Subgroups — C₂×D₄×D₅₅₆ #31819844

Order 17792.

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×D₅₅₆	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄×D₁₃₉	C₂	1	2	yes	maximal, complemented
3	D₄×D₅₅₆	C₂	1	16	yes	maximal, direct product factor
4	C₂×#17984839	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17984817	C₂	1	2	yes	maximal, complemented
6	C₂×#17984816	C₂	1	2	yes	maximal, complemented
7	C₂×#17984812	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×D₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₃₉	C₂²	1	4	yes	complemented
11	C₂²×D₅₅₆	—	2	16	no	—
12	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic
13	C₂²×D₅₅₆	C₂²	1	2	yes	complemented
14	C₂²×D₅₅₆	C₂²	1	4	yes	complemented
15	C₂²×D₅₅₆	C₂²	1	2	yes	complemented
16	C₂²×D₅₅₆	C₂²	1	2	yes	complemented
17	C₂²×D₅₅₆	C₂²	1	16	yes	complemented
18	C₂²×C₂²⋊D₁₃₉	C₂²	1	4	yes	complemented
19	C₂×D₄×D₁₃₉	—	2	16	no	—
20	C₂×D₄×D₁₃₉	C₂²	1	16	yes	complemented
21	C₂²×C₄×D₁₃₉	C₂²	1	2	yes	complemented
22	#17984839	C₂²	1	8	yes	complemented
23	#17984817	C₂²	1	16	yes	complemented
24	#17984816	C₂²	1	16	yes	complemented
25	#17984812	C₂²	1	32	yes	complemented
26	C₄×D₅₅₆	C₂²	1	8	yes	complemented
27	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic, complemented
28	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	2	yes	complemented
29	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	4	yes	complemented
30	C₂²×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
31	C₂³×C₄×C₁₃₉	C₂²	1	2	yes	complemented
32	C₄×D₄×C₁₃₉	C₂²	1	8	yes	complemented
33	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	complemented
35	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
36	C₂³×D₁₃₉	—	2	16	no	—
37	C₂³×D₁₃₉	—	2	16	no	—
38	C₂³×D₁₃₉	—	4	8	no	—
39	C₂³×D₁₃₉	C₂³	1	4	yes	—
40	C₂³×D₁₃₉	C₂³	1	2	yes	—
41	C₂³×D₁₃₉	C₂³	1	16	yes	complemented
42	C₂×D₅₅₆	—	2	16	no	—
43	C₂×D₅₅₆	—	2	64	no	—
44	C₂×D₅₅₆	—	2	8	no	—
45	C₂×D₅₅₆	—	2	8	no	—
46	C₂×D₅₅₆	—	4	4	no	—
47	C₂×D₅₅₆	D₄	1	8	yes	direct product factor
48	C₂×D₅₅₆	C₂³	1	4	yes	—
49	C₂×D₅₅₆	C₂³	1	16	yes	complemented
50	C₂×D₅₅₆	C₂³	1	8	yes	complemented
51	C₂×D₅₅₆	C₂³	1	8	yes	complemented
52	C₂×D₅₅₆	C₂³	1	8	yes	complemented
53	C₂×C₂²⋊D₁₃₉	—	2	16	no	—
54	C₂×C₂²⋊D₁₃₉	C₂³	1	16	yes	complemented
55	D₄×D₁₃₉	—	2	64	no	—
56	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
57	C₂×C₄×D₁₃₉	—	2	8	no	—
58	C₂×C₄×D₁₃₉	C₂³	1	8	yes	complemented
