Subgroups — C₁₃×#7182 #15693521

Order 3328.

258 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×M₃₂	—	2	1	no	—
5	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
6	C₁₃×#427	C₄	1	1	yes	characteristic
7	C₁₃×#405	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃	—	2	1	no	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#364	—	2	1	no	—
11	C₁₃×#364	C₄	1	2	yes	complemented
12	C₁₃×#339	—	2	1	no	—
13	C₁₃×#322	—	2	1	no	—
14	C₁₃×#322	C₄	1	2	yes	complemented
15	C₂×C₁₃×M₁₆	—	2	1	no	—
16	C₂²×C₈×C₁₃	—	2	2	no	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₁₃×#117	—	2	1	no	—
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₁₃×M₃₂	—	2	2	no	—
22	C₁₃×M₃₂	—	4	1	no	—
23	C₂×C₁₃×C₁₆	—	4	1	no	—
24	C₁₃×#108	—	2	1	no	—
25	C₁₃×#108	C₈	1	2	yes	complemented
26	C₁₃×#107	C₈	1	2	yes	complemented
27	C₁₃×#106	—	2	1	no	—
28	C₁₃×#97	—	4	1	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₄×C₈×C₁₃	—	2	2	no	—
31	C₄×C₈×C₁₃	—	4	1	no	—
32	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
36	#7182	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₄×C₁₃	—	2	1	no	—
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₁₃×M₁₆	—	2	2	no	—
40	C₁₃×M₁₆	—	4	1	no	—
41	C₂×C₈×C₁₃	—	2	4	no	—
42	C₂×C₈×C₁₃	—	4	4	no	—
43	C₂×C₈×C₁₃	—	4	1	no	—
44	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
52	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
53	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₂²⋊C₄	—	4	1	no	—
58	C₄²×C₁₃	—	2	1	no	—
59	C₄²×C₁₃	—	4	1	no	—
60	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
62	C₁₃×C₁₆	—	4	2	no	—
63	C₂×#364	C₂×C₁₃	1	1	yes	characteristic
64	#1772	C₂×C₁₃	1	1	yes	characteristic
65	C₂×#322	C₂×C₁₃	1	1	yes	characteristic
66	C₂³×C₁₃	#104	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	4	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
77	C₈×C₁₃	—	4	4	no	—
78	C₈×C₁₃	—	4	2	no	—
79	C₈×C₁₃	C₄×C₈	1	1	yes	characteristic
80	C₈×C₁₃	#97	1	1	yes	characteristic
81	C₈×C₁₃	#102	1	1	yes	characteristic
82	C₈×C₁₃	#103	1	1	yes	characteristic
83	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
84	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
85	C₈×C₁₃	#106	1	1	yes	characteristic
86	C₈×C₁₃	#107	1	1	yes	characteristic
87	C₂×M₃₂	—	2	1	no	—
88	C₂×#108	C₄×C₁₃	1	1	yes	characteristic
89	#427	C₄×C₁₃	1	1	yes	characteristic
90	#405	—	2	1	no	—
91	C₂×C₄×C₈	—	2	1	no	—
92	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
93	#364	—	2	1	no	—
94	#364	C₄×C₁₃	1	2	yes	complemented
95	#339	—	2	1	no	—
96	#322	—	2	1	no	—
97	#322	C₄×C₁₃	1	2	yes	complemented
98	C₂²×C₁₃	—	2	2	no	—
99	C₂²×C₁₃	D₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
101	C₂²×C₁₃	#339	1	1	yes	characteristic
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	2	2	no	—
104	C₄×C₁₃	—	8	2	no	—
105	C₄×C₁₃	D₄⋊C₈	1	1	yes	characteristic
106	C₄×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
107	C₄×C₁₃	#336	1	1	yes	characteristic
108	C₄×C₁₃	#340	1	1	yes	characteristic
109	C₂×M₁₆	—	2	1	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
112	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
113	#117	—	2	1	no	—
114	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	—	4	1	no	—
117	C₂×C₁₆	—	4	1	no	—
118	#108	—	2	1	no	—
119	#108	C₈×C₁₃	1	2	yes	complemented
120	#107	C₈×C₁₃	1	2	yes	complemented
121	#106	—	2	1	no	—
122	#97	—	4	1	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	4	1	no	—
126	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
127	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
128	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
129	C₄×C₈	Q₈×C₁₃	1	1	yes	characteristic
130	C₂×C₁₃	—	2	1	no	—
131	C₂×C₁₃	—	2	1	no	—
132	C₂×C₁₃	#1279	1	1	yes	characteristic
133	C₂×C₁₃	#1336	1	2	yes	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	4	1	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	C₄²×C₁₃	1	1	yes	characteristic
142	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
146	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
147	C₂×C₈	C₁₃×M₁₆	1	1	yes	characteristic
148	C₂×C₈	C₁₃×M₁₆	1	1	yes	characteristic
149	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
150	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
151	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
152	C₂×C₈	C₁₃×Q₁₆	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	—	4	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	4	1	no	—
157	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
158	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
159	C₁₆	—	4	2	no	—
160	C₁₃	#7182	1	1	yes	characteristic, direct product factor, Sylow
161	C₂³	C₁₃×#104	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₁₃×#95	1	1	yes	characteristic
167	C₂×C₄	C₁₃×C₂²⋊C₈	1	1	yes	characteristic
168	C₂×C₄	C₁₃×C₂²⋊C₈	1	1	yes	characteristic
169	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
170	C₂×C₄	C₁₃×#103	1	1	yes	characteristic
171	C₂×C₄	C₁₃×#104	1	1	yes	characteristic
172	C₈	—	4	4	no	—
173	C₈	—	4	2	no	—
174	C₈	C₄×C₈×C₁₃	1	1	yes	characteristic
175	C₈	C₁₃×#97	1	1	yes	characteristic
176	C₈	C₁₃×#102	1	1	yes	characteristic
177	C₈	C₁₃×#103	1	1	yes	characteristic
178	C₈	C₁₃×C₄⋊C₈	1	1	yes	characteristic
179	C₈	C₁₃×C₄⋊C₈	1	1	yes	characteristic
180	C₈	C₁₃×#106	1	1	yes	characteristic
181	C₈	C₁₃×#107	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	C₁₃×D₄⋊C₈	1	1	yes	characteristic
184	C₂²	C₁₃×Q₈⋊C₈	1	1	yes	characteristic
185	C₂²	C₁₃×#339	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	8	2	no	—
189	C₄	C₁₃×D₄⋊C₈	1	1	yes	characteristic
190	C₄	C₁₃×Q₈⋊C₈	1	1	yes	characteristic
191	C₄	C₁₃×#336	1	1	yes	characteristic
192	C₄	C₁₃×#340	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₁₃×#1279	1	1	yes	characteristic
196	C₂	C₁₃×#1336	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	#102
commutator	C₈	C₄×C₈×C₁₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₃
socle	C₂²×C₁₃	D₄⋊C₈
prime radical	C₂³×C₁₃	#104
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₁₃×#364	C₄	semidirect
14. C₁₃×#322	C₄	semidirect
25. C₁₃×#108	C₈	semidirect
26. C₁₃×#107	C₈	semidirect
36. #7182	C₁₃	direct (N × H)
94. #364	C₄×C₁₃	semidirect
97. #322	C₄×C₁₃	semidirect
119. #108	C₈×C₁₃	semidirect
120. #107	C₈×C₁₃	semidirect
160. C₁₃	#7182	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×#364	2
C₁₃×#1772	2
C₂×C₁₃×#322	2
#7182	13

Subgroups — C13×#7182 #15693521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#7182 #15693521