Subgroups — C₂×C₁₃×#2614 #15720937

Order 3328.

5194 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×#123	C₂	1	3	yes	maximal
3	C₁₃×#2614	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₃×#395	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#393	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₃×#381	C₂	1	3	yes	maximal, complemented
7	C₂³×Q₈×C₁₃	—	2	24	no	—
8	C₂³×Q₈×C₁₃	C₂²	1	4	yes	—
9	C₂³×Q₈×C₁₃	C₂²	1	8	yes	complemented
10	C₂³×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
12	C₂×C₁₃×#123	—	2	24	no	—
13	C₂×C₁₃×#123	C₂²	1	24	yes	—
14	C₂²×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
15	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
16	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	6	yes	—
17	C₁₃×#395	C₂²	1	32	yes	complemented
18	C₁₃×#393	C₂²	1	32	yes	complemented
19	C₁₃×#381	C₂²	1	24	yes	complemented
20	C₂×C₁₃×#95	C₂²	1	12	yes	—
21	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
22	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×Q₈×C₁₃	—	2	128	no	—
24	C₂²×Q₈×C₁₃	—	2	48	no	—
25	C₂²×Q₈×C₁₃	—	2	96	no	—
26	C₂²×Q₈×C₁₃	C₂³	1	16	yes	—
27	C₂²×D₄×C₁₃	—	2	12	no	—
28	C₂²×D₄×C₁₃	C₂³	1	4	yes	—
29	C₂³×C₄×C₁₃	—	2	24	no	—
30	C₂³×C₄×C₁₃	D₄	1	24	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
35	C₁₃×#123	—	2	96	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	24	no	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
38	C₂×C₁₃×C₂²⋊C₄	—	2	12	no	—
39	C₂×C₁₃×C₂²⋊C₄	—	2	24	no	—
40	C₂×C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
41	C₂×C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
42	C₁₃×#95	C₂³	1	48	yes	—
43	C₂×#2614	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂⁴×C₁₃	—	2	8	no	—
45	C₂⁴×C₁₃	—	2	6	no	—
46	C₂⁴×C₁₃	—	2	8	no	—
47	C₂⁴×C₁₃	C₂×D₄	1	12	yes	—
48	C₂⁴×C₁₃	C₂⁴	1	2	yes	—
49	C₂⁴×C₁₃	C₂⁴	1	2	yes	—
50	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
51	C₂×Q₈×C₁₃	—	2	128	no	—
52	C₂×Q₈×C₁₃	—	2	256	no	—
53	C₂×Q₈×C₁₃	—	2	96	no	—
54	C₂×D₄×C₁₃	—	2	24	no	—
55	C₂×D₄×C₁₃	—	4	16	no	—
56	C₂²×C₄×C₁₃	—	2	96	no	—
57	C₂²×C₄×C₁₃	—	2	48	no	—
58	C₂²×C₄×C₁₃	—	2	24	no	—
59	C₂²×C₄×C₁₃	—	2	12	no	—
60	C₂²×C₄×C₁₃	—	2	6	no	—
61	C₂²×C₄×C₁₃	—	2	24	no	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	48	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
67	C₁₃×C₄⋊C₄	—	2	48	no	—
68	C₁₃×C₂²⋊C₄	—	2	24	no	—
69	C₁₃×C₂²⋊C₄	—	2	48	no	—
70	C₂⁴×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
71	C₂²×#123	C₂×C₁₃	1	3	yes	—
72	#2614	C₂×C₁₃	1	16	yes	direct product factor
73	C₂×#395	C₂×C₁₃	1	4	yes	complemented
74	C₂×#393	C₂×C₁₃	1	4	yes	complemented
