Subgroups — #13129241

Order 1664.

1258 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719535	C₂	1	4	yes	maximal, complemented
3	#11719439	C₂	1	2	yes	maximal, complemented
4	#11719422	C₂	1	2	yes	maximal, complemented
5	#11719388	C₂	1	4	yes	maximal, complemented
6	#11719312	C₂	1	8	yes	maximal, complemented
7	#11719268	C₂	1	4	yes	maximal, complemented
8	D₄×D₅₂	C₂	1	4	yes	maximal, complemented
9	#11719182	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₃×#41	C₂²	1	2	yes	complemented
11	C₂×D₄×D₁₃	C₂²	1	8	yes	complemented
12	C₂×#6127	C₂²	1	4	yes	complemented
13	C₂²×D₅₂	C₂²	1	4	yes	complemented
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#127	C₂²	1	1	yes	characteristic
16	C₁₃×#124	C₂²	1	2	yes	complemented
17	C₁₃×#121	C₂²	1	4	yes	complemented
18	C₄×D₄×C₁₃	C₂²	1	4	yes	complemented
19	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
20	#88201	C₂²	1	8	yes	complemented
21	#88194	C₂²	1	8	yes	complemented
22	#88192	C₂²	1	8	yes	complemented
23	#88191	C₂²	1	8	yes	complemented
24	#88191	C₂²	1	8	yes	complemented
25	#88184	C₂²	1	4	yes	complemented
26	#88184	C₂²	1	4	yes	complemented
27	#88184	C₂²	1	8	yes	complemented
28	#88180	C₂²	1	4	yes	complemented
29	#88149	C₂²	1	8	yes	complemented
30	#88146	C₂²	1	8	yes	complemented
31	#88139	C₂²	1	16	yes	complemented
32	#88136	C₂²	1	16	yes	complemented
33	#88129	C₂²	1	4	yes	complemented
34	#88128	C₂²	1	2	yes	—
35	C₄×D₅₂	C₂²	1	4	yes	complemented
36	C₄×D₅₂	C₂²	1	4	yes	complemented
37	#88123	C₂²	1	2	yes	complemented
38	C₂³×D₁₃	C₂³	1	8	yes	—
39	C₁₃×#41	—	4	4	no	—
40	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
41	C₂×D₄×C₁₃	C₂³	1	4	yes	—
42	C₂×D₄×C₁₃	C₂³	1	2	yes	—
43	C₂×D₄×C₁₃	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
47	C₂×C₂²⋊D₁₃	C₂³	1	16	yes	—
48	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
49	D₄×D₁₃	—	4	16	no	—
50	#6127	—	4	8	no	—
51	C₂×D₅₂	—	4	8	no	—
52	C₂×D₅₂	C₂³	1	4	yes	—
53	C₂×D₅₂	C₂³	1	4	yes	—
54	C₂×D₅₂	C₂³	1	4	yes	—
55	C₂×D₅₂	C₂³	1	8	yes	—
56	C₂×C₄×D₁₃	C₂³	1	8	yes	—
57	C₂×Dic₂₆	C₂³	1	4	yes	—
58	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
59	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
60	C₄²×C₁₃	C₂³	1	2	yes	—
61	C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
64	#6103	C₂³	1	8	yes	—
65	#6103	C₂³	1	8	yes	—
66	#6103	C₂³	1	16	yes	—
67	C₄⋊Dic₁₃	C₂³	1	4	yes	—
68	C₄⋊Dic₁₃	C₂³	1	4	yes	—
69	C₄×Dic₁₃	C₂³	1	8	yes	—
70	#3549	—	13	1	no	maximal, Sylow
71	C₂³×C₁₃	C₂⁴	1	4	yes	—
72	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²×D₁₃	—	4	8	no	—
74	C₂²×D₁₃	—	4	16	no	—
75	C₂²×D₁₃	C₂⁴	1	8	yes	—
76	Q₈×C₁₃	—	4	2	no	—
77	D₄×C₁₃	—	2	8	no	—
78	D₄×C₁₃	—	4	4	no	—
79	D₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	—	2	8	no	—
