Subgroups — #13128630

Order 1664.

1540 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88256	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719537	C₂	1	3	yes	maximal, complemented
4	#11719535	C₂	1	3	yes	maximal, complemented
5	#11719534	C₂	1	12	yes	maximal, complemented
6	#41×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#11719522	C₂	1	4	yes	maximal, complemented
8	#11719521	C₂	1	3	yes	maximal, complemented
9	C₂×#88201	C₂	1	3	yes	maximal, complemented
10	C₁₃×2₋¹⁺⁴	—	2	8	no	—
11	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#41	C₂²	1	6	yes	complemented
13	C₂×C₁₃×#41	C₂²	1	3	yes	complemented
14	C₂²×Q₈×C₁₃	C₂²	1	3	yes	complemented
15	C₂²×Q₈×C₁₃	C₂²	1	2	yes	complemented
16	#88256	—	2	8	no	—
17	C₂×#6131	C₂²	1	2	yes	—
18	C₂×D₄×D₁₃	C₂²	1	6	yes	—
19	C₂×#6127	C₂²	1	3	yes	—
20	C₂²×D₅₂	C₂²	1	3	yes	—
21	#88201	C₂²	1	3	yes	complemented
22	#88201	C₂²	1	6	yes	complemented
23	#88201	C₂²	1	12	yes	complemented
24	#88200	C₂²	1	12	yes	complemented
25	Q₈×Dic₁₃	C₂²	1	2	yes	complemented
26	#88198	C₂²	1	3	yes	complemented
27	#88194	C₂²	1	3	yes	complemented
28	#88193	C₂²	1	12	yes	complemented
29	#88191	C₂²	1	12	yes	complemented
30	#88190	C₂²	1	1	yes	characteristic, complemented
31	D₄×Dic₁₃	C₂²	1	6	yes	complemented
32	#88184	C₂²	1	12	yes	complemented
33	#88183	C₂²	1	12	yes	complemented
34	C₄×C₂²⋊D₁₃	C₂²	1	12	yes	complemented
35	C₂×#6103	C₂²	1	12	yes	complemented
36	#88180	C₂²	1	3	yes	complemented
37	C₂×C₄×Dic₁₃	C₂²	1	3	yes	complemented
38	C₂³×D₁₃	C₂³	1	6	yes	—
39	C₁₃×#41	—	2	24	no	—
40	C₁₃×#41	—	2	12	no	—
41	C₁₃×#41	D₄	1	8	yes	complemented
42	C₂×Q₈×C₁₃	—	2	8	no	—
43	C₂×Q₈×C₁₃	—	2	12	no	—
44	C₂×Q₈×C₁₃	C₂³	1	3	yes	—
45	C₂×Q₈×C₁₃	C₂³	1	6	yes	—
46	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₃	C₂³	1	6	yes	—
49	C₂×D₄×C₁₃	C₂³	1	3	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
53	C₂×C₂²⋊D₁₃	C₂³	1	6	yes	—
54	C₂×C₂²⋊D₁₃	C₂³	1	12	yes	—
55	C₂²×Dic₁₃	C₂³	1	6	yes	—
56	#6131	—	2	8	no	—
57	D₄×D₁₃	—	2	24	no	—
58	#6127	—	2	12	no	—
59	C₂×D₅₂	—	2	12	no	—
60	C₂×D₅₂	C₂³	1	6	yes	—
61	C₂×D₅₂	C₂³	1	3	yes	—
62	C₂×C₄×D₁₃	C₂³	1	6	yes	—
63	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₁₃	C₂³	1	6	yes	—
65	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
66	#6103	C₂³	1	12	yes	—
67	#6103	C₂³	1	24	yes	—
68	C₄⋊Dic₁₃	C₂³	1	6	yes	—
69	#6101	C₂³	1	12	yes	—
70	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
71	C₄×Dic₁₃	C₂³	1	3	yes	—
72	C₄×Dic₁₃	C₂³	1	6	yes	—
73	#3500	—	13	1	no	maximal, Sylow
74	C₂³×C₁₃	C₂⁴	1	2	yes	—
75	C₂³×C₁₃	C₂⁴	1	3	yes	—
76	C₂²×D₁₃	—	2	12	no	—
77	C₂²×D₁₃	—	4	12	no	—
78	C₂²×D₁₃	C₂⁴	1	6	yes	—
79	Q₈×C₁₃	—	2	6	no	—
80	Q₈×C₁₃	—	2	12	no	—
81	Q₈×C₁₃	C₂×D₄	1	4	yes	—
82	D₄×C₁₃	—	2	12	no	—
83	D₄×C₁₃	—	2	2	no	—
84	D₄×C₁₃	C₂×D₄	1	12	yes	—
