Subgroups — C₂×C₁₃×#2864 #15722973

Order 3328.

2078 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2864	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₁₃×#389	C₂	1	4	yes	maximal
3	C₂×C₁₃×#387	C₂	1	8	yes	maximal
4	C₂×C₁₃×#382	C₂	1	2	yes	maximal
5	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
7	C₂²×C₁₃×C₄⋊C₄	C₂²	1	8	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#389	C₂²	1	32	yes	—
12	C₁₃×#387	C₂²	1	64	yes	—
13	C₁₃×#382	C₂²	1	16	yes	—
14	C₂×C₁₃×#95	C₂²	1	8	yes	—
15	C₂×C₁₃×#95	C₂²	1	8	yes	—
16	C₄³×C₁₃	C₂²	1	8	yes	—
17	C₂³×C₄×C₁₃	C₂³	1	8	yes	—
18	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
19	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	—	2	32	no	—
22	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
23	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
24	C₂×C₁₃×C₄⋊C₄	C₂³	1	32	yes	—
25	C₂×C₄²×C₁₃	—	2	16	no	—
26	C₂×C₄²×C₁₃	—	2	8	no	—
27	C₂×C₄²×C₁₃	Q₈	1	8	yes	—
28	C₂×C₄²×C₁₃	C₂³	1	16	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	8	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
31	C₁₃×#95	C₂³	1	32	yes	—
32	C₁₃×#95	C₂³	1	32	yes	—
33	C₂×#2864	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	—	2	8	no	—
36	C₂²×C₄×C₁₃	—	2	8	no	—
37	C₂²×C₄×C₁₃	—	2	8	no	—
38	C₂²×C₄×C₁₃	—	2	16	no	—
39	C₂²×C₄×C₁₃	—	2	32	no	—
40	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
41	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
42	C₂²×C₄×C₁₃	#41	1	16	yes	—
43	C₂²×C₄×C₁₃	#41	1	8	yes	—
44	C₂²×C₄×C₁₃	C₂⁴	1	16	yes	—
45	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
46	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
47	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
48	C₁₃×C₄⋊C₄	—	2	64	no	—
49	C₁₃×C₄⋊C₄	—	2	32	no	—
50	C₄²×C₁₃	—	2	32	no	—
51	C₄²×C₁₃	—	2	16	no	—
52	C₄²×C₁₃	C₂×Q₈	1	16	yes	—
53	#2864	C₂×C₁₃	1	16	yes	direct product factor
54	C₂×#389	C₂×C₁₃	1	4	yes	—
55	C₂×#387	C₂×C₁₃	1	8	yes	—
56	C₂×#382	C₂×C₁₃	1	2	yes	—
57	C₂×C₄³	C₂×C₁₃	1	1	yes	characteristic
58	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
59	C₂³×C₁₃	C₂×#41	1	4	yes	—
60	C₂³×C₁₃	C₂×#41	1	8	yes	—
61	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
62	C₂×C₄×C₁₃	—	2	32	no	—
63	C₂×C₄×C₁₃	—	2	16	no	—
64	C₂×C₄×C₁₃	—	2	8	no	—
65	C₂×C₄×C₁₃	—	2	8	no	—
66	C₂×C₄×C₁₃	—	2	8	no	—
67	C₂×C₄×C₁₃	—	2	16	no	—
68	C₂×C₄×C₁₃	—	2	32	no	—
69	C₂×C₄×C₁₃	—	4	32	no	—
70	C₂×C₄×C₁₃	#125	1	16	yes	—
71	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
72	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
73	C₂×C₄×C₁₃	C₂×#41	1	16	yes	—
74	C₂×C₄×C₁₃	C₂×#41	1	8	yes	—
75	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
76	C₂²×C₄⋊C₄	C₂²×C₁₃	1	8	yes	—
77	C₂²×C₄²	C₂²×C₁₃	1	4	yes	—
78	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
79	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
80	#389	C₂²×C₁₃	1	32	yes	—
81	#387	C₂²×C₁₃	1	64	yes	—
82	#382	C₂²×C₁₃	1	16	yes	—
83	C₂×#95	C₂²×C₁₃	1	8	yes	—
84	C₂×#95	C₂²×C₁₃	1	8	yes	—
85	C₄³	C₂²×C₁₃	1	8	yes	—
86	C₂²×C₁₃	C₂×#125	1	4	yes	—
