Subgroups — C₂×D₁₃×H(3,𝔽₄) #16769790

Order 3328.

10246 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719299	C₂	1	6	yes	maximal, complemented
2	C₂×#11719388	C₂	1	9	yes	maximal, complemented
3	D₁₃×H(3,𝔽₄)	C₂	1	32	yes	maximal, direct product factor
4	C₂×D₁₃×#124	C₂	1	9	yes	maximal, complemented
5	C₂×D₁₃×#120	C₂	1	6	yes	maximal, complemented
6	C₂×C₁₃×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₃	C₂²	1	2	yes	complemented
8	C₂²×Q₈×D₁₃	C₂²	1	3	yes	complemented
9	C₂²×D₄×D₁₃	C₂²	1	9	yes	—
10	C₁₃×H(3,𝔽₄)	C₂²	1	16	yes	complemented
11	C₂×C₁₃×#124	C₂²	1	9	yes	complemented
12	C₂×C₁₃×#120	C₂²	1	6	yes	complemented
13	C₂×#88207	C₂²	1	6	yes	complemented
14	C₂×#88201	C₂²	1	9	yes	complemented
15	C₂×#88191	C₂²	1	18	yes	complemented
16	C₂×#88190	C₂²	1	9	yes	complemented
17	#11719388	C₂²	1	144	yes	complemented
18	D₁₃×#124	C₂²	1	144	yes	complemented
19	#11719299	C₂²	1	96	yes	complemented
20	D₁₃×#120	C₂²	1	96	yes	complemented
21	C₂×#88139	C₂²	1	36	yes	complemented
22	C₂×#88136	C₂²	1	18	yes	complemented
23	C₂×C₂²⋊C₄×D₁₃	C₂²	1	18	yes	complemented
24	C₂×#88129	C₂²	1	9	yes	complemented
25	C₂×C₄²×D₁₃	C₂²	1	3	yes	complemented
26	C₂⁵×C₁₃	C₂³	1	2	yes	complemented
27	C₂⁴×D₁₃	—	4	24	no	—
28	C₂⁴×D₁₃	C₂³	1	6	yes	—
29	C₂⁴×D₁₃	C₂³	1	16	yes	complemented
30	C₂⁴×D₁₃	C₂³	1	6	yes	complemented
31	C₂²×Q₈×C₁₃	C₂³	1	3	yes	complemented
32	C₂²×D₄×C₁₃	C₂³	1	9	yes	—
33	C₂²×C₂²⋊D₁₃	C₂³	1	18	yes	—
34	C₂×Q₈×D₁₃	—	4	12	no	—
35	C₂×Q₈×D₁₃	C₂³	1	24	yes	complemented
36	C₂×D₄×D₁₃	—	4	36	no	—
37	C₂×D₄×D₁₃	C₂³	1	72	yes	—
38	C₂²×D₅₂	C₂³	1	9	yes	—
39	C₂²×C₄×D₁₃	C₂³	1	9	yes	—
40	C₂²×Dic₂₆	C₂³	1	9	yes	complemented
41	C₁₃×#124	C₂³	1	72	yes	complemented
42	C₁₃×#120	C₂³	1	48	yes	complemented
43	C₂×C₁₃×C₂²⋊C₄	C₂³	1	18	yes	complemented
44	C₂×C₄²×C₁₃	C₂³	1	3	yes	complemented
45	#88207	C₂³	1	48	yes	complemented
46	C₂×C₂²⋊Dic₁₃	C₂³	1	18	yes	complemented
47	#88201	C₂³	1	72	yes	complemented
48	#88191	C₂³	1	144	yes	complemented
49	#88190	C₂³	1	72	yes	complemented
50	C₂×#6103	C₂³	1	36	yes	complemented
51	C₂×C₄×Dic₁₃	C₂³	1	9	yes	complemented
52	#88139	C₂³	1	288	yes	complemented
53	#88136	C₂³	1	144	yes	complemented
