Subgroups — #13128534

Order 1664.

1052 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88183	C₂	1	2	yes	maximal, complemented
3	#11719236	C₂	1	8	yes	maximal, complemented
4	#11719232	C₂	1	8	yes	maximal, complemented
5	#11719220	C₂	1	4	yes	maximal, complemented
6	#11719198	C₂	1	1	yes	maximal, characteristic
7	#11719191	C₂	1	2	yes	maximal, complemented
8	#11719188	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719184	C₂	1	4	yes	maximal, complemented
10	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊D₁₃	C₂²	1	2	yes	—
12	C₁₃×#117	C₂²	1	8	yes	complemented
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	complemented
14	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂²	1	2	yes	complemented
17	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
18	#88184	C₂²	1	4	yes	—
19	#88183	C₂²	1	8	yes	—
20	#88183	C₂²	1	4	yes	complemented
21	#88183	C₂²	1	4	yes	complemented
22	C₄×C₂²⋊D₁₃	C₂²	1	8	yes	complemented
23	C₂×#6103	C₂²	1	4	yes	complemented
24	#88178	C₂²	1	4	yes	complemented
25	C₂×#6101	C₂²	1	4	yes	complemented
26	#88152	C₂²	1	8	yes	complemented
27	#88150	C₂²	1	8	yes	complemented
28	#88148	C₂²	1	8	yes	complemented
29	#88143	C₂²	1	8	yes	complemented
30	#88140	C₂²	1	4	yes	—
31	#88136	C₂²	1	4	yes	—
32	#88135	C₂²	1	8	yes	—
33	C₂²⋊C₄×D₁₃	C₂²	1	4	yes	complemented
34	#88132	C₂²	1	8	yes	complemented
35	#88131	C₂²	1	4	yes	complemented
36	#88130	C₂²	1	8	yes	complemented
37	#88129	C₂²	1	4	yes	complemented
38	C₄×D₅₂	C₂²	1	4	yes	complemented
39	#88126	C₂²	1	8	yes	complemented
40	#88124	C₂²	1	4	yes	complemented
41	C₄×Dic₂₆	C₂²	1	4	yes	complemented
42	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
43	C₂³×D₁₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂×C₂²⋊D₁₃	—	2	8	no	—
51	C₂×C₂²⋊D₁₃	C₂³	1	4	yes	—
52	C₂×C₂²⋊D₁₃	C₂³	1	4	yes	—
53	C₂²×Dic₁₃	C₂³	1	2	yes	—
54	C₂²×Dic₁₃	C₂³	1	4	yes	—
55	C₂×D₅₂	C₂³	1	4	yes	—
56	C₂×C₄×D₁₃	C₂³	1	8	yes	—
57	C₂×Dic₂₆	C₂³	1	4	yes	—
58	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
59	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
60	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
61	C₄²×C₁₃	C₂³	1	4	yes	—
62	C₄²×C₁₃	C₂³	1	4	yes	—
63	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
64	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
65	#6103	C₂³	1	8	yes	—
66	#6103	C₂³	1	8	yes	—
67	#6103	C₂³	1	8	yes	—
68	#6103	C₂³	1	8	yes	—
69	C₄⋊Dic₁₃	C₂³	1	4	yes	—
70	C₄⋊Dic₁₃	C₂³	1	4	yes	—
71	#6101	C₂³	1	8	yes	—
72	#6101	C₂³	1	8	yes	—
73	#6101	C₂³	1	8	yes	—
74	#6101	C₂³	1	8	yes	—
75	C₄×Dic₁₃	C₂³	1	8	yes	—
76	#3491	—	13	1	no	maximal, Sylow
77	C₂³×C₁₃	—	4	2	no	—
78	C₂³×C₁₃	#41	1	4	yes	—
79	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₂³×C₁₃	C₂⁴	1	2	yes	—
81	C₂²×D₁₃	—	2	4	no	—
