	2	no	—
204	C₂²×C₄	—	47	6	no	—
205	C₂²×C₄	—	47	12	no	—
206	C₂²×C₄	—	47	24	no	—
207	C₂²×C₄	—	47	12	no	—
208	C₂²×C₄	—	47	12	no	—
209	C₂²×C₄	—	47	12	no	—
210	C₂²×C₄	—	47	12	no	—
211	C₂²×C₄	—	47	12	no	—
212	C₂²×C₄	—	47	12	no	—
213	C₂²×C₄	—	47	4	no	—
214	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
216	C₂²×C₄	D₄×D₄₇	1	3	yes	—
217	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	Q₈×D₄₇	1	3	yes	—
219	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
220	M₁₆	—	2	12	no	—
221	M₁₆	—	47	64	no	—
222	M₁₆	—	47	8	no	—
223	M₁₆	—	94	32	no	—
224	M₁₆	—	94	12	no	—
225	M₁₆	C₂³×D₄₇	1	8	yes	—
226	C₂×C₈	—	2	12	no	—
227	C₂×C₈	—	47	12	no	—
228	C₂×C₈	—	47	12	no	—
229	C₂×C₈	—	47	4	no	—
230	C₂×C₈	—	94	32	no	—
231	C₂×C₈	—	94	12	no	—
232	C₂×C₈	C₂×C₄×D₄₇	1	12	yes	—
233	C₂×C₈	C₂×C₄×D₄₇	1	12	yes	—
234	C₂×C₈	C₂³×D₄₇	1	4	yes	—
235	C₂³	—	2	4	no	—
236	C₂³	—	47	12	no	—
237	C₂³	—	47	12	no	—
238	C₂³	—	47	4	no	—
239	C₂³	—	94	8	no	—
240	C₂³	—	188	12	no	—
241	C₂³	—	188	12	no	—
242	C₂³	C₄⋊C₄×D₄₇	1	3	yes	—
243	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
244	C₂³	C₂×Q₈×D₄₇	1	3	yes	—
245	C₂×C₄	—	47	12	no	—
246	C₂×C₄	—	47	12	no	—
247	C₂×C₄	—	47	4	no	—
248	C₂×C₄	—	47	24	no	—
249	C₂×C₄	—	47	24	no	—
250	C₂×C₄	—	47	8	no	—
251	C₂×C₄	—	47	8	no	—
252	C₂×C₄	C₄⋊C₄×D₄₇	1	6	yes	—
253	C₂×C₄	C₄⋊C₄×D₄₇	1	3	yes	—
254	C₂×C₄	C₄⋊C₄×D₄₇	1	3	yes	—
255	C₂×C₄	C₂²×C₄×D₄₇	1	3	yes	—
256	C₂×C₄	C₂²×C₄×D₄₇	1	3	yes	—
257	C₂×C₄	C₂×D₄×D₄₇	1	3	yes	—
258	C₂×C₄	C₂×D₄×D₄₇	1	3	yes	—
259	C₂×C₄	C₂×Q₈×D₄₇	1	3	yes	—
260	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
261	C₈	—	2	8	no	—
262	C₈	—	47	8	no	—
263	C₈	—	94	8	no	—
264	C₈	C₂²×C₄×D₄₇	1	8	yes	—
265	C₂²	—	2	6	no	—
266	C₂²	—	2	6	no	—
267	C₂²	—	47	8	no	—
268	C₂²	—	94	24	no	—
269	C₂²	—	188	12	no	—
270	C₂²	—	188	4	no	—
271	C₂²	#15689170	1	4	yes	direct product factor
272	C₂²	C₂×C₄⋊C₄×D₄₇	1	3	yes	—
273	C₂²	C₂×C₄⋊C₄×D₄₇	1	3	yes	—
274	C₂²	C₂²×Q₈×D₄₇	1	1	yes	characteristic
275	C₄	—	47	8	no	—
276	C₄	C₂×C₄⋊C₄×D₄₇	1	3	yes	—
277	C₄	C₂×C₄⋊C₄×D₄₇	1	3	yes	—
278	C₄	C₂³×C₄×D₄₇	1	1	yes	characteristic
279	C₄	C₂²×D₄×D₄₇	1	1	yes	characteristic
280	C₂	—	2	3	no	—
281	C₂	—	2	1	no	—
282	C₂	—	94	8	no	—
283	C₂	C₂×#15689170	1	6	yes	direct product factor
284	C₂	C₂²×C₄⋊C₄×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#108×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×D₄₇	Q₈
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×#108	D₄₇
hypocenter	C₄₇	C₂²×#430
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. C₂×#15689170	C₂	direct (N × H)
4. C₂²×#13093960	C₂	semidirect
5. C₂²×#13093959	C₂	semidirect
6. C₂²×#108×C₄₇	C₂	semidirect
17. #15689170	C₂²	direct (N × H)
18. C₂×#13093960	C₂²	semidirect
19. C₂×#13093959	C₂²	semidirect
22. C₂×#108×C₄₇	C₂²	semidirect
44. #13093960	C₂³	semidirect
45. #13093959	C₂³	semidirect
53. #108×C₄₇	C₂³	semidirect
125. C₂²×C₄₇	#430	semidirect
143. C₂²×#108	D₄₇	semidirect
146. C₂×C₄₇	C₂×#430	semidirect
164. C₂×#108	C₂×D₄₇	semidirect
165. C₄₇	C₂²×#430	semidirect
196. #108	C₂²×D₄₇	semidirect
271. C₂²	#15689170	direct (N × H)
283. C₂	C₂×#15689170	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×#10625324	2
C₂²×M₁₆×D₄₇	2
C₂×#15689170	2
C₂²×#13093960	2
C₂²×#13093959	2
C₂²×#108×C₄₇	2
C₂²×#430	47

Subgroups — C22×#15689170 #27079969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689170 #27079969