Subgroups — C₂³×D₄×#64408 #34032693

Order 18688.

19998 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×#64408	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄×#64408	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊#64408	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₂²⋊#64408	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×#64408	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×#64408	—	2	32	no	—
8	C₂⁴×#64408	C₂²	1	1	yes	characteristic
9	C₂⁴×#64408	C₂²	1	2	yes	—
10	C₂⁴×#64408	C₂²	1	1	yes	characteristic
11	C₂⁴×#64408	C₂²	1	56	yes	complemented
12	C₂×D₄×#64408	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊#64408	C₂²	1	28	yes	complemented
14	C₂²×C₂²⋊#64408	C₂²	1	56	yes	complemented
15	C₂²×C₄×#64408	C₂²	1	28	yes	complemented
16	C₂⁵×D₇₃	C₂²	1	2	yes	—
17	C₂³×D₂₉₂	C₄	1	1	yes	characteristic, complemented
18	C₂³×C₂²⋊D₇₃	C₄	1	2	yes	complemented
19	C₂²×D₄×D₇₃	C₄	1	28	yes	complemented
20	C₂²×D₄×D₇₃	C₂²	1	28	yes	—
21	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
22	C₂³×D₄×C₇₃	C₄	1	1	yes	characteristic, complemented
23	C₂³×#64408	—	2	448	no	—
24	C₂³×#64408	—	2	16	no	—
25	C₂³×#64408	—	2	8	no	—
26	C₂³×#64408	D₄	1	16	yes	direct product factor
27	C₂³×#64408	C₂³	1	14	yes	—
28	C₂³×#64408	C₂³	1	28	yes	—
29	C₂³×#64408	C₂³	1	14	yes	—
30	C₂³×#64408	C₂³	1	224	yes	complemented
31	D₄×#64408	C₂³	1	512	yes	direct product factor
32	C₂×C₄⋊#64408	C₂³	1	112	yes	complemented
33	C₂×C₂²⋊#64408	C₂³	1	224	yes	complemented
34	C₂×C₄×#64408	C₂³	1	112	yes	complemented
35	C₂⁴×D₇₃	—	2	16	no	—
36	C₂⁴×D₇₃	—	2	16	no	—
37	C₂⁴×D₇₃	C₂×C₄	1	2	yes	complemented
38	C₂⁴×D₇₃	C₂×C₄	1	28	yes	complemented
39	C₂⁴×D₇₃	C₂³	1	1	yes	characteristic
40	C₂⁴×D₇₃	C₂³	1	28	yes	—
41	C₂²×D₂₉₂	C₂×C₄	1	28	yes	complemented
42	C₂²×C₂²⋊D₇₃	C₂×C₄	1	56	yes	complemented
43	C₂×D₄×D₇₃	C₂×C₄	1	336	yes	complemented
44	C₂×D₄×D₇₃	C₂³	1	112	yes	—
45	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂²×C₄×D₇₃	C₂×C₄	1	14	yes	complemented
47	C₂²×C₄×D₇₃	C₂³	1	14	yes	—
48	C₂⁵×C₇₃	C₂×C₄	1	2	yes	complemented
