Subgroups — C₄×#88256 #13128043

Order 1664.

1660 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88256	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719532	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719279	C₂	1	2	yes	maximal, complemented
5	C₄×#6131	C₂	1	2	yes	maximal, complemented
6	#11719250	C₂	1	6	yes	maximal, complemented
7	C₄×D₄×D₁₃	C₂	1	6	yes	maximal, complemented
8	#11719228	C₂	1	3	yes	maximal, complemented
9	#11719226	C₂	1	3	yes	maximal, complemented
10	C₄×#6127	C₂	1	3	yes	maximal, complemented
11	C₂×C₄×D₅₂	C₂	1	3	yes	maximal, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	#88256	C₄	1	16	yes	direct product factor
14	C₂×#6131	C₂²	1	2	yes	—
15	C₂×D₄×D₁₃	C₂²	1	6	yes	—
16	C₂×#6127	C₂²	1	3	yes	—
17	C₂²×D₅₂	C₂²	1	3	yes	—
18	C₂²×C₄×D₁₃	C₂²	1	6	yes	complemented
19	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
20	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
21	C₄×D₄×C₁₃	C₂²	1	3	yes	complemented
22	C₄×D₄×C₁₃	C₂²	1	3	yes	complemented
23	C₁₃×#117	C₂²	1	3	yes	complemented
24	C₂×C₄²×C₁₃	C₂²	1	3	yes	complemented
25	Q₈×Dic₁₃	C₂²	1	2	yes	complemented
26	D₄×Dic₁₃	C₂²	1	6	yes	complemented
27	C₄×C₂²⋊D₁₃	C₂²	1	6	yes	complemented
28	C₄×C₂²⋊D₁₃	C₂²	1	6	yes	complemented
29	C₂×#6103	C₂²	1	6	yes	complemented
30	#88180	C₂²	1	3	yes	complemented
31	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	complemented
32	#88147	C₂²	1	6	yes	complemented
33	#88147	C₂²	1	6	yes	complemented
34	#88146	C₂²	1	6	yes	complemented
35	C₄⋊C₄×D₁₃	C₂²	1	6	yes	complemented
36	#88135	C₂²	1	12	yes	complemented
37	C₂²⋊C₄×D₁₃	C₂²	1	12	yes	complemented
38	C₄×D₅₂	C₂²	1	3	yes	complemented
39	C₄×D₅₂	C₂²	1	6	yes	complemented
40	C₄×D₅₂	C₂²	1	6	yes	complemented
41	C₄×D₅₂	C₂²	1	6	yes	complemented
42	C₄×D₅₂	C₂²	1	3	yes	complemented
43	#88126	C₂²	1	6	yes	complemented
44	C₄²×D₁₃	C₂²	1	6	yes	complemented
45	C₄×Dic₂₆	C₂²	1	1	yes	characteristic, complemented
46	C₄×Dic₂₆	C₂²	1	3	yes	complemented
47	C₂³×D₁₃	C₂³	1	6	yes	—
48	C₁₃×#41	C₂×C₄	1	8	yes	complemented
49	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₃	C₂³	1	3	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
53	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
54	C₂×C₂²⋊D₁₃	C₂³	1	6	yes	—
55	C₂²×Dic₁₃	C₂³	1	6	yes	—
56	#6131	C₂×C₄	1	16	yes	complemented
57	D₄×D₁₃	C₂×C₄	1	48	yes	complemented
58	#6127	C₂×C₄	1	24	yes	complemented
59	C₂×D₅₂	C₂×C₄	1	24	yes	complemented
60	C₂×D₅₂	C₂³	1	3	yes	—
61	C₂×D₅₂	C₂³	1	6	yes	—
62	C₂×C₄×D₁₃	—	4	12	no	—
63	C₂×C₄×D₁₃	C₂³	1	6	yes	—
64	C₂×C₄×D₁₃	C₂³	1	6	yes	—
65	C₂×C₄×D₁₃	C₂³	1	12	yes	—
66	C₂×C₄×D₁₃	C₂³	1	6	yes	—
67	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
69	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
70	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
71	C₄²×C₁₃	C₂³	1	1	yes	characteristic