54	C₂²⋊C₄×D₁₃	C₂³	1	144	yes	complemented
55	#88129	C₂³	1	72	yes	complemented
56	C₄²×D₁₃	C₂³	1	24	yes	complemented
57	C₂²×H(3,𝔽₄)	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	—	4	12	no	—
59	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
60	C₂⁴×C₁₃	C₂⁴	1	8	yes	complemented
61	C₂³×D₁₃	—	2	72	no	—
62	C₂³×D₁₃	—	4	96	no	—
63	C₂³×D₁₃	—	4	36	no	—
64	C₂³×D₁₃	—	4	128	no	—
65	C₂³×D₁₃	C₂⁴	1	24	yes	—
66	C₂³×D₁₃	C₂⁴	1	6	yes	—
67	C₂³×D₁₃	C₂⁴	1	1	yes	characteristic
68	C₂³×D₁₃	C₂⁴	1	24	yes	complemented
69	C₂×Q₈×C₁₃	—	4	6	no	—
70	C₂×Q₈×C₁₃	C₂⁴	1	12	yes	complemented
71	C₂×D₄×C₁₃	—	4	18	no	—
72	C₂×D₄×C₁₃	C₂⁴	1	36	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	9	yes	—
74	C₂×C₂²⋊D₁₃	—	4	36	no	—
75	C₂×C₂²⋊D₁₃	C₂⁴	1	72	yes	—
76	C₂²×Dic₁₃	C₂⁴	1	9	yes	—
77	Q₈×D₁₃	—	4	48	no	—
78	D₄×D₁₃	—	4	144	no	—
79	C₂×D₅₂	—	4	18	no	—
80	C₂×D₅₂	C₂⁴	1	36	yes	—
81	C₂×C₄×D₁₃	—	2	36	no	—
82	C₂×C₄×D₁₃	C₂⁴	1	36	yes	—
83	C₂×Dic₂₆	—	4	18	no	—
84	C₂×Dic₂₆	C₂⁴	1	36	yes	complemented
85	C₁₃×C₂²⋊C₄	C₂⁴	1	72	yes	complemented
86	C₄²×C₁₃	C₂⁴	1	12	yes	complemented
87	C₂²⋊Dic₁₃	C₂⁴	1	72	yes	complemented
88	#6103	C₂⁴	1	144	yes	complemented
89	C₄×Dic₁₃	C₂⁴	1	36	yes	complemented
90	C₂×H(3,𝔽₄)	—	13	32	no	—
91	C₂×H(3,𝔽₄)	—	13	9	no	—
92	C₂×H(3,𝔽₄)	—	13	6	no	—
93	C₂×H(3,𝔽₄)	D₁₃	1	1	yes	characteristic, direct product factor
94	C₂²×#124	—	13	9	no	—
95	C₂²×#120	—	13	6	no	—
96	C₂³×C₁₃	—	2	36	no	—
97	C₂³×C₁₃	—	4	24	no	—
98	C₂³×C₁₃	—	4	32	no	—
99	C₂³×C₁₃	C₂⁵	1	12	yes	—
100	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
101	C₂²×D₁₃	—	2	36	no	—
102	C₂²×D₁₃	—	2	144	no	—
103	C₂²×D₁₃	—	4	72	no	—
104	C₂²×D₁₃	—	4	96	no	—
105	C₂²×D₁₃	—	4	256	no	—
106	C₂²×D₁₃	—	4	96	no	—
107	C₂²×D₁₃	H(5,𝔽₂)	1	12	yes	—
108	C₂²×D₁₃	C₂⁵	1	12	yes	—
109	C₂²×D₁₃	C₂⁵	1	2	yes	—
110	Q₈×C₁₃	—	4	12	no	—
111	D₄×C₁₃	—	4	36	no	—
112	C₂×C₄×C₁₃	—	2	18	no	—
113	C₂×C₄×C₁₃	C₂⁵	1	18	yes	—
114	C₂²⋊D₁₃	—	4	72	no	—
115	C₂×Dic₁₃	—	2	18	no	—
116	C₂×Dic₁₃	C₂⁵	1	18	yes	—
117	D₅₂	—	4	36	no	—
118	C₄×D₁₃	—	2	72	no	—
119	Dic₂₆	—	4	36	no	—
120	C₂⁶	—	13	2	no	—
121	C₂³×Q₈	—	13	3	no	—
122	C₂³×D₄	—	13	9	no	—
123	H(3,𝔽₄)	—	13	144	no	—
124	H(3,𝔽₄)	—	13	96	no	—