49	C₂²×D₄×C₇₃	C₂×C₄	1	28	yes	complemented
50	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂²×#64408	—	2	112	no	—
52	C₂²×#64408	—	2	56	no	—
53	C₂²×#64408	—	2	896	no	—
54	C₂²×#64408	C₂×D₄	1	112	yes	direct product factor
55	C₂²×#64408	C₂⁴	1	56	yes	—
56	C₂²×#64408	C₂⁴	1	28	yes	—
57	C₂²×#64408	C₂⁴	1	28	yes	—
58	C₂²×#64408	C₂⁴	1	128	yes	complemented
59	C₄⋊#64408	C₂⁴	1	64	yes	complemented
60	C₂²⋊#64408	C₂⁴	1	128	yes	complemented
61	C₄×#64408	C₂⁴	1	64	yes	complemented
62	C₂³×D₇₃	—	2	16	no	—
63	C₂³×D₇₃	—	2	224	no	—
64	C₂³×D₇₃	—	2	112	no	—
65	C₂³×D₇₃	—	2	112	no	—
66	C₂³×D₇₃	C₂²×C₄	1	7	yes	—
67	C₂³×D₇₃	C₂²×C₄	1	168	yes	complemented
68	C₂³×D₇₃	C₂²×C₄	1	28	yes	complemented
69	C₂³×D₇₃	C₂×D₄	1	8	yes	—
70	C₂³×D₇₃	#41	1	8	yes	—
71	C₂³×D₇₃	C₂⁴	1	56	yes	—
72	C₂³×D₇₃	C₂⁴	1	7	yes	—
73	C₂×D₂₉₂	C₂²×C₄	1	112	yes	complemented
74	C₂×C₂²⋊D₇₃	C₂²×C₄	1	224	yes	complemented
75	D₄×D₇₃	C₂²×C₄	1	448	yes	complemented
76	D₄×D₇₃	C₂⁴	1	64	yes	—
77	C₂²×Dic₇₃	C₂²×C₄	1	14	yes	complemented
78	C₂×C₄×D₇₃	C₂²×C₄	1	84	yes	complemented
79	C₂×C₄×D₇₃	C₂⁴	1	28	yes	—
80	C₂⁴×C₇₃	—	2	16	no	—
81	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₂⁴×C₇₃	C₂²×C₄	1	28	yes	complemented
83	C₂×D₄×C₇₃	C₂²×C₄	1	112	yes	complemented
84	C₂²×C₄×C₇₃	C₂²×C₄	1	14	yes	complemented
85	C₂³×C₇₃	—	2	112	no	—
86	C₂³×C₇₃	C₄×D₄	1	8	yes	complemented
87	C₂³×C₇₃	C₂³×C₄	1	7	yes	—
88	C₂³×C₇₃	C₂³×C₄	1	56	yes	complemented
89	C₂²×D₇₃	—	2	112	no	—
90	C₂²×D₇₃	—	2	672	no	—
91	C₂²×D₇₃	—	2	112	no	—
92	C₂²×D₇₃	—	2	112	no	—
93	C₂²×D₇₃	C₄×D₄	1	56	yes	complemented
94	C₂²×D₇₃	C₂³×C₄	1	21	yes	—
95	C₂²×D₇₃	C₂³×C₄	1	112	yes	complemented
96	C₂²×D₇₃	C₂³×C₄	1	56	yes	complemented
97	C₂²×D₇₃	C₂²×D₄	1	28	yes	—
98	C₂²×D₇₃	C₂×#41	1	28	yes	—
99	C₂²×D₇₃	C₂⁵	1	16	yes	—
100	C₂²×D₇₃	C₂⁵	1	7	yes	—
101	C₂×#64408	—	2	112	no	—
102	C₂×#64408	—	2	56	no	—
103	C₂×#64408	—	2	256	no	—
104	C₂×#64408	C₂²×D₄	1	112	yes	direct product factor
105	C₂×#64408	C₂⁵	1	8	yes	—
106	C₂×#64408	C₂⁵	1	16	yes	—
107	C₂×#64408	C₂⁵	1	8	yes	—
108	D₄×C₇₃	C₂³×C₄	1	64	yes	complemented
