Subgroups — #13123571

Order 1664.

544 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×M₁₆×D₁₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718787	C₂	1	2	yes	maximal, complemented
5	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718499	C₂	1	2	yes	maximal, complemented
7	#11718497	C₂	1	4	yes	maximal, complemented
8	#11718495	C₂	1	2	yes	maximal, complemented
9	C₂²×D₅₂	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
11	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6103	C₄	1	2	yes	—
14	C₂×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
15	C₂×#6099	C₂²	1	1	yes	characteristic
16	C₂×#6099	C₂²	1	1	yes	characteristic
17	M₁₆×D₁₃	—	2	8	no	—
18	C₂×#6094	C₂²	1	4	yes	complemented
19	C₂×C₈×D₁₃	C₂²	1	4	yes	complemented
20	C₄×D₅₂	C₄	1	2	yes	—
21	C₄×D₅₂	C₄	1	1	yes	characteristic
22	C₄×D₅₂	C₄	1	1	yes	characteristic
23	C₄×D₅₂	C₂²	1	1	yes	characteristic
24	C₄×D₅₂	C₂²	1	2	yes	—
25	C₄×D₅₂	C₂²	1	1	yes	characteristic
26	C₁₃×C₄⋊C₈	C₂²	1	2	yes	complemented
27	C₁₃×C₄⋊C₈	C₂²	1	2	yes	complemented
28	#88060	C₂²	1	4	yes	complemented
29	#88060	C₂²	1	4	yes	complemented
30	#88043	C₂²	1	2	yes	complemented
31	C₄×#1055	C₂²	1	2	yes	complemented
32	C₂³×D₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
37	C₂×D₅₂	—	2	4	no	—
38	C₂×D₅₂	C₂×C₄	1	2	yes	—
39	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
40	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₃	—	2	4	no	—
42	C₂×C₄×D₁₃	—	2	4	no	—
43	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₃	C₂³	1	2	yes	—
46	C₂×C₄×D₁₃	C₂³	1	2	yes	—
47	C₁₃×M₁₆	—	2	4	no	—
48	C₂×C₈×C₁₃	C₂³	1	4	yes	—
49	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	#6103	C₂×C₄	1	2	yes	—
54	#6103	C₂×C₄	1	2	yes	—
55	#6103	C₂×C₄	1	2	yes	—
56	#6103	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
60	#6099	—	2	2	no	—
61	#6099	D₄	1	4	yes	complemented
62	C₂×#1055	C₂³	1	2	yes	—
63	C₂×#1055	C₂³	1	2	yes	—
64	#6094	—	2	8	no	—
65	C₈×D₁₃	—	2	8	no	—
66	D₄×M₁₆	—	13	1	no	maximal, Sylow
67	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₁₃	—	2	4	no	—
69	C₂²×D₁₃	—	2	4	no	—
70	C₂²×D₁₃	—	4	2	no	—
71	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
72	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	4	1	no	—
75	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	#41	1	2	yes	—
82	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₁₃	—	2	4	no	—
84	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
86	D₅₂	—	2	4	no	—
87	D₅₂	—	2	2	no	—
88	D₅₂	M₁₆	1	4	yes	complemented
89	C₄×D₁₃	—	2	4	no	—
90	C₄×D₁₃	—	2	4	no	—
91	C₄×D₁₃	—	2	2	no	—
92	C₄×D₁₃	—	2	2	no	—
93	C₄×D₁₃	—	2	2	no	—
94	C₄×D₁₃	—	2	2	no	—
95	C₈×C₁₃	—	2	4	no	—
96	#1055	—	2	2	no	—
97	#1055	C₂×D₄	1	4	yes	complemented
98	C₂²×M₁₆	—	13	2	no	—
99	C₂×C₄×D₄	—	13	1	no	—
100	#436	—	13	2	no	—
101	#435	—	13	4	no	—
102	C₈×D₄	—	13	2	no	—
103	#423	D₁₃	1	1	yes	characteristic, complemented
104	#407	—	13	2	no	—
105	C₄×M₁₆	—	13	1	no	—
106	C₂²×C₁₃	—	2	1	no	—
107	C₂²×C₁₃	—	2	1	no	—
108	C₂²×C₁₃	C₄×D₄	1	2	yes	—
109	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×D₁₃	—	2	2	no	—
111	C₂×D₁₃	—	2	2	no	—
112	C₂×D₁₃	—	2	2	no	—
113	C₂×D₁₃	—	2	2	no	—
114	C₂×D₁₃	—	2	2	no	—
115	C₂×D₁₃	—	4	8	no	—
116	C₂×D₁₃	C₂×M₁₆	1	4	yes	complemented
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₄×C₁₃	C₂×M₁₆	1	2	yes	complemented
122	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	Dic₁₃	—	2	2	no	—
125	Dic₁₃	—	2	2	no	—
126	C₂²×D₄	—	13	1	no	—
127	C₂³×C₄	—	13	2	no	—
128	C₂×M₁₆	—	13	1	no	—
129	C₂×M₁₆	—	13	1	no	—
130	C₂×M₁₆	—	13	4	no	—
131	C₂×M₁₆	—	26	8	no	—
132	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
133	C₂²×C₈	—	13	4	no	—
134	C₄×D₄	—	13	1	no	—
135	C₄×D₄	—	13	2	no	—
136	C₄×D₄	—	13	1	no	—
137	C₄×D₄	—	13	1	no	—