59	C₄⋊Dic₁₃₉	C₂³	1	4	yes	complemented
60	(C₂×D₁₃₉)⋊C₄	C₂³	1	8	yes	complemented
61	(C₂×D₁₃₉)⋊C₄	C₂³	1	16	yes	complemented
62	C₂⁴×C₁₃₉	C₂³	1	2	yes	—
63	C₂×D₄×C₁₃₉	D₄	1	8	yes	complemented
64	C₂×D₄×C₁₃₉	C₂³	1	4	yes	—
65	C₂²×C₄×C₁₃₉	—	2	8	no	—
66	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
68	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₉	C₂³	1	8	yes	complemented
71	C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	complemented
72	C₂²⋊C₄×C₁₃₉	C₂³	1	8	yes	complemented
73	C₄²×C₁₃₉	C₂³	1	4	yes	complemented
74	C₂²×D₁₃₉	—	2	4	no	—
75	C₂²×D₁₃₉	—	2	4	no	—
76	C₂²×D₁₃₉	—	2	8	no	—
77	C₂²×D₁₃₉	—	2	8	no	—
78	C₂²×D₁₃₉	—	2	32	no	—
79	C₂²×D₁₃₉	—	2	8	no	—
80	C₂²×D₁₃₉	—	2	32	no	—
81	C₂²×D₁₃₉	—	4	16	no	—
82	C₂²×D₁₃₉	—	4	64	no	—
83	C₂²×D₁₃₉	C₂×D₄	1	8	yes	complemented
84	C₂²×D₁₃₉	C₂⁴	1	8	yes	—
85	C₂²×D₁₃₉	C₂⁴	1	4	yes	—
86	D₅₅₆	—	2	32	no	—
87	D₅₅₆	—	2	16	no	—
88	D₅₅₆	—	2	16	no	—
89	D₅₅₆	—	4	8	no	—
90	D₅₅₆	C₂×D₄	1	16	yes	direct product factor
91	C₂²⋊D₁₃₉	—	2	32	no	—
92	C₂×Dic₁₃₉	—	2	4	no	—
93	C₂×Dic₁₃₉	C₂⁴	1	4	yes	—
94	C₄×D₁₃₉	—	2	16	no	—
95	C₂³×C₁₃₉	—	2	4	no	—
96	C₂³×C₁₃₉	—	2	4	no	—
97	C₂³×C₁₃₉	C₂×D₄	1	8	yes	complemented
98	C₂³×C₁₃₉	C₂⁴	1	4	yes	—
99	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
100	D₄×C₁₃₉	C₂×D₄	1	16	yes	complemented
101	C₂×C₄×C₁₃₉	—	2	4	no	—
102	C₂×C₄×C₁₃₉	—	2	2	no	—
103	C₂×C₄×C₁₃₉	—	2	16	no	—
104	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	complemented
105	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	complemented
106	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
107	C₂×C₄×C₁₃₉	C₂⁴	1	4	yes	—
108	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
109	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
110	C₂²×C₁₃₉	—	2	16	no	—
111	C₂²×C₁₃₉	—	2	4	no	—
112	C₂²×C₁₃₉	—	2	4	no	—
113	C₂²×C₁₃₉	C₂²×D₄	1	2	yes	—
114	C₂²×C₁₃₉	C₂²×D₄	1	2	yes	—
115	C₂²×C₁₃₉	C₂²×D₄	1	8	yes	complemented
116	C₂²×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
117	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
118	C₂×D₁₃₉	—	2	16	no	—
119	C₂×D₁₃₉	—	2	4	no	—
120	C₂×D₁₃₉	—	2	4	no	—
121	C₂×D₁₃₉	—	2	8	no	—
122	C₂×D₁₃₉	—	2	8	no	—
123	C₂×D₁₃₉	—	2	8	no	—
124	C₂×D₁₃₉	—	4	64	no	—
125	C₂×D₁₃₉	—	4	16	no	—
126	C₂×D₁₃₉	C₂²×D₄	1	8	yes	complemented
127	C₄×C₁₃₉	—	2	4	no	—
128	C₄×C₁₃₉	—	2	2	no	—
129	C₄×C₁₃₉	C₂²×D₄	1	4	yes	complemented
130	C₄×C₁₃₉	C₂²×D₄	1	4	yes	complemented
