Subgroups — C₂×#20242212 #26185677

Order 12032.

894 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
2	#20242212	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688874	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
5	C₂×#15688844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	—	2	2	no	—
8	D₄×C₄₇⋊C₈	—	2	4	no	—
9	D₄×C₄₇⋊C₈	C₂²	1	8	yes	complemented
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	#15688874	C₂²	1	4	yes	complemented
15	D₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	complemented
16	#15688844	C₂²	1	4	yes	complemented
17	C₂²×D₈×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
19	C₄×D₈×C₄₇	C₄	1	4	yes	—
20	C₄×D₈×C₄₇	C₂²	1	4	yes	—
21	C₂×#107×C₄₇	C₄	1	1	yes	characteristic
22	C₂×#102×C₄₇	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	—	2	1	no	—
26	C₂²×C₄₇⋊C₈	—	2	8	no	—
27	C₂²×C₄₇⋊C₈	—	4	8	no	—
28	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
29	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
31	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
32	C₄×C₄₇⋊C₈	—	2	1	no	—
33	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
34	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₄₇	—	2	2	no	—
38	C₂×D₈×C₄₇	C₈	1	8	yes	complemented
39	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₄×D₄×C₄₇	—	2	2	no	—
43	C₄×D₄×C₄₇	—	2	2	no	—
44	C₄×D₄×C₄₇	C₂×C₄	1	4	yes	—
45	C₄×D₄×C₄₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	#107×C₄₇	C₂×C₄	1	4	yes	—
50	#102×C₄₇	C₂×C₄	1	8	yes	—
51	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂⁴×C₄₇	—	2	2	no	—
54	C₂×D₄×C₄₇	—	2	2	no	—
55	C₂×D₄×C₄₇	—	2	4	no	—
56	C₂×D₄×C₄₇	C₂×C₈	1	8	yes	complemented
57	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	4	4	no	—
62	C₂²×C₄×C₄₇	—	4	4	no	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
66	D₈×C₄₇	C₂×C₈	1	16	yes	complemented
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	complemented
69	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
71	C₄⋊C₄×C₄₇	—	2	2	no	—
72	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
73	C₂²⋊C₄×C₄₇	—	2	4	no	—
74	C₂²⋊C₄×C₄₇	—	2	4	no	—
75	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₄₇	#41	1	1	yes	characteristic
78	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂×C₄₇⋊C₈	—	2	2	no	—
82	C₂×C₄₇⋊C₈	—	4	16	no	—
83	C₂×C₄₇⋊C₈	—	4	2	no	—
84	C₂×C₄₇⋊C₈	—	4	4	no	—
85	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
86	C₂³×C₄₇	—	2	4	no	—
87	C₂³×C₄₇	—	2	8	no	—
88	C₂³×C₄₇	—	4	4	no	—
89	C₂³×C₄₇	—	4	4	no	—
90	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
91	D₄×C₄₇	—	2	4	no	—
92	D₄×C₄₇	—	2	8	no	—
93	D₄×C₄₇	C₂²×C₈	1	8	yes	complemented
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	—	4	4	no	—
98	C₂×C₄×C₄₇	—	4	4	no	—
99	C₂×C₄×C₄₇	—	4	8	no	—
100	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₄₇	#131	1	2	yes	—
105	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
109	C₈×C₄₇	—	2	2	no	—
110	C₈×C₄₇	C₂²×C₈	1	4	yes	complemented
111	C₄₇⋊C₈	—	2	4	no	—
112	C₄₇⋊C₈	—	4	2	no	—
113	C₄₇⋊C₈	—	4	4	no	—
114	C₂×#1580	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	8	no	—
116	C₂²×C₄₇	—	4	16	no	—
117	C₂²×C₄₇	—	4	4	no	—
118	C₂²×C₄₇	—	4	4	no	—
119	C₂²×C₄₇	C₈×D₄	1	2	yes	—
120	C₂²×C₄₇	#442	1	2	yes	—
121	C₂²×C₄₇	#444	1	2	yes	—
122	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₄₇	—	2	1	no	—
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	—	4	4	no	—
126	C₄×C₄₇	C₈×D₄	1	2	yes	—
127	C₄×C₄₇	#442	1	2	yes	—
128	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
129	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
130	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic
132	C₂×C₈×D₄	—	47	2	no	—
133	#1580	—	47	8	no	—
134	C₂×#334	—	47	1	no	—
135	C₂×D₄⋊C₈	—	47	2	no	—
136	C₂×#322	—	47	1	no	—
137	C₂×C₄₇	—	4	8	no	—
138	C₂×C₄₇	#1580	1	4	yes	complemented
139	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
140	C₂×C₄₇	C₂×#442	1	1	yes	characteristic
141	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
142	C₂²×D₈	Dic₄₇	1	1	yes	characteristic
143	C₂³×C₈	—	94	2	no	—
144	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
145	C₄×D₈	Dic₄₇	1	4	yes	—
146	C₄×D₈	C₂×D₄₇	1	4	yes	—
147	C₈×D₄	—	47	8	no	—
148	C₈×D₄	—	94	4	no	—
149	C₂×#107	Dic₄₇	1	1	yes	characteristic
150	C₂×C₄⋊C₈	—	47	2	no	—
151	C₂×#102	Dic₄₇	1	2	yes	—
152	C₂×C₂²⋊C₈	—	94	2	no	—
153	C₂×C₄×C₈	—	47	1	no	—
154	C₂×C₄×C₈	—	47	1	no	—