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
84	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
86	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
88	C₂²⋊D₁₃	—	4	8	no	—
89	C₂²⋊D₁₃	—	4	16	no	—
90	C₂×Dic₁₃	C₂⁴	1	8	yes	—
91	D₅₂	—	2	8	no	—
92	D₅₂	—	2	8	no	—
93	D₅₂	—	4	8	no	—
94	D₅₂	—	4	4	no	—
95	C₄×D₁₃	—	2	16	no	—
96	Dic₂₆	—	4	4	no	—
97	H(3,𝔽₄)	—	13	4	no	—
98	#555	—	13	2	no	—
99	#555	—	13	4	no	—
100	D₄²	—	13	2	no	—
101	D₄²	—	13	4	no	—
102	#535	—	13	4	no	—
103	#535	—	13	8	no	—
104	#532	—	13	2	no	—
105	#532	D₁₃	1	1	yes	characteristic, complemented
106	C₂²×C₁₃	—	2	2	no	—
107	C₂²×C₁₃	—	2	8	no	—
108	C₂²×C₁₃	—	2	1	no	—
109	C₂²×C₁₃	—	2	4	no	—
110	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
111	C₂×D₁₃	—	2	8	no	—
112	C₂×D₁₃	—	2	8	no	—
113	C₂×D₁₃	—	2	8	no	—
114	C₂×D₁₃	—	8	16	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	—	2	4	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
119	C₄×C₁₃	H(5,𝔽₂)	1	4	yes	—
120	Dic₁₃	—	2	8	no	—
121	C₂×#41	—	13	4	no	—
122	C₂×#41	C₂×D₁₃	1	2	yes	complemented
123	C₂²×D₄	—	13	8	no	—
124	C₂²×D₄	—	13	4	no	—
125	C₄⋊Q₈	—	13	2	no	—
126	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
127	#127	—	13	8	no	—
128	#127	—	13	2	no	—
129	#127	C₂×D₁₃	1	1	yes	characteristic
130	#124	—	13	8	no	—
131	#124	—	13	16	no	—
132	#124	—	13	4	no	—
133	#124	C₂×D₁₃	1	2	yes	complemented
134	#121	—	13	8	no	—
135	#121	—	13	4	no	—
136	#121	—	13	4	no	—
137	#121	—	13	8	no	—
138	#121	—	13	8	no	—
139	#121	C₂×D₁₃	1	4	yes	complemented
140	#120	—	13	8	no	—
141	#120	—	13	8	no	—
142	#120	—	13	16	no	—
143	C₄×D₄	—	13	4	no	—
144	C₄×D₄	—	13	4	no	—
145	C₄×D₄	C₂×D₁₃	1	4	yes	complemented
146	#117	—	13	4	no	—
147	#117	—	13	8	no	—
148	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic, complemented
149	C₂×C₁₃	—	4	4	no	—
150	C₂×C₁₃	—	4	1	no	—
151	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
153	D₁₃	—	4	8	no	—
154	C₂⁴	—	13	8	no	—
155	#41	—	4	4	no	—
156	#41	—	52	8	no	—
157	C₂×Q₈	—	13	4	no	—
158	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
159	C₂×D₄	—	13	16	no	—
160	C₂×D₄	—	13	8	no	—
161	C₂×D₄	—	13	4	no	—
162	C₂×D₄	—	13	4	no	—
163	C₂×D₄	—	13	4	no	—
164	C₂×D₄	—	13	8	no	—
165	C₂×D₄	—	52	8	no	—
166	C₂×D₄	—	52	16	no	—
167	C₂×D₄	C₂²×D₁₃	1	4	yes	—
168	C₂×D₄	C₂²×D₁₃	1	2	yes	—
169	C₂×D₄	C₂²×D₁₃	1	4	yes	—
170	C₂²×C₄	—	13	8	no	—
171	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
173	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
174	C₄⋊C₄	—	13	4	no	—
175	C₄⋊C₄	—	13	4	no	—
176	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
177	C₂²⋊C₄	—	13	8	no	—
178	C₂²⋊C₄	—	13	16	no	—
179	C₂²⋊C₄	—	13	8	no	—
180	C₂²⋊C₄	—	13	8	no	—
181	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