109	C₂×C₄×C₇₃	C₂³×C₄	1	28	yes	complemented
110	C₂²⋊D₇₃	C₂³×C₄	1	128	yes	complemented
111	C₂×Dic₇₃	C₂³×C₄	1	28	yes	complemented
112	D₂₉₂	C₂³×C₄	1	64	yes	complemented
113	C₄×D₇₃	C₂³×C₄	1	56	yes	complemented
114	C₄×D₇₃	C₂⁵	1	8	yes	—
115	C₂²×C₇₃	—	2	112	no	—
116	C₂²×C₇₃	C₂×C₄×D₄	1	28	yes	complemented
117	C₂²×C₇₃	C₂⁴×C₄	1	7	yes	—
118	C₂²×C₇₃	C₂⁴×C₄	1	16	yes	complemented
119	C₂×D₇₃	—	2	112	no	—
120	C₂×D₇₃	—	2	16	no	—
121	C₂×D₇₃	—	2	16	no	—
122	C₂×D₇₃	—	2	224	no	—
123	C₂×D₇₃	C₂×C₄×D₄	1	84	yes	complemented
124	C₂×D₇₃	C₂⁴×C₄	1	7	yes	—
125	C₂×D₇₃	C₂⁴×C₄	1	16	yes	complemented
126	C₂×D₇₃	C₂³×D₄	1	14	yes	—
127	C₂×D₇₃	C₂²×#41	1	14	yes	—
128	C₂×D₇₃	C₂⁶	1	1	yes	characteristic
129	#64408	—	2	16	no	—
130	#64408	—	2	8	no	—
131	#64408	C₂³×D₄	1	16	yes	direct product factor
132	C₄×C₇₃	C₂⁴×C₄	1	8	yes	complemented
133	Dic₇₃	C₂⁴×C₄	1	8	yes	complemented
134	C₂³×C₄×D₄	—	73	1	no	maximal, Sylow
135	C₂×C₇₃	—	2	16	no	—
136	C₂×C₇₃	C₂²×C₄×D₄	1	14	yes	complemented
137	C₂×C₇₃	C₂⁵×C₄	1	1	yes	characteristic
138	D₇₃	—	2	16	no	—
139	D₇₃	C₂²×C₄×D₄	1	14	yes	complemented
140	D₇₃	C₂⁴×D₄	1	1	yes	characteristic
141	D₇₃	C₂³×#41	1	1	yes	characteristic
142	C₂⁴×D₄	—	73	1	no	—
143	C₂⁵×C₄	—	73	2	no	—
144	C₂²×C₄×D₄	—	73	56	no	—
145	C₂³×C₄⋊C₄	—	73	1	no	—
146	C₂³×C₂²⋊C₄	—	73	2	no	—
147	C₂³×C₄²	—	73	1	no	—
148	C₇₃	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
149	C₂⁶	—	73	2	no	—
150	C₂³×D₄	—	73	2	no	—
151	C₂³×D₄	—	73	28	no	—
152	C₂³×D₄	—	73	28	no	—
153	C₂³×D₄	—	73	1	no	—
154	C₂³×D₄	#64408	1	1	yes	characteristic, direct product factor
155	C₂⁴×C₄	—	73	1	no	—
156	C₂⁴×C₄	—	73	1	no	—
157	C₂⁴×C₄	—	73	2	no	—
158	C₂⁴×C₄	—	73	56	no	—
159	C₂⁴×C₄	—	73	1	no	—
160	C₂⁴×C₄	—	146	32	no	—
161	C₂×C₄×D₄	—	73	448	no	—
162	C₂²×C₄⋊C₄	—	73	28	no	—
163	C₂²×C₂²⋊C₄	—	73	56	no	—
164	C₂²×C₄²	—	73	28	no	—
165	C₂⁵	—	73	1	no	—
166	C₂⁵	—	73	2	no	—
167	C₂⁵	—	73	28	no	—
168	C₂⁵	—	73	28	no	—
169	C₂⁵	—	146	16	no	—
170	C₂⁵	—	146	16	no	—
171	C₂⁵	C₂×#64408	1	2	yes	complemented
172	C₂²×D₄	—	73	56	no	—
173	C₂²×D₄	—	73	112	no	—
