Subgroups — C₂×D₁₆×#64408 #34030252

Order 18688.

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×#64408	C₂	1	2	yes	maximal, complemented
2	D₁₆×#64408	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989513	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17989499	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×#64408	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×D₄×#64408	—	2	2	no	—
8	C₂×#14556664	C₂²	1	1	yes	characteristic
9	D₈×#64408	—	2	4	no	—
10	D₈×#64408	C₂²	1	8	yes	complemented
11	C₂×#14556660	C₂²	1	2	yes	—
12	C₂×#14556653	—	2	2	no	—
13	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
14	#17989513	C₂²	1	4	yes	complemented
15	#17989499	C₂²	1	8	yes	complemented
16	C₁₆×#64408	C₂²	1	4	yes	complemented
17	C₂×D₈×D₇₃	C₂²	1	2	yes	—
18	C₂×D₁₁₆₈	C₄	1	1	yes	characteristic, complemented
19	C₂×#14556532	C₄	1	2	yes	complemented
20	D₁₆×D₇₃	C₄	1	4	yes	complemented
21	D₁₆×D₇₃	C₂²	1	4	yes	—
22	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
23	C₂×D₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
24	C₂³×#64408	—	4	2	no	—
25	C₂×#11924036	D₄	1	1	yes	characteristic
26	D₄×#64408	—	2	8	no	—
27	D₄×#64408	—	4	4	no	—
28	C₂×C₄⋊#64408	—	2	2	no	—
29	C₂×C₂²⋊#64408	—	4	2	no	—
30	C₂×C₄×#64408	D₄	1	1	yes	characteristic
31	#14556664	—	2	1	no	—
32	#14556664	C₂³	1	2	yes	—
33	C₂×#11924031	—	2	1	no	—
34	#14556660	—	2	2	no	—
35	#14556660	C₂³	1	4	yes	—
36	#14556653	—	2	8	no	—
37	C₈×#64408	D₄	1	2	yes	—
38	C₈×#64408	C₂³	1	2	yes	—
39	C₂×D₄×D₇₃	—	2	2	no	—
40	C₂×D₅₈₄	C₂×C₄	1	2	yes	complemented
41	C₂×#11924013	—	2	2	no	—
42	D₈×D₇₃	—	2	2	no	—
43	D₈×D₇₃	—	2	2	no	—
44	D₈×D₇₃	C₂×C₄	1	4	yes	complemented
45	D₈×D₇₃	C₂³	1	4	yes	—
46	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
47	D₁₁₆₈	C₂×C₄	1	4	yes	complemented
48	#14556532	C₂×C₄	1	8	yes	complemented
49	C₂×#11924003	C₂×C₄	1	1	yes	characteristic, complemented
50	C₁₆×D₇₃	C₂×C₄	1	2	yes	complemented
51	C₁₆×D₇₃	C₂³	1	2	yes	—
52	C₂×D₈×C₇₃	C₂×C₄	1	2	yes	complemented
53	D₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
54	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂²×#64408	—	4	2	no	—
56	C₂²×#64408	—	4	8	no	—
57	C₂²×#64408	—	8	8	no	—
58	C₂²×#64408	D₈	1	1	yes	characteristic
59	C₂²×#64408	D₈	1	1	yes	characteristic
60	#11924036	—	2	1	no	—
61	#11924036	C₂×D₄	1	2	yes	—
62	C₂×#10596833	—	2	1	no	—
63	C₄⋊#64408	—	2	4	no	—
64	C₄⋊#64408	—	4	2	no	—
65	C₂²⋊#64408	—	4	8	no	—
66	C₄×#64408	D₈	1	2	yes	—
67	C₄×#64408	C₂×D₄	1	2	yes	—
68	#11924031	—	2	2	no	—
69	C₂³×D₇₃	—	4	2	no	—
70	C₂×D₂₉₂	—	2	2	no	—
71	C₂×C₂²⋊D₇₃	—	4	2	no	—
72	D₄×D₇₃	—	2	4	no	—
73	D₄×D₇₃	—	2	4	no	—
74	D₄×D₇₃	—	4	2	no	—
75	D₄×D₇₃	—	4	2	no	—
76	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
