Subgroups — C₂×D₄×D₇₆ #14577075

Order 2432.

2536 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₇₆	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄×D₁₉	C₂	1	2	yes	maximal, complemented
3	D₄×D₇₆	C₂	1	16	yes	maximal, direct product factor
4	C₂×#10597320	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10597341	C₂	1	2	yes	maximal, complemented
6	C₂×#10597376	C₂	1	2	yes	maximal, complemented
7	C₂×#10597328	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₉	C₂²	1	4	yes	complemented
11	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₉	C₂²	1	2	yes	complemented
13	C₂²×C₂²⋊D₁₉	C₂²	1	4	yes	complemented
14	C₂×D₄×D₁₉	—	2	16	no	—
15	C₂×D₄×D₁₉	C₂²	1	16	yes	complemented
16	C₂²×D₇₆	—	2	16	no	—
17	C₂²×D₇₆	C₂²	1	1	yes	characteristic
18	C₂²×D₇₆	C₂²	1	16	yes	complemented
19	C₂²×D₇₆	C₂²	1	2	yes	complemented
20	C₂²×D₇₆	C₂²	1	2	yes	complemented
21	C₂²×D₇₆	C₂²	1	4	yes	complemented
22	C₂²×D₇₆	C₂²	1	2	yes	complemented
23	C₂²×C₄×D₁₉	C₂²	1	2	yes	complemented
24	C₄×D₄×C₁₉	C₂²	1	8	yes	complemented
25	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	complemented
27	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic, complemented
28	#10597376	C₂²	1	16	yes	complemented
29	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	complemented
30	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	complemented
31	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic, complemented
32	#10597341	C₂²	1	16	yes	complemented
33	#10597328	C₂²	1	32	yes	complemented
34	#10597320	C₂²	1	8	yes	complemented
35	C₄×D₇₆	C₂²	1	8	yes	complemented
36	C₂⁴×C₁₉	C₂³	1	2	yes	—
37	C₂³×D₁₉	—	2	16	no	—
38	C₂³×D₁₉	—	2	16	no	—
39	C₂³×D₁₉	—	4	8	no	—
40	C₂³×D₁₉	C₂³	1	4	yes	—
41	C₂³×D₁₉	C₂³	1	2	yes	—
42	C₂³×D₁₉	C₂³	1	16	yes	complemented
43	C₂×D₄×C₁₉	D₄	1	8	yes	complemented
44	C₂×D₄×C₁₉	C₂³	1	4	yes	—
45	C₂²×C₄×C₁₉	—	2	8	no	—
46	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂³	1	8	yes	complemented
51	C₂×C₂²⋊D₁₉	—	2	16	no	—
52	C₂×C₂²⋊D₁₉	C₂³	1	16	yes	complemented
53	C₂²×Dic₁₉	C₂³	1	2	yes	—
54	D₄×D₁₉	—	2	64	no	—
55	C₂×D₇₆	—	2	16	no	—
56	C₂×D₇₆	—	2	8	no	—
57	C₂×D₇₆	—	2	8	no	—
58	C₂×D₇₆	—	2	64	no	—
59	C₂×D₇₆	—	4	4	no	—
60	C₂×D₇₆	D₄	1	8	yes	direct product factor
61	C₂×D₇₆	C₂³	1	4	yes	—
62	C₂×D₇₆	C₂³	1	8	yes	complemented
63	C₂×D₇₆	C₂³	1	16	yes	complemented