174	C₂²×D₄	—	73	336	no	—
175	C₂²×D₄	—	73	28	no	—
176	C₂²×D₄	C₂×#64408	1	28	yes	direct product factor
177	C₂³×C₄	—	73	16	no	—
178	C₂³×C₄	—	73	14	no	—
179	C₂³×C₄	—	73	1	no	—
180	C₂³×C₄	—	73	14	no	—
181	C₂³×C₄	—	73	14	no	—
182	C₂³×C₄	—	73	14	no	—
183	C₂³×C₄	—	73	28	no	—
184	C₂³×C₄	—	73	224	no	—
185	C₂³×C₄	—	146	448	no	—
186	C₂³×C₄	—	146	8	no	—
187	C₂³×C₄	—	146	16	no	—
188	C₂³×C₄	C₂×#64408	1	1	yes	characteristic, complemented
189	C₄×D₄	—	73	512	no	—
190	C₂×C₄⋊C₄	—	73	112	no	—
191	C₂×C₂²⋊C₄	—	73	224	no	—
192	C₂×C₄²	—	73	112	no	—
193	C₂⁴	—	2	16	no	—
194	C₂⁴	—	73	7	no	—
195	C₂⁴	—	73	7	no	—
196	C₂⁴	—	73	8	no	—
197	C₂⁴	—	73	8	no	—
198	C₂⁴	—	73	28	no	—
199	C₂⁴	—	73	168	no	—
200	C₂⁴	—	73	56	no	—
201	C₂⁴	—	146	112	no	—
202	C₂⁴	—	146	112	no	—
203	C₂⁴	—	146	224	no	—
204	C₂⁴	—	146	16	no	—
205	C₂⁴	C₂²×#64408	1	1	yes	characteristic
206	C₂⁴	C₂²×#64408	1	28	yes	complemented
207	C₂×D₄	—	73	224	no	—
208	C₂×D₄	—	73	448	no	—
209	C₂×D₄	—	73	64	no	—
210	C₂×D₄	—	73	112	no	—
211	C₂×D₄	C₂²×#64408	1	112	yes	direct product factor
212	C₂²×C₄	—	73	112	no	—
213	C₂²×C₄	—	73	28	no	—
214	C₂²×C₄	—	73	14	no	—
215	C₂²×C₄	—	73	84	no	—
216	C₂²×C₄	—	73	28	no	—
217	C₂²×C₄	—	73	28	no	—
218	C₂²×C₄	—	73	56	no	—
219	C₂²×C₄	—	73	128	no	—
220	C₂²×C₄	—	146	896	no	—
221	C₂²×C₄	—	146	56	no	—
222	C₂²×C₄	—	146	112	no	—
223	C₂²×C₄	C₂²×#64408	1	14	yes	complemented
224	C₄⋊C₄	—	73	64	no	—
225	C₂²⋊C₄	—	73	128	no	—
226	C₄²	—	73	64	no	—
227	C₂³	—	2	112	no	—
228	C₂³	—	73	28	no	—
229	C₂³	—	73	28	no	—
230	C₂³	—	73	56	no	—
231	C₂³	—	73	7	no	—
232	C₂³	—	73	21	no	—
233	C₂³	—	73	56	no	—
234	C₂³	—	73	16	no	—
235	C₂³	—	73	112	no	—
236	C₂³	—	146	112	no	—
237	C₂³	—	146	112	no	—
238	C₂³	—	146	672	no	—
239	C₂³	—	146	112	no	—
240	C₂³	D₄×#64408	1	8	yes	direct product factor
241	C₂³	C₂³×#64408	1	7	yes	—
242	C₂³	C₂³×#64408	1	56	yes	complemented
243	D₄	—	73	128	no	—
244	D₄	—	73	64	no	—
245	D₄	C₂³×#64408	1	64	yes	direct product factor
246	C₂×C₄	—	73	8	no	—
247	C₂×C₄	—	73	112	no	—
248	C₂×C₄	—	73	28	no	—
249	C₂×C₄	—	73	8	no	—