125	H(3,𝔽₄)	C₂×D₁₃	1	16	yes	direct product factor
126	C₂×#124	—	13	9	no	—
127	C₂×#124	—	13	9	no	—
128	C₂×#124	—	13	144	no	—
129	C₂×#124	—	13	36	no	—
130	C₂×#124	—	13	9	no	—
131	C₂×#124	C₂×D₁₃	1	9	yes	complemented
132	C₂×#120	—	13	6	no	—
133	C₂×#120	—	13	18	no	—
134	C₂×#120	—	13	96	no	—
135	C₂×#120	—	13	18	no	—
136	C₂×#120	C₂×D₁₃	1	6	yes	complemented
137	C₂²×C₂²⋊C₄	—	13	18	no	—
138	C₂²×C₄²	—	13	3	no	—
139	C₂²×C₁₃	—	2	36	no	—
140	C₂²×C₁₃	—	4	32	no	—
141	C₂²×C₁₃	—	4	24	no	—
142	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	6	yes	—
143	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
144	C₂×D₁₃	—	2	36	no	—
145	C₂×D₁₃	—	4	96	no	—
146	C₂×D₁₃	—	4	24	no	—
147	C₂×D₁₃	—	4	96	no	—
148	C₂×D₁₃	H(3,𝔽₄)	1	16	yes	direct product factor
149	C₂×D₁₃	C₂×H(5,𝔽₂)	1	12	yes	—
150	C₄×C₁₃	—	2	18	no	—
151	Dic₁₃	—	2	18	no	—
152	C₂⁵	—	13	6	no	—
153	C₂⁵	—	13	6	no	—
154	C₂⁵	—	13	16	no	—
155	C₂⁵	—	52	24	no	—
156	C₂⁵	C₂²×D₁₃	1	2	yes	complemented
157	C₂²×Q₈	—	13	24	no	—
158	C₂²×Q₈	—	13	9	no	—
159	C₂²×Q₈	—	52	12	no	—
160	C₂²×Q₈	C₂²×D₁₃	1	3	yes	complemented
161	C₂²×D₄	—	13	18	no	—
162	C₂²×D₄	—	13	72	no	—
163	C₂²×D₄	—	13	9	no	—
164	C₂²×D₄	—	52	36	no	—
165	C₂²×D₄	C₂²×D₁₃	1	9	yes	—
166	C₂³×C₄	—	13	9	no	—
167	#124	—	13	72	no	—
168	#124	—	13	72	no	—
169	#124	—	13	288	no	—
170	#124	—	13	72	no	—
171	#124	C₂²×D₁₃	1	72	yes	complemented
172	#120	—	13	48	no	—
173	#120	—	13	144	no	—
174	#120	—	13	144	no	—
175	#120	C₂²×D₁₃	1	48	yes	complemented
176	C₂×C₂²⋊C₄	—	13	18	no	—
177	C₂×C₂²⋊C₄	—	13	36	no	—
178	C₂×C₂²⋊C₄	—	13	144	no	—
179	C₂×C₂²⋊C₄	C₂²×D₁₃	1	18	yes	complemented
180	C₂×C₄²	—	13	9	no	—
181	C₂×C₄²	—	13	24	no	—
182	C₂×C₄²	C₂²×D₁₃	1	3	yes	complemented
183	C₂×C₁₃	—	4	12	no	—
184	C₂×C₁₃	C₂×H(3,𝔽₄)	1	4	yes	complemented
185	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	3	yes	—
186	D₁₃	—	4	12	no	—
187	D₁₃	C₂×H(3,𝔽₄)	1	8	yes	direct product factor
188	C₂⁴	—	4	12	no	—
189	C₂⁴	—	13	1	no	—
190	C₂⁴	—	13	24	no	—
191	C₂⁴	—	13	6	no	—
192	C₂⁴	—	13	24	no	—
193	C₂⁴	—	26	72	no	—
194	C₂⁴	—	52	36	no	—
195	C₂⁴	—	52	96	no	—
196	C₂⁴	—	52	128	no	—
197	C₂⁴	C₂³×D₁₃	1	6	yes	—
198	C₂⁴	C₂³×D₁₃	1	8	yes	complemented
199	C₂×Q₈	—	4	6	no	—
