Subgroups — C₂×C₁₃×#2151 #15719735

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2151	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×C₄⋊C₁₆	C₂	1	4	yes	maximal, complemented
6	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	—	2	4	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₃×C₁₆	—	2	4	no	—
12	C₂²×C₁₃×C₁₆	C₂²	1	4	yes	—
13	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
14	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
15	C₁₃×#405	C₄	1	4	yes	—
16	C₁₃×#405	C₂²	1	4	yes	—
17	C₂×C₁₃×#97	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
19	C₁₃×C₄⋊C₁₆	C₂²	1	16	yes	complemented
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×M₁₆	—	2	4	no	—
22	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	D₄	1	4	yes	—
27	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₁₃×#117	C₈	1	4	yes	—
31	C₁₃×#117	C₂×C₄	1	4	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
33	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₁₃×M₃₂	—	2	8	no	—
36	C₂×C₁₃×C₁₆	—	2	4	no	—
37	C₂×C₁₃×C₁₆	—	2	8	no	—
38	C₂×C₁₃×C₁₆	—	2	4	no	—
39	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
40	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
41	C₁₃×#97	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₁₃	C₂³	1	4	yes	—
44	C₂×#2151	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
53	C₁₃×M₁₆	—	2	8	no	—
54	C₂×C₈×C₁₃	—	2	4	no	—
55	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
56	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
62	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
64	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
66	C₁₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
67	C₄²×C₁₃	C₂×C₈	1	4	yes	—
68	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
69	C₁₃×C₁₆	—	2	4	no	—
70	C₁₃×C₁₆	—	2	4	no	—
71	C₂²×M₃₂	C₂×C₁₃	1	1	yes	characteristic
72	C₂³×C₁₆	C₂×C₁₃	1	1	yes	characteristic
73	C₂×#405	C₂×C₁₃	1	1	yes	characteristic
74	#2151	C₂×C₁₃	1	8	yes	direct product factor
75	C₂×C₄⋊C₁₆	C₂×C₁₃	1	4	yes	complemented
76	C₂³×C₁₃	—	2	4	no	—
77	C₂³×C₁₃	C₄⋊C₈	1	4	yes	—
78	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
79	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	—	2	8	no	—
81	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
82	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
86	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
90	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₁₃	—	2	4	no	—
92	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
94	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
95	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂²×M₁₆	C₄×C₁₃	1	1	yes	characteristic
97	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
98	C₂×#117	C₄×C₁₃	1	1	yes	characteristic
99	C₂×M₃₂	—	2	4	no	—
100	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
101	C₂²×C₁₆	—	2	4	no	—
102	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
103	C₂²×C₁₆	C₂²×C₁₃	1	4	yes	—
104	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
105	#405	C₄×C₁₃	1	4	yes	—
106	#405	C₂²×C₁₃	1	4	yes	—
107	C₂×#97	C₄×C₁₃	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
109	C₄⋊C₁₆	C₂²×C₁₃	1	16	yes	complemented
110	C₂²×C₁₃	—	2	4	no	—
111	C₂²×C₁₃	—	2	8	no	—
112	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
113	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
114	C₂²×C₁₃	#504	1	4	yes	—
115	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
116	C₄×C₁₃	—	2	8	no	—
117	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
118	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
119	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
121	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
122	C₂×M₁₆	—	2	4	no	—
123	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
124	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
125	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
126	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
127	C₂²×C₈	D₄×C₁₃	1	4	yes	—
128	C₂²×C₈	Q₈×C₁₃	1	4	yes	—
129	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
131	#117	C₈×C₁₃	1	4	yes	—
132	#117	C₂×C₄×C₁₃	1	4	yes	—
133	C₂×C₄⋊C₄	C₈×C₁₃	1	2	yes	—
134	C₂×C₂²⋊C₄	C₈×C₁₃	1	2	yes	—
135	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
136	M₃₂	—	2	8	no	—
137	C₂×C₁₆	—	2	4	no	—
138	C₂×C₁₆	—	2	8	no	—
139	C₂×C₁₆	—	2	4	no	—
140	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
141	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
142	#97	C₂×C₄×C₁₃	1	8	yes	—
143	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
144	C₄×C₈	C₂³×C₁₃	1	4	yes	—
145	C₂×C₁₃	—	2	4	no	—
146	C₂×C₁₃	#2151	1	4	yes	direct product factor
147	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
148	C₂×C₁₃	C₂×#504	1	2	yes	—
149	C₂⁴	C₂×C₈×C₁₃	1	1	yes	characteristic
150	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
151	C₂²×C₄	C₂×C₈×C₁₃	1	4	yes	—
152	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
154	C₂²×C₄	C₁₃×M₁₆	1	4	yes	—
155	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
156	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
157	M₁₆	—	2	8	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	C₁₃×C₄⋊C₄	1	4	yes	—
160	C₂×C₈	C₁₃×C₄⋊C₄	1	8	yes	—
161	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
162	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
163	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
164	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
166	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
167	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
168	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
169	C₄⋊C₄	C₂×C₈×C₁₃	1	8	yes	—
170	C₂²⋊C₄	C₂×C₈×C₁₃	1	8	yes	—
171	C₄²	C₂×C₈×C₁₃	1	4	yes	—
172	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
173	C₁₆	—	2	4	no	—
174	C₁₆	—	2	4	no	—
175	C₁₃	C₂×#2151	1	1	yes	characteristic, direct product factor, Sylow
176	C₂³	—	2	4	no	—
177	C₂³	C₁₃×C₄⋊C₈	1	4	yes	—
178	C₂³	C₂²×C₈×C₁₃	1	2	yes	—
179	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	C₁₃×C₄⋊C₈	1	8	yes	—
182	C₂×C₄	C₁₃×C₄⋊C₈	1	4	yes	—
183	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
184	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
185	C₂×C₄	C₂²×C₈×C₁₃	1	8	yes	—
186	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
188	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
189	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
190	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
191	C₈	—	2	4	no	—
192	C₈	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
193	C₈	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
194	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
195	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
196	C₂²	—	2	4	no	—
197	C₂²	—	2	8	no	—
198	C₂²	C₂×C₁₃×C₄⋊C₈	1	4	yes	—
199	C₂²	C₂×C₁₃×C₄⋊C₈	1	2	yes	—
200	C₂²	C₁₃×#504	1	4	yes	—
201	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
202	C₄	—	2	8	no	—
203	C₄	C₂×C₁₃×C₄⋊C₈	1	2	yes	—
204	C₄	C₂×C₁₃×C₄⋊C₈	1	4	yes	—
205	C₄	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
206	C₄	C₂²×C₁₃×M₁₆	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	C₁₃×#2151	1	4	yes	direct product factor
209	C₂	C₂²×C₁₃×C₄⋊C₈	1	1	yes	characteristic
210	C₂	C₂×C₁₃×#504	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₃	C₂³
commutator	C₂²	C₂³×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×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
4. C₁₃×#2151	C₂	direct (N × H)
5. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
19. C₁₃×C₄⋊C₁₆	C₂²	semidirect
44. C₂×#2151	C₁₃	direct (N × H)
74. #2151	C₂×C₁₃	direct (N × H)
75. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
109. C₄⋊C₁₆	C₂²×C₁₃	semidirect
146. C₂×C₁₃	#2151	direct (N × H)
175. C₁₃	C₂×#2151	direct (N × H)
208. C₂	C₁₃×#2151	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×M₃₂	2
C₂³×C₁₃×C₁₆	2
C₂×C₁₃×#405	2
C₁₃×#2151	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×#2151	13

Subgroups — C2×C13×#2151 #15719735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2151 #15719735