Subgroups — C₂²×#15689451 #27084006

Order 12032.

2662 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689451	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13093955	C₂	1	2	yes	maximal, complemented
4	C₂²×#13093948	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂²⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₇	—	2	1	no	—
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
10	C₂²×C₄⋊Dic₄₇	—	2	6	no	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂²×C₄⋊Dic₄₇	C₂²	1	12	yes	—
13	#15689451	C₂²	1	64	yes	direct product factor
14	C₂×#13093955	C₂²	1	24	yes	complemented
15	C₂×#13093948	C₂²	1	24	yes	complemented
16	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	12	yes	—
17	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₄₇	—	2	6	no	—
20	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	12	yes	—
21	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₄₇	—	2	1	no	—
24	C₂³×Dic₄₇	—	2	12	no	—
25	C₂³×Dic₄₇	—	2	2	no	—
26	C₂³×Dic₄₇	—	4	8	no	—
27	C₂²×C₄₇⋊C₈	—	2	8	no	—
28	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
29	C₂×C₄⋊Dic₄₇	—	2	12	no	—
30	C₂×C₄⋊Dic₄₇	—	2	24	no	—
31	C₂×C₄⋊Dic₄₇	—	2	6	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
34	#13093955	C₂³	1	32	yes	complemented
35	#13093948	C₂³	1	32	yes	complemented
36	C₂²⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
37	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
38	C₂²×Q₈×C₄₇	—	2	3	no	—
39	C₂²×Q₈×C₄₇	—	2	8	no	—
40	C₂²×Q₈×C₄₇	C₂³	1	6	yes	—
41	C₂³×C₄×C₄₇	—	2	8	no	—
42	C₂³×C₄×C₄₇	—	2	1	no	—
43	C₂³×C₄×C₄₇	—	2	1	no	—
44	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	D₄	1	6	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²⋊Q₈×C₄₇	—	2	24	no	—
49	C₂²⋊Q₈×C₄₇	C₂³	1	16	yes	—
50	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
51	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	6	yes	—
53	C₂×C₂²⋊C₄×C₄₇	—	2	12	no	—
54	C₂⁴×C₄₇	—	2	4	no	—
55	C₂⁴×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	SD₁₆	1	8	yes	—
57	C₂⁴×C₄₇	C₂×D₄	1	6	yes	—
58	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₄₇	—	2	6	no	—
60	C₂×Q₈×C₄₇	—	2	24	no	—
61	C₂×Q₈×C₄₇	—	4	12	no	—
62	C₂×Q₈×C₄₇	C₂⁴	1	4	yes	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	24	no	—
65	C₂²×C₄×C₄₇	—	2	24	no	—
66	C₂²×C₄×C₄₇	—	2	6	no	—
67	C₂²×C₄×C₄₇	—	2	6	no	—
68	C₂²×C₄×C₄₇	—	4	4	no	—
69	C₂²×C₄×C₄₇	—	4	4	no	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₄₇	#41	1	8	yes	—
74	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
76	C₂²×Dic₄₇	—	2	16	no	—
77	C₂²×Dic₄₇	—	2	12	no	—
78	C₂²×Dic₄₇	—	4	48	no	—
79	C₂²×Dic₄₇	—	4	8	no	—
80	C₄⋊C₄×C₄₇	—	2	16	no	—
81	C₄⋊C₄×C₄₇	—	2	6	no	—
82	C₄⋊C₄×C₄₇	C₂⁴	1	4	yes	—
83	C₂²⋊C₄×C₄₇	—	2	16	no	—
84	C₄⋊Dic₄₇	—	2	16	no	—
85	C₄⋊Dic₄₇	—	2	12	no	—
86	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
87	C₂×C₄₇⋊C₈	—	2	24	no	—
88	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
89	C₂³×C₄₇	—	2	12	no	—
90	C₂³×C₄₇	—	2	12	no	—
91	C₂³×C₄₇	—	2	32	no	—
92	C₂³×C₄₇	#123	1	4	yes	—
93	C₂³×C₄₇	C₂×SD₁₆	1	24	yes	—
94	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