138	C₄×D₄	—	13	2	no	—
139	C₄×D₄	—	13	1	no	—
140	C₂×C₄⋊C₄	—	13	1	no	—
141	C₂×C₂²⋊C₄	—	13	2	no	—
142	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
143	C₄⋊C₈	C₂×D₁₃	1	2	yes	complemented
144	C₄⋊C₈	C₂×D₁₃	1	2	yes	complemented
145	C₂²⋊C₈	—	13	4	no	—
146	C₂²⋊C₈	—	13	4	no	—
147	#97	—	13	2	no	—
148	C₄×C₈	—	13	2	no	—
149	C₂×C₁₃	—	2	2	no	—
150	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
152	C₂×C₁₃	#568	1	1	yes	characteristic
153	D₁₃	—	2	4	no	—
154	D₁₃	—	4	2	no	—
155	C₂⁴	—	13	2	no	—
156	C₂×D₄	—	13	1	no	—
157	C₂×D₄	—	13	2	no	—
158	C₂×D₄	—	13	1	no	—
159	C₂×D₄	—	26	4	no	—
160	C₂²×C₄	—	13	2	no	—
161	C₂²×C₄	—	13	2	no	—
162	C₂²×C₄	—	13	2	no	—
163	C₂²×C₄	—	13	2	no	—
164	C₂²×C₄	—	13	2	no	—
165	C₂²×C₄	—	26	4	no	—
166	C₂²×C₄	—	26	4	no	—
167	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
170	M₁₆	—	2	4	no	—
171	M₁₆	—	13	4	no	—
172	M₁₆	—	26	8	no	—
173	M₁₆	—	26	2	no	—
174	C₂×C₈	—	13	2	no	—
175	C₂×C₈	—	13	2	no	—
176	C₂×C₈	—	26	8	no	—
177	C₂×C₈	C₂²×D₁₃	1	4	yes	—
178	C₄⋊C₄	—	13	2	no	—
179	C₄⋊C₄	—	13	1	no	—
180	C₄⋊C₄	—	13	1	no	—
181	C₂²⋊C₄	—	13	2	no	—
182	C₂²⋊C₄	—	13	2	no	—
183	C₂²⋊C₄	—	13	2	no	—
184	C₂²⋊C₄	—	13	2	no	—
185	C₄²	C₄×D₁₃	1	1	yes	characteristic
186	C₄²	C₄×D₁₃	1	1	yes	characteristic
187	C₄²	C₂²×D₁₃	1	1	yes	characteristic
188	C₄²	C₂²×D₁₃	1	1	yes	characteristic
189	C₁₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
190	C₂³	—	13	2	no	—
191	C₂³	—	13	2	no	—
192	C₂³	—	26	4	no	—
193	C₂³	—	26	4	no	—
194	C₂³	—	52	2	no	—
195	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
196	D₄	—	13	4	no	—
197	D₄	—	26	4	no	—
198	D₄	—	26	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	13	2	no	—
202	C₂×C₄	—	13	2	no	—
203	C₂×C₄	—	26	4	no	—
204	C₂×C₄	—	26	4	no	—
205	C₂×C₄	—	26	4	no	—
206	C₂×C₄	—	26	2	no	—
207	C₂×C₄	—	26	2	no	—
208	C₂×C₄	—	26	2	no	—
209	C₂×C₄	—	26	2	no	—
210	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
215	C₂×C₄	D₄×D₁₃	1	2	yes	—
216	C₂×C₄	#6131	1	2	yes	—
217	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	—	13	4	no	—
220	C₈	—	26	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	13	4	no	—
224	C₂²	—	26	2	no	—
225	C₂²	—	26	2	no	—
226	C₂²	—	26	2	no	—
227	C₂²	—	26	2	no	—
228	C₂²	—	26	2	no	—
229	C₂²	—	52	8	no	—
230	C₂²	#88147	1	2	yes	—
231	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	26	2	no	—
236	C₄	—	26	2	no	—
237	C₄	#88147	1	2	yes	—
238	C₄	M₁₆×D₁₃	1	2	yes	complemented
239	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
240	C₄	C₂×#6131	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	26	4	no	—
243	C₂	—	52	2	no	—
244	C₂	C₂×#88147	1	1	yes	characteristic
245	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic
246	C₂	#11719450	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#423	C₂
prime radical	C₂²×D₅₂	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#423	D₁₃
hypocenter	C₁₃	D₄×M₁₆
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₁₃	C₂	semidirect
3. C₁₃×#423	C₂	semidirect
4. #11718787	C₂	semidirect
5. C₄×#6099	C₂	semidirect
6. #11718499	C₂	semidirect
7. #11718497	C₂	semidirect
8. #11718495	C₂	semidirect
18. C₂×#6094	C₂²	semidirect
19. C₂×C₈×D₁₃	C₂²	semidirect
26. C₁₃×C₄⋊C₈	C₂²	semidirect
27. C₁₃×C₄⋊C₈	C₂²	semidirect
28. #88060	C₂²	semidirect
29. #88060	C₂²	semidirect
30. #88043	C₂²	semidirect
31. C₄×#1055	C₂²	semidirect
61. #6099	D₄	semidirect
88. D₅₂	M₁₆	semidirect
97. #1055	C₂×D₄	semidirect
103. #423	D₁₃	semidirect
116. C₂×D₁₃	C₂×M₁₆	semidirect
121. C₄×C₁₃	C₂×M₁₆	semidirect
143. C₄⋊C₈	C₂×D₁₃	semidirect
144. C₄⋊C₈	C₂×D₁₃	semidirect
189. C₁₃	D₄×M₁₆	semidirect
238. C₄	M₁₆×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₃	2
C₂×C₄×D₅₂	2
C₁₃×#423	2
#11718787	2
C₄×#6099	2
#11718499	2
#11718497	2
#11718495	2
D₄×M₁₆	13