Subgroups — #13128610

Order 1664.

1748 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88256	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719539	C₂	1	4	yes	maximal, complemented
4	#11719537	C₂	1	3	yes	maximal, complemented
5	#11719535	C₂	1	3	yes	maximal, complemented
6	#41×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#11719510	C₂	1	3	yes	maximal, complemented
8	C₂×#88194	C₂	1	3	yes	maximal, complemented
9	D₄×C₂²⋊D₁₃	C₂	1	12	yes	maximal, complemented
10	C₁₃×H(5,𝔽₂)	—	2	8	no	—
11	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#41	C₂²	1	3	yes	complemented
13	C₂×C₁₃×#41	C₂²	1	2	yes	complemented
14	C₂²×D₄×C₁₃	C₂²	1	6	yes	complemented
15	C₂²×D₄×C₁₃	C₂²	1	3	yes	complemented
16	C₂²×C₂²⋊D₁₃	C₂²	1	12	yes	complemented
17	#88256	—	2	8	no	—
18	C₂×#6131	C₂²	1	2	yes	complemented
19	C₂×D₄×D₁₃	C₂²	1	6	yes	complemented
20	C₂×#6127	C₂²	1	3	yes	complemented
21	C₂²×D₅₂	C₂²	1	3	yes	complemented
22	#88207	C₂²	1	12	yes	complemented
23	#88201	C₂²	1	3	yes	complemented
24	Q₈×Dic₁₃	C₂²	1	2	yes	complemented
25	#88198	C₂²	1	1	yes	characteristic, complemented
26	#88194	C₂²	1	12	yes	complemented
27	#88194	C₂²	1	3	yes	complemented
28	#88194	C₂²	1	6	yes	complemented
29	#88193	C₂²	1	12	yes	complemented
30	#88192	C₂²	1	12	yes	complemented
31	#88191	C₂²	1	12	yes	complemented
32	#88190	C₂²	1	3	yes	complemented
33	D₄×Dic₁₃	C₂²	1	6	yes	complemented
34	#88184	C₂²	1	12	yes	complemented
35	C₄×C₂²⋊D₁₃	C₂²	1	12	yes	complemented
36	#88180	C₂²	1	3	yes	complemented
37	C₂×C₄×Dic₁₃	C₂²	1	3	yes	complemented
38	C₂⁴×C₁₃	C₂³	1	6	yes	—
39	C₂³×D₁₃	C₂³	1	6	yes	—
40	C₁₃×#41	—	2	8	no	—
41	C₁₃×#41	—	2	12	no	—
42	C₁₃×#41	D₄	1	8	yes	complemented
43	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃	—	2	24	no	—
46	C₂×D₄×C₁₃	—	2	12	no	—
47	C₂×D₄×C₁₃	C₂³	1	3	yes	—
48	C₂×D₄×C₁₃	C₂³	1	6	yes	—
49	C₂×D₄×C₁₃	C₂³	1	6	yes	—
50	C₂×D₄×C₁₃	C₂³	1	3	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
53	C₂×C₂²⋊D₁₃	—	4	24	no	—
54	C₂×C₂²⋊D₁₃	C₂³	1	12	yes	—
55	C₂×C₂²⋊D₁₃	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₁₃	C₂³	1	24	yes	—
57	C₂²×Dic₁₃	C₂³	1	6	yes	—
58	#6131	—	2	8	no	—
59	D₄×D₁₃	—	2	24	no	—
60	#6127	—	2	12	no	—
61	C₂×D₅₂	—	2	12	no	—
62	C₂×D₅₂	C₂³	1	6	yes	—
