Subgroups — #15768619

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114569	C₂	1	2	yes	maximal
2	#13114553	C₂	1	2	yes	maximal
3	C₂×#11718149	C₂	1	2	yes	maximal
4	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
8	C₂×#88180	C₂²	1	4	yes	—
9	C₁₃×#405	—	2	4	no	—
10	C₂×C₁₃×#97	C₄	1	2	yes	complemented
11	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	complemented
12	#11718149	C₄	1	8	yes	complemented
13	#11718143	—	2	4	no	—
14	#11718118	—	2	4	no	—
15	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₁₃×M₁₆	—	2	2	no	—
17	C₂×C₁₃×M₁₆	—	2	2	no	—
18	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	D₄	1	4	yes	—
25	C₁₃×#117	—	2	2	no	—
26	C₁₃×#117	—	2	2	no	—
27	C₂×C₁₃×C₄⋊C₄	Q₈	1	2	yes	—
28	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
31	#88180	—	2	4	no	—
32	#88180	—	2	4	no	—
33	#88180	—	2	4	no	—
34	#88180	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊Dic₁₃	—	2	4	no	—
36	C₂×C₄×Dic₁₃	—	2	4	no	—
37	C₁₃×#97	—	2	4	no	—
38	C₄×C₈×C₁₃	—	2	4	no	—
39	#10814	—	13	1	no	maximal, Sylow
40	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	#41	1	2	yes	—
47	C₂²×C₄×C₁₃	#41	1	2	yes	—
48	C₂²×C₄×C₁₃	#41	1	2	yes	—
49	C₂²×C₄×C₁₃	#41	1	2	yes	—
50	C₂²×Dic₁₃	—	4	4	no	—
51	C₁₃×M₁₆	—	2	8	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	1	no	—
54	C₂×C₈×C₁₃	—	2	1	no	—
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	—	2	8	no	—
59	C₂×C₈×C₁₃	C₄²	1	4	yes	—
60	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₁₃×C₄⋊C₄	—	2	4	no	—
63	C₁₃×C₂²⋊C₄	—	2	4	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	—	2	2	no	—
66	C₂²⋊Dic₁₃	—	2	8	no	—
67	C₂²⋊Dic₁₃	—	4	4	no	—
68	C₄⋊Dic₁₃	—	2	8	no	—
69	C₄⋊Dic₁₃	—	4	4	no	—
70	C₄×Dic₁₃	—	2	8	no	—
71	C₄×Dic₁₃	—	4	4	no	—
72	C₂×#405	D₁₃	1	1	yes	characteristic
73	#1740	—	13	2	no	—
74	#1735	—	13	2	no	—
75	C₂×H(3,ℤ/4)	—	13	2	no	—
76	C₂³×C₁₃	—	4	2	no	—
77	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₁₃	C₄×D₄	1	2	yes	—
79	C₂³×C₁₃	#121	1	2	yes	—
80	C₂³×C₁₃	#124	1	2	yes	—
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	4	4	no	—
83	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#117	1	4	yes	—
88	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
92	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
93	C₂×C₄×C₁₃	#123	1	4	yes	—
94	C₂×C₄×C₁₃	#125	1	2	yes	—
95	C₂×C₄×C₁₃	#126	1	4	yes	—
96	C₂×Dic₁₃	—	4	8	no	—
97	C₂×Dic₁₃	—	8	8	no	—
98	C₈×C₁₃	—	2	4	no	—
99	C₈×C₁₃	—	2	2	no	—
100	C₈×C₁₃	—	2	2	no	—
101	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
102	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
103	C₂×#117	—	13	4	no	—
104	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
105	#405	—	2	4	no	—
106	C₂×#97	Dic₁₃	1	2	yes	complemented
107	C₂×C₄×C₈	Dic₁₃	1	2	yes	complemented
108	#342	—	26	4	no	—
109	#340	—	26	4	no	—
110	H(3,ℤ/4)	—	13	8	no	—
111	C₂²×C₁₃	—	2	4	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₁₃	#380	1	1	yes	characteristic
117	C₂²×C₁₃	#382	1	1	yes	characteristic
118	C₂²×C₁₃	#388	1	4	yes	—
119	C₄×C₁₃	—	4	4	no	—
120	C₄×C₁₃	#376	1	1	yes	characteristic
121	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₁₃	#381	1	1	yes	characteristic
123	C₄×C₁₃	#383	1	1	yes	characteristic
124	C₄×C₁₃	#387	1	4	yes	—
125	Dic₁₃	—	8	8	no	—
126	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