85	C₂×C₄×C₁₃	—	2	12	no	—
86	C₂×C₄×C₁₃	—	2	12	no	—
87	C₂×C₄×C₁₃	C₂×D₄	1	12	yes	—
88	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
89	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
91	C₂²⋊D₁₃	—	2	12	no	—
92	C₂²⋊D₁₃	—	4	12	no	—
93	C₂×Dic₁₃	—	4	6	no	—
94	C₂×Dic₁₃	C₂⁴	1	2	yes	—
95	C₂×Dic₁₃	C₂⁴	1	6	yes	—
96	C₂×Dic₁₃	C₂⁴	1	2	yes	—
97	D₅₂	—	2	6	no	—
98	D₅₂	—	2	12	no	—
99	C₄×D₁₃	—	2	12	no	—
100	Dic₂₆	—	2	2	no	—
101	C₂×2₋¹⁺⁴	D₁₃	1	1	yes	characteristic, complemented
102	C₂×H(5,𝔽₂)	—	13	1	no	—
103	#548	—	13	4	no	—
104	#546	—	13	12	no	—
105	#536	—	13	3	no	—
106	#535	—	13	3	no	—
107	#532	—	13	3	no	—
108	C₂×#124	—	13	3	no	—
109	C₄×#41	—	13	1	no	—
110	C₂²×C₁₃	—	2	6	no	—
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	—	2	4	no	—
113	C₂²×C₁₃	C₂²×D₄	1	6	yes	—
114	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
115	C₂×D₁₃	—	2	6	no	—
116	C₂×D₁₃	—	2	12	no	—
117	C₂×D₁₃	—	8	12	no	—
118	C₄×C₁₃	—	2	6	no	—
119	C₄×C₁₃	C₂²×D₄	1	2	yes	—
120	C₄×C₁₃	C₂²×D₄	1	6	yes	—
121	Dic₁₃	—	2	2	no	—
122	Dic₁₃	—	2	2	no	—
123	Dic₁₃	—	2	6	no	—
124	2₋¹⁺⁴	—	2	8	no	—
125	H(5,𝔽₂)	—	26	8	no	—
126	C₂×#41	—	13	2	no	—
127	C₂×#41	—	13	3	no	—
128	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
129	C₂×#41	C₂×D₁₃	1	3	yes	complemented
130	C₂×#41	C₂×D₁₃	1	6	yes	complemented
131	C₂²×Q₈	C₂×D₁₃	1	3	yes	complemented
132	C₂²×Q₈	C₂×D₁₃	1	2	yes	complemented
133	C₂²×D₄	—	13	6	no	—
134	C₂²×D₄	—	13	3	no	—
135	C₄⋊Q₈	—	13	3	no	—
136	#127	—	13	3	no	—
137	#124	—	13	3	no	—
138	#124	—	13	6	no	—
139	#124	—	13	12	no	—
140	#124	—	13	1	no	—
141	#123	—	13	12	no	—
142	C₂²⋊Q₈	—	13	12	no	—
143	#121	—	13	12	no	—
144	#121	—	13	12	no	—
145	#120	—	13	12	no	—
146	C₄×Q₈	—	13	2	no	—
147	C₄×D₄	—	13	6	no	—
148	C₄×D₄	—	13	12	no	—
149	#117	—	13	3	no	—
150	C₂×C₂²⋊C₄	—	13	12	no	—
151	C₂×C₄²	—	13	3	no	—
152	C₂×C₁₃	—	2	6	no	—
153	C₂×C₁₃	—	4	2	no	—
154	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
155	C₂×C₁₃	#585	1	2	yes	—
156	D₁₃	—	4	6	no	—
157	C₂⁴	—	13	6	no	—
158	#41	—	2	24	no	—
159	#41	—	2	12	no	—
160	#41	—	26	8	no	—
161	#41	—	26	12	no	—
162	#41	C₂²⋊D₁₃	1	8	yes	complemented
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	—	2	12	no	—
165	C₂×Q₈	—	13	1	no	—
166	C₂×Q₈	C₂²×D₁₃	1	6	yes	—
167	C₂×Q₈	C₂²×D₁₃	1	3	yes	—
168	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×D₄	—	13	6	no	—
170	C₂×D₄	—	13	12	no	—
171	C₂×D₄	—	13	6	no	—
172	C₂×D₄	—	13	3	no	—
173	C₂×D₄	—	26	24	no	—
174	C₂×D₄	—	26	12	no	—
175	C₂×D₄	C₂²×D₁₃	1	6	yes	—
176	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×D₄	C₂²×D₁₃	1	3	yes	—
178	C₂²×C₄	—	13	6	no	—
179	C₂²×C₄	—	13	6	no	—
180	C₂²×C₄	C₂²×D₁₃	1	6	yes	—
181	C₂²×C₄	C₂²×D₁₃	1	6	yes	—
182	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
183	C₄⋊C₄	—	13	6	no	—