64	C₂×D₇₆	C₂³	1	8	yes	complemented
65	C₂×D₇₆	C₂³	1	8	yes	complemented
66	C₂×C₄×D₁₉	—	2	8	no	—
67	C₂×C₄×D₁₉	C₂³	1	8	yes	complemented
68	C₄⋊C₄×C₁₉	C₂³	1	4	yes	complemented
69	C₂²⋊C₄×C₁₉	C₂³	1	8	yes	complemented
70	C₄²×C₁₉	C₂³	1	4	yes	complemented
71	(C₂×D₁₉)⋊C₄	C₂³	1	16	yes	complemented
72	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	complemented
73	C₄⋊Dic₁₉	C₂³	1	4	yes	complemented
74	C₂³×C₁₉	—	2	4	no	—
75	C₂³×C₁₉	—	2	4	no	—
76	C₂³×C₁₉	C₂×D₄	1	8	yes	complemented
77	C₂³×C₁₉	C₂⁴	1	4	yes	—
78	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
79	C₂²×D₁₉	—	2	4	no	—
80	C₂²×D₁₉	—	2	4	no	—
81	C₂²×D₁₉	—	2	8	no	—
82	C₂²×D₁₉	—	2	8	no	—
83	C₂²×D₁₉	—	2	32	no	—
84	C₂²×D₁₉	—	2	8	no	—
85	C₂²×D₁₉	—	2	32	no	—
86	C₂²×D₁₉	—	4	16	no	—
87	C₂²×D₁₉	—	4	64	no	—
88	C₂²×D₁₉	C₂×D₄	1	8	yes	complemented
89	C₂²×D₁₉	C₂⁴	1	8	yes	—
90	C₂²×D₁₉	C₂⁴	1	4	yes	—
91	D₄×C₁₉	C₂×D₄	1	16	yes	complemented
92	C₂×C₄×C₁₉	—	2	4	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	16	no	—
95	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	complemented
96	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	complemented
97	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
98	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
99	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
100	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
101	C₂²⋊D₁₉	—	2	32	no	—
102	C₂×Dic₁₉	—	2	4	no	—
103	C₂×Dic₁₉	C₂⁴	1	4	yes	—
104	D₇₆	—	2	32	no	—
105	D₇₆	—	2	16	no	—
106	D₇₆	—	2	16	no	—
107	D₇₆	—	4	8	no	—
108	D₇₆	C₂×D₄	1	16	yes	direct product factor
109	C₄×D₁₉	—	2	16	no	—
110	C₂×D₄²	—	19	1	no	maximal, Sylow
111	C₂²×C₁₉	—	2	16	no	—
112	C₂²×C₁₉	—	2	4	no	—
113	C₂²×C₁₉	—	2	4	no	—
114	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
115	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
116	C₂²×C₁₉	C₂²×D₄	1	8	yes	complemented
117	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
118	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
119	C₂×D₁₉	—	2	16	no	—
120	C₂×D₁₉	—	2	4	no	—
121	C₂×D₁₉	—	2	4	no	—
122	C₂×D₁₉	—	2	8	no	—
123	C₂×D₁₉	—	2	8	no	—
124	C₂×D₁₉	—	2	8	no	—
125	C₂×D₁₉	—	4	64	no	—
126	C₂×D₁₉	—	4	16	no	—
127	C₂×D₁₉	C₂²×D₄	1	8	yes	complemented
128	C₄×C₁₉	—	2	4	no	—
129	C₄×C₁₉	—	2	2	no	—
130	C₄×C₁₉	C₂²×D₄	1	4	yes	complemented
131	C₄×C₁₉	C₂²×D₄	1	4	yes	complemented