182	C₄²	—	13	8	no	—
183	C₄²	C₂²×D₁₃	1	1	yes	characteristic
184	C₄²	C₂²×D₁₃	1	2	yes	—
185	C₄²	C₂²×D₁₃	1	1	yes	characteristic
186	C₁₃	#3549	1	1	yes	characteristic, complemented, Sylow
187	C₂³	—	13	8	no	—
188	C₂³	—	52	8	no	—
189	C₂³	—	52	16	no	—
190	C₂³	C₂³×D₁₃	1	4	yes	—
191	C₂³	C₂³×D₁₃	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	52	4	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	2	no	—
196	D₄	—	4	4	no	—
197	D₄	—	26	8	no	—
198	D₄	—	26	8	no	—
199	D₄	—	52	8	no	—
200	D₄	—	52	4	no	—
201	D₄	—	52	16	no	—
202	D₄	—	52	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	13	8	no	—
207	C₂×C₄	—	26	16	no	—
208	C₂×C₄	C₂³×D₁₃	1	4	yes	—
209	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₁₃	1	2	yes	—
211	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
212	C₂×C₄	C₂³×D₁₃	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	26	8	no	—
218	C₂²	—	26	8	no	—
219	C₂²	—	26	8	no	—
220	C₂²	—	104	16	no	—
221	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	2	no	—
225	C₄	—	26	8	no	—
226	C₄	#88251	1	2	yes	—
227	C₄	#88256	1	4	yes	—
228	C₂	—	4	4	no	—
229	C₂	—	4	1	no	—
230	C₂	—	52	8	no	—
231	C₂	C₂×#88251	1	1	yes	characteristic
232	C₂	C₂×#88256	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#532	C₂
hypercenter	#532	D₁₃
hypocenter	C₁₃	#3549
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₁₃×#532	C₂	semidirect
2. #11719535	C₂	semidirect
3. #11719439	C₂	semidirect
4. #11719422	C₂	semidirect
5. #11719388	C₂	semidirect
6. #11719312	C₂	semidirect
7. #11719268	C₂	semidirect
8. D₄×D₅₂	C₂	semidirect
9. #11719182	C₂	semidirect
10. C₂×C₁₃×#41	C₂²	semidirect
11. C₂×D₄×D₁₃	C₂²	semidirect
12. C₂×#6127	C₂²	semidirect
13. C₂²×D₅₂	C₂²	semidirect
14. C₁₃×C₄⋊Q₈	C₂²	semidirect
16. C₁₃×#124	C₂²	semidirect
17. C₁₃×#121	C₂²	semidirect
18. C₄×D₄×C₁₃	C₂²	semidirect
19. C₂×C₄²×C₁₃	C₂²	semidirect
20. #88201	C₂²	semidirect
21. #88194	C₂²	semidirect
22. #88192	C₂²	semidirect
23. #88191	C₂²	semidirect
24. #88191	C₂²	semidirect
25. #88184	C₂²	semidirect
26. #88184	C₂²	semidirect
27. #88184	C₂²	semidirect
28. #88180	C₂²	semidirect
29. #88149	C₂²	semidirect
30. #88146	C₂²	semidirect
31. #88139	C₂²	semidirect
32. #88136	C₂²	semidirect
33. #88129	C₂²	semidirect
35. C₄×D₅₂	C₂²	semidirect
36. C₄×D₅₂	C₂²	semidirect
37. #88123	C₂²	semidirect
105. #532	D₁₃	semidirect
122. C₂×#41	C₂×D₁₃	semidirect
126. C₄⋊Q₈	C₂×D₁₃	semidirect
133. #124	C₂×D₁₃	semidirect
139. #121	C₂×D₁₃	semidirect
145. C₄×D₄	C₂×D₁₃	semidirect
148. C₂×C₄²	C₂×D₁₃	semidirect
186. C₁₃	#3549	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#532	2
#11719535	2
#11719439	2
#11719422	2
#11719388	2
#11719312	2
#11719268	2
D₄×D₅₂	2
#11719182	2
#3549	13