Subgroups — #13128710

Order 1664.

1232 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×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719511	C₂	1	8	yes	maximal, complemented
3	#11719510	C₂	1	2	yes	maximal, complemented
4	#11719503	C₂	1	2	yes	maximal, complemented
5	#11719424	C₂	1	4	yes	maximal, complemented
6	#11719423	C₂	1	4	yes	maximal, complemented
7	#11719421	C₂	1	4	yes	maximal, complemented
8	#11719418	C₂	1	4	yes	maximal, complemented
9	C₄×#6127	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719175	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
12	C₂²×D₄×C₁₃	C₂²	1	4	yes	—
13	C₂×#6129	C₂²	1	8	yes	complemented
14	C₂×#6127	C₂²	1	1	yes	characteristic
15	C₁₃×#127	C₂²	1	2	yes	—
16	C₁₃×#127	C₂²	1	4	yes	—
17	C₁₃×#127	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	#88207	C₂²	1	8	yes	—
20	C₂×C₂²⋊Dic₁₃	C₂²	1	8	yes	—
21	#88194	C₂²	1	4	yes	complemented
22	#88193	C₂²	1	16	yes	complemented
23	#88192	C₂²	1	8	yes	complemented
24	#88192	C₂²	1	8	yes	complemented
25	#88190	C₂²	1	4	yes	complemented
26	#88190	C₂²	1	8	yes	complemented
27	#88189	C₂²	1	16	yes	complemented
28	D₄×Dic₁₃	C₂²	1	8	yes	complemented
29	D₄×Dic₁₃	C₂²	1	8	yes	complemented
30	D₄×Dic₁₃	C₂²	1	8	yes	complemented
31	C₄×C₂²⋊D₁₃	C₂²	1	4	yes	—
32	#88180	C₂²	1	2	yes	—
33	#88180	C₂²	1	1	yes	characteristic
34	#88178	C₂²	1	4	yes	—
35	C₄×D₅₂	C₂²	1	2	yes	complemented
36	#88126	C₂²	1	2	yes	complemented
37	C₄²×D₁₃	C₂²	1	2	yes	complemented
38	#88124	C₂²	1	2	yes	complemented
39	#88123	C₂²	1	2	yes	complemented
40	C₄×Dic₂₆	C₂²	1	4	yes	complemented
41	C₄×Dic₂₆	C₂²	1	2	yes	complemented
42	C₂⁴×C₁₃	C₂³	1	4	yes	—
43	C₂×D₄×C₁₃	—	2	8	no	—
44	C₂×D₄×C₁₃	—	4	8	no	—
45	C₂×D₄×C₁₃	C₂³	1	4	yes	—
46	C₂×D₄×C₁₃	C₂³	1	4	yes	—
47	C₂×D₄×C₁₃	C₂³	1	8	yes	—
48	C₂×D₄×C₁₃	C₂³	1	8	yes	—
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₃	C₂³	1	2	yes	—
52	C₂×C₂²⋊D₁₃	C₂³	1	16	yes	—
53	C₂²×Dic₁₃	C₂³	1	16	yes	—
54	#6129	—	4	16	no	—
55	#6127	—	2	4	no	—
56	C₂×D₅₂	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₃	C₂³	1	2	yes	—
58	C₂×C₄×D₁₃	C₂³	1	4	yes	—
59	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
60	C₂×Dic₂₆	C₂³	1	4	yes	—
61	C₄²×C₁₃	C₂³	1	2	yes	—
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₄²×C₁₃	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
65	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
66	C₂²⋊Dic₁₃	C₂³	1	16	yes	—
67	C₂²⋊Dic₁₃	C₂³	1	16	yes	—
68	#6103	C₂³	1	4	yes	—
69	C₄⋊Dic₁₃	C₂³	1	4	yes	—
70	C₄⋊Dic₁₃	C₂³	1	2	yes	—
71	C₄⋊Dic₁₃	C₂³	1	4	yes	—
72	#6101	C₂³	1	8	yes	—
73	#6101	C₂³	1	4	yes	—
74	C₄×Dic₁₃	C₂³	1	4	yes	—
75	C₄×Dic₁₃	C₂³	1	4	yes	—
76	C₄×Dic₁₃	C₂³	1	2	yes	—
77	#3508	—	13	1	no	maximal, Sylow
78	C₂³×C₁₃	—	2	8	no	—
79	C₂³×C₁₃	—	4	8	no	—
80	C₂³×C₁₃	C₂⁴	1	8	yes	—