184	C₄⋊C₄	—	13	12	no	—
185	C₂²⋊C₄	—	13	6	no	—
186	C₂²⋊C₄	—	13	4	no	—
187	C₂²⋊C₄	—	13	12	no	—
188	C₂²⋊C₄	—	13	24	no	—
189	C₄²	—	13	1	no	—
190	C₄²	—	13	3	no	—
191	C₄²	—	13	6	no	—
192	C₁₃	#3500	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	13	6	no	—
194	C₂³	—	26	12	no	—
195	C₂³	—	52	12	no	—
196	C₂³	C₂³×D₁₃	1	2	yes	—
197	C₂³	C₂³×D₁₃	1	3	yes	—
198	Q₈	—	2	12	no	—
199	Q₈	—	2	6	no	—
200	Q₈	—	26	2	no	—
201	Q₈	C₂×C₂²⋊D₁₃	1	4	yes	—
202	D₄	—	2	12	no	—
203	D₄	—	2	2	no	—
204	D₄	—	26	12	no	—
205	D₄	—	26	12	no	—
206	D₄	—	26	6	no	—
207	D₄	—	52	12	no	—
208	D₄	C₂×C₂²⋊D₁₃	1	12	yes	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	2	12	no	—
211	C₂×C₄	—	13	2	no	—
212	C₂×C₄	—	13	6	no	—
213	C₂×C₄	—	13	2	no	—
214	C₂×C₄	—	26	12	no	—
215	C₂×C₄	—	52	6	no	—
216	C₂×C₄	C₂×C₂²⋊D₁₃	1	12	yes	—
217	C₂×C₄	C₂³×D₁₃	1	6	yes	—
218	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂³×D₁₃	1	3	yes	—
220	C₂²	—	2	6	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	26	12	no	—
224	C₂²	—	26	6	no	—
225	C₂²	—	104	12	no	—
226	C₂²	C₂²×C₂²⋊D₁₃	1	6	yes	—
227	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
228	C₄	—	2	6	no	—
229	C₄	—	26	2	no	—
230	C₄	—	26	6	no	—
231	C₄	—	26	2	no	—
232	C₄	C₂²×C₂²⋊D₁₃	1	6	yes	—
233	C₄	C₂²×C₂²⋊D₁₃	1	2	yes	—
234	C₂	—	2	6	no	—
235	C₂	—	4	2	no	—
236	C₂	—	52	6	no	—
237	C₂	#11719652	1	2	yes	—
238	C₂	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₂×C₁₃×2₋¹⁺⁴	C₂
hypercenter	C₂×2₋¹⁺⁴	D₁₃
hypocenter	C₁₃	#3500
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×2₋¹⁺⁴	C₂	semidirect
2. C₂×#88256	C₂	semidirect
3. #11719537	C₂	semidirect
4. #11719535	C₂	semidirect
5. #11719534	C₂	semidirect
6. #41×Dic₁₃	C₂	semidirect
7. #11719522	C₂	semidirect
8. #11719521	C₂	semidirect
9. C₂×#88201	C₂	semidirect
12. C₂×C₁₃×#41	C₂²	semidirect
13. C₂×C₁₃×#41	C₂²	semidirect
14. C₂²×Q₈×C₁₃	C₂²	semidirect
15. C₂²×Q₈×C₁₃	C₂²	semidirect
21. #88201	C₂²	semidirect
22. #88201	C₂²	semidirect
23. #88201	C₂²	semidirect
24. #88200	C₂²	semidirect
25. Q₈×Dic₁₃	C₂²	semidirect
26. #88198	C₂²	semidirect
27. #88194	C₂²	semidirect
28. #88193	C₂²	semidirect
29. #88191	C₂²	semidirect
30. #88190	C₂²	semidirect
31. D₄×Dic₁₃	C₂²	semidirect
32. #88184	C₂²	semidirect
33. #88183	C₂²	semidirect
34. C₄×C₂²⋊D₁₃	C₂²	semidirect
35. C₂×#6103	C₂²	semidirect
36. #88180	C₂²	semidirect
37. C₂×C₄×Dic₁₃	C₂²	semidirect
41. C₁₃×#41	D₄	semidirect
101. C₂×2₋¹⁺⁴	D₁₃	semidirect
129. C₂×#41	C₂×D₁₃	semidirect
130. C₂×#41	C₂×D₁₃	semidirect
131. C₂²×Q₈	C₂×D₁₃	semidirect
132. C₂²×Q₈	C₂×D₁₃	semidirect
162. #41	C₂²⋊D₁₃	semidirect
192. C₁₃	#3500	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×2₋¹⁺⁴	2
C₂×#88256	2
#11719537	2
#11719535	2
#11719534	2
#41×Dic₁₃	2
#11719522	2
#11719521	2
C₂×#88201	2
#3500	13