155	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
156	#334	—	47	4	no	—
157	D₄⋊C₈	—	47	8	no	—
158	#322	—	47	4	no	—
159	C₄₇	C₂×#1580	1	1	yes	characteristic, complemented, Sylow
160	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
161	C₂³×C₄	—	2	2	no	—
162	C₂×D₈	C₄₇⋊C₈	1	8	yes	complemented
163	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
164	C₂²×C₈	—	47	2	no	—
165	C₂²×C₈	—	94	8	no	—
166	C₂²×C₈	—	94	2	no	—
167	C₂²×C₈	—	94	1	no	—
168	C₂²×C₈	—	188	8	no	—
169	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
170	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
171	C₄×D₄	—	2	2	no	—
172	C₄×D₄	—	2	2	no	—
173	C₄×D₄	C₂×Dic₄₇	1	4	yes	—
174	C₄×D₄	C₂²×D₄₇	1	4	yes	—
175	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
176	C₂×C₂²⋊C₄	—	2	2	no	—
177	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
178	#107	C₂×Dic₄₇	1	4	yes	—
179	C₄⋊C₈	—	47	4	no	—
180	C₄⋊C₈	—	94	2	no	—
181	#102	C₂×Dic₄₇	1	8	yes	—
182	C₂²⋊C₈	—	94	8	no	—
183	C₄×C₈	—	47	2	no	—
184	C₄×C₈	—	47	2	no	—
185	C₄×C₈	—	47	2	no	—
186	C₄×C₈	—	94	1	no	—
187	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
188	C₄×C₈	C₂²×D₄₇	1	2	yes	—
189	C₂⁴	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	C₂×C₄₇⋊C₈	1	8	yes	complemented
193	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	#10625352	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
202	D₈	C₂×C₄₇⋊C₈	1	16	yes	complemented
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	47	4	no	—
205	C₂×C₈	—	94	4	no	—
206	C₂×C₈	—	94	4	no	—
207	C₂×C₈	—	94	2	no	—
208	C₂×C₈	—	188	16	no	—
209	C₂×C₈	—	188	2	no	—
210	C₂×C₈	—	188	4	no	—
211	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	complemented
212	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
213	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	D₄×D₄₇	1	1	yes	characteristic
219	C₄²	#10625352	1	1	yes	characteristic
220	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
221	C₄²	C₂³×D₄₇	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	—	2	8	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	4	no	—
226	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	2	8	no	—
229	D₄	C₂²×C₄₇⋊C₈	1	8	yes	complemented
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
238	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
240	C₂×C₄	#13094031	1	2	yes	—
241	C₂×C₄	#13094048	1	4	yes	—
242	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
244	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	94	4	no	—
247	C₈	—	188	2	no	—
248	C₈	—	188	4	no	—
249	C₈	C₂²×C₄₇⋊C₈	1	4	yes	complemented
250	C₂²	—	2	8	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	16	no	—
253	C₂²	—	4	4	no	—
254	C₂²	D₄×C₄₇⋊C₈	1	2	yes	—
255	C₂²	#15689235	1	2	yes	—
256	C₂²	#15689245	1	2	yes	—
257	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	4	no	—
261	C₄	D₄×C₄₇⋊C₈	1	2	yes	—
262	C₄	#15689235	1	2	yes	—
263	C₄	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
264	C₄	C₂×#13094031	1	1	yes	characteristic
265	C₄	C₂×#13094048	1	2	yes	—
266	C₂	—	4	8	no	—
267	C₂	#20242212	1	4	yes	direct product factor
268	C₂	C₂×D₄×C₄₇⋊C₈	1	1	yes	characteristic
269	C₂	C₂×#15689235	1	1	yes	characteristic
270	C₂	C₂×#15689245	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂²×D₈×C₄₇	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	C₂×#1580
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×C₄₇⋊C₈	C₂	semidirect
2. #20242212	C₂	direct (N × H)
3. C₂×#15688874	C₂	semidirect
4. C₂×D₄⋊(C₄₇⋊C₈)	C₂	semidirect
5. C₂×#15688844	C₂	semidirect
9. D₄×C₄₇⋊C₈	C₂²	semidirect
14. #15688874	C₂²	semidirect
15. D₄⋊(C₄₇⋊C₈)	C₂²	semidirect
16. #15688844	C₂²	semidirect
38. C₂×D₈×C₄₇	C₈	semidirect
56. C₂×D₄×C₄₇	C₂×C₈	semidirect
66. D₈×C₄₇	C₂×C₈	semidirect
68. C₂×C₈×C₄₇	C₂×C₈	semidirect
93. D₄×C₄₇	C₂²×C₈	semidirect
110. C₈×C₄₇	C₂²×C₈	semidirect
138. C₂×C₄₇	#1580	semidirect
159. C₄₇	C₂×#1580	semidirect
162. C₂×D₈	C₄₇⋊C₈	semidirect
192. C₂×D₄	C₂×C₄₇⋊C₈	semidirect
202. D₈	C₂×C₄₇⋊C₈	semidirect
211. C₂×C₈	C₂×C₄₇⋊C₈	semidirect
229. D₄	C₂²×C₄₇⋊C₈	semidirect
249. C₈	C₂²×C₄₇⋊C₈	semidirect
267. C₂	#20242212	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₄×C₄₇⋊C₈	2
#20242212	2
C₂×#15688874	2
C₂×D₄⋊(C₄₇⋊C₈)	2
C₂×#15688844	2
C₂×C₄×D₈×C₄₇	2
C₂×#1580	47

Subgroups — C2×#20242212 #26185677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242212 #26185677