Subgroups — #261×#346 #16789012

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131037	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718216	C₂	1	2	yes	maximal
3	C₂×C₁₆×#261	C₂	1	2	yes	maximal
4	C₄×C₈×#261	C₂	1	1	yes	maximal, characteristic
5	D₁₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×#261	C₄	1	1	yes	characteristic
7	C₂×C₈×#261	C₄	1	1	yes	characteristic
8	C₂×C₈×#261	C₂²	1	1	yes	characteristic
9	C₂×C₈×#261	C₂²	1	1	yes	characteristic
10	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
11	C₄²×#261	C₄	1	1	yes	characteristic
12	C₄×D₁₃⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
14	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
15	#11718258	C₄	1	2	yes	complemented
16	#11718231	C₄	1	2	yes	complemented
17	#11718216	C₄	1	8	yes	complemented
18	C₁₆×#261	C₄	1	8	yes	complemented
19	C₁₃×#346	C₄	1	1	yes	characteristic, complemented
20	#11718094	C₄	1	2	yes	complemented
21	#11718053	C₄	1	1	yes	characteristic, complemented
22	C₂×C₄×#261	C₈	1	4	yes	—
23	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
25	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
29	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×#1057	C₈	1	4	yes	—
31	C₂×#6092	C₂×C₄	1	4	yes	—
32	C₈×#261	C₈	1	4	yes	—
33	C₈×#261	C₈	1	4	yes	—
34	C₈×#261	C₂×C₄	1	4	yes	—
35	C₈×#261	C₂×C₄	1	4	yes	—
36	D₁₃⋊C₁₆	C₂×C₄	1	4	yes	—
37	D₁₃⋊C₁₆	C₂×C₄	1	4	yes	—
38	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×#6090	C₂×C₄	1	2	yes	—
43	C₄×#1055	C₂×C₄	1	1	yes	characteristic
44	C₁₆×D₁₃	C₂×C₄	1	4	yes	—
45	C₁₆×D₁₃	C₂×C₄	1	4	yes	—
46	C₄×#346	—	13	1	no	maximal, Sylow
47	C₂²×#261	C₂×C₈	1	2	yes	—
48	C₂²×#261	C₂×C₈	1	2	yes	—
49	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
50	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×#1057	C₂×C₈	1	2	yes	—
52	C₂×#1057	C₂×C₈	1	2	yes	—
53	C₄×#261	—	2	8	no	—
54	C₄×#261	C₄²	1	2	yes	—
55	C₄×#261	C₄²	1	2	yes	—
56	C₄×#261	C₂×C₈	1	2	yes	—
57	C₄×#261	C₂×C₈	1	2	yes	—
58	D₁₃⋊C₈	C₄²	1	2	yes	—
59	D₁₃⋊C₈	C₄²	1	2	yes	—
60	D₁₃⋊C₈	C₂×C₈	1	2	yes	—
61	D₁₃⋊C₈	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₁₃	C₄²	1	1	yes	characteristic
65	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
66	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
67	C₂×#1055	C₄²	1	1	yes	characteristic
68	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
69	C₈×D₁₃	C₂×C₈	1	2	yes	—
70	C₈×D₁₃	C₂×C₈	1	2	yes	—
71	C₈×D₁₃	C₂²×C₄	1	2	yes	—
72	C₈×D₁₃	C₂²×C₄	1	2	yes	—
73	#6092	C₄²	1	8	yes	complemented
74	C₁₃×C₁₆	C₄²	1	4	yes	complemented