133	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
136	#117	—	2	2	no	—
137	#117	—	2	2	no	—
138	#117	—	13	8	no	—
139	#117	—	26	4	no	—
140	#117	—	26	4	no	—
141	#117	—	26	4	no	—
142	C₂×C₄⋊C₄	—	26	4	no	—
143	C₂×C₄⋊C₄	Dic₂₆	1	2	yes	—
144	C₂×C₂²⋊C₄	—	26	4	no	—
145	C₂×C₂²⋊C₄	D₅₂	1	2	yes	—
146	C₂×C₄²	—	26	4	no	—
147	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
148	#97	—	2	4	no	—
149	C₄×C₈	—	2	4	no	—
150	C₂×C₁₃	—	2	4	no	—
151	C₂×C₁₃	—	2	1	no	—
152	C₂×C₁₃	—	2	1	no	—
153	C₂×C₁₃	#1434	1	1	yes	characteristic
154	C₂×C₁₃	#1812	1	2	yes	—
155	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
156	C₂²×C₄	—	52	4	no	—
157	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
158	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
160	C₂²×C₄	#6127	1	2	yes	—
161	C₂²×C₄	#6127	1	2	yes	—
162	C₂²×C₄	#6127	1	2	yes	—
163	C₂²×C₄	#6127	1	2	yes	—
164	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
166	M₁₆	—	2	8	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	8	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
175	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
176	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	26	8	no	—
179	C₄⋊C₄	—	52	4	no	—
180	C₂²⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	26	8	no	—
182	C₂²⋊C₄	—	52	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	26	8	no	—
186	C₄²	—	52	4	no	—
187	C₁₃	#10814	1	1	yes	characteristic, complemented, Sylow
188	C₂³	—	4	2	no	—
189	C₂³	C₄×D₅₂	1	2	yes	—
190	C₂³	#88129	1	2	yes	—
191	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
192	C₂³	#88184	1	2	yes	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	52	8	no	—
196	C₂×C₄	—	104	8	no	—
197	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
198	C₂×C₄	#88124	1	2	yes	—
199	C₂×C₄	#88126	1	4	yes	—
200	C₂×C₄	#88130	1	4	yes	—
201	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
202	C₂×C₄	#88178	1	2	yes	—
203	C₂×C₄	#88180	1	1	yes	characteristic
204	C₂×C₄	#88180	1	1	yes	characteristic
205	C₂×C₄	#88180	1	1	yes	characteristic
206	C₂×C₄	#88180	1	1	yes	characteristic
207	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
208	C₂×C₄	#88183	1	4	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
210	C₈	—	2	4	no	—
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
218	C₂²	#11718597	1	1	yes	characteristic
219	C₂²	#11718601	1	4	yes	—
220	C₂²	#11718873	1	1	yes	characteristic
221	C₄	—	4	4	no	—
222	C₄	—	104	8	no	—
223	C₄	#11718593	1	1	yes	characteristic
224	C₄	#11718594	1	4	yes	—
225	C₄	#11718598	1	1	yes	characteristic
226	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
227	C₄	#11718872	1	1	yes	characteristic
228	C₂	—	2	4	no	—
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	#13113063	1	1	yes	characteristic
232	C₂	#13114882	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88180
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#10814
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
10. C₂×C₁₃×#97	C₄	semidirect
11. C₂×C₄×C₈×C₁₃	C₄	semidirect
12. #11718149	C₄	semidirect
106. C₂×#97	Dic₁₃	semidirect
107. C₂×C₄×C₈	Dic₁₃	semidirect
187. C₁₃	#10814	semidirect

Maximal subgroups

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

subgroup	index
#13114569	2
#13114553	2
C₂×#11718149	2
C₂×C₁₃×#405	2
#10814	13