200	C₂×Q₈	—	13	36	no	—
201	C₂×Q₈	—	52	18	no	—
202	C₂×Q₈	—	52	48	no	—
203	C₂×Q₈	C₂³×D₁₃	1	12	yes	complemented
204	C₂×D₄	—	4	18	no	—
205	C₂×D₄	—	13	72	no	—
206	C₂×D₄	—	13	36	no	—
207	C₂×D₄	—	52	18	no	—
208	C₂×D₄	—	52	144	no	—
209	C₂×D₄	—	52	36	no	—
210	C₂×D₄	C₂³×D₁₃	1	36	yes	—
211	C₂²×C₄	—	13	9	no	—
212	C₂²×C₄	—	13	36	no	—
213	C₂²×C₄	—	26	36	no	—
214	C₂²×C₄	C₂³×D₁₃	1	9	yes	—
215	C₂²⋊C₄	—	13	72	no	—
216	C₂²⋊C₄	—	13	144	no	—
217	C₂²⋊C₄	C₂³×D₁₃	1	72	yes	complemented
218	C₄²	—	13	36	no	—
219	C₄²	C₂³×D₁₃	1	12	yes	complemented
220	C₁₃	C₂²×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
221	C₂³	—	2	36	no	—
222	C₂³	—	4	24	no	—
223	C₂³	—	4	32	no	—
224	C₂³	—	13	12	no	—
225	C₂³	—	13	2	no	—
226	C₂³	—	13	12	no	—
227	C₂³	—	26	144	no	—
228	C₂³	—	26	36	no	—
229	C₂³	—	52	72	no	—
230	C₂³	—	52	96	no	—
231	C₂³	—	52	256	no	—
232	C₂³	—	52	96	no	—
233	C₂³	C₂⁴×D₁₃	1	12	yes	—
234	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
235	Q₈	—	4	12	no	—
236	Q₈	—	52	36	no	—
237	D₄	—	4	36	no	—
238	D₄	—	52	36	no	—
239	D₄	—	52	72	no	—
240	C₂×C₄	—	2	18	no	—
241	C₂×C₄	—	13	18	no	—
242	C₂×C₄	—	26	72	no	—
243	C₂×C₄	—	26	18	no	—
244	C₂×C₄	C₂⁴×D₁₃	1	18	yes	—
245	C₂²	—	2	36	no	—
246	C₂²	—	4	32	no	—
247	C₂²	—	4	24	no	—
248	C₂²	—	13	12	no	—
249	C₂²	—	13	16	no	—
250	C₂²	—	26	36	no	—
251	C₂²	—	52	96	no	—
252	C₂²	—	52	96	no	—
253	C₂²	—	52	24	no	—
254	C₂²	D₁₃×H(5,𝔽₂)	1	6	yes	—
255	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
256	C₄	—	2	18	no	—
257	C₄	—	26	18	no	—
258	C₂	—	4	12	no	—
259	C₂	—	13	8	no	—
260	C₂	—	52	12	no	—
261	C₂	D₁₃×H(3,𝔽₄)	1	4	yes	direct product factor
262	C₂	C₂×D₁₃×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂×C₁₃×H(3,𝔽₄)	C₂
socle	C₂³×C₁₃	C₂⁵
hypercenter	C₂×H(3,𝔽₄)	D₁₃
hypocenter	C₁₃	C₂²×H(3,𝔽₄)
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. C₂×#11719299	C₂	semidirect
2. C₂×#11719388	C₂	semidirect
3. D₁₃×H(3,𝔽₄)	C₂	direct (N × H)
4. C₂×D₁₃×#124	C₂	semidirect
5. C₂×D₁₃×#120	C₂	semidirect
6. C₂×C₁₃×H(3,𝔽₄)	C₂	semidirect
7. C₂⁵×D₁₃	C₂²	semidirect
8. C₂²×Q₈×D₁₃	C₂²	semidirect
10. C₁₃×H(3,𝔽₄)	C₂²	semidirect
11. C₂×C₁₃×#124	C₂²	semidirect
12. C₂×C₁₃×#120	C₂²	semidirect
