Subgroups — #16789291

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718259	C₂	1	2	yes	maximal
2	C₂×#11718231	C₂	1	2	yes	maximal, complemented
3	#13129875	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
6	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
8	C₂²×#6092	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₄×C₈×D₁₃	—	2	4	no	—
14	#11718259	C₄	1	8	yes	complemented
15	#11718231	—	2	4	no	—
16	C₂²×C₄×D₁₃	C₈	1	2	yes	—
17	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
22	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂³	1	1	yes	characteristic
25	C₂×C₈×D₁₃	—	2	2	no	—
26	C₂×C₈×D₁₃	—	2	2	no	—
27	C₂×C₈×D₁₃	—	2	2	no	—
28	C₂×C₈×D₁₃	—	2	2	no	—
29	C₂×C₈×D₁₃	C₈	1	4	yes	—
30	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
31	C₄²×D₁₃	—	2	2	no	—
32	C₄²×D₁₃	—	2	2	no	—
33	C₂×#6092	—	2	4	no	—
34	C₂×#6092	—	2	4	no	—
35	C₂×#6092	C₂×C₄	1	8	yes	complemented
36	C₄×C₈×C₁₃	—	2	2	no	—
37	C₈×Dic₁₃	—	2	2	no	—
38	C₈×Dic₁₃	—	2	2	no	—
39	C₄×#1055	D₄	1	4	yes	—
40	#12599	—	13	1	no	maximal, Sylow
41	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
46	C₂×C₄×D₁₃	—	2	4	no	—
47	C₂×C₄×D₁₃	—	2	4	no	—
48	C₂×C₄×D₁₃	—	2	1	no	—
49	C₂×C₄×D₁₃	—	2	1	no	—
50	C₂×C₄×D₁₃	—	2	1	no	—
51	C₂×C₄×D₁₃	—	2	1	no	—
52	C₂×C₄×D₁₃	—	2	4	no	—
53	C₂×C₄×D₁₃	C₄²	1	2	yes	—
54	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	C₄²	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
61	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₁₃	M₁₆	1	2	yes	—
63	C₄×Dic₁₃	—	2	4	no	—
64	C₄×Dic₁₃	—	2	1	no	—
65	C₄×Dic₁₃	—	2	1	no	—
66	C₂×#1055	C₂²⋊C₄	1	2	yes	—
67	C₂×#1055	C₂×C₈	1	2	yes	—
68	C₂×#1055	M₁₆	1	2	yes	—
69	C₂×#1055	C₂²×C₄	1	2	yes	—
70	C₂×#1055	C₂×D₄	1	2	yes	—
71	C₂×#1055	#41	1	2	yes	—
72	C₈×D₁₃	—	2	8	no	—
73	C₈×D₁₃	—	2	8	no	—
74	#6092	—	2	8	no	—
75	C₂²×C₄×C₈	—	13	1	no	—
76	C₂×C₂²⋊C₁₆	—	13	2	no	—
77	C₂×#346	—	13	2	no	—
78	#1374	—	13	2	no	—
79	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂²×D₁₃	—	2	1	no	—
81	C₂²×D₁₃	—	2	1	no	—
82	C₂²×D₁₃	—	2	2	no	—
83	C₂²×D₁₃	C₄×C₈	1	2	yes	—
84	C₂²×D₁₃	M₃₂	1	4	yes	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
88	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×Dic₁₃	—	2	2	no	—
95	C₂×Dic₁₃	—	2	1	no	—
96	C₂×Dic₁₃	—	2	1	no	—
97	C₂×Dic₁₃	—	2	2	no	—
98	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
99	C₂×Dic₁₃	M₃₂	1	4	yes	—
100	C₄×D₁₃	—	2	16	no	—
101	C₄×D₁₃	—	2	2	no	—
102	C₄×D₁₃	—	2	2	no	—
103	C₄×D₁₃	—	2	2	no	—
104	C₄×D₁₃	—	2	2	no	—
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	—	2	2	no	—
107	#1055	C₂²⋊C₈	1	4	yes	—
108	#1055	C₄×D₄	1	4	yes	—
109	C₂³×C₈	—	13	1	no	—
110	C₂³×C₈	—	13	1	no	—
111	C₂²×C₄²	—	13	1	no	—
112	C₂²×C₁₆	—	13	4	no	—
113	C₂×C₄×C₈	—	13	1	no	—
114	C₂×C₄×C₈	—	13	1	no	—
115	C₂×C₄×C₈	—	13	1	no	—
116	C₂×C₄×C₈	—	26	4	no	—
117	C₂×C₄×C₈	#261	1	1	yes	characteristic
118	C₂²⋊C₁₆	—	13	8	no	—
119	#346	—	26	4	no	—
120	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
121	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₁₃	#431	1	1	yes	characteristic
123	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
124	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
125	C₂×D₁₃	—	2	2	no	—
126	C₂×D₁₃	—	2	4	no	—
127	C₂×D₁₃	—	2	4	no	—
128	C₂×D₁₃	—	2	2	no	—
129	C₂×D₁₃	#346	1	4	yes	complemented
130	C₄×C₁₃	—	2	4	no	—
131	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
133	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₄×C₁₃	#431	1	1	yes	characteristic
135	Dic₁₃	—	2	4	no	—
136	Dic₁₃	—	2	2	no	—
137	C₂³×C₄	—	13	1	no	—
138	C₂³×C₄	—	13	2	no	—