75	#6090	C₄²	1	4	yes	complemented
76	C₂×#346	—	13	2	no	—
77	C₂×#346	—	13	1	no	—
78	C₂×#346	—	13	1	no	—
79	C₂×C₄×C₁₆	—	13	2	no	—
80	C₄²×C₈	—	13	1	no	—
81	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×#261	—	2	4	no	—
83	C₂×#261	C₄×C₈	1	4	yes	—
84	C₂×#261	C₄×C₈	1	4	yes	—
85	C₂×#261	M₃₂	1	8	yes	—
86	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
87	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
89	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
90	C₄×D₁₃	—	2	4	no	—
91	C₄×D₁₃	C₂×C₄²	1	1	yes	characteristic
92	C₄×D₁₃	C₂×C₄²	1	1	yes	characteristic
93	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
94	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
95	#1057	C₄×C₈	1	4	yes	—
96	#1057	C₄×C₈	1	4	yes	—
97	C₈×C₁₃	C₄×C₈	1	2	yes	—
98	C₈×C₁₃	C₂×C₄²	1	2	yes	—
99	#1055	C₄×C₈	1	2	yes	—
100	#1055	C₂×C₄²	1	2	yes	—
101	C₂²×C₁₆	—	13	2	no	—
102	C₂²×C₁₆	—	13	2	no	—
103	C₂×C₄×C₈	—	13	1	no	—
104	C₂×C₄×C₈	—	13	1	no	—
105	C₂×C₄×C₈	—	13	1	no	—
106	C₂×C₄×C₈	—	13	1	no	—
107	C₂×C₄×C₈	—	13	1	no	—
108	C₂×C₄×C₈	—	13	1	no	—
109	C₄³	—	13	1	no	—
110	#346	—	13	8	no	—
111	#346	—	13	2	no	—
112	#346	—	13	2	no	—
113	#346	—	13	2	no	—
114	#346	—	13	1	no	—
115	#346	#261	1	1	yes	characteristic, direct product factor
116	C₄×C₁₆	—	13	8	no	—
117	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂×D₁₃	C₂×C₄×C₈	1	1	yes	characteristic
119	C₂×D₁₃	C₂×C₄×C₈	1	1	yes	characteristic
120	C₂×D₁₃	C₂×M₃₂	1	4	yes	—
121	#261	—	2	4	no	—
122	#261	#346	1	8	yes	direct product factor
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	C₄³	1	1	yes	characteristic
125	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	C₂×C₄×C₈	1	1	yes	characteristic
128	Dic₁₃	C₂×C₄×C₈	1	1	yes	characteristic
129	C₂²×C₈	—	13	1	no	—
130	C₂²×C₈	—	13	1	no	—
131	C₂²×C₈	—	13	1	no	—
132	C₂²×C₈	—	13	1	no	—
133	C₂×C₄²	—	13	1	no	—
134	C₂×C₄²	—	13	1	no	—
135	C₂×C₄²	—	13	1	no	—
136	C₂×C₄²	—	13	4	no	—
137	C₂×C₁₆	—	13	4	no	—
138	C₂×C₁₆	—	13	4	no	—
139	C₂×C₁₆	—	13	4	no	—
140	C₂×C₁₆	—	13	2	no	—
141	C₂×C₁₆	—	13	4	no	—
142	C₂×C₁₆	—	13	4	no	—
143	C₂×C₁₆	C₂×#261	1	2	yes	—
144	C₄×C₈	—	13	4	no	—
145	C₄×C₈	—	13	4	no	—
146	C₄×C₈	—	13	4	no	—
147	C₄×C₈	—	13	4	no	—
148	C₄×C₈	—	13	4	no	—
149	C₄×C₈	—	13	1	no	—
150	C₄×C₈	—	13	1	no	—
151	C₄×C₈	—	13	1	no	—
152	C₄×C₈	C₂×#261	1	1	yes	characteristic
153	C₂×C₁₃	C₄²×C₈	1	1	yes	characteristic
154	C₂×C₁₃	C₄×M₃₂	1	2	yes	—
155	D₁₃	C₂×#346	1	2	yes	—
156	D₁₃	C₄×M₃₂	1	2	yes	—
157	C₂²×C₄	—	13	2	no	—
158	C₂²×C₄	—	13	1	no	—
159	C₂²×C₄	—	13	2	no	—
