Subgroups — C₂×C₁₃×#1526 #15703976

Order 3328.

1122 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₃×M₁₆	C₂	1	2	yes	maximal
3	C₁₃×#1526	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#330	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×M₁₆	—	2	2	no	—
6	C₁₃×#533	—	2	4	no	—
7	C₁₃×#533	C₄	1	4	yes	—
8	C₁₃×#533	C₂²	1	4	yes	—
9	C₂×C₁₃×C₄⋊Q₈	C₄	1	2	yes	—
10	C₂×C₁₃×#125	C₂²	1	2	yes	—
11	C₂×C₁₃×C₂²⋊Q₈	—	2	2	no	—
12	C₂×C₄×Q₈×C₁₃	—	2	2	no	—
13	C₂×C₁₃×#117	C₄	1	2	yes	—
14	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×M₁₆	—	2	4	no	—
16	C₄×C₁₃×M₁₆	C₂²	1	8	yes	—
17	C₂×C₁₃×#97	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₁₃	—	2	2	no	—
19	C₁₃×#330	C₂²	1	16	yes	complemented
20	C₂²×Q₈×C₁₃	—	2	2	no	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	D₄	1	2	yes	—
23	C₂×C₁₃×M₁₆	—	2	16	no	—
24	C₂×C₁₃×M₁₆	—	2	8	no	—
25	C₂²×C₈×C₁₃	—	2	4	no	—
26	C₁₃×C₄⋊Q₈	—	2	4	no	—
27	C₁₃×C₄⋊Q₈	C₂×C₄	1	8	yes	—
28	C₁₃×#125	—	2	4	no	—
29	C₁₃×#125	C₂×C₄	1	4	yes	—
30	C₁₃×#125	C₂³	1	4	yes	—
31	C₁₃×C₂²⋊Q₈	—	2	8	no	—
32	C₁₃×C₂²⋊Q₈	—	2	8	no	—
33	C₄×Q₈×C₁₃	—	2	8	no	—
34	C₄×Q₈×C₁₃	—	2	8	no	—
35	C₁₃×#117	—	2	4	no	—
36	C₁₃×#117	C₂×C₄	1	8	yes	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
40	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
41	C₂×C₄²×C₁₃	—	2	2	no	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₃	D₄	1	4	yes	—
45	C₂×C₄²×C₁₃	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
48	C₁₃×#97	—	2	4	no	—
49	C₁₃×#97	C₂³	1	8	yes	—
50	C₄×C₈×C₁₃	—	2	8	no	—
51	C₂×#1526	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Q₈×C₁₃	—	2	8	no	—
54	C₂×Q₈×C₁₃	—	4	8	no	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	8	no	—
57	C₂²×C₄×C₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
66	C₁₃×M₁₆	—	2	32	no	—
67	C₂×C₈×C₁₃	—	2	16	no	—
68	C₂×C₈×C₁₃	—	2	8	no	—
69	C₁₃×C₄⋊C₄	—	2	8	no	—
70	C₁₃×C₄⋊C₄	—	2	8	no	—
71	C₁₃×C₄⋊C₄	—	2	4	no	—
72	C₁₃×C₄⋊C₄	—	2	8	no	—
73	C₁₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
74	C₁₃×C₂²⋊C₄	—	2	8	no	—
75	C₄²×C₁₃	—	2	8	no	—
76	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
77	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₁₃	C₂×D₄	1	4	yes	—
81	C₄²×C₁₃	C₂×D₄	1	2	yes	—
82	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₂×#533	C₂×C₁₃	1	1	yes	characteristic
84	C₂×C₄×M₁₆	C₂×C₁₃	1	2	yes	—
85	#1526	C₂×C₁₃	1	8	yes	direct product factor
86	C₂×#330	C₂×C₁₃	1	4	yes	complemented
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	2	4	no	—
89	C₂³×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
90	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	Q₈×C₁₃	—	4	16	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	8	no	—
97	C₂×C₄×C₁₃	—	2	8	no	—
98	C₂×C₄×C₁₃	—	2	8	no	—
99	C₂×C₄×C₁₃	—	4	8	no	—
100	C₂×C₄×C₁₃	#101	1	4	yes	—
101	C₂×C₄×C₁₃	#104	1	8	yes	—
102	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
108	C₈×C₁₃	—	2	16	no	—
109	C₂²×M₁₆	—	2	2	no	—
110	#533	—	2	4	no	—
111	#533	C₄×C₁₃	1	4	yes	—
112	#533	C₂²×C₁₃	1	4	yes	—
113	C₂×C₄⋊Q₈	C₄×C₁₃	1	2	yes	—
114	C₂×#125	C₂²×C₁₃	1	2	yes	—
115	C₂×C₂²⋊Q₈	—	2	2	no	—
116	C₂×C₄×Q₈	—	2	2	no	—
117	C₂×#117	C₄×C₁₃	1	2	yes	—
118	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
119	C₄×M₁₆	—	2	4	no	—
120	C₄×M₁₆	C₂²×C₁₃	1	8	yes	—
121	C₂×#97	C₂²×C₁₃	1	4	yes	—
122	C₂×C₄×C₈	—	2	2	no	—
123	#330	C₂²×C₁₃	1	16	yes	complemented
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	2	4	no	—
126	C₂²×C₁₃	—	4	4	no	—
127	C₂²×C₁₃	C₂×#101	1	2	yes	—
128	C₂²×C₁₃	C₂×#104	1	4	yes	—
129	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	4	8	no	—
135	C₄×C₁₃	C₂×#101	1	4	yes	—
136	C₄×C₁₃	C₂×#104	1	8	yes	—