132	Dic₁₉	—	2	4	no	—
133	C₂³×D₄	—	19	2	no	—
134	C₂³×D₄	—	19	2	no	—
135	D₄²	—	19	16	no	—
136	C₂×#127	—	19	1	no	—
137	C₂×#121	—	19	2	no	—
138	C₂×#121	—	19	2	no	—
139	C₂×#120	—	19	4	no	—
140	C₂×C₄×D₄	—	19	1	no	—
141	C₂×C₄×D₄	D₁₉	1	1	yes	characteristic, complemented
142	C₂×C₁₉	—	2	8	no	—
143	C₂×C₁₉	D₄²	1	4	yes	complemented
144	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
145	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
146	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
147	D₁₉	—	2	8	no	—
148	D₁₉	—	4	8	no	—
149	C₂⁵	—	19	4	no	—
150	C₂²×D₄	—	19	4	no	—
151	C₂²×D₄	—	19	16	no	—
152	C₂²×D₄	—	19	1	no	—
153	C₂²×D₄	—	19	4	no	—
154	C₂²×D₄	—	19	2	no	—
155	C₂²×D₄	—	19	2	no	—
156	C₂²×D₄	—	19	2	no	—
157	C₂²×D₄	—	19	16	no	—
158	C₂²×D₄	—	38	16	no	—
159	C₂²×D₄	—	38	16	no	—
160	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
161	C₂³×C₄	—	19	2	no	—
162	C₂³×C₄	C₂×D₁₉	1	2	yes	complemented
163	#127	—	19	8	no	—
164	#121	—	19	16	no	—
165	#121	—	19	16	no	—
166	#120	—	19	32	no	—
167	C₄×D₄	—	19	8	no	—
168	C₄×D₄	C₂×D₁₉	1	8	yes	complemented
169	C₂×C₄⋊C₄	—	19	1	no	—
170	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic, complemented
171	C₂×C₂²⋊C₄	—	19	2	no	—
172	C₂×C₂²⋊C₄	—	19	4	no	—
173	C₂×C₂²⋊C₄	C₂×D₁₉	1	2	yes	complemented
174	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic, complemented
175	C₁₉	C₂×D₄²	1	1	yes	characteristic, complemented, Sylow
176	C₂⁴	—	19	2	no	—
177	C₂⁴	—	19	4	no	—
178	C₂⁴	—	19	16	no	—
179	C₂⁴	—	38	16	no	—
180	C₂⁴	—	38	16	no	—
181	C₂⁴	—	76	8	no	—
182	C₂⁴	C₂²×D₁₉	1	2	yes	—
183	C₂×D₄	—	19	16	no	—
184	C₂×D₄	—	19	8	no	—
185	C₂×D₄	—	19	4	no	—
186	C₂×D₄	—	19	16	no	—
187	C₂×D₄	—	19	8	no	—
188	C₂×D₄	—	19	8	no	—
189	C₂×D₄	—	19	8	no	—
190	C₂×D₄	—	38	64	no	—
191	C₂×D₄	—	38	16	no	—
192	C₂×D₄	—	38	16	no	—
193	C₂×D₄	—	38	64	no	—
194	C₂×D₄	—	38	8	no	—
195	C₂×D₄	—	38	8	no	—
196	C₂×D₄	—	76	4	no	—
197	C₂×D₄	D₇₆	1	8	yes	direct product factor
198	C₂×D₄	C₂²×D₁₉	1	4	yes	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	19	2	no	—
201	C₂²×C₄	—	19	8	no	—
202	C₂²×C₄	—	38	8	no	—
203	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
205	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₉	1	8	yes	complemented
208	C₄⋊C₄	—	19	4	no	—
209	C₄⋊C₄	C₂²×D₁₉	1	4	yes	complemented
210	C₂²⋊C₄	—	19	8	no	—