250	C₂×C₄	—	73	56	no	—
251	C₂×C₄	—	73	8	no	—
252	C₂×C₄	—	73	16	no	—
253	C₂×C₄	—	146	256	no	—
254	C₂×C₄	—	146	112	no	—
255	C₂×C₄	—	146	56	no	—
256	C₂×C₄	C₂³×#64408	1	28	yes	complemented
257	C₂²	—	2	112	no	—
258	C₂²	—	73	14	no	—
259	C₂²	—	73	14	no	—
260	C₂²	—	73	84	no	—
261	C₂²	—	73	1	no	—
262	C₂²	—	73	7	no	—
263	C₂²	—	73	16	no	—
264	C₂²	—	146	16	no	—
265	C₂²	—	146	16	no	—
266	C₂²	—	146	224	no	—
267	C₂²	—	146	112	no	—
268	C₂²	C₂×D₄×#64408	1	28	yes	direct product factor
269	C₂²	C₂⁴×#64408	1	7	yes	—
270	C₂²	C₂⁴×#64408	1	16	yes	complemented
271	C₄	—	73	16	no	—
272	C₄	—	73	8	no	—
273	C₄	—	146	16	no	—
274	C₄	—	146	8	no	—
275	C₄	C₂⁴×#64408	1	8	yes	complemented
276	C₂	—	2	16	no	—
277	C₂	—	73	1	no	—
278	C₂	—	73	1	no	—
279	C₂	—	73	14	no	—
280	C₂	—	146	16	no	—
281	C₂	C₂²×D₄×#64408	1	14	yes	direct product factor
282	C₂	C₂⁵×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×#64408
commutator	C₂×C₇₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#64408
Fitting	C₂³×D₄×C₇₃	C₄
socle	C₂⁴×C₇₃	C₂²×C₄
prime radical	C₂³×D₄×D₇₃	C₂
cosocle	C₂×D₇₃	C₂⁶
hypercenter	C₂³×D₄	#64408
hypocenter	C₇₃	C₂³×C₄×D₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×#64408	C₂	semidirect
2. C₂²×D₄×#64408	C₂	direct (N × H)
3. C₂³×C₄⋊#64408	C₂	semidirect
4. C₂³×C₂²⋊#64408	C₂	semidirect
5. C₂³×C₄×#64408	C₂	semidirect
11. C₂⁴×#64408	C₂²	semidirect
12. C₂×D₄×#64408	C₂²	direct (N × H)
13. C₂²×C₄⋊#64408	C₂²	semidirect
14. C₂²×C₂²⋊#64408	C₂²	semidirect
15. C₂²×C₄×#64408	C₂²	semidirect
17. C₂³×D₂₉₂	C₄	semidirect
18. C₂³×C₂²⋊D₇₃	C₄	semidirect
19. C₂²×D₄×D₇₃	C₄	semidirect
22. C₂³×D₄×C₇₃	C₄	semidirect
26. C₂³×#64408	D₄	direct (N × H)
30. C₂³×#64408	C₂³	semidirect
31. D₄×#64408	C₂³	direct (N × H)
32. C₂×C₄⋊#64408	C₂³	semidirect
33. C₂×C₂²⋊#64408	C₂³	semidirect
34. C₂×C₄×#64408	C₂³	semidirect
37. C₂⁴×D₇₃	C₂×C₄	semidirect
38. C₂⁴×D₇₃	C₂×C₄	semidirect
41. C₂²×D₂₉₂	C₂×C₄	semidirect
42. C₂²×C₂²⋊D₇₃	C₂×C₄	semidirect
43. C₂×D₄×D₇₃	C₂×C₄	semidirect
45. C₂³×Dic₇₃	C₂×C₄	semidirect
46. C₂²×C₄×D₇₃	C₂×C₄	semidirect