72	C₄²×C₁₃	C₂³	1	3	yes	—
73	C₄²×C₁₃	C₂³	1	3	yes	—
74	C₄²×C₁₃	C₂³	1	3	yes	—
75	C₂²⋊Dic₁₃	C₂³	1	6	yes	—
76	#6103	C₂³	1	6	yes	—
77	#6103	C₂³	1	6	yes	—
78	#6103	C₂³	1	12	yes	—
79	C₄⋊Dic₁₃	C₂³	1	3	yes	—
80	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₃	C₂³	1	6	yes	—
82	#6101	C₂³	1	2	yes	—
83	#6101	C₂³	1	6	yes	—
84	C₄×Dic₁₃	C₂³	1	2	yes	—
85	C₄×Dic₁₃	C₂³	1	6	yes	—
86	C₄×Dic₁₃	C₂³	1	6	yes	—
87	C₄×H(5,𝔽₂)	—	13	1	no	maximal, Sylow
88	C₂³×C₁₃	C₂⁴	1	3	yes	—
89	C₂²×D₁₃	—	4	6	no	—
90	C₂²×D₁₃	—	4	6	no	—
91	C₂²×D₁₃	C₂²×C₄	1	24	yes	complemented
92	C₂²×D₁₃	C₂⁴	1	6	yes	—
93	Q₈×C₁₃	C₂²×C₄	1	4	yes	complemented
94	D₄×C₁₃	C₂²×C₄	1	12	yes	complemented
95	C₂×C₄×C₁₃	—	2	6	no	—
96	C₂×C₄×C₁₃	—	4	3	no	—
97	C₂×C₄×C₁₃	C₂²×C₄	1	12	yes	complemented
98	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
99	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
102	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₁₃	C₂²×C₄	1	24	yes	complemented
105	C₂×Dic₁₃	—	4	6	no	—
106	C₂×Dic₁₃	C₂⁴	1	2	yes	—
107	C₂×Dic₁₃	C₂⁴	1	6	yes	—
108	C₂×Dic₁₃	C₂⁴	1	2	yes	—
109	D₅₂	C₂²×C₄	1	12	yes	complemented
110	D₅₂	C₂²×C₄	1	24	yes	complemented
111	C₄×D₁₃	—	2	12	no	—
112	C₄×D₁₃	—	4	12	no	—
113	C₄×D₁₃	—	4	6	no	—
114	C₄×D₁₃	C₂²×C₄	1	24	yes	complemented
115	Dic₂₆	C₂²×C₄	1	4	yes	complemented
116	C₂×H(5,𝔽₂)	—	13	1	no	—
117	#520	—	13	1	no	—
118	#520	—	13	2	no	—
119	#520	—	13	3	no	—
120	#518	—	13	6	no	—
121	#518	—	13	3	no	—
122	C₄×#41	—	13	2	no	—
123	C₄×#41	—	13	3	no	—
124	C₄×#41	D₁₃	1	1	yes	characteristic, complemented
125	C₂×C₄×D₄	—	13	6	no	—
126	C₂×C₄×D₄	—	13	3	no	—
127	C₂²×C₁₃	—	2	3	no	—
128	C₂²×C₁₃	—	2	3	no	—
129	C₂²×C₁₃	C₂³×C₄	1	6	yes	—
130	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
131	C₂×D₁₃	—	2	6	no	—
132	C₂×D₁₃	—	2	6	no	—
133	C₂×D₁₃	—	4	24	no	—
134	C₂×D₁₃	C₂³×C₄	1	12	yes	—
135	C₄×C₁₃	—	2	3	no	—
136	C₄×C₁₃	—	2	3	no	—
137	C₄×C₁₃	—	2	1	no	—
138	C₄×C₁₃	C₂³×C₄	1	2	yes	—
139	C₄×C₁₃	C₂³×C₄	1	6	yes	—
140	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	complemented
141	Dic₁₃	—	2	6	no	—
142	Dic₁₃	—	2	2	no	—
143	Dic₁₃	C₂³×C₄	1	4	yes	—
144	H(5,𝔽₂)	—	13	16	no	—
145	C₂×#41	—	13	2	no	—
146	C₂×#41	—	13	3	no	—
147	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
148	C₂²×D₄	—	13	6	no	—
149	C₂²×D₄	—	13	3	no	—
150	C₂³×C₄	—	13	6	no	—
151	C₄×Q₈	—	13	2	no	—
152	C₄×Q₈	—	13	1	no	—
153	C₄×Q₈	—	13	3	no	—
154	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
155	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
156	C₄×D₄	—	13	6	no	—
157	C₄×D₄	—	13	6	no	—
158	C₄×D₄	—	13	6	no	—
159	C₄×D₄	—	13	6	no	—
160	C₄×D₄	—	13	6	no	—
161	C₄×D₄	—	13	12	no	—
162	C₄×D₄	—	13	6	no	—
163	C₄×D₄	—	13	3	no	—
164	C₄×D₄	—	13	6	no	—
165	C₄×D₄	—	13	6	no	—
166	C₄×D₄	—	13	3	no	—
167	C₄×D₄	C₂×D₁₃	1	3	yes	complemented
168	C₄×D₄	C₂×D₁₃	1	3	yes	complemented