211	C₂²⋊C₄	—	19	16	no	—
212	C₂²⋊C₄	C₂²×D₁₉	1	8	yes	complemented
213	C₄²	C₂²×D₁₉	1	4	yes	complemented
214	C₂³	—	2	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	19	8	no	—
217	C₂³	—	19	4	no	—
218	C₂³	—	19	8	no	—
219	C₂³	—	38	32	no	—
220	C₂³	—	38	4	no	—
221	C₂³	—	38	4	no	—
222	C₂³	—	38	8	no	—
223	C₂³	—	38	32	no	—
224	C₂³	—	38	8	no	—
225	C₂³	—	38	8	no	—
226	C₂³	—	76	16	no	—
227	C₂³	—	76	64	no	—
228	C₂³	C₂×D₇₆	1	8	yes	complemented
229	C₂³	C₂³×D₁₉	1	4	yes	—
230	C₂³	C₂³×D₁₉	1	1	yes	characteristic
231	D₄	—	19	16	no	—
232	D₄	—	38	32	no	—
233	D₄	—	38	32	no	—
234	D₄	—	38	16	no	—
235	D₄	—	38	16	no	—
236	D₄	—	76	8	no	—
237	D₄	C₂×D₇₆	1	16	yes	direct product factor
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	16	no	—
241	C₂×C₄	—	19	4	no	—
242	C₂×C₄	—	38	4	no	—
243	C₂×C₄	—	38	16	no	—
244	C₂×C₄	C₂×D₇₆	1	4	yes	complemented
245	C₂×C₄	D₄×D₁₉	1	4	yes	complemented
246	C₂×C₄	C₂³×D₁₉	1	2	yes	—
247	C₂×C₄	C₂³×D₁₉	1	4	yes	—
248	C₂×C₄	C₂³×D₁₉	1	2	yes	—
249	C₂×C₄	C₂³×D₁₉	1	2	yes	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	16	no	—
253	C₂²	—	19	8	no	—
254	C₂²	—	38	4	no	—
255	C₂²	—	38	4	no	—
256	C₂²	—	38	16	no	—
257	C₂²	—	38	8	no	—
258	C₂²	—	38	8	no	—
259	C₂²	—	38	8	no	—
260	C₂²	—	76	64	no	—
261	C₂²	—	76	16	no	—
262	C₂²	C₂²×D₇₆	1	2	yes	—
263	C₂²	C₂²×D₇₆	1	8	yes	complemented
264	C₂²	C₂×D₄×D₁₉	1	2	yes	—
265	C₂²	#10597435	1	2	yes	—
266	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	2	2	no	—
269	C₄	—	38	4	no	—
270	C₄	C₂²×D₇₆	1	4	yes	complemented
271	C₄	C₂×D₄×D₁₉	1	4	yes	complemented
272	C₂	—	2	8	no	—
273	C₂	—	38	8	no	—
274	C₂	—	76	8	no	—
275	C₂	D₄×D₇₆	1	4	yes	direct product factor
276	C₂	C₂²×D₄×D₁₉	1	1	yes	characteristic
277	C₂	C₂³×D₇₆	1	1	yes	characteristic
278	C₂	C₂×#10597435	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₉
commutator, cosocle	C₂²×C₁₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₂×C₄×D₄×C₁₉	C₂
socle	C₂³×C₁₉	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₉
hypocenter	C₁₉	C₂×D₄²
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₇₆	C₂	semidirect
2. C₂²×D₄×D₁₉	C₂	semidirect
3. D₄×D₇₆	C₂	direct (N × H)
4. C₂×#10597320	C₂	semidirect
5. C₂×#10597341	C₂	semidirect
6. C₂×#10597376	C₂	semidirect
7. C₂×#10597328	C₂	semidirect
8. C₂×C₄×D₇₆	C₂	semidirect
9. C₂×C₄×D₄×C₁₉	C₂	semidirect