137	C₂²×Q₈	—	2	2	no	—
138	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
139	C₂³×C₄	D₄×C₁₃	1	2	yes	—
140	C₂×M₁₆	—	2	16	no	—
141	C₂×M₁₆	—	2	8	no	—
142	C₂²×C₈	—	2	4	no	—
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	C₂×C₄×C₁₃	1	8	yes	—
145	#125	—	2	4	no	—
146	#125	C₂×C₄×C₁₃	1	4	yes	—
147	#125	C₂³×C₁₃	1	4	yes	—
148	C₂²⋊Q₈	—	2	8	no	—
149	C₂²⋊Q₈	—	2	8	no	—
150	C₄×Q₈	—	2	8	no	—
151	C₄×Q₈	—	2	8	no	—
152	#117	—	2	4	no	—
153	#117	C₂×C₄×C₁₃	1	8	yes	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
157	C₂×C₂²⋊C₄	—	2	2	no	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
161	C₂×C₄²	D₄×C₁₃	1	2	yes	—
162	C₂×C₄²	D₄×C₁₃	1	4	yes	—
163	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
164	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
165	#97	—	2	4	no	—
166	#97	C₂³×C₁₃	1	8	yes	—
167	C₄×C₈	—	2	8	no	—
168	C₂×C₁₃	—	4	2	no	—
169	C₂×C₁₃	#1526	1	4	yes	direct product factor
170	C₂×C₁₃	C₂²×#101	1	1	yes	characteristic
171	C₂×C₁₃	C₂²×#104	1	2	yes	—
172	C₂⁴	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×Q₈	—	2	8	no	—
174	C₂×Q₈	—	4	8	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
179	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
180	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
182	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
184	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
185	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
186	M₁₆	—	2	32	no	—
187	C₂×C₈	—	2	16	no	—
188	C₂×C₈	—	2	8	no	—
189	C₄⋊C₄	—	2	8	no	—
190	C₄⋊C₄	—	2	8	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	8	no	—
193	C₄⋊C₄	C₂²×C₄×C₁₃	1	8	yes	—
194	C₂²⋊C₄	—	2	8	no	—
195	C₄²	—	2	8	no	—
196	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
197	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
198	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
199	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
200	C₄²	C₂×D₄×C₁₃	1	4	yes	—
201	C₄²	C₂×D₄×C₁₃	1	2	yes	—
202	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
203	C₁₃	C₂×#1526	1	1	yes	characteristic, direct product factor, Sylow
204	C₂³	—	2	2	no	—
205	C₂³	—	2	4	no	—
206	C₂³	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
207	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
208	Q₈	—	4	16	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	C₁₃×#101	1	4	yes	—
218	C₂×C₄	C₁₃×#104	1	8	yes	—
219	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
220	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
222	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	4	yes	—
223	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
225	C₈	—	2	16	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	4	4	no	—
229	C₂²	C₂×C₁₃×#101	1	2	yes	—
230	C₂²	C₂×C₁₃×#104	1	4	yes	—
231	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	4	8	no	—
237	C₄	C₂×C₁₃×#101	1	4	yes	—
238	C₄	C₂×C₁₃×#104	1	8	yes	—
239	C₂	—	4	2	no	—
240	C₂	C₁₃×#1526	1	4	yes	direct product factor
241	C₂	C₂²×C₁₃×#101	1	1	yes	characteristic
242	C₂	C₂²×C₁₃×#104	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1526	C₂	direct (N × H)
4. C₂×C₁₃×#330	C₂	semidirect
19. C₁₃×#330	C₂²	semidirect
51. C₂×#1526	C₁₃	direct (N × H)
85. #1526	C₂×C₁₃	direct (N × H)
86. C₂×#330	C₂×C₁₃	semidirect
123. #330	C₂²×C₁₃	semidirect
169. C₂×C₁₃	#1526	direct (N × H)
203. C₁₃	C₂×#1526	direct (N × H)
240. C₂	C₁₃×#1526	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#533	2
C₂×C₄×C₁₃×M₁₆	2
C₁₃×#1526	2
C₂×C₁₃×#330	2
C₂×#1526	13

Subgroups — C2×C13×#1526 #15703976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#1526 #15703976