48. C₂⁵×C₇₃	C₂×C₄	semidirect
49. C₂²×D₄×C₇₃	C₂×C₄	semidirect
50. C₂³×C₄×C₇₃	C₂×C₄	semidirect
54. C₂²×#64408	C₂×D₄	direct (N × H)
58. C₂²×#64408	C₂⁴	semidirect
59. C₄⋊#64408	C₂⁴	semidirect
60. C₂²⋊#64408	C₂⁴	semidirect
61. C₄×#64408	C₂⁴	semidirect
67. C₂³×D₇₃	C₂²×C₄	semidirect
68. C₂³×D₇₃	C₂²×C₄	semidirect
73. C₂×D₂₉₂	C₂²×C₄	semidirect
74. C₂×C₂²⋊D₇₃	C₂²×C₄	semidirect
75. D₄×D₇₃	C₂²×C₄	semidirect
77. C₂²×Dic₇₃	C₂²×C₄	semidirect
78. C₂×C₄×D₇₃	C₂²×C₄	semidirect
82. C₂⁴×C₇₃	C₂²×C₄	semidirect
83. C₂×D₄×C₇₃	C₂²×C₄	semidirect
84. C₂²×C₄×C₇₃	C₂²×C₄	semidirect
86. C₂³×C₇₃	C₄×D₄	semidirect
88. C₂³×C₇₃	C₂³×C₄	semidirect
93. C₂²×D₇₃	C₄×D₄	semidirect
95. C₂²×D₇₃	C₂³×C₄	semidirect
96. C₂²×D₇₃	C₂³×C₄	semidirect
104. C₂×#64408	C₂²×D₄	direct (N × H)
108. D₄×C₇₃	C₂³×C₄	semidirect
109. C₂×C₄×C₇₃	C₂³×C₄	semidirect
110. C₂²⋊D₇₃	C₂³×C₄	semidirect
111. C₂×Dic₇₃	C₂³×C₄	semidirect
112. D₂₉₂	C₂³×C₄	semidirect
113. C₄×D₇₃	C₂³×C₄	semidirect
116. C₂²×C₇₃	C₂×C₄×D₄	semidirect
118. C₂²×C₇₃	C₂⁴×C₄	semidirect
123. C₂×D₇₃	C₂×C₄×D₄	semidirect
125. C₂×D₇₃	C₂⁴×C₄	semidirect
131. #64408	C₂³×D₄	direct (N × H)
132. C₄×C₇₃	C₂⁴×C₄	semidirect
133. Dic₇₃	C₂⁴×C₄	semidirect
136. C₂×C₇₃	C₂²×C₄×D₄	semidirect
139. D₇₃	C₂²×C₄×D₄	semidirect
148. C₇₃	C₂³×C₄×D₄	semidirect
154. C₂³×D₄	#64408	direct (N × H)
171. C₂⁵	C₂×#64408	semidirect
176. C₂²×D₄	C₂×#64408	direct (N × H)
188. C₂³×C₄	C₂×#64408	semidirect
206. C₂⁴	C₂²×#64408	semidirect
211. C₂×D₄	C₂²×#64408	direct (N × H)
223. C₂²×C₄	C₂²×#64408	semidirect
240. C₂³	D₄×#64408	direct (N × H)
242. C₂³	C₂³×#64408	semidirect
245. D₄	C₂³×#64408	direct (N × H)
256. C₂×C₄	C₂³×#64408	semidirect
268. C₂²	C₂×D₄×#64408	direct (N × H)
270. C₂²	C₂⁴×#64408	semidirect
275. C₄	C₂⁴×#64408	semidirect
281. C₂	C₂²×D₄×#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
C₂⁵×#64408	2
C₂²×D₄×#64408	2
C₂³×C₄⋊#64408	2
C₂³×C₂²⋊#64408	2
C₂³×C₄×#64408	2
C₂³×D₄×D₇₃	2
C₂³×C₄×D₄	73

Subgroups — C23×D4×#64408 #34032693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₄×#64408 #34032693