Subgroups — C₂×C₁₃×#2183 #15719812

Order 3328.

542 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2183	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×C₄⋊C₁₆	C₂	1	2	yes	maximal
5	C₂×C₁₃×#346	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
8	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₁₃	C₄	1	4	yes	—
12	C₈×Q₈×C₁₃	C₂²	1	4	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
17	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
18	C₁₃×C₄⋊C₁₆	C₂²	1	8	yes	—
19	C₁₃×#346	C₂²	1	8	yes	—
20	C₄×C₁₃×C₁₆	C₂²	1	4	yes	—
21	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₄×Q₈×C₁₃	C₈	1	4	yes	—
26	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
27	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×C₁₆	—	2	2	no	—
32	C₂×C₁₃×C₁₆	—	2	4	no	—
33	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
34	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
35	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
36	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
37	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
38	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₁₃	C₂³	1	2	yes	—
42	C₄×C₈×C₁₃	C₂³	1	4	yes	—
43	C₂×#2183	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×Q₈×C₁₃	—	2	4	no	—
45	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
46	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	—	2	4	no	—
53	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₁₃	#41	1	2	yes	—
58	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
59	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
60	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
61	C₄²×C₁₃	C₂×C₈	1	4	yes	—
62	C₄²×C₁₃	C₂×C₈	1	2	yes	—
63	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
64	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
65	C₁₃×C₁₆	—	2	2	no	—
66	C₁₃×C₁₆	—	2	4	no	—
67	C₁₃×C₁₆	C₂×Q₈	1	4	yes	complemented
68	C₂×C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic
69	#2183	C₂×C₁₃	1	8	yes	direct product factor
70	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
71	C₂×C₄⋊C₁₆	C₂×C₁₃	1	2	yes	—
72	C₂×#346	C₂×C₁₃	1	2	yes	—
73	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
74	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	Q₈×C₁₃	—	2	8	no	—
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
80	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
82	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₁₃	#131	1	2	yes	—
87	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
88	C₈×C₁₃	—	2	2	no	—
89	C₈×C₁₃	—	2	4	no	—
90	C₈×C₁₃	C₄×Q₈	1	2	yes	—
91	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
93	C₂×C₄×Q₈	C₄×C₁₃	1	1	yes	characteristic
94	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
95	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
96	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
97	C₈×Q₈	C₄×C₁₃	1	4	yes	—
98	C₈×Q₈	C₂²×C₁₃	1	4	yes	—
99	C₂×C₄⋊C₈	C₄×C₁₃	1	2	yes	—
100	C₂×C₄⋊C₈	C₄×C₁₃	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
103	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	—
104	C₄⋊C₁₆	C₂²×C₁₃	1	8	yes	—
105	#346	C₂²×C₁₃	1	8	yes	—
106	C₄×C₁₆	C₂²×C₁₃	1	4	yes	—
107	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
108	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
109	C₂²×C₁₃	#504	1	2	yes	—
110	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	—	2	4	no	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	C₈×Q₈	1	2	yes	—
115	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
116	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
117	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
118	C₂²×Q₈	C₈×C₁₃	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
120	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
121	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
122	C₄×Q₈	C₈×C₁₃	1	4	yes	—
123	C₄×Q₈	C₂×C₄×C₁₃	1	4	yes	—
124	C₂×C₄⋊C₄	C₈×C₁₃	1	2	yes	—
125	C₂×C₄⋊C₄	C₈×C₁₃	1	1	yes	characteristic
126	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
127	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	4	no	—
130	C₂×C₁₆	Q₈×C₁₃	1	4	yes	complemented
131	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
132	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
133	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
134	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
135	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
136	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
137	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
138	C₄×C₈	C₂³×C₁₃	1	2	yes	—
139	C₄×C₈	C₂³×C₁₃	1	4	yes	—
140	C₂×C₁₃	#2183	1	4	yes	direct product factor
141	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
142	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
143	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	C₂×C₈×C₁₃	1	4	yes	—
146	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
147	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
149	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
154	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
155	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
156	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
157	C₂×C₈	C₁₃×#41	1	2	yes	—
158	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
159	C₄⋊C₄	C₂×C₈×C₁₃	1	8	yes	—
160	C₄⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
161	C₄²	C₂×C₈×C₁₃	1	4	yes	—
162	C₄²	C₂×C₈×C₁₃	1	2	yes	—
163	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
164	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
165	C₁₆	—	2	2	no	—
166	C₁₆	—	2	4	no	—
167	C₁₆	C₂×Q₈×C₁₃	1	4	yes	complemented
168	C₁₃	C₂×#2183	1	1	yes	characteristic, direct product factor, Sylow
169	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
170	Q₈	—	2	8	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₁₃×M₃₂	1	4	yes	—
175	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
176	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
177	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
178	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
179	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
181	C₂×C₄	C₁₃×#131	1	2	yes	—
182	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	4	no	—
185	C₈	C₄×Q₈×C₁₃	1	2	yes	—
186	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
187	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
188	C₂²	C₈×Q₈×C₁₃	1	2	yes	—
189	C₂²	C₂×C₁₃×M₃₂	1	2	yes	—
190	C₂²	C₁₃×#504	1	2	yes	—
191	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	2	no	—
195	C₄	C₈×Q₈×C₁₃	1	2	yes	—
196	C₄	C₂×C₁₃×M₃₂	1	4	yes	—
197	C₄	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
198	C₄	C₂×C₁₃×#131	1	1	yes	characteristic
199	C₂	C₁₃×#2183	1	4	yes	direct product factor
200	C₂	C₂×C₈×Q₈×C₁₃	1	1	yes	characteristic
201	C₂	C₂²×C₁₃×M₃₂	1	1	yes	characteristic
202	C₂	C₂×C₁₃×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₁₃	C₂³
commutator	C₂²	C₂³×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle, prime radical	C₂³×C₁₃	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2183	C₂	direct (N × H)
33. C₂×C₁₃×C₁₆	Q₈	semidirect
43. C₂×#2183	C₁₃	direct (N × H)
67. C₁₃×C₁₆	C₂×Q₈	semidirect
69. #2183	C₂×C₁₃	direct (N × H)
130. C₂×C₁₆	Q₈×C₁₃	semidirect
140. C₂×C₁₃	#2183	direct (N × H)
167. C₁₆	C₂×Q₈×C₁₃	semidirect
168. C₁₃	C₂×#2183	direct (N × H)
199. C₂	C₁₃×#2183	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×C₈×Q₈×C₁₃	2
C₁₃×#2183	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#346	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#2183	13

Subgroups — C2×C13×#2183 #15719812

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2183 #15719812