63	C₂×D₅₂	C₂³	1	3	yes	—
64	C₂×C₄×D₁₃	C₂³	1	6	yes	—
65	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
66	C₂²⋊Dic₁₃	C₂³	1	6	yes	—
67	C₂²⋊Dic₁₃	C₂³	1	12	yes	—
68	#6103	C₂³	1	12	yes	—
69	C₄⋊Dic₁₃	C₂³	1	6	yes	—
70	#6101	C₂³	1	4	yes	—
71	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
72	C₄×Dic₁₃	C₂³	1	3	yes	—
73	C₄×Dic₁₃	C₂³	1	6	yes	—
74	#3498	—	13	1	no	maximal, Sylow
75	C₂³×C₁₃	—	2	12	no	—
76	C₂³×C₁₃	—	4	12	no	—
77	C₂³×C₁₃	C₂⁴	1	6	yes	—
78	C₂³×C₁₃	C₂⁴	1	3	yes	—
79	C₂²×D₁₃	—	2	12	no	—
80	C₂²×D₁₃	—	4	12	no	—
81	C₂²×D₁₃	C₂⁴	1	6	yes	—
82	Q₈×C₁₃	—	2	2	no	—
83	Q₈×C₁₃	C₂×D₄	1	4	yes	complemented
84	D₄×C₁₃	—	2	12	no	—
85	D₄×C₁₃	—	2	6	no	—
86	D₄×C₁₃	—	2	12	no	—
87	D₄×C₁₃	C₂×D₄	1	12	yes	complemented
88	C₂×C₄×C₁₃	—	2	12	no	—
89	C₂×C₄×C₁₃	C₂×D₄	1	12	yes	complemented
90	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
91	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
93	C₂²⋊D₁₃	—	2	12	no	—
94	C₂²⋊D₁₃	—	2	24	no	—
95	C₂²⋊D₁₃	—	4	24	no	—
96	C₂×Dic₁₃	—	2	12	no	—
97	C₂×Dic₁₃	C₂⁴	1	6	yes	—
98	C₂×Dic₁₃	C₂⁴	1	2	yes	—
99	C₂×Dic₁₃	C₂⁴	1	2	yes	—
100	D₅₂	—	2	6	no	—
101	D₅₂	—	2	12	no	—
102	C₄×D₁₃	—	2	12	no	—
103	Dic₂₆	—	2	2	no	—
104	C₂×H(5,𝔽₂)	—	13	1	no	—
105	C₂×H(5,𝔽₂)	D₁₃	1	1	yes	characteristic, complemented
106	#550	—	13	4	no	—
107	D₄²	—	13	12	no	—
108	#535	—	13	3	no	—
109	#535	—	13	3	no	—
110	#532	—	13	3	no	—
111	C₂×#127	—	13	3	no	—
112	C₄×#41	—	13	1	no	—
113	C₂²×C₁₃	—	2	6	no	—
114	C₂²×C₁₃	—	2	6	no	—
115	C₂²×C₁₃	—	2	12	no	—
116	C₂²×C₁₃	—	8	12	no	—
117	C₂²×C₁₃	C₂²×D₄	1	6	yes	—
118	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
119	C₂×D₁₃	—	2	6	no	—
120	C₂×D₁₃	—	2	12	no	—
121	C₂×D₁₃	—	8	12	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	C₂²×D₄	1	2	yes	—
124	C₄×C₁₃	C₂²×D₄	1	6	yes	—
125	Dic₁₃	—	2	2	no	—
126	Dic₁₃	—	2	6	no	—
127	Dic₁₃	H(5,𝔽₂)	1	4	yes	complemented
128	H(5,𝔽₂)	—	2	8	no	—
129	H(5,𝔽₂)	—	26	8	no	—
130	C₂×#41	—	13	2	no	—
131	C₂×#41	—	13	3	no	—
132	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
133	C₂×#41	C₂×D₁₃	1	3	yes	complemented
134	C₂×#41	C₂×D₁₃	1	2	yes	complemented
135	C₂²×D₄	—	13	12	no	—
136	C₂²×D₄	—	13	6	no	—
137	C₂²×D₄	—	13	3	no	—
138	C₂²×D₄	C₂×D₁₃	1	3	yes	complemented
139	C₂²×D₄	C₂×D₁₃	1	6	yes	complemented