160	C₂²×C₄	—	13	2	no	—
161	C₂×C₈	—	13	2	no	—
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	—	13	2	no	—
164	C₂×C₈	—	13	2	no	—
165	C₂×C₈	—	13	2	no	—
166	C₂×C₈	—	13	2	no	—
167	C₂×C₈	—	13	1	no	—
168	C₂×C₈	—	13	1	no	—
169	C₂×C₈	—	13	2	no	—
170	C₂×C₈	—	13	2	no	—
171	C₂×C₈	—	13	2	no	—
172	C₂×C₈	—	13	2	no	—
173	C₂×C₈	C₄×#261	1	1	yes	characteristic
174	C₂×C₈	C₂²×#261	1	1	yes	characteristic
175	C₄²	—	13	2	no	—
176	C₄²	—	13	2	no	—
177	C₄²	—	13	2	no	—
178	C₄²	—	13	2	no	—
179	C₄²	—	13	2	no	—
180	C₄²	—	13	1	no	—
181	C₄²	—	26	8	no	—
182	C₄²	C₄×#261	1	1	yes	characteristic
183	C₁₆	—	13	8	no	—
184	C₁₆	—	13	4	no	—
185	C₁₆	C₄×#261	1	4	yes	complemented
186	C₁₃	C₄×#346	1	1	yes	characteristic, complemented, Sylow
187	C₂³	—	13	1	no	—
188	C₂×C₄	—	13	4	no	—
189	C₂×C₄	—	13	8	no	—
190	C₂×C₄	—	13	1	no	—
191	C₂×C₄	—	13	1	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	13	1	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	4	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	26	4	no	—
198	C₂×C₄	—	26	4	no	—
199	C₂×C₄	C₈×#261	1	2	yes	—
200	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
201	C₈	—	13	4	no	—
202	C₈	—	13	4	no	—
203	C₈	—	13	2	no	—
204	C₈	—	13	2	no	—
205	C₈	C₈×#261	1	2	yes	—
206	C₈	C₂×C₄×#261	1	2	yes	—
207	C₂²	—	13	4	no	—
208	C₂²	—	13	1	no	—
209	C₂²	—	13	1	no	—
210	C₂²	C₂×C₈×#261	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	13	8	no	—
213	C₄	—	13	1	no	—
214	C₄	—	13	1	no	—
215	C₄	—	26	4	no	—
216	C₄	—	26	2	no	—
217	C₄	C₄²×#261	1	1	yes	characteristic
218	C₄	C₂×C₈×#261	1	1	yes	characteristic
219	C₂	—	13	2	no	—
220	C₂	—	13	2	no	—
221	C₂	C₄×C₈×#261	1	1	yes	characteristic
222	C₂	M₃₂×#261	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#261
commutator	C₂×C₁₃	C₄²×C₈
Fitting	C₁₃×#346	C₄
socle	C₂²×C₁₃	C₂×C₄×C₈
prime radical	C₂²×D₁₃	C₂²×C₈
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	#346	#261
hypocenter	C₁₃	C₄×#346
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
15. #11718258	C₄	semidirect
16. #11718231	C₄	semidirect
17. #11718216	C₄	semidirect
18. C₁₆×#261	C₄	semidirect
19. C₁₃×#346	C₄	semidirect
20. #11718094	C₄	semidirect
21. #11718053	C₄	semidirect
73. #6092	C₄²	semidirect
74. C₁₃×C₁₆	C₄²	semidirect
75. #6090	C₄²	semidirect
115. #346	#261	direct (N × H)
122. #261	#346	direct (N × H)
185. C₁₆	C₄×#261	semidirect
186. C₁₃	C₄×#346	semidirect

Maximal subgroups

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

subgroup	index
#13131037	2
C₂×#11718216	2
C₂×C₁₆×#261	2
C₄×C₈×#261	2
D₁₃×#346	2
C₄×#346	13