139	C₂²×C₈	—	13	1	no	—
140	C₂²×C₈	—	13	1	no	—
141	C₂²×C₈	—	13	4	no	—
142	C₂²×C₈	—	13	4	no	—
143	C₂²×C₈	—	26	2	no	—
144	C₂²×C₈	—	26	2	no	—
145	C₂²×C₈	—	26	2	no	—
146	C₂²×C₈	—	26	2	no	—
147	C₂²×C₈	C₂×#261	1	1	yes	characteristic
148	C₂²×C₈	C₂×#261	1	1	yes	characteristic
149	C₂×C₄²	—	13	2	no	—
150	C₂×C₄²	—	13	1	no	—
151	C₂×C₄²	—	26	2	no	—
152	C₂×C₄²	—	26	2	no	—
153	C₂×C₄²	C₂×#261	1	1	yes	characteristic
154	C₂×C₁₆	—	13	8	no	—
155	C₂×C₁₆	—	26	4	no	—
156	C₂×C₁₆	—	26	4	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	13	4	no	—
159	C₄×C₈	—	26	2	no	—
160	C₄×C₈	—	26	2	no	—
161	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
162	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
163	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
164	C₂×C₁₃	#2112	1	2	yes	—
165	C₂×C₁₃	#2163	1	2	yes	—
166	D₁₃	—	2	4	no	—
167	C₂⁴	—	13	1	no	—
168	C₂²×C₄	—	13	1	no	—
169	C₂²×C₄	—	13	2	no	—
170	C₂²×C₄	—	13	2	no	—
171	C₂²×C₄	—	13	2	no	—
172	C₂²×C₄	—	26	4	no	—
173	C₂²×C₄	—	26	4	no	—
174	C₂²×C₄	—	26	4	no	—
175	C₂²×C₄	—	26	1	no	—
176	C₂²×C₄	—	26	1	no	—
177	C₂²×C₄	—	26	1	no	—
178	C₂²×C₄	—	26	1	no	—
179	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
180	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	13	2	no	—
186	C₂×C₈	—	13	2	no	—
187	C₂×C₈	—	13	2	no	—
188	C₂×C₈	—	13	2	no	—
189	C₂×C₈	—	13	2	no	—
190	C₂×C₈	—	13	2	no	—
191	C₂×C₈	—	26	8	no	—
192	C₂×C₈	—	26	8	no	—
193	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
194	C₂×C₈	C₄×#261	1	2	yes	—
195	C₄²	—	26	4	no	—
196	C₄²	—	26	1	no	—
197	C₄²	—	26	1	no	—
198	C₄²	#6118	1	2	yes	—
199	C₄²	C₂²⋊#261	1	2	yes	—
200	C₁₆	—	26	8	no	—
201	C₁₃	#12599	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	13	2	no	—
203	C₂³	—	13	4	no	—
204	C₂³	—	26	1	no	—
205	C₂³	—	26	2	no	—
206	C₂³	—	26	1	no	—
207	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	13	2	no	—
211	C₂×C₄	—	13	4	no	—
212	C₂×C₄	—	26	16	no	—
213	C₂×C₄	—	26	1	no	—
214	C₂×C₄	—	26	2	no	—
215	C₂×C₄	—	26	2	no	—
216	C₂×C₄	—	26	1	no	—
217	C₂×C₄	—	26	2	no	—
218	C₂×C₄	—	26	2	no	—
219	C₂×C₄	—	26	2	no	—
220	C₂×C₄	—	26	2	no	—
221	C₂×C₄	#88111	1	4	yes	—
222	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
223	C₂×C₄	C₂×#6118	1	1	yes	characteristic
224	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
226	C₂×C₄	#88237	1	1	yes	characteristic
227	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	13	4	no	—
231	C₈	—	13	4	no	—
232	C₂²	—	13	4	no	—
233	C₂²	—	26	2	no	—
234	C₂²	—	26	4	no	—
235	C₂²	—	26	2	no	—
236	C₂²	—	26	4	no	—
237	C₂²	C₄×D₁₃⋊C₈	1	1	yes	characteristic
238	C₂²	#11719049	1	1	yes	characteristic
239	C₂²	D₁₃⋊M₃₂	1	2	yes	—
240	C₂²	D₁₃⋊M₃₂	1	2	yes	—
241	C₂²	C₂×#88111	1	1	yes	characteristic
242	C₄	—	2	4	no	—
243	C₄	—	26	4	no	—
244	C₄	—	26	2	no	—
245	C₄	C₄×#6118	1	1	yes	characteristic
246	C₄	#11719051	1	1	yes	characteristic
247	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
248	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
249	C₂	—	26	4	no	—
250	C₂	C₄×#88111	1	1	yes	characteristic
251	C₂	#13131037	1	1	yes	characteristic
252	C₂	#13131043	1	1	yes	characteristic
253	C₂	#13131065	1	2	yes	—
254	C₂	#13131199	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#12599
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718231	C₂	semidirect
3. #13129875	C₂	semidirect
14. #11718259	C₄	semidirect
35. C₂×#6092	C₂×C₄	semidirect
129. C₂×D₁₃	#346	semidirect
201. C₁₃	#12599	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718259	2
C₂×#11718231	2
#13129875	2
C₂×C₄×C₈×D₁₃	2
#12599	13