81	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²×D₁₃	C₂⁴	1	2	yes	—
83	D₄×C₁₃	—	2	8	no	—
84	D₄×C₁₃	—	2	8	no	—
85	D₄×C₁₃	—	2	16	no	—
86	D₄×C₁₃	—	4	8	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	#41	1	4	yes	complemented
89	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
90	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
93	C₂²⋊D₁₃	—	2	4	no	—
94	C₂²⋊D₁₃	—	4	16	no	—
95	C₂×Dic₁₃	—	4	16	no	—
96	C₂×Dic₁₃	C₂⁴	1	2	yes	—
97	C₂×Dic₁₃	C₂⁴	1	4	yes	—
98	C₂×Dic₁₃	C₂⁴	1	8	yes	—
99	D₅₂	—	2	2	no	—
100	C₄×D₁₃	—	2	4	no	—
101	C₄×D₁₃	—	2	8	no	—
102	Dic₂₆	—	2	2	no	—
103	Dic₂₆	—	2	8	no	—
104	#559	—	13	4	no	—
105	#550	—	13	4	no	—
106	#546	—	13	8	no	—
107	#540	—	13	4	no	—
108	#535	—	13	2	no	—
109	#533	—	13	1	no	—
110	#532	—	13	4	no	—
111	C₂×#127	D₁₃	1	1	yes	characteristic, complemented
112	#518	—	13	2	no	—
113	C₄×#41	—	13	1	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	2	8	no	—
116	C₂²×C₁₃	—	2	16	no	—
117	C₂²×C₁₃	—	8	8	no	—
118	C₂²×C₁₃	C₂×#41	1	2	yes	—
119	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
120	C₂×D₁₃	—	2	2	no	—
121	C₂×D₁₃	—	2	4	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	C₂×#41	1	2	yes	—
124	C₄×C₁₃	C₂×#41	1	2	yes	—
125	C₄×C₁₃	H(5,𝔽₂)	1	4	yes	—
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	—	2	8	no	—
128	Dic₁₃	—	2	4	no	—
129	C₂×#41	—	13	8	no	—
130	C₂×#41	—	13	1	no	—
131	C₂²×D₄	C₂×D₁₃	1	2	yes	—
132	C₂²×D₄	C₂×D₁₃	1	4	yes	—
133	C₄⋊Q₈	—	13	2	no	—
134	#127	—	13	4	no	—
135	#127	C₂×D₁₃	1	2	yes	—
136	#127	C₂×D₁₃	1	2	yes	—
137	#127	C₂×D₁₃	1	4	yes	—
138	#125	—	13	2	no	—
139	#124	—	13	8	no	—
140	#124	—	13	4	no	—
141	#123	—	13	16	no	—
142	C₂²⋊Q₈	—	13	4	no	—
143	#121	—	13	8	no	—
144	#121	—	13	8	no	—
145	#121	—	13	16	no	—
146	#120	—	13	8	no	—
147	C₄×Q₈	—	13	2	no	—
148	C₄×Q₈	—	13	4	no	—
149	C₄×D₄	—	13	8	no	—
150	C₄×D₄	—	13	8	no	—
151	C₄×D₄	—	13	8	no	—
152	C₄×D₄	—	13	4	no	—
153	C₄×D₄	—	13	2	no	—
154	#117	—	13	1	no	—
155	#117	—	13	2	no	—
156	#117	—	13	2	no	—
157	C₂×C₂²⋊C₄	—	13	8	no	—
158	C₂×C₄²	—	13	2	no	—
159	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
160	C₂×C₁₃	—	2	2	no	—
161	C₂×C₁₃	—	4	8	no	—
162	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
163	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
164	D₁₃	—	4	2	no	—
165	C₂⁴	C₂²×D₁₃	1	4	yes	—
166	#41	—	26	4	no	—
167	#41	—	52	16	no	—
168	C₂×Q₈	—	13	1	no	—
169	C₂×Q₈	—	13	4	no	—
170	C₂×D₄	—	2	8	no	—
171	C₂×D₄	—	4	8	no	—
172	C₂×D₄	—	13	2	no	—
173	C₂×D₄	—	13	16	no	—
174	C₂×D₄	—	13	1	no	—
175	C₂×D₄	C₂²×D₁₃	1	4	yes	—
176	C₂×D₄	C₂²×D₁₃	1	4	yes	—
177	C₂×D₄	C₂²×D₁₃	1	8	yes	—
178	C₂×D₄	C₂²×D₁₃	1	8	yes	—
179	C₂²×C₄	—	13	16	no	—
180	C₂²×C₄	—	13	2	no	—
181	C₂²×C₄	—	13	4	no	—
182	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
183	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
184	C₄⋊C₄	—	13	2	no	—