131	Dic₁₃₉	—	2	4	no	—
132	C₂×C₁₃₉	—	2	8	no	—
133	C₂×C₁₃₉	D₄²	1	4	yes	complemented
134	C₂×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
135	C₂×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
136	C₂×C₁₃₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
137	D₁₃₉	—	2	8	no	—
138	D₁₃₉	—	4	8	no	—
139	C₁₃₉	C₂×D₄²	1	1	yes	characteristic, complemented, Sylow
140	C₂×D₄²	—	139	1	no	maximal, Sylow
141	C₂³×D₄	—	139	2	no	—
142	C₂³×D₄	—	139	2	no	—
143	D₄²	—	139	16	no	—
144	C₂×#127	—	139	1	no	—
145	C₂×#121	—	139	2	no	—
146	C₂×#121	—	139	2	no	—
147	C₂×#120	—	139	4	no	—
148	C₂×C₄×D₄	—	139	1	no	—
149	C₂×C₄×D₄	D₁₃₉	1	1	yes	characteristic, complemented
150	C₂⁵	—	139	4	no	—
151	C₂²×D₄	—	278	16	no	—
152	C₂²×D₄	—	278	16	no	—
153	C₂²×D₄	—	139	4	no	—
154	C₂²×D₄	—	139	16	no	—
155	C₂²×D₄	—	139	4	no	—
156	C₂²×D₄	—	139	2	no	—
157	C₂²×D₄	—	139	2	no	—
158	C₂²×D₄	—	139	2	no	—
159	C₂²×D₄	—	139	1	no	—
160	C₂²×D₄	—	139	16	no	—
161	C₂²×D₄	C₂×D₁₃₉	1	1	yes	characteristic
162	C₂³×C₄	—	139	2	no	—
163	C₂³×C₄	C₂×D₁₃₉	1	2	yes	complemented
164	#127	—	139	8	no	—
165	#121	—	139	16	no	—
166	#121	—	139	16	no	—
167	#120	—	139	32	no	—
168	C₄×D₄	—	139	8	no	—
169	C₄×D₄	C₂×D₁₃₉	1	8	yes	complemented
170	C₂×C₄⋊C₄	—	139	1	no	—
171	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic, complemented
172	C₂×C₂²⋊C₄	—	139	2	no	—
173	C₂×C₂²⋊C₄	—	139	4	no	—
174	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	2	yes	complemented
175	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic, complemented
176	C₂⁴	—	278	16	no	—
177	C₂⁴	—	278	16	no	—
178	C₂⁴	—	556	8	no	—
179	C₂⁴	—	139	4	no	—
180	C₂⁴	—	139	16	no	—
181	C₂⁴	—	139	2	no	—
182	C₂⁴	C₂²×D₁₃₉	1	2	yes	—
183	C₂×D₄	—	278	16	no	—
184	C₂×D₄	—	278	64	no	—
185	C₂×D₄	—	278	16	no	—
186	C₂×D₄	—	278	64	no	—
187	C₂×D₄	—	278	8	no	—
188	C₂×D₄	—	278	8	no	—
189	C₂×D₄	—	556	4	no	—
190	C₂×D₄	—	139	16	no	—
191	C₂×D₄	—	139	8	no	—
192	C₂×D₄	—	139	16	no	—
193	C₂×D₄	—	139	8	no	—
194	C₂×D₄	—	139	8	no	—
195	C₂×D₄	—	139	8	no	—
196	C₂×D₄	—	139	4	no	—
197	C₂×D₄	D₅₅₆	1	8	yes	direct product factor
198	C₂×D₄	C₂²×D₁₃₉	1	4	yes	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	278	8	no	—
201	C₂²×C₄	—	139	2	no	—
202	C₂²×C₄	—	139	8	no	—
203	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
205	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₃₉	1	8	yes	complemented
208	C₄⋊C₄	—	139	4	no	—
209	C₄⋊C₄	C₂²×D₁₃₉	1	4	yes	complemented