95	C₂³×C₄₇	#137	1	4	yes	—
96	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
97	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
98	Q₈×C₄₇	—	2	8	no	—
99	Q₈×C₄₇	—	4	12	no	—
100	C₂×C₄×C₄₇	—	2	12	no	—
101	C₂×C₄×C₄₇	—	2	24	no	—
102	C₂×C₄×C₄₇	—	2	8	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	4	12	no	—
106	C₂×C₄×C₄₇	—	4	12	no	—
107	C₂×C₄×C₄₇	#123	1	12	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	12	yes	—
111	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
112	C₂×Dic₄₇	—	2	8	no	—
113	C₂×Dic₄₇	—	4	32	no	—
114	C₂×Dic₄₇	—	4	24	no	—
115	C₄₇⋊C₈	—	2	8	no	—
116	C₂²×#483	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	48	no	—
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	—	2	4	no	—
120	C₂²×C₄₇	#483	1	16	yes	complemented
121	C₂²×C₄₇	C₂×#123	1	6	yes	—
122	C₂²×C₄₇	C₂²×SD₁₆	1	8	yes	—
123	C₂²×C₄₇	C₂²×SD₁₆	1	6	yes	—
124	C₂²×C₄₇	C₂×#137	1	6	yes	—
125	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	4	4	no	—
128	C₄×C₄₇	—	4	4	no	—
129	C₄×C₄₇	C₂×#123	1	6	yes	—
130	C₄×C₄₇	C₂²×#41	1	2	yes	—
131	Dic₄₇	—	4	8	no	—
132	C₂²×C₂²⋊Q₈	D₄₇	1	1	yes	characteristic
133	C₂³×C₄⋊C₄	—	47	1	no	—
134	C₂×#483	—	47	24	no	—
135	C₂²×#106	—	47	2	no	—
136	C₂²×#103	—	47	2	no	—
137	C₂²×C₂²⋊C₈	—	47	1	no	—
138	C₂×C₄₇	—	2	8	no	—
139	C₂×C₄₇	C₂×#483	1	12	yes	complemented
140	C₂×C₄₇	C₂²×#123	1	1	yes	characteristic
141	C₂×C₄₇	C₂³×SD₁₆	1	1	yes	characteristic
142	C₂×C₄₇	C₂²×#137	1	1	yes	characteristic
143	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
144	C₂⁴×C₄	—	94	1	no	—
145	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
146	C₂³×C₈	—	47	2	no	—
147	C₂×C₂²⋊Q₈	—	2	6	no	—
148	C₂×C₂²⋊Q₈	C₂×D₄₇	1	12	yes	—
149	C₂²×C₄⋊C₄	—	2	1	no	—
150	C₂²×C₄⋊C₄	—	47	12	no	—
151	C₂²×C₄⋊C₄	—	47	2	no	—
152	C₂²×C₄⋊C₄	—	94	6	no	—
153	C₂²×C₄⋊C₄	—	94	1	no	—
154	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
155	C₂²×C₂²⋊C₄	—	2	1	no	—
156	#483	—	47	64	no	—
157	C₂×#106	—	47	24	no	—
158	C₂×#103	—	47	24	no	—
159	C₂×C₂²⋊C₈	—	47	12	no	—
160	C₄₇	C₂²×#483	1	1	yes	characteristic, complemented, Sylow
161	C₂⁵	C₂²⋊D₄₇	1	1	yes	characteristic
162	C₂²×Q₈	—	2	3	no	—
163	C₂²×Q₈	—	2	8	no	—
164	C₂²×Q₈	C₂²×D₄₇	1	6	yes	—
165	C₂³×C₄	—	2	8	no	—
166	C₂³×C₄	—	2	1	no	—
167	C₂³×C₄	—	2	1	no	—
168	C₂³×C₄	—	94	12	no	—
169	C₂³×C₄	—	94	2	no	—
170	C₂³×C₄	—	188	8	no	—
171	C₂³×C₄	C₂²⋊D₄₇	1	6	yes	—
172	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
173	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
174	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
175	C₂²×C₈	—	47	12	no	—
176	C₂²×C₈	—	94	8	no	—
177	C₂²⋊Q₈	—	2	24	no	—
178	C₂²⋊Q₈	C₂²×D₄₇	1	16	yes	—
179	C₂×C₄⋊C₄	—	2	12	no	—
180	C₂×C₄⋊C₄	—	2	3	no	—
181	C₂×C₄⋊C₄	—	47	12	no	—
182	C₂×C₄⋊C₄	—	47	16	no	—
183	C₂×C₄⋊C₄	—	94	12	no	—
184	C₂×C₄⋊C₄	—	94	24	no	—
185	C₂×C₄⋊C₄	—	94	6	no	—
186	C₂×C₄⋊C₄	C₂²×D₄₇	1	6	yes	—
187	C₂×C₂²⋊C₄	—	2	12	no	—
188	#106	—	47	32	no	—
189	#103	—	47	32	no	—
190	C₂²⋊C₈	—	47	16	no	—
191	C₂⁴	—	2	4	no	—
192	C₂⁴	—	2	4	no	—
193	C₂⁴	#10625336	1	8	yes	—
194	C₂⁴	C₂×C₂²⋊D₄₇	1	6	yes	—
195	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂×Q₈	—	2	6	no	—