87	C₂²×C₁₃	#529	1	16	yes	—
88	C₂²×C₁₃	#533	1	8	yes	—
89	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
90	C₂²×C₁₃	C₂²×#41	1	2	yes	—
91	C₂²×C₁₃	C₂²×#41	1	4	yes	—
92	C₄×C₁₃	—	2	16	no	—
93	C₄×C₁₃	—	2	8	no	—
94	C₄×C₁₃	—	4	16	no	—
95	C₄×C₁₃	C₂×#125	1	8	yes	—
96	C₂³×C₄	C₂³×C₁₃	1	8	yes	—
97	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
98	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
99	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	2	32	no	—
101	C₂×C₄⋊C₄	—	2	16	no	—
102	C₂×C₄⋊C₄	C₂³×C₁₃	1	16	yes	—
103	C₂×C₄⋊C₄	C₂³×C₁₃	1	32	yes	—
104	C₂×C₄²	—	2	16	no	—
105	C₂×C₄²	—	2	8	no	—
106	C₂×C₄²	Q₈×C₁₃	1	8	yes	—
107	C₂×C₄²	C₂³×C₁₃	1	16	yes	—
108	C₂×C₄²	C₂³×C₁₃	1	8	yes	—
109	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
110	#95	C₂³×C₁₃	1	32	yes	—
111	#95	C₂³×C₁₃	1	32	yes	—
112	C₂×C₁₃	#2864	1	8	yes	direct product factor
113	C₂×C₁₃	C₂²×#125	1	1	yes	characteristic
114	C₂×C₁₃	C₂×#529	1	4	yes	—
115	C₂×C₁₃	C₂×#533	1	2	yes	—
116	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
117	C₂²×C₄	—	2	8	no	—
118	C₂²×C₄	—	2	8	no	—
119	C₂²×C₄	—	2	8	no	—
120	C₂²×C₄	—	2	16	no	—
121	C₂²×C₄	—	2	32	no	—
122	C₂²×C₄	C₂×Q₈×C₁₃	1	8	yes	—
123	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
124	C₂²×C₄	C₁₃×#41	1	16	yes	—
125	C₂²×C₄	C₁₃×#41	1	8	yes	—
126	C₂²×C₄	C₂⁴×C₁₃	1	16	yes	—
127	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
128	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
129	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
130	C₄⋊C₄	—	2	64	no	—
131	C₄⋊C₄	—	2	32	no	—
132	C₄²	—	2	32	no	—
133	C₄²	—	2	16	no	—
134	C₄²	C₂×Q₈×C₁₃	1	16	yes	—
135	C₁₃	C₂×#2864	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³	C₂²×Q₈×C₁₃	1	2	yes	—
137	C₂³	C₂×C₁₃×#41	1	4	yes	—
138	C₂³	C₂×C₁₃×#41	1	8	yes	—
139	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
140	C₂×C₄	—	2	32	no	—
141	C₂×C₄	—	2	16	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	16	no	—
146	C₂×C₄	—	2	32	no	—
147	C₂×C₄	—	4	32	no	—
148	C₂×C₄	C₁₃×#125	1	16	yes	—
149	C₂×C₄	C₂²×Q₈×C₁₃	1	8	yes	—
150	C₂×C₄	C₂²×Q₈×C₁₃	1	4	yes	—
151	C₂×C₄	C₂×C₁₃×#41	1	16	yes	—
152	C₂×C₄	C₂×C₁₃×#41	1	8	yes	—
153	C₂²	C₂×C₁₃×#125	1	4	yes	—
154	C₂²	C₁₃×#529	1	16	yes	—
155	C₂²	C₁₃×#533	1	8	yes	—
156	C₂²	C₂³×Q₈×C₁₃	1	1	yes	characteristic
157	C₂²	C₂²×C₁₃×#41	1	2	yes	—
158	C₂²	C₂²×C₁₃×#41	1	4	yes	—
159	C₄	—	2	16	no	—
160	C₄	—	2	8	no	—
161	C₄	—	4	16	no	—
162	C₄	C₂×C₁₃×#125	1	8	yes	—
163	C₂	C₁₃×#2864	1	8	yes	direct product factor
164	C₂	C₂²×C₁₃×#125	1	1	yes	characteristic
165	C₂	C₂×C₁₃×#529	1	4	yes	—
166	C₂	C₂×C₁₃×#533	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2864	C₂	direct (N × H)
33. C₂×#2864	C₁₃	direct (N × H)
53. #2864	C₂×C₁₃	direct (N × H)
112. C₂×C₁₃	#2864	direct (N × H)
135. C₁₃	C₂×#2864	direct (N × H)
163. C₂	C₁₃×#2864	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×#2864	2
C₂×C₁₃×#389	2
C₂×C₁₃×#387	2
C₂×C₁₃×#382	2
C₂×C₄³×C₁₃	2
C₂×#2864	13

Subgroups — C2×C13×#2864 #15722973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2864 #15722973