210	C₂²⋊C₄	—	139	8	no	—
211	C₂²⋊C₄	—	139	16	no	—
212	C₂²⋊C₄	C₂²×D₁₃₉	1	8	yes	complemented
213	C₄²	C₂²×D₁₃₉	1	4	yes	complemented
214	C₂³	—	2	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	278	32	no	—
217	C₂³	—	278	4	no	—
218	C₂³	—	278	4	no	—
219	C₂³	—	278	8	no	—
220	C₂³	—	278	8	no	—
221	C₂³	—	278	32	no	—
222	C₂³	—	278	8	no	—
223	C₂³	—	556	16	no	—
224	C₂³	—	556	64	no	—
225	C₂³	—	139	8	no	—
226	C₂³	—	139	8	no	—
227	C₂³	—	139	4	no	—
228	C₂³	C₂×D₅₅₆	1	8	yes	complemented
229	C₂³	C₂³×D₁₃₉	1	4	yes	—
230	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
231	D₄	—	278	32	no	—
232	D₄	—	278	32	no	—
233	D₄	—	278	16	no	—
234	D₄	—	278	16	no	—
235	D₄	—	556	8	no	—
236	D₄	—	139	16	no	—
237	D₄	C₂×D₅₅₆	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	—	278	4	no	—
242	C₂×C₄	—	278	16	no	—
243	C₂×C₄	—	139	4	no	—
244	C₂×C₄	D₄×D₁₃₉	1	4	yes	complemented
245	C₂×C₄	C₂×D₅₅₆	1	4	yes	complemented
246	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
247	C₂×C₄	C₂³×D₁₃₉	1	4	yes	—
248	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
249	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	16	no	—
253	C₂²	—	278	4	no	—
254	C₂²	—	278	4	no	—
255	C₂²	—	278	16	no	—
256	C₂²	—	278	8	no	—
257	C₂²	—	278	8	no	—
258	C₂²	—	278	8	no	—
259	C₂²	—	556	64	no	—
260	C₂²	—	556	16	no	—
261	C₂²	—	139	8	no	—
262	C₂²	C₂×D₄×D₁₃₉	1	2	yes	—
263	C₂²	C₂²×D₅₅₆	1	2	yes	—
264	C₂²	C₂²×D₅₅₆	1	8	yes	complemented
265	C₂²	#17984888	1	2	yes	—
266	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	2	2	no	—
269	C₄	—	278	4	no	—
270	C₄	C₂×D₄×D₁₃₉	1	4	yes	complemented
271	C₄	C₂²×D₅₅₆	1	4	yes	complemented
272	C₂	—	2	8	no	—
273	C₂	—	278	8	no	—
274	C₂	—	556	8	no	—
275	C₂	D₄×D₅₅₆	1	4	yes	direct product factor
276	C₂	C₂²×D₄×D₁₃₉	1	1	yes	characteristic
277	C₂	C₂³×D₅₅₆	1	1	yes	characteristic
278	C₂	C₂×#17984888	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃₉
commutator, cosocle	C₂²×C₁₃₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₂×C₄×D₄×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₃₉
hypocenter	C₁₃₉	C₂×D₄²
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₅₅₆	C₂	semidirect
2. C₂²×D₄×D₁₃₉	C₂	semidirect
3. D₄×D₅₅₆	C₂	direct (N × H)
4. C₂×#17984839	C₂	semidirect
5. C₂×#17984817	C₂	semidirect
6. C₂×#17984816	C₂	semidirect
7. C₂×#17984812	C₂	semidirect
8. C₂×C₄×D₅₅₆	C₂	semidirect
9. C₂×C₄×D₄×C₁₃₉	C₂	semidirect