77	D₅₈₄	—	2	2	no	—
78	D₅₈₄	C₂²×C₄	1	4	yes	complemented
79	#11924013	—	2	4	no	—
80	#11924013	—	2	4	no	—
81	C₂×#10596831	#41	1	1	yes	characteristic
82	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
84	C₈×D₇₃	#41	1	1	yes	characteristic
85	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
86	#11924003	C₂²×C₄	1	2	yes	complemented
87	C₂×D₄×C₇₃	—	2	2	no	—
88	D₈×C₇₃	—	2	2	no	—
89	D₈×C₇₃	C₂²×C₄	1	4	yes	complemented
90	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₁₆×C₇₃	C₂²×C₄	1	2	yes	complemented
92	C₂³×C₇₃	—	4	2	no	—
93	C₂²×D₇₃	—	4	2	no	—
94	C₂²×D₇₃	—	4	4	no	—
95	C₂²×D₇₃	—	4	4	no	—
96	C₂²×D₇₃	—	8	4	no	—
97	C₂²×D₇₃	—	8	4	no	—
98	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
99	C₂×#64408	—	2	2	no	—
100	C₂×#64408	—	4	4	no	—
101	C₂×#64408	—	8	4	no	—
102	C₂×#64408	—	8	16	no	—
103	C₂×#64408	D₁₆	1	4	yes	direct product factor
104	C₂×#64408	C₂×D₈	1	2	yes	—
105	C₂×#64408	C₂×D₈	1	2	yes	—
106	D₄×C₇₃	—	2	4	no	—
107	D₄×C₇₃	—	4	2	no	—
108	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
109	C₂²⋊D₇₃	—	4	4	no	—
110	C₂²⋊D₇₃	—	4	4	no	—
111	C₂×Dic₇₃	#135	1	1	yes	characteristic
112	D₂₉₂	—	2	4	no	—
113	D₂₉₂	—	4	2	no	—
114	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
115	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
116	C₄×D₇₃	#135	1	1	yes	characteristic
117	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
118	#10596833	—	2	2	no	—
119	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
120	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
121	#10596831	C₄×D₄	1	1	yes	characteristic
122	#10596831	C₂×#41	1	1	yes	characteristic
123	C₂²×C₇₃	—	4	4	no	—
124	C₂²×C₇₃	—	8	4	no	—
125	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
126	C₂×D₇₃	—	4	4	no	—
127	C₂×D₇₃	—	8	4	no	—
128	C₂×D₇₃	—	8	4	no	—
129	C₂×D₇₃	—	8	4	no	—
130	C₂×D₇₃	—	8	8	no	—
131	C₂×D₇₃	C₄×D₈	1	1	yes	characteristic
132	C₂×D₇₃	C₂×D₁₆	1	2	yes	—
133	C₂×D₇₃	#508	1	2	yes	—
134	C₂×D₇₃	C₂²×D₈	1	1	yes	characteristic
135	#64408	—	2	2	no	—
136	#64408	—	8	4	no	—
137	#64408	C₂×D₁₆	1	4	yes	direct product factor
138	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
139	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
140	Dic₇₃	C₄×D₈	1	1	yes	characteristic
141	Dic₇₃	C₂×#135	1	1	yes	characteristic
142	C₂×C₄×D₁₆	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	—	8	4	no	—
144	C₂×C₇₃	C₄×D₁₆	1	2	yes	complemented
145	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
146	D₇₃	—	8	4	no	—
147	D₇₃	C₄×D₁₆	1	2	yes	complemented
148	D₇₃	C₂²×D₁₆	1	1	yes	characteristic
149	D₇₃	C₂×#508	1	1	yes	characteristic
150	C₂²×D₁₆	—	73	1	no	—
151	C₂×C₄×D₈	—	73	2	no	—
152	C₄×D₁₆	—	73	8	no	—
153	C₂×#366	—	73	1	no	—
154	C₂×#357	—	73	2	no	—
155	C₂×C₄×C₁₆	—	73	1	no	—