169	#117	—	13	3	no	—
170	#117	—	13	6	no	—
171	#117	—	13	6	no	—
172	#117	C₂×D₁₃	1	3	yes	complemented
173	C₂×C₄⋊C₄	—	13	3	no	—
174	C₂×C₄⋊C₄	—	13	6	no	—
175	C₂×C₂²⋊C₄	—	13	6	no	—
176	C₂×C₂²⋊C₄	—	13	12	no	—
177	C₂×C₄²	—	13	6	no	—
178	C₂×C₄²	C₂×D₁₃	1	3	yes	complemented
179	C₂×C₁₃	—	2	6	no	—
180	C₂×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
181	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
182	C₂×C₁₃	#585	1	1	yes	characteristic
183	D₁₃	—	2	12	no	—
184	C₂⁴	—	13	6	no	—
185	#41	—	13	16	no	—
186	#41	—	13	24	no	—
187	#41	C₄×D₁₃	1	8	yes	complemented
188	C₂×Q₈	—	13	1	no	—
189	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
190	C₂×D₄	—	13	6	no	—
191	C₂×D₄	—	13	48	no	—
192	C₂×D₄	—	13	24	no	—
193	C₂×D₄	—	13	6	no	—
194	C₂×D₄	—	13	3	no	—
195	C₂×D₄	C₂²×D₁₃	1	3	yes	—
196	C₂²×C₄	—	13	6	no	—
197	C₂²×C₄	—	13	6	no	—
198	C₂²×C₄	—	13	6	no	—
199	C₂²×C₄	—	13	6	no	—
200	C₂²×C₄	—	13	12	no	—
201	C₂²×C₄	—	52	12	no	—
202	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
203	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
204	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
205	C₄⋊C₄	—	13	6	no	—
206	C₄⋊C₄	—	13	6	no	—
207	C₄⋊C₄	—	13	3	no	—
208	C₄⋊C₄	—	13	2	no	—
209	C₄⋊C₄	—	13	1	no	—
210	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
211	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
212	C₂²⋊C₄	—	13	6	no	—
213	C₂²⋊C₄	—	13	6	no	—
214	C₂²⋊C₄	—	13	12	no	—
215	C₂²⋊C₄	—	13	6	no	—
216	C₂²⋊C₄	C₂²×D₁₃	1	6	yes	—
217	C₄²	—	13	6	no	—
218	C₄²	—	13	6	no	—
219	C₄²	—	13	2	no	—
220	C₄²	C₂²×D₁₃	1	3	yes	—
221	C₄²	C₂²×D₁₃	1	3	yes	—
222	C₄²	C₂²×D₁₃	1	3	yes	—
223	C₄²	C₂²×D₁₃	1	1	yes	characteristic
224	C₁₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	13	6	no	—
226	C₂³	—	13	24	no	—
227	C₂³	—	52	6	no	—
228	C₂³	—	52	6	no	—
229	C₂³	C₂³×D₁₃	1	3	yes	—
230	Q₈	—	13	4	no	—
231	Q₈	C₂×C₄×D₁₃	1	4	yes	complemented
232	D₄	—	13	24	no	—
233	D₄	—	13	24	no	—
234	D₄	—	13	12	no	—
235	D₄	C₂×C₄×D₁₃	1	12	yes	complemented
236	C₂×C₄	—	2	6	no	—
237	C₂×C₄	—	4	3	no	—
238	C₂×C₄	—	13	2	no	—
239	C₂×C₄	—	13	24	no	—
240	C₂×C₄	—	13	2	no	—
241	C₂×C₄	—	13	6	no	—
242	C₂×C₄	—	26	12	no	—
243	C₂×C₄	—	52	6	no	—
244	C₂×C₄	—	52	12	no	—
245	C₂×C₄	—	52	6	no	—
246	C₂×C₄	C₂×C₄×D₁₃	1	12	yes	complemented
247	C₂×C₄	C₂³×D₁₃	1	3	yes	—
248	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂³×D₁₃	1	3	yes	—
250	C₂×C₄	C₂³×D₁₃	1	3	yes	—
251	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
253	C₂²	—	2	3	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	13	12	no	—
256	C₂²	—	26	6	no	—
257	C₂²	—	26	6	no	—
258	C₂²	—	52	24	no	—
259	C₂²	C₂²×C₄×D₁₃	1	6	yes	—
260	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
261	C₄	—	2	3	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	3	no	—
264	C₄	—	13	4	no	—
265	C₄	—	26	6	no	—
