Subgroups — C₂×D₁₆×#6923 #30624281

Order 15616.

806 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×#6923	C₂	1	2	yes	maximal, complemented
2	D₁₆×#6923	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16840688	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16840674	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×#6923	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂×D₄×#6923	—	2	2	no	—
8	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
9	D₈×#6923	—	2	4	no	—
10	D₈×#6923	C₂²	1	8	yes	complemented
11	C₂×#14519846	C₂²	1	2	yes	—
12	C₂×#14519839	—	2	2	no	—
13	C₂×C₈×#6923	C₂²	1	1	yes	characteristic
14	#16840688	C₂²	1	4	yes	complemented
15	#16840674	C₂²	1	8	yes	complemented
16	C₁₆×#6923	C₂²	1	4	yes	complemented
17	C₂×D₈×D₆₁	C₂²	1	2	yes	—
18	C₂×D₉₇₆	C₄	1	1	yes	characteristic, complemented
19	C₂×#14519718	C₄	1	2	yes	complemented
20	D₁₆×D₆₁	C₄	1	4	yes	complemented
21	D₁₆×D₆₁	C₂²	1	4	yes	—
22	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
23	C₂×D₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
24	C₂³×#6923	—	4	2	no	—
25	C₂×D₆₁⋊C₈	D₄	1	1	yes	characteristic
26	D₄×#6923	—	2	8	no	—
27	D₄×#6923	—	4	4	no	—
28	C₂×C₄⋊#6923	—	2	2	no	—
29	C₂×C₂²⋊#6923	—	4	2	no	—
30	C₂×C₄×#6923	D₄	1	1	yes	characteristic
31	D₆₁⋊C₁₆	—	2	1	no	—
32	D₆₁⋊C₁₆	C₂³	1	2	yes	—
33	C₂×#11758187	—	2	1	no	—
34	#14519846	—	2	2	no	—
35	#14519846	C₂³	1	4	yes	—
36	#14519839	—	2	8	no	—
37	C₈×#6923	D₄	1	2	yes	—
38	C₈×#6923	C₂³	1	2	yes	—
39	C₂×D₄×D₆₁	—	2	2	no	—
40	C₂×D₄₈₈	C₂×C₄	1	2	yes	complemented
41	C₂×#11758199	—	2	2	no	—
42	D₈×D₆₁	—	2	2	no	—
43	D₈×D₆₁	—	2	2	no	—
44	D₈×D₆₁	C₂×C₄	1	4	yes	complemented
45	D₈×D₆₁	C₂³	1	4	yes	—
46	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
47	D₉₇₆	C₂×C₄	1	4	yes	complemented
48	#14519718	C₂×C₄	1	8	yes	complemented
49	C₂×#11758185	C₂×C₄	1	1	yes	characteristic, complemented
50	C₁₆×D₆₁	C₂×C₄	1	2	yes	complemented
51	C₁₆×D₆₁	C₂³	1	2	yes	—
52	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	complemented
53	D₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
54	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂³×D₆₁	—	4	2	no	—
56	C₂²×#6923	—	4	2	no	—
57	C₂²×#6923	—	4	8	no	—
58	C₂²×#6923	—	8	8	no	—
59	C₂²×#6923	D₈	1	1	yes	characteristic
60	C₂²×#6923	D₈	1	1	yes	characteristic
61	C₂×D₄×C₆₁	—	2	2	no	—
62	C₂×C₂²⋊D₆₁	—	4	2	no	—
63	D₄×D₆₁	—	2	4	no	—
64	D₄×D₆₁	—	2	4	no	—
65	D₄×D₆₁	—	4	2	no	—
66	D₄×D₆₁	—	4	2	no	—
67	C₂×D₂₄₄	—	2	2	no	—
68	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
69	C₂²⋊#6923	—	4	8	no	—
70	C₂×#92420	—	2	1	no	—
71	C₄⋊#6923	—	2	4	no	—
72	C₄⋊#6923	—	4	2	no	—
73	C₄×#6923	D₈	1	2	yes	—
74	C₄×#6923	C₂×D₄	1	2	yes	—
75	D₆₁⋊C₈	—	2	1	no	—