156	C₇₃	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
157	C₂²×D₈	—	73	2	no	—
158	C₂×C₄×D₄	—	146	2	no	—
159	C₂×D₁₆	—	73	2	no	—
160	C₂×D₁₆	—	73	4	no	—
161	C₂×D₁₆	—	73	4	no	—
162	C₂×D₁₆	—	73	1	no	—
163	C₂×D₁₆	#64408	1	1	yes	characteristic, direct product factor
164	C₂²×C₁₆	—	73	1	no	—
165	C₂²×C₁₆	—	73	1	no	—
166	C₄×D₈	—	73	8	no	—
167	C₄×D₈	—	146	4	no	—
168	C₂×#107	—	73	2	no	—
169	C₂×#102	—	146	2	no	—
170	C₂×C₄×C₈	—	73	1	no	—
171	#366	—	73	4	no	—
172	#357	—	73	8	no	—
173	C₄×C₁₆	—	73	4	no	—
174	C₂²×D₄	—	146	2	no	—
175	C₂³×C₄	—	292	2	no	—
176	C₂×D₈	—	73	4	no	—
177	C₂×D₈	—	73	4	no	—
178	C₂×D₈	—	73	2	no	—
179	C₂×D₈	—	146	2	no	—
180	C₂×D₈	—	146	2	no	—
181	C₂×D₈	—	146	2	no	—
182	C₂×D₈	C₂×#64408	1	2	yes	complemented
183	C₂²×C₈	—	73	1	no	—
184	C₂²×C₈	—	73	1	no	—
185	C₄×D₄	—	292	4	no	—
186	C₄×D₄	—	146	8	no	—
187	C₂×C₄⋊C₄	—	146	2	no	—
188	C₂×C₂²⋊C₄	—	292	2	no	—
189	C₂×C₄²	—	73	1	no	—
190	D₁₆	—	73	8	no	—
191	D₁₆	—	73	4	no	—
192	D₁₆	C₂×#64408	1	4	yes	direct product factor
193	C₂×C₁₆	—	73	1	no	—
194	C₂×C₁₆	—	73	2	no	—
195	C₂×C₁₆	—	73	2	no	—
196	C₂×C₁₆	—	73	2	no	—
197	C₂×C₁₆	—	146	1	no	—
198	C₂×C₁₆	—	146	1	no	—
199	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic, complemented
200	#107	—	73	4	no	—
201	#107	—	146	2	no	—
202	#102	—	146	8	no	—
203	C₄×C₈	—	73	2	no	—
204	C₄×C₈	—	73	2	no	—
205	C₂⁴	—	292	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	292	2	no	—
208	C₂×D₄	—	292	2	no	—
209	C₂×D₄	—	292	2	no	—
210	C₂×D₄	—	146	4	no	—
211	C₂×D₄	—	146	4	no	—
212	C₂×D₄	—	146	2	no	—
213	C₂²×C₄	—	292	2	no	—
214	C₂²×C₄	—	292	8	no	—
215	C₂²×C₄	—	584	8	no	—
216	C₂²×C₄	—	73	1	no	—
217	C₂²×C₄	—	73	1	no	—
218	C₂²×C₄	—	73	1	no	—
219	D₈	—	2	2	no	—
220	D₈	—	73	4	no	—
221	D₈	—	146	4	no	—
222	D₈	—	146	4	no	—
223	D₈	—	146	2	no	—
224	D₈	C₂²×#64408	1	4	yes	complemented
225	C₂×C₈	—	73	1	no	—
226	C₂×C₈	—	73	1	no	—
227	C₂×C₈	—	73	1	no	—
228	C₂×C₈	—	73	1	no	—
229	C₂×C₈	—	73	1	no	—
230	C₂×C₈	—	73	2	no	—
231	C₂×C₈	—	146	1	no	—
232	C₂×C₈	—	146	1	no	—
233	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
234	C₄⋊C₄	—	292	2	no	—
235	C₄⋊C₄	—	146	4	no	—
236	C₂²⋊C₄	—	292	8	no	—
237	C₄²	—	73	2	no	—
238	C₄²	—	73	2	no	—
239	C₁₆	—	73	2	no	—
240	C₁₆	—	146	2	no	—
241	C₁₆	C₂²×#64408	1	2	yes	complemented
242	C₂³	—	4	2	no	—
243	C₂³	—	292	4	no	—
244	C₂³	—	292	4	no	—
245	C₂³	—	292	2	no	—
246	C₂³	—	584	4	no	—
247	C₂³	—	584	4	no	—
248	C₂³	—	73	1	no	—
249	D₄	—	2	4	no	—
250	D₄	—	4	2	no	—
251	D₄	—	292	4	no	—
252	D₄	—	292	4	no	—
253	D₄	—	292	2	no	—
254	D₄	—	146	4	no	—
255	C₂×C₄	—	292	4	no	—
256	C₂×C₄	—	584	4	no	—
257	C₂×C₄	—	584	16	no	—
258	C₂×C₄	—	73	2	no	—
259	C₂×C₄	—	73	4	no	—
260	C₂×C₄	—	73	1	no	—
261	C₂×C₄	—	73	1	no	—
262	C₂×C₄	—	73	1	no	—
263	C₂×C₄	—	73	1	no	—
