Subgroups — C₂×#20245505 #26370832

Order 12032.

678 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689539	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689390	C₂	1	2	yes	maximal, complemented
3	#20245505	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688953	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688936	C₂	1	2	yes	maximal, complemented
6	C₂²×D₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13094050	—	2	2	no	—
8	C₂×#13094030	C₂²	1	1	yes	characteristic
9	#15689539	—	2	2	no	—
10	#15689539	C₂²	1	4	yes	complemented
11	C₂×#13093964	C₂²	1	1	yes	characteristic
12	C₂×#13093963	—	2	1	no	—
13	C₂×C₁₆×D₄₇	—	2	1	no	—
14	#15689390	—	2	4	no	—
15	#15689390	C₂²	1	8	yes	complemented
16	C₂×#13093960	C₂²	1	2	yes	—
17	C₂×#13093941	—	2	2	no	—
18	#15688953	C₂²	1	4	yes	complemented
19	#15688936	C₂²	1	8	yes	complemented
20	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
21	C₂×D₁₆×C₄₇	—	2	1	no	—
22	C₂×D₁₆×C₄₇	—	2	1	no	—
23	C₂×D₁₆×C₄₇	—	2	2	no	—
24	C₂×D₁₆×C₄₇	C₂²	1	4	yes	complemented
25	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#10625350	D₄	1	1	yes	characteristic
27	#13094050	—	2	8	no	—
28	#13094050	—	4	4	no	—
29	C₂×#10625335	—	4	2	no	—
30	C₂×#10625334	—	4	2	no	—
31	#13094030	—	2	1	no	—
32	#13094030	—	2	1	no	—
33	#13094030	D₄	1	2	yes	—
34	#13094030	C₂³	1	2	yes	—
35	C₂×#10625329	—	2	2	no	—
36	C₂×#10625329	D₄	1	1	yes	characteristic
37	C₂×#10625324	—	2	1	no	—
38	C₂×C₈×D₄₇	—	2	1	no	—
39	#13093964	—	2	1	no	—
40	#13093964	C₂³	1	2	yes	—
41	#13093963	—	2	4	no	—
42	C₂×C₄₇⋊C₁₆	—	2	1	no	—
43	C₁₆×D₄₇	—	2	4	no	—
44	#13093960	—	2	2	no	—
45	#13093960	C₂³	1	4	yes	—
46	#13093941	—	2	8	no	—
47	C₂²×D₄×C₄₇	—	2	2	no	—
48	C₂×D₈×C₄₇	—	2	2	no	—
49	C₂×D₈×C₄₇	—	2	2	no	—
50	C₂×D₈×C₄₇	—	2	2	no	—
51	C₂×D₈×C₄₇	—	2	2	no	—
52	C₂×D₈×C₄₇	—	2	2	no	—
53	C₂×D₈×C₄₇	C₂³	1	4	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	D₁₆×C₄₇	—	2	4	no	—
56	D₁₆×C₄₇	—	2	4	no	—
57	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
60	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
61	C₂⁴×C₄₇	—	4	2	no	—
62	C₂×D₄×C₄₇	—	2	4	no	—
63	C₂×D₄×C₄₇	—	4	2	no	—
64	C₂×D₄×C₄₇	—	4	2	no	—
65	C₂×D₄×C₄₇	—	4	2	no	—
66	C₂×D₄×C₄₇	—	4	2	no	—
67	C₂×D₄×C₄₇	—	4	2	no	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₄₇	—	2	1	no	—
70	#10625350	—	2	1	no	—
71	#10625350	—	2	1	no	—
72	#10625350	D₈	1	2	yes	—
73	#10625350	C₂×D₄	1	2	yes	—
74	C₂×D₁₈₈	D₈	1	1	yes	characteristic
75	C₂×C₄×D₄₇	—	2	1	no	—
76	C₂×Dic₉₄	D₈	1	1	yes	characteristic
77	D₈×C₄₇	—	2	4	no	—
78	D₈×C₄₇	—	2	4	no	—
79	D₈×C₄₇	—	4	2	no	—
80	D₈×C₄₇	—	4	2	no	—
81	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
85	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
87	#10625335	—	4	8	no	—
88	#10625334	—	4	8	no	—
89	#10625329	—	2	1	no	—
90	#10625329	—	2	4	no	—
91	#10625329	—	4	2	no	—
92	#10625329	C₂×D₄	1	2	yes	—
93	C₂×C₄₇⋊C₈	—	2	1	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	#10625324	—	2	2	no	—
96	#10625324	—	2	2	no	—
97	C₈×D₄₇	—	2	2	no	—
98	C₈×D₄₇	—	2	2	no	—
99	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
100	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
101	C₄₇⋊C₁₆	—	2	2	no	—
102	C₂³×C₄₇	—	4	4	no	—
103	C₂³×C₄₇	—	8	2	no	—
104	C₂³×C₄₇	—	8	2	no	—
105	C₂³×C₄₇	—	16	4	no	—