76	D₆₁⋊C₈	C₂×D₄	1	2	yes	—
77	D₈×C₆₁	—	2	2	no	—
78	D₈×C₆₁	C₂²×C₄	1	4	yes	complemented
79	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	#11758199	—	2	4	no	—
81	#11758199	—	2	4	no	—
82	C₂×#92418	#41	1	1	yes	characteristic
83	D₄₈₈	—	2	2	no	—
84	D₄₈₈	C₂²×C₄	1	4	yes	complemented
85	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
87	C₈×D₆₁	#41	1	1	yes	characteristic
88	C₈×D₆₁	C₂⁴	1	1	yes	characteristic
89	#11758187	—	2	2	no	—
90	C₁₆×C₆₁	C₂²×C₄	1	2	yes	complemented
91	#11758185	C₂²×C₄	1	2	yes	complemented
92	C₂³×C₆₁	—	4	2	no	—
93	C₂²×D₆₁	—	4	2	no	—
94	C₂²×D₆₁	—	4	4	no	—
95	C₂²×D₆₁	—	4	4	no	—
96	C₂²×D₆₁	—	8	4	no	—
97	C₂²×D₆₁	—	8	4	no	—
98	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
99	C₂×#6923	—	2	2	no	—
100	C₂×#6923	—	4	4	no	—
101	C₂×#6923	—	8	4	no	—
102	C₂×#6923	—	8	16	no	—
103	C₂×#6923	D₁₆	1	4	yes	direct product factor
104	C₂×#6923	C₂×D₈	1	2	yes	—
105	C₂×#6923	C₂×D₈	1	2	yes	—
106	D₄×C₆₁	—	2	4	no	—
107	D₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
109	C₂²⋊D₆₁	—	4	4	no	—
110	C₂²⋊D₆₁	—	4	4	no	—
111	C₂×Dic₆₁	#135	1	1	yes	characteristic
112	D₂₄₄	—	2	4	no	—
113	D₂₄₄	—	4	2	no	—
114	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
115	C₄×D₆₁	C₂×D₈	1	1	yes	characteristic
116	C₄×D₆₁	#135	1	1	yes	characteristic
117	C₄×D₆₁	C₂²×D₄	1	1	yes	characteristic
118	#92420	—	2	2	no	—
119	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
120	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
121	#92418	C₄×D₄	1	1	yes	characteristic
122	#92418	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×D₁₆	—	61	1	no	maximal, Sylow
124	C₂²×C₆₁	—	4	4	no	—
125	C₂²×C₆₁	—	8	4	no	—
126	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
127	C₂×D₆₁	—	4	4	no	—
128	C₂×D₆₁	—	8	4	no	—
129	C₂×D₆₁	—	8	4	no	—
130	C₂×D₆₁	—	8	4	no	—
131	C₂×D₆₁	—	8	8	no	—
132	C₂×D₆₁	C₄×D₈	1	1	yes	characteristic
133	C₂×D₆₁	C₂×D₁₆	1	2	yes	—
134	C₂×D₆₁	#508	1	2	yes	—
135	C₂×D₆₁	C₂²×D₈	1	1	yes	characteristic
136	#6923	—	2	2	no	—
137	#6923	—	8	4	no	—
138	#6923	C₂×D₁₆	1	4	yes	direct product factor
139	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
140	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
141	Dic₆₁	C₄×D₈	1	1	yes	characteristic
142	Dic₆₁	C₂×#135	1	1	yes	characteristic
143	C₂²×D₁₆	—	61	1	no	—
144	C₂×C₄×D₈	—	61	2	no	—
145	C₄×D₁₆	—	61	8	no	—
146	C₂×#366	—	61	1	no	—
147	C₂×#357	—	61	2	no	—
148	C₂×C₄×C₁₆	—	61	1	no	—
149	C₂×C₆₁	—	8	4	no	—
150	C₂×C₆₁	C₄×D₁₆	1	2	yes	complemented
151	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
152	D₆₁	—	8	4	no	—
153	D₆₁	C₄×D₁₆	1	2	yes	complemented
154	D₆₁	C₂²×D₁₆	1	1	yes	characteristic
155	D₆₁	C₂×#508	1	1	yes	characteristic