140	C₄⋊Q₈	—	13	1	no	—
141	#127	—	13	3	no	—
142	#127	—	13	6	no	—
143	#127	—	13	12	no	—
144	#124	—	13	3	no	—
145	#124	—	13	3	no	—
146	#121	—	13	12	no	—
147	#121	—	13	12	no	—
148	#121	—	13	12	no	—
149	#120	—	13	12	no	—
150	#120	—	13	12	no	—
151	C₄×Q₈	—	13	2	no	—
152	C₄×D₄	—	13	6	no	—
153	C₄×D₄	—	13	12	no	—
154	#117	—	13	3	no	—
155	C₂×C₄²	—	13	3	no	—
156	C₂×C₁₃	—	2	6	no	—
157	C₂×C₁₃	—	4	6	no	—
158	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
159	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
160	D₁₃	—	4	6	no	—
161	C₂⁴	—	13	6	no	—
162	C₂⁴	C₂²×D₁₃	1	6	yes	—
163	#41	—	2	8	no	—
164	#41	—	2	12	no	—
165	#41	—	26	8	no	—
166	#41	—	26	12	no	—
167	#41	C₂²⋊D₁₃	1	8	yes	complemented
168	C₂×Q₈	—	13	1	no	—
169	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
170	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×D₄	—	2	24	no	—
172	C₂×D₄	—	2	12	no	—
173	C₂×D₄	—	13	12	no	—
174	C₂×D₄	—	13	6	no	—
175	C₂×D₄	—	13	24	no	—
176	C₂×D₄	—	13	6	no	—
177	C₂×D₄	—	13	3	no	—
178	C₂×D₄	—	26	24	no	—
179	C₂×D₄	—	26	12	no	—
180	C₂×D₄	—	52	24	no	—
181	C₂×D₄	C₂²×D₁₃	1	6	yes	—
182	C₂×D₄	C₂²×D₁₃	1	6	yes	—
183	C₂×D₄	C₂²×D₁₃	1	3	yes	—
184	C₂×D₄	C₂²×D₁₃	1	3	yes	—
185	C₂²×C₄	—	13	6	no	—
186	C₂²×C₄	—	13	6	no	—
187	C₂²×C₄	C₂²×D₁₃	1	6	yes	—
188	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
189	C₄⋊C₄	—	13	6	no	—
190	C₄⋊C₄	—	13	4	no	—
191	C₂²⋊C₄	—	13	6	no	—
192	C₂²⋊C₄	—	13	12	no	—
193	C₂²⋊C₄	—	13	12	no	—
194	C₄²	—	13	1	no	—
195	C₄²	—	13	3	no	—
196	C₄²	—	13	6	no	—
197	C₁₃	#3498	1	1	yes	characteristic, complemented, Sylow
198	C₂³	—	2	12	no	—
199	C₂³	—	4	12	no	—
200	C₂³	—	13	6	no	—
201	C₂³	—	26	12	no	—
202	C₂³	—	52	12	no	—
203	C₂³	C₂³×D₁₃	1	6	yes	—
204	C₂³	C₂³×D₁₃	1	3	yes	—
205	Q₈	—	2	2	no	—
206	Q₈	—	26	2	no	—
207	Q₈	C₂×C₂²⋊D₁₃	1	4	yes	complemented
208	D₄	—	2	12	no	—
209	D₄	—	2	12	no	—
210	D₄	—	2	6	no	—
211	D₄	—	26	24	no	—
212	D₄	—	26	12	no	—
213	D₄	—	26	12	no	—
214	D₄	—	26	6	no	—
215	D₄	—	52	24	no	—
216	D₄	C₂×C₂²⋊D₁₃	1	12	yes	complemented
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	13	2	no	—
219	C₂×C₄	—	13	6	no	—
220	C₂×C₄	—	13	2	no	—
221	C₂×C₄	—	26	12	no	—
222	C₂×C₄	—	26	12	no	—
223	C₂×C₄	C₂×C₂²⋊D₁₃	1	12	yes	complemented
224	C₂×C₄	C₂³×D₁₃	1	2	yes	—
