Subgroups — C₁₃×#10837 #15697176

Order 3328.

842 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2285	C₂	1	2	yes	maximal
3	C₂×C₁₃×H(3,ℤ/4)	C₂	1	4	yes	maximal
4	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
5	C₂×C₁₃×#117	C₂²	1	4	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₄×C₁₃×M₁₆	—	2	2	no	—
8	C₄×C₁₃×M₁₆	—	2	2	no	—
9	C₂×C₁₃×#97	C₄	1	2	yes	complemented
10	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	complemented
11	C₁₃×#384	—	2	2	no	—
12	C₁₃×#382	—	2	2	no	—
13	C₁₃×#380	—	2	4	no	—
14	C₄×C₁₃×C₄⋊C₄	C₄	1	8	yes	complemented
15	C₁₃×H(3,ℤ/4)	—	2	8	no	—
16	C₂³×C₄×C₁₃	D₄	1	2	yes	—
17	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₁₃×M₁₆	—	2	4	no	—
19	C₂×C₁₃×M₁₆	—	2	4	no	—
20	C₂×C₁₃×M₁₆	D₄	1	4	yes	—
21	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
23	C₁₃×#117	—	2	8	no	—
24	C₁₃×#117	—	2	8	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
27	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	8	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
35	C₁₃×#97	—	2	4	no	—
36	C₄×C₈×C₁₃	—	2	2	no	—
37	C₄×C₈×C₁₃	—	2	2	no	—
38	C₁₃×#95	—	2	8	no	—
39	#10837	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	—	2	8	no	—
42	C₂²×C₄×C₁₃	—	4	4	no	—
43	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
53	C₂²×C₄×C₁₃	#41	1	4	yes	—
54	C₁₃×M₁₆	—	2	8	no	—
55	C₁₃×M₁₆	—	2	8	no	—
56	C₂×C₈×C₁₃	—	2	4	no	—
57	C₂×C₈×C₁₃	—	2	4	no	—
58	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	complemented
59	C₁₃×C₄⋊C₄	—	4	16	no	—
60	C₁₃×C₄⋊C₄	—	4	8	no	—
61	C₁₃×C₂²⋊C₄	—	4	4	no	—
62	C₁₃×C₂²⋊C₄	—	4	4	no	—
63	C₄²×C₁₃	—	2	16	no	—
64	C₄²×C₁₃	C₄²	1	2	yes	—
65	C₄²×C₁₃	C₄²	1	2	yes	—
66	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
70	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
71	C₂×C₄×M₁₆	C₂×C₁₃	1	1	yes	characteristic
72	#2285	C₂×C₁₃	1	2	yes	—
73	C₂×H(3,ℤ/4)	C₂×C₁₃	1	4	yes	—
74	C₂³×C₁₃	—	4	2	no	—
75	C₂³×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
76	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₁₃	#124	1	2	yes	—
78	C₂³×C₁₃	#127	1	1	yes	characteristic
79	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	—	2	8	no	—
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	4	4	no	—
83	C₂×C₄×C₁₃	—	4	2	no	—
84	C₂×C₄×C₁₃	—	4	16	no	—
85	C₂×C₄×C₁₃	—	4	8	no	—
86	C₂×C₄×C₁₃	#95	1	2	yes	—
87	C₂×C₄×C₁₃	#95	1	4	yes	—
88	C₂×C₄×C₁₃	#95	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₁₃	#117	1	2	yes	—
94	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
95	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
96	C₂×C₄×C₁₃	#121	1	4	yes	—
97	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
98	C₂×C₄×C₁₃	#125	1	2	yes	—
99	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
100	C₈×C₁₃	—	2	8	no	—
101	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
102	C₂×#117	C₂²×C₁₃	1	4	yes	—
103	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
104	C₄×M₁₆	—	2	2	no	—
105	C₄×M₁₆	—	2	2	no	—
106	C₂×#97	C₄×C₁₃	1	2	yes	complemented
107	C₂×C₄×C₈	C₄×C₁₃	1	2	yes	complemented
108	#384	—	2	2	no	—
109	#382	—	2	2	no	—
110	#380	—	2	4	no	—
111	C₄×C₄⋊C₄	C₄×C₁₃	1	8	yes	complemented
112	H(3,ℤ/4)	—	2	8	no	—
113	C₂²×C₁₃	—	2	4	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	4	2	no	—
116	C₂²×C₁₃	—	4	1	no	—
117	C₂²×C₁₃	—	4	1	no	—
118	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
119	C₂²×C₁₃	#382	1	1	yes	characteristic
120	C₂²×C₁₃	#384	1	1	yes	characteristic
121	C₂²×C₁₃	#390	1	2	yes	—
122	C₂²×C₁₃	#391	1	2	yes	—
123	C₄×C₁₃	—	2	4	no	—
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	8	8	no	—
126	C₄×C₁₃	H(3,ℤ/4)	1	4	yes	complemented
127	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
128	C₄×C₁₃	#380	1	2	yes	—
129	C₄×C₁₃	#389	1	4	yes	—
130	C₂³×C₄	D₄×C₁₃	1	2	yes	—
131	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	—	2	4	no	—
134	C₂×M₁₆	D₄×C₁₃	1	4	yes	—