10. C₂⁴×D₁₃₉	C₂²	semidirect
13. C₂²×D₅₅₆	C₂²	semidirect
14. C₂²×D₅₅₆	C₂²	semidirect
15. C₂²×D₅₅₆	C₂²	semidirect
16. C₂²×D₅₅₆	C₂²	semidirect
17. C₂²×D₅₅₆	C₂²	semidirect
18. C₂²×C₂²⋊D₁₃₉	C₂²	semidirect
20. C₂×D₄×D₁₃₉	C₂²	semidirect
21. C₂²×C₄×D₁₃₉	C₂²	semidirect
22. #17984839	C₂²	semidirect
23. #17984817	C₂²	semidirect
24. #17984816	C₂²	semidirect
25. #17984812	C₂²	semidirect
26. C₄×D₅₅₆	C₂²	semidirect
27. C₂×C₄⋊Dic₁₃₉	C₂²	semidirect
28. C₂×(C₂×D₁₃₉)⋊C₄	C₂²	semidirect
29. C₂×(C₂×D₁₃₉)⋊C₄	C₂²	semidirect
31. C₂³×C₄×C₁₃₉	C₂²	semidirect
32. C₄×D₄×C₁₃₉	C₂²	semidirect
33. C₂×C₄⋊C₄×C₁₃₉	C₂²	semidirect
34. C₂×C₂²⋊C₄×C₁₃₉	C₂²	semidirect
35. C₂×C₄²×C₁₃₉	C₂²	semidirect
41. C₂³×D₁₃₉	C₂³	semidirect
47. C₂×D₅₅₆	D₄	direct (N × H)
49. C₂×D₅₅₆	C₂³	semidirect
50. C₂×D₅₅₆	C₂³	semidirect
51. C₂×D₅₅₆	C₂³	semidirect
52. C₂×D₅₅₆	C₂³	semidirect
54. C₂×C₂²⋊D₁₃₉	C₂³	semidirect
58. C₂×C₄×D₁₃₉	C₂³	semidirect
59. C₄⋊Dic₁₃₉	C₂³	semidirect
60. (C₂×D₁₃₉)⋊C₄	C₂³	semidirect
61. (C₂×D₁₃₉)⋊C₄	C₂³	semidirect
63. C₂×D₄×C₁₃₉	D₄	semidirect
70. C₂²×C₄×C₁₃₉	C₂³	semidirect
71. C₄⋊C₄×C₁₃₉	C₂³	semidirect
72. C₂²⋊C₄×C₁₃₉	C₂³	semidirect
73. C₄²×C₁₃₉	C₂³	semidirect
83. C₂²×D₁₃₉	C₂×D₄	semidirect
90. D₅₅₆	C₂×D₄	direct (N × H)
97. C₂³×C₁₃₉	C₂×D₄	semidirect
100. D₄×C₁₃₉	C₂×D₄	semidirect
104. C₂×C₄×C₁₃₉	C₂×D₄	semidirect
105. C₂×C₄×C₁₃₉	C₂×D₄	semidirect
115. C₂²×C₁₃₉	C₂²×D₄	semidirect
126. C₂×D₁₃₉	C₂²×D₄	semidirect
129. C₄×C₁₃₉	C₂²×D₄	semidirect
130. C₄×C₁₃₉	C₂²×D₄	semidirect
133. C₂×C₁₃₉	D₄²	semidirect
139. C₁₃₉	C₂×D₄²	semidirect
149. C₂×C₄×D₄	D₁₃₉	semidirect
163. C₂³×C₄	C₂×D₁₃₉	semidirect
169. C₄×D₄	C₂×D₁₃₉	semidirect
171. C₂×C₄⋊C₄	C₂×D₁₃₉	semidirect
174. C₂×C₂²⋊C₄	C₂×D₁₃₉	semidirect
175. C₂×C₄²	C₂×D₁₃₉	semidirect
197. C₂×D₄	D₅₅₆	direct (N × H)
207. C₂²×C₄	C₂²×D₁₃₉	semidirect
209. C₄⋊C₄	C₂²×D₁₃₉	semidirect
212. C₂²⋊C₄	C₂²×D₁₃₉	semidirect
213. C₄²	C₂²×D₁₃₉	semidirect
228. C₂³	C₂×D₅₅₆	semidirect
237. D₄	C₂×D₅₅₆	direct (N × H)
244. C₂×C₄	D₄×D₁₃₉	semidirect
245. C₂×C₄	C₂×D₅₅₆	semidirect
264. C₂²	C₂²×D₅₅₆	semidirect
270. C₄	C₂×D₄×D₁₃₉	semidirect
271. C₄	C₂²×D₅₅₆	semidirect
275. C₂	D₄×D₅₅₆	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×D₅₅₆	2
C₂²×D₄×D₁₃₉	2
D₄×D₅₅₆	2
C₂×#17984839	2
C₂×#17984817	2
C₂×#17984816	2
C₂×#17984812	2
C₂×C₄×D₅₅₆	2
C₂×C₄×D₄×C₁₃₉	2
C₂×D₄²	139

Subgroups — C2×D4×D556 #31819844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×D₅₅₆ #31819844