264	C₂×C₄	—	73	1	no	—
265	C₂×C₄	—	73	2	no	—
266	C₂×C₄	—	146	2	no	—
267	C₂×C₄	D₄×#64408	1	1	yes	characteristic
268	C₈	—	73	1	no	—
269	C₈	—	73	1	no	—
270	C₈	—	146	2	no	—
271	C₈	D₄×#64408	1	1	yes	characteristic
272	C₈	C₂³×#64408	1	1	yes	characteristic
273	C₂²	—	4	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	292	4	no	—
276	C₂²	—	584	4	no	—
277	C₂²	—	584	4	no	—
278	C₂²	—	584	8	no	—
279	C₂²	—	584	4	no	—
280	C₂²	—	73	2	no	—
281	C₂²	—	73	2	no	—
282	C₂²	—	73	1	no	—
283	C₂²	—	73	1	no	—
284	C₂²	D₈×#64408	1	1	yes	characteristic
285	C₄	—	584	4	no	—
286	C₄	—	73	4	no	—
287	C₄	—	73	1	no	—
288	C₄	—	73	1	no	—
289	C₄	—	146	2	no	—
290	C₄	D₈×#64408	1	1	yes	characteristic
291	C₄	C₂×D₄×#64408	1	1	yes	characteristic
292	C₂	—	8	4	no	—
293	C₂	—	584	4	no	—
294	C₂	—	73	1	no	—
295	C₂	—	73	1	no	—
296	C₂	—	73	2	no	—
297	C₂	D₁₆×#64408	1	2	yes	direct product factor
298	C₂	C₂×D₈×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×#64408
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₈	C₂³×#64408
Fitting	C₂×D₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₄×D₈
prime radical	C₂×D₁₆×D₇₃	C₂
cosocle	C₈×D₇₃	C₂⁴
hypercenter	C₂×D₁₆	#64408
hypocenter	C₇₃	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
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×#64408	C₂	semidirect
2. D₁₆×#64408	C₂	direct (N × H)
3. C₂×#17989513	C₂	semidirect
4. C₂×#17989499	C₂	semidirect
5. C₂×C₁₆×#64408	C₂	semidirect
10. D₈×#64408	C₂²	semidirect
14. #17989513	C₂²	semidirect
15. #17989499	C₂²	semidirect
16. C₁₆×#64408	C₂²	semidirect
18. C₂×D₁₁₆₈	C₄	semidirect
19. C₂×#14556532	C₄	semidirect
20. D₁₆×D₇₃	C₄	semidirect
23. C₂×D₁₆×C₇₃	C₄	semidirect
40. C₂×D₅₈₄	C₂×C₄	semidirect
44. D₈×D₇₃	C₂×C₄	semidirect
47. D₁₁₆₈	C₂×C₄	semidirect
48. #14556532	C₂×C₄	semidirect
49. C₂×#11924003	C₂×C₄	semidirect
50. C₁₆×D₇₃	C₂×C₄	semidirect
52. C₂×D₈×C₇₃	C₂×C₄	semidirect
53. D₁₆×C₇₃	C₂×C₄	semidirect
54. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
78. D₅₈₄	C₂²×C₄	semidirect
86. #11924003	C₂²×C₄	semidirect
89. D₈×C₇₃	C₂²×C₄	semidirect
91. C₁₆×C₇₃	C₂²×C₄	semidirect
103. C₂×#64408	D₁₆	direct (N × H)
137. #64408	C₂×D₁₆	direct (N × H)
144. C₂×C₇₃	C₄×D₁₆	semidirect
147. D₇₃	C₄×D₁₆	semidirect
156. C₇₃	C₂×C₄×D₁₆	semidirect
163. C₂×D₁₆	#64408	direct (N × H)
182. C₂×D₈	C₂×#64408	semidirect
192. D₁₆	C₂×#64408	direct (N × H)
199. C₂×C₁₆	C₂×#64408	semidirect
224. D₈	C₂²×#64408	semidirect
241. C₁₆	C₂²×#64408	semidirect
297. C₂	D₁₆×#64408	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×#64408	2
D₁₆×#64408	2
C₂×#17989513	2
C₂×#17989499	2
C₂×C₁₆×#64408	2
C₂×D₁₆×D₇₃	2
C₂×C₄×D₁₆	73

Subgroups — C2×D16×#64408 #34030252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₆×#64408 #34030252