106	C₂³×C₄₇	#135	1	1	yes	characteristic
107	C₂²×D₄₇	—	2	1	no	—
108	D₄×C₄₇	—	4	4	no	—
109	D₄×C₄₇	—	4	4	no	—
110	D₄×C₄₇	—	8	2	no	—
111	D₄×C₄₇	—	8	2	no	—
112	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂²⋊D₄₇	—	2	2	no	—
119	C₂²⋊D₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	2	1	no	—
121	D₁₈₈	—	2	1	no	—
122	D₁₈₈	C₂×D₈	1	2	yes	—
123	C₄×D₄₇	—	2	2	no	—
124	C₄×D₄₇	—	2	2	no	—
125	Dic₉₄	—	2	1	no	—
126	Dic₉₄	C₂×D₈	1	2	yes	—
127	C₈×C₄₇	#121	1	1	yes	characteristic
128	C₈×C₄₇	#121	1	1	yes	characteristic
129	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
131	C₄₇⋊C₈	—	2	2	no	—
132	C₄₇⋊C₈	—	4	4	no	—
133	C₂×#2221	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	8	4	no	—
136	C₂²×C₄₇	—	16	4	no	—
137	C₂²×C₄₇	—	16	4	no	—
138	C₂²×C₄₇	—	16	4	no	—
139	C₂²×C₄₇	#466	1	1	yes	characteristic
140	C₂²×C₄₇	#466	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
142	C₂×D₄₇	—	2	2	no	—
143	C₂×D₄₇	—	4	2	no	—
144	C₄×C₄₇	#466	1	1	yes	characteristic
145	C₄×C₄₇	#466	1	1	yes	characteristic
146	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
147	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
148	Dic₄₇	—	2	2	no	—
149	C₂²×D₁₆	D₄₇	1	1	yes	characteristic, complemented
150	C₂×#504	—	47	1	no	—
151	C₂×#472	—	47	2	no	—
152	#2221	—	47	8	no	—
153	C₂×#368	—	47	1	no	—
154	C₂×#361	—	47	2	no	—
155	C₂×C₄₇	—	2	1	no	—
156	C₂×C₄₇	—	2	1	no	—
157	C₂×C₄₇	—	16	4	no	—
158	C₂×C₄₇	#2221	1	2	yes	complemented
159	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
160	D₄₇	—	4	2	no	—
161	C₂×AGL(1,ℤ/8)	—	94	2	no	—
162	C₂²×D₈	C₂×D₄₇	1	2	yes	—
163	C₂×#131	—	47	1	no	—
164	C₂×D₁₆	—	2	1	no	—
165	C₂×D₁₆	—	2	1	no	—
166	C₂×D₁₆	—	2	2	no	—
167	C₂×D₁₆	C₂×D₄₇	1	4	yes	complemented
168	#504	—	47	4	no	—
169	#504	—	94	2	no	—
170	C₂×M₃₂	—	47	1	no	—
171	C₂×M₃₂	—	94	1	no	—
172	C₂²×C₁₆	—	94	1	no	—
173	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
174	#472	—	47	8	no	—
175	#472	—	94	4	no	—
176	C₂×#108	—	47	2	no	—
177	C₂×#100	—	94	2	no	—
178	#368	—	47	4	no	—
179	#361	—	47	8	no	—
180	C₄₇	C₂×#2221	1	1	yes	characteristic, complemented, Sylow
181	C₂×#41	—	47	1	no	—
182	C₂²×D₄	—	2	2	no	—
183	AGL(1,ℤ/8)	—	94	8	no	—
184	AGL(1,ℤ/8)	—	188	4	no	—
185	C₂×SD₁₆	—	188	2	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	2	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	188	2	no	—
192	C₂×D₈	C₂²×D₄₇	1	4	yes	—
193	#131	—	47	2	no	—
194	#131	—	47	2	no	—
195	#131	—	94	1	no	—
196	#131	—	94	1	no	—
197	C₂×M₁₆	—	47	1	no	—
198	C₂×M₁₆	—	94	1	no	—
199	C₂×M₁₆	—	94	2	no	—
200	C₂²×C₈	—	94	1	no	—
201	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
202	D₁₆	—	2	4	no	—
203	D₁₆	—	2	4	no	—
204	M₃₂	—	47	2	no	—
205	M₃₂	—	94	1	no	—
206	M₃₂	—	94	4	no	—
207	C₂×C₁₆	—	94	1	no	—
208	C₂×C₁₆	—	94	4	no	—
209	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
210	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
211	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
212	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
213	#108	—	47	4	no	—
214	#108	—	94	2	no	—
215	#100	—	94	8	no	—
216	C₂⁴	—	4	2	no	—
217	#41	—	47	2	no	—
218	#41	—	47	2	no	—
219	#41	—	94	1	no	—
220	#41	—	94	1	no	—
221	C₂×Q₈	—	47	1	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	47	1	no	—
229	C₂×D₄	—	94	1	no	—
230	C₂²×C₄	—	94	1	no	—