10. C₂⁴×D₁₉	C₂²	semidirect
12. C₂³×C₄×C₁₉	C₂²	semidirect
13. C₂²×C₂²⋊D₁₉	C₂²	semidirect
15. C₂×D₄×D₁₉	C₂²	semidirect
18. C₂²×D₇₆	C₂²	semidirect
19. C₂²×D₇₆	C₂²	semidirect
20. C₂²×D₇₆	C₂²	semidirect
21. C₂²×D₇₆	C₂²	semidirect
22. C₂²×D₇₆	C₂²	semidirect
23. C₂²×C₄×D₁₉	C₂²	semidirect
24. C₄×D₄×C₁₉	C₂²	semidirect
25. C₂×C₄⋊C₄×C₁₉	C₂²	semidirect
26. C₂×C₂²⋊C₄×C₁₉	C₂²	semidirect
27. C₂×C₄²×C₁₉	C₂²	semidirect
28. #10597376	C₂²	semidirect
29. C₂×(C₂×D₁₉)⋊C₄	C₂²	semidirect
30. C₂×(C₂×D₁₉)⋊C₄	C₂²	semidirect
31. C₂×C₄⋊Dic₁₉	C₂²	semidirect
32. #10597341	C₂²	semidirect
33. #10597328	C₂²	semidirect
34. #10597320	C₂²	semidirect
35. C₄×D₇₆	C₂²	semidirect
42. C₂³×D₁₉	C₂³	semidirect
43. C₂×D₄×C₁₉	D₄	semidirect
50. C₂²×C₄×C₁₉	C₂³	semidirect
52. C₂×C₂²⋊D₁₉	C₂³	semidirect
60. C₂×D₇₆	D₄	direct (N × H)
62. C₂×D₇₆	C₂³	semidirect
63. C₂×D₇₆	C₂³	semidirect
64. C₂×D₇₆	C₂³	semidirect
65. C₂×D₇₆	C₂³	semidirect
67. C₂×C₄×D₁₉	C₂³	semidirect
68. C₄⋊C₄×C₁₉	C₂³	semidirect
69. C₂²⋊C₄×C₁₉	C₂³	semidirect
70. C₄²×C₁₉	C₂³	semidirect
71. (C₂×D₁₉)⋊C₄	C₂³	semidirect
72. (C₂×D₁₉)⋊C₄	C₂³	semidirect
73. C₄⋊Dic₁₉	C₂³	semidirect
76. C₂³×C₁₉	C₂×D₄	semidirect
88. C₂²×D₁₉	C₂×D₄	semidirect
91. D₄×C₁₉	C₂×D₄	semidirect
95. C₂×C₄×C₁₉	C₂×D₄	semidirect
96. C₂×C₄×C₁₉	C₂×D₄	semidirect
108. D₇₆	C₂×D₄	direct (N × H)
116. C₂²×C₁₉	C₂²×D₄	semidirect
127. C₂×D₁₉	C₂²×D₄	semidirect
130. C₄×C₁₉	C₂²×D₄	semidirect
131. C₄×C₁₉	C₂²×D₄	semidirect
141. C₂×C₄×D₄	D₁₉	semidirect
143. C₂×C₁₉	D₄²	semidirect
162. C₂³×C₄	C₂×D₁₉	semidirect
168. C₄×D₄	C₂×D₁₉	semidirect
170. C₂×C₄⋊C₄	C₂×D₁₉	semidirect
173. C₂×C₂²⋊C₄	C₂×D₁₉	semidirect
174. C₂×C₄²	C₂×D₁₉	semidirect
175. C₁₉	C₂×D₄²	semidirect
197. C₂×D₄	D₇₆	direct (N × H)
207. C₂²×C₄	C₂²×D₁₉	semidirect
209. C₄⋊C₄	C₂²×D₁₉	semidirect
212. C₂²⋊C₄	C₂²×D₁₉	semidirect
213. C₄²	C₂²×D₁₉	semidirect
228. C₂³	C₂×D₇₆	semidirect
237. D₄	C₂×D₇₆	direct (N × H)
244. C₂×C₄	C₂×D₇₆	semidirect
245. C₂×C₄	D₄×D₁₉	semidirect
263. C₂²	C₂²×D₇₆	semidirect
270. C₄	C₂²×D₇₆	semidirect
271. C₄	C₂×D₄×D₁₉	semidirect
275. C₂	D₄×D₇₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₇₆	2
C₂²×D₄×D₁₉	2
D₄×D₇₆	2
C₂×#10597320	2
C₂×#10597341	2
C₂×#10597376	2
C₂×#10597328	2
C₂×C₄×D₇₆	2
C₂×C₄×D₄×C₁₉	2
C₂×D₄²	19

Subgroups — C2×D4×D76 #14577075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×D₇₆ #14577075