135	C₂×M₁₆	Q₈×C₁₃	1	4	yes	—
136	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
137	#117	—	2	8	no	—
138	#117	—	2	8	no	—
139	C₂×C₄⋊C₄	—	2	8	no	—
140	C₂×C₄⋊C₄	—	2	4	no	—
141	C₂×C₂²⋊C₄	—	2	4	no	—
142	C₂×C₄²	C₂×C₄×C₁₃	1	8	yes	—
143	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
144	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
146	C₂×C₄²	D₄×C₁₃	1	2	yes	—
147	C₂×C₄²	D₄×C₁₃	1	2	yes	—
148	C₂×C₄²	Q₈×C₁₃	1	2	yes	—
149	#97	—	2	4	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	#95	—	2	8	no	—
153	C₂×C₁₃	—	2	2	no	—
154	C₂×C₁₃	—	2	1	no	—
155	C₂×C₁₃	—	2	1	no	—
156	C₂×C₁₃	—	2	2	no	—
157	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
158	C₂×C₁₃	C₂×H(3,ℤ/4)	1	2	yes	—
159	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
163	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
164	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
165	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
166	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
168	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
170	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
172	C₂²×C₄	C₁₃×#41	1	4	yes	—
173	M₁₆	—	2	8	no	—
174	M₁₆	—	2	8	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	C₁₃×C₄⋊C₄	1	8	yes	complemented
178	C₄⋊C₄	—	4	16	no	—
179	C₄⋊C₄	—	4	8	no	—
180	C₂²⋊C₄	—	4	4	no	—
181	C₂²⋊C₄	—	4	4	no	—
182	C₄²	—	2	16	no	—
183	C₄²	C₄²×C₁₃	1	2	yes	—
184	C₄²	C₄²×C₁₃	1	2	yes	—
185	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
186	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
187	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
188	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
189	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
190	C₁₃	#10837	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³	—	4	2	no	—
192	C₂³	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
193	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
194	C₂³	C₁₃×#124	1	2	yes	—
195	C₂³	C₁₃×#127	1	1	yes	characteristic
196	C₂³	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	16	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	C₁₃×#95	1	2	yes	—
204	C₂×C₄	C₁₃×#95	1	4	yes	—
205	C₂×C₄	C₁₃×#95	1	2	yes	—
206	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
207	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
210	C₂×C₄	C₁₃×#117	1	2	yes	—
211	C₂×C₄	C₄×D₄×C₁₃	1	4	yes	—
212	C₂×C₄	C₄×Q₈×C₁₃	1	4	yes	—
213	C₂×C₄	C₁₃×#121	1	4	yes	—
214	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
215	C₂×C₄	C₁₃×#125	1	2	yes	—
216	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
217	C₈	—	2	8	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	C₂×C₁₃×#95	1	1	yes	characteristic
224	C₂²	C₁₃×#382	1	1	yes	characteristic
225	C₂²	C₁₃×#384	1	1	yes	characteristic
226	C₂²	C₁₃×#390	1	2	yes	—
227	C₂²	C₁₃×#391	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	8	8	no	—
231	C₄	C₁₃×H(3,ℤ/4)	1	4	yes	complemented
232	C₄	C₄×C₁₃×C₄⋊C₄	1	2	yes	—
233	C₄	C₁₃×#380	1	2	yes	—
234	C₄	C₁₃×#389	1	4	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	C₁₃×C₄⋊#95	1	1	yes	characteristic
240	C₂	C₂×C₁₃×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#95
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₁₃×#97	C₄	semidirect
10. C₂×C₄×C₈×C₁₃	C₄	semidirect
14. C₄×C₁₃×C₄⋊C₄	C₄	semidirect
39. #10837	C₁₃	direct (N × H)
58. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
106. C₂×#97	C₄×C₁₃	semidirect
107. C₂×C₄×C₈	C₄×C₁₃	semidirect
111. C₄×C₄⋊C₄	C₄×C₁₃	semidirect
126. C₄×C₁₃	H(3,ℤ/4)	semidirect
177. C₂×C₈	C₁₃×C₄⋊C₄	semidirect
190. C₁₃	#10837	direct (N × H)
231. C₄	C₁₃×H(3,ℤ/4)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₃×M₁₆	2
C₁₃×#2285	2
C₂×C₁₃×H(3,ℤ/4)	2
#10837	13

Subgroups — C13×#10837 #15697176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10837 #15697176