185	C₄⋊C₄	—	13	4	no	—
186	C₄⋊C₄	—	13	4	no	—
187	C₄⋊C₄	—	13	4	no	—
188	C₄⋊C₄	—	13	8	no	—
189	C₂²⋊C₄	—	13	2	no	—
190	C₂²⋊C₄	—	13	4	no	—
191	C₂²⋊C₄	—	13	16	no	—
192	C₂²⋊C₄	—	13	16	no	—
193	C₂²⋊C₄	—	13	4	no	—
194	C₄²	—	13	2	no	—
195	C₄²	—	13	4	no	—
196	C₄²	—	13	4	no	—
197	C₄²	C₂²×D₁₃	1	1	yes	characteristic
198	C₄²	C₂²×D₁₃	1	1	yes	characteristic
199	C₄²	C₂²×D₁₃	1	2	yes	—
200	C₁₃	#3508	1	1	yes	characteristic, complemented, Sylow
201	C₂³	—	2	8	no	—
202	C₂³	—	4	8	no	—
203	C₂³	—	13	2	no	—
204	C₂³	C₂³×D₁₃	1	8	yes	—
205	C₂³	C₂³×D₁₃	1	1	yes	characteristic
206	Q₈	—	26	2	no	—
207	Q₈	—	26	8	no	—
208	D₄	—	2	8	no	—
209	D₄	—	2	8	no	—
210	D₄	—	2	16	no	—
211	D₄	—	4	8	no	—
212	D₄	—	26	4	no	—
213	D₄	—	26	2	no	—
214	D₄	—	52	16	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	13	2	no	—
217	C₂×C₄	—	13	4	no	—
218	C₂×C₄	—	13	8	no	—
219	C₂×C₄	—	26	4	no	—
220	C₂×C₄	—	26	8	no	—
221	C₂×C₄	—	52	16	no	—
222	C₂×C₄	#6129	1	4	yes	complemented
223	C₂×C₄	C₂³×D₁₃	1	2	yes	—
224	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
225	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₁₃	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	8	no	—
229	C₂²	—	2	16	no	—
230	C₂²	—	8	8	no	—
231	C₂²	—	26	2	no	—
232	C₂²	—	26	4	no	—
233	C₂²	C₂×#6129	1	2	yes	—
234	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	26	2	no	—
237	C₄	—	26	4	no	—
238	C₄	—	26	8	no	—
239	C₄	C₂×#6129	1	2	yes	—
240	C₄	C₂×#6129	1	2	yes	—
241	C₄	#88251	1	4	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	8	no	—
244	C₂	—	52	2	no	—
245	C₂	C₂²×#6129	1	1	yes	characteristic
246	C₂	C₂×#88251	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₂×C₁₃×#127	C₂
hypercenter	C₂×#127	D₁₃
hypocenter	C₁₃	#3508
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#127	C₂	semidirect
2. #11719511	C₂	semidirect
3. #11719510	C₂	semidirect
4. #11719503	C₂	semidirect
5. #11719424	C₂	semidirect
6. #11719423	C₂	semidirect
7. #11719421	C₂	semidirect
8. #11719418	C₂	semidirect
9. C₄×#6127	C₂	semidirect
10. #11719175	C₂	semidirect
13. C₂×#6129	C₂²	semidirect
21. #88194	C₂²	semidirect
22. #88193	C₂²	semidirect
23. #88192	C₂²	semidirect
24. #88192	C₂²	semidirect
25. #88190	C₂²	semidirect
26. #88190	C₂²	semidirect
27. #88189	C₂²	semidirect
28. D₄×Dic₁₃	C₂²	semidirect
29. D₄×Dic₁₃	C₂²	semidirect
30. D₄×Dic₁₃	C₂²	semidirect
35. C₄×D₅₂	C₂²	semidirect
36. #88126	C₂²	semidirect
37. C₄²×D₁₃	C₂²	semidirect
38. #88124	C₂²	semidirect
39. #88123	C₂²	semidirect
40. C₄×Dic₂₆	C₂²	semidirect
41. C₄×Dic₂₆	C₂²	semidirect
88. C₂×C₄×C₁₃	#41	semidirect
111. C₂×#127	D₁₃	semidirect
200. C₁₃	#3508	semidirect
222. C₂×C₄	#6129	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#127	2
#11719511	2
#11719510	2
#11719503	2
#11719424	2
#11719423	2
#11719421	2
#11719418	2
C₄×#6127	2
#11719175	2
#3508	13