156	C₂²×D₈	—	61	2	no	—
157	C₂×C₄×D₄	—	122	2	no	—
158	C₂×D₁₆	—	61	2	no	—
159	C₂×D₁₆	—	61	4	no	—
160	C₂×D₁₆	—	61	4	no	—
161	C₂×D₁₆	—	61	1	no	—
162	C₂×D₁₆	#6923	1	1	yes	characteristic, direct product factor
163	C₂²×C₁₆	—	61	1	no	—
164	C₂²×C₁₆	—	61	1	no	—
165	C₄×D₈	—	61	8	no	—
166	C₄×D₈	—	122	4	no	—
167	C₂×#107	—	61	2	no	—
168	C₂×#102	—	122	2	no	—
169	C₂×C₄×C₈	—	61	1	no	—
170	#366	—	61	4	no	—
171	#357	—	61	8	no	—
172	C₄×C₁₆	—	61	4	no	—
173	C₆₁	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
174	C₂²×D₄	—	122	2	no	—
175	C₂³×C₄	—	244	2	no	—
176	C₂×D₈	—	61	4	no	—
177	C₂×D₈	—	61	4	no	—
178	C₂×D₈	—	61	2	no	—
179	C₂×D₈	—	122	2	no	—
180	C₂×D₈	—	122	2	no	—
181	C₂×D₈	—	122	2	no	—
182	C₂×D₈	C₂×#6923	1	2	yes	complemented
183	C₂²×C₈	—	61	1	no	—
184	C₂²×C₈	—	61	1	no	—
185	C₄×D₄	—	122	8	no	—
186	C₄×D₄	—	244	4	no	—
187	C₂×C₄⋊C₄	—	122	2	no	—
188	C₂×C₂²⋊C₄	—	244	2	no	—
189	C₂×C₄²	—	61	1	no	—
190	D₁₆	—	61	8	no	—
191	D₁₆	—	61	4	no	—
192	D₁₆	C₂×#6923	1	4	yes	direct product factor
193	C₂×C₁₆	—	61	1	no	—
194	C₂×C₁₆	—	61	2	no	—
195	C₂×C₁₆	—	61	2	no	—
196	C₂×C₁₆	—	61	2	no	—
197	C₂×C₁₆	—	122	1	no	—
198	C₂×C₁₆	—	122	1	no	—
199	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic, complemented
200	#107	—	61	4	no	—
201	#107	—	122	2	no	—
202	#102	—	122	8	no	—
203	C₄×C₈	—	61	2	no	—
204	C₄×C₈	—	61	2	no	—
205	C₂⁴	—	244	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	122	4	no	—
208	C₂×D₄	—	122	4	no	—
209	C₂×D₄	—	122	2	no	—
210	C₂×D₄	—	244	2	no	—
211	C₂×D₄	—	244	2	no	—
212	C₂×D₄	—	244	2	no	—
213	C₂²×C₄	—	61	1	no	—
214	C₂²×C₄	—	61	1	no	—
215	C₂²×C₄	—	61	1	no	—
216	C₂²×C₄	—	488	8	no	—
217	C₂²×C₄	—	244	8	no	—
218	C₂²×C₄	—	244	2	no	—
219	D₈	—	2	2	no	—
220	D₈	—	61	4	no	—
221	D₈	—	122	4	no	—
222	D₈	—	122	4	no	—
223	D₈	—	122	2	no	—
224	D₈	C₂²×#6923	1	4	yes	complemented
225	C₂×C₈	—	61	1	no	—
226	C₂×C₈	—	61	1	no	—
227	C₂×C₈	—	61	1	no	—
228	C₂×C₈	—	61	1	no	—
229	C₂×C₈	—	61	1	no	—
230	C₂×C₈	—	61	2	no	—
231	C₂×C₈	—	122	1	no	—
232	C₂×C₈	—	122	1	no	—
233	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
234	C₄⋊C₄	—	122	4	no	—
235	C₄⋊C₄	—	244	2	no	—
236	C₂²⋊C₄	—	244	8	no	—
237	C₄²	—	61	2	no	—
238	C₄²	—	61	2	no	—
239	C₁₆	—	61	2	no	—
240	C₁₆	—	122	2	no	—
241	C₁₆	C₂²×#6923	1	2	yes	complemented
242	C₂³	—	4	2	no	—
243	C₂³	—	61	1	no	—
244	C₂³	—	488	4	no	—
245	C₂³	—	488	4	no	—
246	C₂³	—	244	4	no	—
247	C₂³	—	244	4	no	—
248	C₂³	—	244	2	no	—
249	D₄	—	2	4	no	—
250	D₄	—	4	2	no	—
251	D₄	—	122	4	no	—
252	D₄	—	244	4	no	—
253	D₄	—	244	4	no	—
254	D₄	—	244	2	no	—
255	C₂×C₄	—	61	4	no	—
256	C₂×C₄	—	61	2	no	—
257	C₂×C₄	—	61	1	no	—
258	C₂×C₄	—	61	1	no	—
259	C₂×C₄	—	61	1	no	—
260	C₂×C₄	—	61	1	no	—
261	C₂×C₄	—	61	1	no	—
262	C₂×C₄	—	61	2	no	—
