Subgroups — C₁₃×#3078 #13112136

Order 1664.

376 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#479	C₂	1	4	yes	maximal, complemented
4	C₁₃×#477	C₂	1	2	yes	maximal, complemented
5	C₁₃×#475	C₂	1	2	yes	maximal, complemented
6	C₁₃×#428	C₂	1	2	yes	maximal
7	C₁₃×#423	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#419	C₂	1	2	yes	maximal
9	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
11	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₁₃×#125	C₂²	1	2	yes	—
14	C₁₃×C₂²⋊Q₈	—	2	1	no	—
15	C₁₃×C₂²⋊Q₈	—	2	1	no	—
16	C₄×Q₈×C₁₃	—	2	1	no	—
17	C₄×Q₈×C₁₃	—	2	1	no	—
18	C₄×Q₈×C₁₃	—	2	1	no	—
19	C₄×Q₈×C₁₃	—	2	1	no	—
20	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
21	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
22	C₁₃×#117	C₂²	1	2	yes	—
23	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
24	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	—	2	1	no	—
26	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#107	C₂²	1	4	yes	—
28	C₁₃×#106	C₂²	1	4	yes	—
29	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₁₃×#103	C₂²	1	4	yes	—
32	C₁₃×#103	C₂²	1	4	yes	—
33	C₂×Q₈×C₁₃	—	2	1	no	—
34	C₂×Q₈×C₁₃	—	2	1	no	—
35	C₂×Q₈×C₁₃	—	2	1	no	—
36	C₂×Q₈×C₁₃	—	2	2	no	—
37	C₂×Q₈×C₁₃	Q₈	1	4	yes	complemented
38	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	—	2	1	no	—
40	C₂²×C₄×C₁₃	—	2	1	no	—
41	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×M₁₆	—	2	4	no	—
45	C₂×C₈×C₁₃	C₂³	1	4	yes	—
46	C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₁₃×C₄⋊C₄	—	2	1	no	—
49	C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	—	2	1	no	—
52	C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₁₃×C₂²⋊C₄	—	2	2	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	2	2	no	—
60	C₄²×C₁₃	D₄	1	1	yes	characteristic
61	C₄²×C₁₃	D₄	1	1	yes	characteristic
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₄²×C₁₃	C₂³	1	1	yes	characteristic
64	#3078	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₁₃	—	2	4	no	—
67	Q₈×C₁₃	—	2	2	no	—
68	Q₈×C₁₃	—	4	2	no	—
69	Q₈×C₁₃	—	4	2	no	—
70	Q₈×C₁₃	C₂×Q₈	1	4	yes	—
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	4	1	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
85	C₂×C₄×C₁₃	#41	1	2	yes	—
86	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₈×C₁₃	—	2	4	no	—
88	#533	C₂×C₁₃	1	1	yes	characteristic
89	C₂×C₄×Q₈	C₂×C₁₃	1	1	yes	characteristic
90	#479	C₂×C₁₃	1	4	yes	complemented
91	#477	C₂×C₁₃	1	2	yes	complemented
92	#475	C₂×C₁₃	1	2	yes	complemented
93	#428	C₂×C₁₃	1	2	yes	—
94	#423	C₂×C₁₃	1	1	yes	characteristic
95	#419	C₂×C₁₃	1	2	yes	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	—	2	1	no	—
98	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
99	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₁₃	—	2	4	no	—
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	—	4	2	no	—
105	C₄×C₁₃	—	4	2	no	—
106	C₄×C₁₃	—	4	2	no	—
107	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
108	C₄×C₁₃	#137	1	2	yes	—
109	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
110	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
111	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
112	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
113	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
114	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
115	#125	C₂²×C₁₃	1	2	yes	—
116	C₂²⋊Q₈	—	2	1	no	—
117	C₂²⋊Q₈	—	2	1	no	—
118	C₄×Q₈	—	2	1	no	—
119	C₄×Q₈	—	2	1	no	—
120	C₄×Q₈	—	2	1	no	—
121	C₄×Q₈	—	2	1	no	—
122	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
123	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
124	#117	C₂²×C₁₃	1	2	yes	—
125	C₂×C₄⋊C₄	—	2	1	no	—
126	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×C₄²	—	2	1	no	—
128	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
129	#107	C₂²×C₁₃	1	4	yes	—
130	#106	C₂²×C₁₃	1	4	yes	—
131	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
132	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
133	#103	C₂²×C₁₃	1	4	yes	—
134	#103	C₂²×C₁₃	1	4	yes	—
135	C₂×C₁₃	—	2	2	no	—
136	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
138	C₂×C₁₃	#575	1	1	yes	characteristic
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	Q₈×C₁₃	1	4	yes	complemented
144	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
148	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
150	M₁₆	—	2	4	no	—
151	C₂×C₈	C₂³×C₁₃	1	4	yes	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
160	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
161	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	D₄×C₁₃	1	1	yes	characteristic
167	C₄²	D₄×C₁₃	1	1	yes	characteristic
168	C₄²	C₂³×C₁₃	1	1	yes	characteristic
169	C₄²	C₂³×C₁₃	1	1	yes	characteristic
170	C₁₃	#3078	1	1	yes	characteristic, direct product factor, Sylow
171	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
172	Q₈	—	2	4	no	—
173	Q₈	—	2	2	no	—
174	Q₈	—	4	2	no	—
175	Q₈	—	4	2	no	—
176	Q₈	C₂×Q₈×C₁₃	1	4	yes	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
187	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
188	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
189	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
190	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
191	C₂×C₄	C₁₃×#41	1	2	yes	—
192	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
193	C₈	—	2	4	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	C₁₃×C₂²⋊Q₈	1	2	yes	—
197	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
206	C₄	C₁₃×#137	1	2	yes	—
207	C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
208	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
211	C₂	C₂×C₁₃×#137	1	1	yes	characteristic
212	C₂	C₁₃×#575	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₃×#479	C₂	semidirect
4. C₁₃×#477	C₂	semidirect
5. C₁₃×#475	C₂	semidirect
37. C₂×Q₈×C₁₃	Q₈	semidirect
64. #3078	C₁₃	direct (N × H)
90. #479	C₂×C₁₃	semidirect
91. #477	C₂×C₁₃	semidirect
92. #475	C₂×C₁₃	semidirect
143. C₂×Q₈	Q₈×C₁₃	semidirect
170. C₁₃	#3078	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#533	2
C₂×C₄×Q₈×C₁₃	2
C₁₃×#479	2
C₁₃×#477	2
C₁₃×#475	2
C₁₃×#428	2
C₁₃×#423	2
C₁₃×#419	2
#3078	13

Subgroups — C13×#3078 #13112136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3078 #13112136