266	C₄	—	26	2	no	—
267	C₄	C₂²×C₄×D₁₃	1	6	yes	—
268	C₄	C₂²×C₄×D₁₃	1	2	yes	—
269	C₄	#88256	1	2	yes	direct product factor
270	C₂	—	2	6	no	—
271	C₂	—	26	12	no	—
272	C₂	C₂³×C₄×D₁₃	1	1	yes	characteristic
273	C₂	C₂×#88256	1	1	yes	characteristic
274	C₂	#11719648	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₃
commutator	C₂×C₁₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₄×C₁₃×#41	C₂
socle, cosocle	C₂²×C₁₃	C₂⁵
prime radical	C₂×#88256	C₂
hypercenter	C₄×#41	D₁₃
hypocenter	C₁₃	C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₃×#41	C₂	semidirect
3. #11719532	C₂	semidirect
4. #11719279	C₂	semidirect
5. C₄×#6131	C₂	semidirect
6. #11719250	C₂	semidirect
7. C₄×D₄×D₁₃	C₂	semidirect
8. #11719228	C₂	semidirect
9. #11719226	C₂	semidirect
10. C₄×#6127	C₂	semidirect
11. C₂×C₄×D₅₂	C₂	semidirect
13. #88256	C₄	direct (N × H)
18. C₂²×C₄×D₁₃	C₂²	semidirect
19. C₄×Q₈×C₁₃	C₂²	semidirect
20. C₄×Q₈×C₁₃	C₂²	semidirect
21. C₄×D₄×C₁₃	C₂²	semidirect
22. C₄×D₄×C₁₃	C₂²	semidirect
23. C₁₃×#117	C₂²	semidirect
24. C₂×C₄²×C₁₃	C₂²	semidirect
25. Q₈×Dic₁₃	C₂²	semidirect
26. D₄×Dic₁₃	C₂²	semidirect
27. C₄×C₂²⋊D₁₃	C₂²	semidirect
28. C₄×C₂²⋊D₁₃	C₂²	semidirect
29. C₂×#6103	C₂²	semidirect
30. #88180	C₂²	semidirect
31. C₂×C₄⋊Dic₁₃	C₂²	semidirect
32. #88147	C₂²	semidirect
33. #88147	C₂²	semidirect
34. #88146	C₂²	semidirect
35. C₄⋊C₄×D₁₃	C₂²	semidirect
36. #88135	C₂²	semidirect
37. C₂²⋊C₄×D₁₃	C₂²	semidirect
38. C₄×D₅₂	C₂²	semidirect
39. C₄×D₅₂	C₂²	semidirect
40. C₄×D₅₂	C₂²	semidirect
41. C₄×D₅₂	C₂²	semidirect
42. C₄×D₅₂	C₂²	semidirect
43. #88126	C₂²	semidirect
44. C₄²×D₁₃	C₂²	semidirect
45. C₄×Dic₂₆	C₂²	semidirect
46. C₄×Dic₂₆	C₂²	semidirect
48. C₁₃×#41	C₂×C₄	semidirect
56. #6131	C₂×C₄	semidirect
57. D₄×D₁₃	C₂×C₄	semidirect
58. #6127	C₂×C₄	semidirect
59. C₂×D₅₂	C₂×C₄	semidirect
91. C₂²×D₁₃	C₂²×C₄	semidirect
93. Q₈×C₁₃	C₂²×C₄	semidirect
94. D₄×C₁₃	C₂²×C₄	semidirect
97. C₂×C₄×C₁₃	C₂²×C₄	semidirect
104. C₂²⋊D₁₃	C₂²×C₄	semidirect
109. D₅₂	C₂²×C₄	semidirect
110. D₅₂	C₂²×C₄	semidirect
114. C₄×D₁₃	C₂²×C₄	semidirect
115. Dic₂₆	C₂²×C₄	semidirect
124. C₄×#41	D₁₃	semidirect
140. C₄×C₁₃	H(5,𝔽₂)	semidirect
154. C₄×Q₈	C₂×D₁₃	semidirect
155. C₄×Q₈	C₂×D₁₃	semidirect
167. C₄×D₄	C₂×D₁₃	semidirect
168. C₄×D₄	C₂×D₁₃	semidirect
172. #117	C₂×D₁₃	semidirect
178. C₂×C₄²	C₂×D₁₃	semidirect
187. #41	C₄×D₁₃	semidirect
224. C₁₃	C₄×H(5,𝔽₂)	semidirect
231. Q₈	C₂×C₄×D₁₃	semidirect
235. D₄	C₂×C₄×D₁₃	semidirect
246. C₂×C₄	C₂×C₄×D₁₃	semidirect
269. C₄	#88256	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×#88256	2
C₄×C₁₃×#41	2
#11719532	2
#11719279	2
C₄×#6131	2
#11719250	2
C₄×D₄×D₁₃	2
#11719228	2
#11719226	2
C₄×#6127	2
C₂×C₄×D₅₂	2
C₄×H(5,𝔽₂)	13

Subgroups — C4×#88256 #13128043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88256 #13128043