225	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₁₃	1	3	yes	—
227	C₂²	—	2	6	no	—
228	C₂²	—	2	12	no	—
229	C₂²	—	2	6	no	—
230	C₂²	—	8	12	no	—
231	C₂²	—	26	12	no	—
232	C₂²	—	26	6	no	—
233	C₂²	—	104	12	no	—
234	C₂²	C₂²×C₂²⋊D₁₃	1	6	yes	—
235	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	13	4	no	—
238	C₄	—	26	6	no	—
239	C₄	—	26	2	no	—
240	C₄	C₂²×C₂²⋊D₁₃	1	6	yes	—
241	C₄	C₂²×C₂²⋊D₁₃	1	2	yes	—
242	C₂	—	2	6	no	—
243	C₂	—	4	6	no	—
244	C₂	—	52	6	no	—
245	C₂	D₁₃×H(5,𝔽₂)	1	2	yes	—
246	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×H(5,𝔽₂)	C₂
hypercenter	C₂×H(5,𝔽₂)	D₁₃
hypocenter	C₁₃	#3498
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×H(5,𝔽₂)	C₂	semidirect
2. C₂×#88256	C₂	semidirect
3. #11719539	C₂	semidirect
4. #11719537	C₂	semidirect
5. #11719535	C₂	semidirect
6. #41×Dic₁₃	C₂	semidirect
7. #11719510	C₂	semidirect
8. C₂×#88194	C₂	semidirect
9. D₄×C₂²⋊D₁₃	C₂	semidirect
12. C₂×C₁₃×#41	C₂²	semidirect
13. C₂×C₁₃×#41	C₂²	semidirect
14. C₂²×D₄×C₁₃	C₂²	semidirect
15. C₂²×D₄×C₁₃	C₂²	semidirect
16. C₂²×C₂²⋊D₁₃	C₂²	semidirect
18. C₂×#6131	C₂²	semidirect
19. C₂×D₄×D₁₃	C₂²	semidirect
20. C₂×#6127	C₂²	semidirect
21. C₂²×D₅₂	C₂²	semidirect
22. #88207	C₂²	semidirect
23. #88201	C₂²	semidirect
24. Q₈×Dic₁₃	C₂²	semidirect
25. #88198	C₂²	semidirect
26. #88194	C₂²	semidirect
27. #88194	C₂²	semidirect
28. #88194	C₂²	semidirect
29. #88193	C₂²	semidirect
30. #88192	C₂²	semidirect
31. #88191	C₂²	semidirect
32. #88190	C₂²	semidirect
33. D₄×Dic₁₃	C₂²	semidirect
34. #88184	C₂²	semidirect
35. C₄×C₂²⋊D₁₃	C₂²	semidirect
36. #88180	C₂²	semidirect
37. C₂×C₄×Dic₁₃	C₂²	semidirect
42. C₁₃×#41	D₄	semidirect
83. Q₈×C₁₃	C₂×D₄	semidirect
87. D₄×C₁₃	C₂×D₄	semidirect
89. C₂×C₄×C₁₃	C₂×D₄	semidirect
105. C₂×H(5,𝔽₂)	D₁₃	semidirect
127. Dic₁₃	H(5,𝔽₂)	semidirect
133. C₂×#41	C₂×D₁₃	semidirect
134. C₂×#41	C₂×D₁₃	semidirect
138. C₂²×D₄	C₂×D₁₃	semidirect
139. C₂²×D₄	C₂×D₁₃	semidirect
167. #41	C₂²⋊D₁₃	semidirect
197. C₁₃	#3498	semidirect
207. Q₈	C₂×C₂²⋊D₁₃	semidirect
216. D₄	C₂×C₂²⋊D₁₃	semidirect
223. C₂×C₄	C₂×C₂²⋊D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×H(5,𝔽₂)	2
C₂×#88256	2
#11719539	2
#11719537	2
#11719535	2
#41×Dic₁₃	2
#11719510	2
C₂×#88194	2
D₄×C₂²⋊D₁₃	2
#3498	13