197	C₂×Q₈	—	2	24	no	—
198	C₂×Q₈	—	4	12	no	—
199	C₂×Q₈	C₂³×D₄₇	1	4	yes	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	24	no	—
202	C₂²×C₄	—	2	24	no	—
203	C₂²×C₄	—	2	6	no	—
204	C₂²×C₄	—	2	6	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	94	16	no	—
208	C₂²×C₄	—	94	12	no	—
209	C₂²×C₄	—	188	48	no	—
210	C₂²×C₄	—	188	8	no	—
211	C₂²×C₄	#10625352	1	8	yes	—
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
215	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
216	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
217	C₂×C₈	—	47	8	no	—
218	C₂×C₈	—	94	24	no	—
219	C₄⋊C₄	—	2	16	no	—
220	C₄⋊C₄	—	2	6	no	—
221	C₄⋊C₄	—	47	8	no	—
222	C₄⋊C₄	—	94	16	no	—
223	C₄⋊C₄	—	94	12	no	—
224	C₄⋊C₄	C₂³×D₄₇	1	4	yes	—
225	C₂²⋊C₄	—	2	16	no	—
226	C₂³	—	2	12	no	—
227	C₂³	—	2	12	no	—
228	C₂³	—	2	32	no	—
229	C₂³	#13094005	1	4	yes	—
230	C₂³	C₂×#10625336	1	24	yes	—
231	C₂³	C₂×#10625336	1	4	yes	—
232	C₂³	#13094054	1	4	yes	—
233	C₂³	C₂²×C₂²⋊D₄₇	1	4	yes	—
234	C₂³	C₂²×C₂²⋊D₄₇	1	3	yes	—
235	Q₈	—	2	8	no	—
236	Q₈	—	4	12	no	—
237	C₂×C₄	—	2	12	no	—
238	C₂×C₄	—	2	24	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	12	no	—
243	C₂×C₄	—	4	12	no	—
244	C₂×C₄	—	94	8	no	—
245	C₂×C₄	—	188	32	no	—
246	C₂×C₄	—	188	24	no	—
247	C₂×C₄	#13094005	1	12	yes	—
248	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
249	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
250	C₂×C₄	C₂×#10625352	1	12	yes	—
251	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
252	C₈	—	94	8	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	48	no	—
255	C₂²	—	2	4	no	—
256	C₂²	#15689451	1	16	yes	direct product factor
257	C₂²	C₂×#13094005	1	6	yes	—
258	C₂²	C₂²×#10625336	1	8	yes	—
259	C₂²	C₂²×#10625336	1	6	yes	—
260	C₂²	C₂×#13094054	1	6	yes	—
261	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	4	4	no	—
264	C₄	—	4	4	no	—
265	C₄	—	188	8	no	—
266	C₄	C₂×#13094005	1	6	yes	—
267	C₄	C₂²×#10625352	1	2	yes	—
268	C₂	—	2	8	no	—
269	C₂	C₂×#15689451	1	12	yes	direct product factor
270	C₂	C₂²×#13094005	1	1	yes	characteristic
271	C₂	C₂³×#10625336	1	1	yes	characteristic
272	C₂	C₂²×#13094054	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×C₂²⋊Q₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
prime radical	C₂⁵×C₄₇	D₄
hypercenter	C₂²×C₂²⋊Q₈	D₄₇
hypocenter	C₄₇	C₂²×#483
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂²×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#15689451	C₂	direct (N × H)
3. C₂²×#13093955	C₂	semidirect
4. C₂²×#13093948	C₂	semidirect
13. #15689451	C₂²	direct (N × H)
14. C₂×#13093955	C₂²	semidirect
15. C₂×#13093948	C₂²	semidirect
34. #13093955	C₂³	semidirect
35. #13093948	C₂³	semidirect
120. C₂²×C₄₇	#483	semidirect
139. C₂×C₄₇	C₂×#483	semidirect
160. C₄₇	C₂²×#483	semidirect
256. C₂²	#15689451	direct (N × H)
269. C₂	C₂×#15689451	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×C₄⋊Dic₄₇	2
C₂×#15689451	2
C₂²×#13093955	2
C₂²×#13093948	2
C₂²×C₂²⋊(C₄₇⋊C₈)	2
C₂²×C₂²⋊Q₈×C₄₇	2
C₂²×#483	47

Subgroups — C22×#15689451 #27084006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689451 #27084006