231	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
232	SD₁₆	—	188	8	no	—
233	D₈	—	2	4	no	—
234	D₈	—	2	4	no	—
235	D₈	—	4	2	no	—
236	D₈	—	4	2	no	—
237	D₈	—	188	8	no	—
238	M₁₆	—	47	2	no	—
239	M₁₆	—	94	1	no	—
240	M₁₆	—	94	2	no	—
241	M₁₆	—	94	2	no	—
242	M₁₆	—	94	4	no	—
243	M₁₆	—	188	2	no	—
244	C₂×C₈	—	94	1	no	—
245	C₂×C₈	—	94	2	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	—	188	2	no	—
248	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
249	C₂×C₈	#10625352	1	1	yes	characteristic
250	C₂×C₈	#10625352	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
252	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
254	C₁₆	—	94	2	no	—
255	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
256	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
257	C₂³	—	4	4	no	—
258	C₂³	—	8	2	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	16	4	no	—
261	C₂³	—	94	1	no	—
262	C₂³	#13094043	1	1	yes	characteristic
263	Q₈	—	47	2	no	—
264	Q₈	—	94	1	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	8	2	no	—
268	D₄	—	8	2	no	—
269	D₄	—	47	2	no	—
270	D₄	—	94	2	no	—
271	D₄	—	94	2	no	—
272	D₄	—	94	1	no	—
273	C₂×C₄	—	94	1	no	—
274	C₂×C₄	—	94	2	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	#13093995	1	1	yes	characteristic
277	C₂×C₄	#13093995	1	1	yes	characteristic
278	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
279	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
280	C₂×C₄	#13094043	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
282	C₈	—	94	2	no	—
283	C₈	—	188	4	no	—
284	C₈	#13093995	1	1	yes	characteristic
285	C₈	#13093995	1	1	yes	characteristic
286	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
287	C₈	C₂×#10625352	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	16	4	no	—
291	C₂²	—	16	4	no	—
292	C₂²	—	16	4	no	—
293	C₂²	—	94	2	no	—
294	C₂²	—	188	2	no	—
295	C₂²	#15689356	1	1	yes	characteristic
296	C₂²	#15689356	1	1	yes	characteristic
297	C₂²	C₂×#13094043	1	1	yes	characteristic
298	C₄	—	94	2	no	—
299	C₄	#15689356	1	1	yes	characteristic
300	C₄	#15689356	1	1	yes	characteristic
301	C₄	C₂×#13093995	1	1	yes	characteristic
302	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	16	4	no	—
306	C₂	—	188	2	no	—
307	C₂	#20245505	1	2	yes	direct product factor
308	C₂	C₂×#15689356	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689356
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂²×D₁₆×C₄₇	C₂
socle	C₂²×C₄₇	#466
hypercenter	C₂²×D₁₆	D₄₇
hypocenter	C₄₇	C₂×#2221
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15689539	C₂	semidirect
2. C₂×#15689390	C₂	semidirect
3. #20245505	C₂	direct (N × H)
4. C₂×#15688953	C₂	semidirect
5. C₂×#15688936	C₂	semidirect
6. C₂²×D₁₆×C₄₇	C₂	semidirect
10. #15689539	C₂²	semidirect
15. #15689390	C₂²	semidirect
18. #15688953	C₂²	semidirect
19. #15688936	C₂²	semidirect
24. C₂×D₁₆×C₄₇	C₂²	semidirect
149. C₂²×D₁₆	D₄₇	semidirect
158. C₂×C₄₇	#2221	semidirect
167. C₂×D₁₆	C₂×D₄₇	semidirect
180. C₄₇	C₂×#2221	semidirect
307. C₂	#20245505	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×#15689539	2
C₂×#15689390	2
#20245505	2
C₂×#15688953	2
C₂×#15688936	2
C₂²×D₁₆×C₄₇	2
C₂×#2221	47

Subgroups — C2×#20245505 #26370832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245505 #26370832