82	C₂²×D₁₃	—	4	4	no	—
83	C₂²×D₁₃	C₂⁴	1	4	yes	—
84	C₂×C₄×C₁₃	—	2	8	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
90	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
91	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
92	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
93	C₂²⋊D₁₃	—	2	8	no	—
94	C₂²⋊D₁₃	—	2	8	no	—
95	C₂×Dic₁₃	—	2	4	no	—
96	C₂×Dic₁₃	—	4	4	no	—
97	C₂×Dic₁₃	C₂⁴	1	4	yes	—
98	C₂×Dic₁₃	C₂⁴	1	8	yes	—
99	D₅₂	—	4	4	no	—
100	C₄×D₁₃	—	4	8	no	—
101	Dic₂₆	—	4	4	no	—
102	#552	—	13	8	no	—
103	#551	—	13	8	no	—
104	#543	—	13	1	no	—
105	#539	—	13	4	no	—
106	#534	—	13	1	no	—
107	#529	—	13	4	no	—
108	C₂×#123	—	13	2	no	—
109	#518	—	13	2	no	—
110	C₂×#117	D₁₃	1	1	yes	characteristic, complemented
111	C₂²×C₁₃	—	2	4	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	4	4	no	—
114	C₂²×C₁₃	C₂×#41	1	4	yes	—
115	C₂²×C₁₃	C₂×#41	1	2	yes	—
116	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
117	C₂×D₁₃	—	2	4	no	—
118	C₂×D₁₃	—	2	8	no	—
119	C₂×D₁₃	—	8	4	no	—
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	2	4	no	—
123	C₄×C₁₃	—	2	4	no	—
124	Dic₁₃	—	2	4	no	—
125	Dic₁₃	—	2	8	no	—
126	C₂²×D₄	—	13	2	no	—
127	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
128	#126	—	13	8	no	—
129	#126	—	13	8	no	—
130	#125	—	13	8	no	—
131	#125	—	13	4	no	—
132	#124	—	13	4	no	—
133	#123	—	13	4	no	—
134	#123	—	13	4	no	—
135	#123	—	13	8	no	—
136	#123	—	13	8	no	—
137	#123	—	13	4	no	—
138	C₂²⋊Q₈	—	13	4	no	—
139	C₂²⋊Q₈	—	13	8	no	—
140	C₂²⋊Q₈	—	13	8	no	—
141	#121	—	13	4	no	—
142	#120	—	13	4	no	—
143	C₄×Q₈	—	13	4	no	—
144	C₄×D₄	—	13	8	no	—
145	C₄×D₄	—	13	8	no	—
146	C₄×D₄	—	13	4	no	—
147	#117	—	13	4	no	—
148	#117	—	13	8	no	—
149	#117	C₂×D₁₃	1	8	yes	complemented
150	C₂×C₄⋊C₄	—	13	4	no	—
151	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	complemented
152	C₂×C₂²⋊C₄	—	13	2	no	—
153	C₂×C₂²⋊C₄	—	13	4	no	—
154	C₂×C₂²⋊C₄	—	13	4	no	—
155	C₂×C₂²⋊C₄	C₂×D₁₃	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	C₂×D₁₃	1	1	yes	characteristic
157	C₂×C₄²	C₂×D₁₃	1	2	yes	complemented
158	C₂×C₁₃	—	2	4	no	—
159	C₂×C₁₃	—	2	2	no	—
160	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
161	C₂×C₁₃	#585	1	2	yes	—
162	D₁₃	—	4	4	no	—
163	C₂⁴	—	13	2	no	—
164	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
165	C₂×Q₈	—	13	4	no	—
166	C₂×D₄	—	13	4	no	—
167	C₂×D₄	—	13	4	no	—
168	C₂×D₄	—	13	4	no	—
169	C₂×D₄	—	26	8	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	13	2	no	—
172	C₂²×C₄	—	13	4	no	—
173	C₂²×C₄	—	13	8	no	—
174	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
176	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
178	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
179	C₄⋊C₄	—	13	8	no	—
180	C₄⋊C₄	—	13	8	no	—
181	C₄⋊C₄	—	13	8	no	—
182	C₄⋊C₄	—	13	8	no	—