13. C₂×#88207	C₂²	semidirect
14. C₂×#88201	C₂²	semidirect
15. C₂×#88191	C₂²	semidirect
16. C₂×#88190	C₂²	semidirect
17. #11719388	C₂²	semidirect
18. D₁₃×#124	C₂²	semidirect
19. #11719299	C₂²	semidirect
20. D₁₃×#120	C₂²	semidirect
21. C₂×#88139	C₂²	semidirect
22. C₂×#88136	C₂²	semidirect
23. C₂×C₂²⋊C₄×D₁₃	C₂²	semidirect
24. C₂×#88129	C₂²	semidirect
25. C₂×C₄²×D₁₃	C₂²	semidirect
26. C₂⁵×C₁₃	C₂³	semidirect
29. C₂⁴×D₁₃	C₂³	semidirect
30. C₂⁴×D₁₃	C₂³	semidirect
31. C₂²×Q₈×C₁₃	C₂³	semidirect
35. C₂×Q₈×D₁₃	C₂³	semidirect
40. C₂²×Dic₂₆	C₂³	semidirect
41. C₁₃×#124	C₂³	semidirect
42. C₁₃×#120	C₂³	semidirect
43. C₂×C₁₃×C₂²⋊C₄	C₂³	semidirect
44. C₂×C₄²×C₁₃	C₂³	semidirect
45. #88207	C₂³	semidirect
46. C₂×C₂²⋊Dic₁₃	C₂³	semidirect
47. #88201	C₂³	semidirect
48. #88191	C₂³	semidirect
49. #88190	C₂³	semidirect
50. C₂×#6103	C₂³	semidirect
51. C₂×C₄×Dic₁₃	C₂³	semidirect
52. #88139	C₂³	semidirect
53. #88136	C₂³	semidirect
54. C₂²⋊C₄×D₁₃	C₂³	semidirect
55. #88129	C₂³	semidirect
56. C₄²×D₁₃	C₂³	semidirect
60. C₂⁴×C₁₃	C₂⁴	semidirect
68. C₂³×D₁₃	C₂⁴	semidirect
70. C₂×Q₈×C₁₃	C₂⁴	semidirect
84. C₂×Dic₂₆	C₂⁴	semidirect
85. C₁₃×C₂²⋊C₄	C₂⁴	semidirect
86. C₄²×C₁₃	C₂⁴	semidirect
87. C₂²⋊Dic₁₃	C₂⁴	semidirect
88. #6103	C₂⁴	semidirect
89. C₄×Dic₁₃	C₂⁴	semidirect
93. C₂×H(3,𝔽₄)	D₁₃	direct (N × H)
125. H(3,𝔽₄)	C₂×D₁₃	direct (N × H)
131. C₂×#124	C₂×D₁₃	semidirect
136. C₂×#120	C₂×D₁₃	semidirect
148. C₂×D₁₃	H(3,𝔽₄)	direct (N × H)
156. C₂⁵	C₂²×D₁₃	semidirect
160. C₂²×Q₈	C₂²×D₁₃	semidirect
171. #124	C₂²×D₁₃	semidirect
175. #120	C₂²×D₁₃	semidirect
179. C₂×C₂²⋊C₄	C₂²×D₁₃	semidirect
182. C₂×C₄²	C₂²×D₁₃	semidirect
184. C₂×C₁₃	C₂×H(3,𝔽₄)	semidirect
187. D₁₃	C₂×H(3,𝔽₄)	direct (N × H)
198. C₂⁴	C₂³×D₁₃	semidirect
203. C₂×Q₈	C₂³×D₁₃	semidirect
217. C₂²⋊C₄	C₂³×D₁₃	semidirect
219. C₄²	C₂³×D₁₃	semidirect
220. C₁₃	C₂²×H(3,𝔽₄)	semidirect
261. C₂	D₁₃×H(3,𝔽₄)	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×#11719299	2
C₂×#11719388	2
D₁₃×H(3,𝔽₄)	2
C₂×D₁₃×#124	2
C₂×D₁₃×#120	2
C₂×C₁₃×H(3,𝔽₄)	2
C₂²×H(3,𝔽₄)	13

Subgroups — C2×D13×H(3,𝔽4) #16769790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₃×H(3,𝔽₄) #16769790