263	C₂×C₄	—	122	2	no	—
264	C₂×C₄	—	488	16	no	—
265	C₂×C₄	—	488	4	no	—
266	C₂×C₄	—	244	4	no	—
267	C₂×C₄	D₄×#6923	1	1	yes	characteristic
268	C₈	—	61	1	no	—
269	C₈	—	61	1	no	—
270	C₈	—	122	2	no	—
271	C₈	D₄×#6923	1	1	yes	characteristic
272	C₈	C₂³×#6923	1	1	yes	characteristic
273	C₂²	—	4	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	61	1	no	—
276	C₂²	—	61	1	no	—
277	C₂²	—	61	2	no	—
278	C₂²	—	61	2	no	—
279	C₂²	—	488	4	no	—
280	C₂²	—	488	4	no	—
281	C₂²	—	488	8	no	—
282	C₂²	—	488	4	no	—
283	C₂²	—	244	4	no	—
284	C₂²	D₈×#6923	1	1	yes	characteristic
285	C₄	—	61	4	no	—
286	C₄	—	61	1	no	—
287	C₄	—	61	1	no	—
288	C₄	—	122	2	no	—
289	C₄	—	488	4	no	—
290	C₄	D₈×#6923	1	1	yes	characteristic
291	C₄	C₂×D₄×#6923	1	1	yes	characteristic
292	C₂	—	8	4	no	—
293	C₂	—	61	1	no	—
294	C₂	—	61	1	no	—
295	C₂	—	61	2	no	—
296	C₂	—	488	4	no	—
297	C₂	D₁₆×#6923	1	2	yes	direct product factor
298	C₂	C₂×D₈×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×#6923
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₈	C₂³×#6923
Fitting	C₂×D₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₄×D₈
prime radical	C₂×D₁₆×D₆₁	C₂
cosocle	C₈×D₆₁	C₂⁴
hypercenter	C₂×D₁₆	#6923
hypocenter	C₆₁	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×#6923	C₂	semidirect
2. D₁₆×#6923	C₂	direct (N × H)
3. C₂×#16840688	C₂	semidirect
4. C₂×#16840674	C₂	semidirect
5. C₂×C₁₆×#6923	C₂	semidirect
10. D₈×#6923	C₂²	semidirect
14. #16840688	C₂²	semidirect
15. #16840674	C₂²	semidirect
16. C₁₆×#6923	C₂²	semidirect
18. C₂×D₉₇₆	C₄	semidirect
19. C₂×#14519718	C₄	semidirect
20. D₁₆×D₆₁	C₄	semidirect
23. C₂×D₁₆×C₆₁	C₄	semidirect
40. C₂×D₄₈₈	C₂×C₄	semidirect
44. D₈×D₆₁	C₂×C₄	semidirect
47. D₉₇₆	C₂×C₄	semidirect
48. #14519718	C₂×C₄	semidirect
49. C₂×#11758185	C₂×C₄	semidirect
50. C₁₆×D₆₁	C₂×C₄	semidirect
52. C₂×D₈×C₆₁	C₂×C₄	semidirect
53. D₁₆×C₆₁	C₂×C₄	semidirect
54. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
78. D₈×C₆₁	C₂²×C₄	semidirect
84. D₄₈₈	C₂²×C₄	semidirect
90. C₁₆×C₆₁	C₂²×C₄	semidirect
91. #11758185	C₂²×C₄	semidirect
103. C₂×#6923	D₁₆	direct (N × H)
138. #6923	C₂×D₁₆	direct (N × H)
150. C₂×C₆₁	C₄×D₁₆	semidirect
153. D₆₁	C₄×D₁₆	semidirect
162. C₂×D₁₆	#6923	direct (N × H)
173. C₆₁	C₂×C₄×D₁₆	semidirect
182. C₂×D₈	C₂×#6923	semidirect
192. D₁₆	C₂×#6923	direct (N × H)
199. C₂×C₁₆	C₂×#6923	semidirect
224. D₈	C₂²×#6923	semidirect
241. C₁₆	C₂²×#6923	semidirect
297. C₂	D₁₆×#6923	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×D₈×#6923	2
D₁₆×#6923	2
C₂×#16840688	2
C₂×#16840674	2
C₂×C₁₆×#6923	2
C₂×D₁₆×D₆₁	2
C₂×C₄×D₁₆	61

Subgroups — C2×D16×#6923 #30624281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₆×#6923 #30624281