183	C₄⋊C₄	—	13	4	no	—
184	C₄⋊C₄	—	13	4	no	—
185	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
186	C₂²⋊C₄	—	13	4	no	—
187	C₂²⋊C₄	—	13	4	no	—
188	C₂²⋊C₄	—	13	8	no	—
189	C₂²⋊C₄	—	13	8	no	—
190	C₂²⋊C₄	—	13	8	no	—
191	C₂²⋊C₄	—	13	8	no	—
192	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
193	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
194	C₄²	—	13	8	no	—
195	C₄²	C₂²×D₁₃	1	4	yes	—
196	C₄²	C₂²×D₁₃	1	4	yes	—
197	C₁₃	#3491	1	1	yes	characteristic, complemented, Sylow
198	C₂³	—	4	2	no	—
199	C₂³	—	13	4	no	—
200	C₂³	—	26	4	no	—
201	C₂³	—	52	4	no	—
202	C₂³	#6127	1	4	yes	—
203	C₂³	C₂³×D₁₃	1	1	yes	characteristic
204	C₂³	C₂³×D₁₃	1	2	yes	—
205	Q₈	—	52	4	no	—
206	D₄	—	26	8	no	—
207	D₄	—	26	8	no	—
208	D₄	—	52	4	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	13	4	no	—
215	C₂×C₄	—	13	8	no	—
216	C₂×C₄	—	26	4	no	—
217	C₂×C₄	—	52	8	no	—
218	C₂×C₄	—	52	4	no	—
219	C₂×C₄	C₂³×D₁₃	1	4	yes	—
220	C₂×C₄	C₂³×D₁₃	1	2	yes	—
221	C₂×C₄	C₂³×D₁₃	1	4	yes	—
222	C₂×C₄	C₂³×D₁₃	1	2	yes	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	4	4	no	—
226	C₂²	—	26	4	no	—
227	C₂²	—	26	8	no	—
228	C₂²	—	104	4	no	—
229	C₂²	C₂×#6127	1	4	yes	—
230	C₂²	C₂×#6127	1	2	yes	—
231	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	4	no	—
236	C₄	—	26	4	no	—
237	C₄	—	26	8	no	—
238	C₂	—	2	4	no	—
239	C₂	—	2	2	no	—
240	C₂	—	52	4	no	—
241	C₂	C₂²×#6127	1	1	yes	characteristic
242	C₂	#11719648	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₂×C₁₃×#117	C₂
prime radical	#11719198	C₂
hypercenter	C₂×#117	D₁₃
hypocenter	C₁₃	#3491
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#117	C₂	semidirect
2. C₂×#88183	C₂	semidirect
3. #11719236	C₂	semidirect
4. #11719232	C₂	semidirect
5. #11719220	C₂	semidirect
7. #11719191	C₂	semidirect
8. #11719188	C₂	semidirect
9. #11719184	C₂	semidirect
12. C₁₃×#117	C₂²	semidirect
13. C₂×C₁₃×C₄⋊C₄	C₂²	semidirect
16. C₂×C₄²×C₁₃	C₂²	semidirect
20. #88183	C₂²	semidirect
21. #88183	C₂²	semidirect
22. C₄×C₂²⋊D₁₃	C₂²	semidirect
23. C₂×#6103	C₂²	semidirect
24. #88178	C₂²	semidirect
25. C₂×#6101	C₂²	semidirect
26. #88152	C₂²	semidirect
27. #88150	C₂²	semidirect
28. #88148	C₂²	semidirect
29. #88143	C₂²	semidirect
33. C₂²⋊C₄×D₁₃	C₂²	semidirect
34. #88132	C₂²	semidirect
35. #88131	C₂²	semidirect
36. #88130	C₂²	semidirect
37. #88129	C₂²	semidirect
38. C₄×D₅₂	C₂²	semidirect
39. #88126	C₂²	semidirect
40. #88124	C₂²	semidirect
41. C₄×Dic₂₆	C₂²	semidirect
110. C₂×#117	D₁₃	semidirect
149. #117	C₂×D₁₃	semidirect
151. C₂×C₄⋊C₄	C₂×D₁₃	semidirect
157. C₂×C₄²	C₂×D₁₃	semidirect
197. C₁₃	#3491	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#117	2
C₂×#88183	2
#11719236	2
#11719232	2
#11719220	2
#11719198	2
#11719191	2
#11719188	2
#11719184	2
#3491	13