75	C₂×#381	C₂×C₁₃	1	3	yes	complemented
76	C₂³×C₁₃	—	2	48	no	—
77	C₂³×C₁₃	—	2	8	no	—
78	C₂³×C₁₃	—	2	6	no	—
79	C₂³×C₁₃	—	2	8	no	—
80	C₂³×C₁₃	—	4	4	no	—
81	C₂³×C₁₃	—	4	24	no	—
82	C₂³×C₁₃	#120	1	16	yes	—
83	C₂³×C₁₃	C₂²×D₄	1	12	yes	—
84	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
85	C₂³×C₁₃	2₋¹⁺⁴	1	8	yes	—
86	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
87	Q₈×C₁₃	—	2	128	no	—
88	D₄×C₁₃	—	4	16	no	—
89	C₂×C₄×C₁₃	—	2	96	no	—
90	C₂×C₄×C₁₃	—	2	48	no	—
91	C₂×C₄×C₁₃	—	2	24	no	—
92	C₂×C₄×C₁₃	—	2	12	no	—
93	C₂×C₄×C₁₃	—	2	6	no	—
94	C₂×C₄×C₁₃	—	2	24	no	—
95	C₂×C₄×C₁₃	—	4	4	no	—
96	C₂×C₄×C₁₃	—	4	24	no	—
97	C₂×C₄×C₁₃	C₂²×D₄	1	24	yes	—
98	C₂³×Q₈	—	2	24	no	—
99	C₂³×Q₈	C₂²×C₁₃	1	4	yes	—
100	C₂³×Q₈	C₂²×C₁₃	1	8	yes	complemented
101	C₂³×D₄	C₂²×C₁₃	1	1	yes	characteristic
102	C₂⁴×C₄	C₂²×C₁₃	1	3	yes	—
103	C₂×#123	—	2	24	no	—
104	C₂×#123	C₂²×C₁₃	1	24	yes	—
105	C₂²×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
106	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	6	yes	—
107	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	3	yes	—
108	#395	C₂²×C₁₃	1	32	yes	complemented
109	#393	C₂²×C₁₃	1	32	yes	complemented
110	#381	C₂²×C₁₃	1	24	yes	complemented
111	C₂×#95	C₂²×C₁₃	1	12	yes	—
112	C₂²×C₁₃	—	2	32	no	—
113	C₂²×C₁₃	—	2	24	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	—	4	12	no	—
116	C₂²×C₁₃	—	8	8	no	—
117	C₂²×C₁₃	C₂×#120	1	8	yes	—
118	C₂²×C₁₃	C₂×#120	1	4	yes	—
119	C₂²×C₁₃	#536	1	24	yes	—
120	C₂²×C₁₃	C₂³×D₄	1	3	yes	—
121	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	4	yes	—
122	C₄×C₁₃	—	2	24	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	—	4	12	no	—
125	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
126	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
127	C₂²×Q₈	—	2	128	no	—
128	C₂²×Q₈	—	2	48	no	—
129	C₂²×Q₈	—	2	96	no	—
130	C₂²×Q₈	C₂³×C₁₃	1	16	yes	—
131	C₂²×D₄	—	2	12	no	—
132	C₂²×D₄	C₂³×C₁₃	1	4	yes	—
133	C₂³×C₄	—	2	24	no	—
134	C₂³×C₄	D₄×C₁₃	1	24	yes	—
135	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
136	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
137	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
138	C₂³×C₄	C₂³×C₁₃	1	12	yes	—
139	#123	—	2	96	no	—
140	C₂×C₄⋊C₄	—	2	24	no	—
141	C₂×C₄⋊C₄	C₂³×C₁₃	1	24	yes	—
142	C₂×C₂²⋊C₄	—	2	12	no	—
143	C₂×C₂²⋊C₄	—	2	24	no	—
144	C₂×C₂²⋊C₄	C₂³×C₁₃	1	12	yes	—
145	C₂×C₂²⋊C₄	C₂³×C₁₃	1	24	yes	—
146	#95	C₂³×C₁₃	1	48	yes	—
147	C₂×C₁₃	—	2	8	no	—
148	C₂×C₁₃	—	8	2	no	—
149	C₂×C₁₃	#2614	1	8	yes	direct product factor
150	C₂×C₁₃	C₂²×#120	1	1	yes	characteristic
151	C₂×C₁₃	C₂×#536	1	6	yes	—
152	C₂⁴	—	2	8	no	—
153	C₂⁴	—	2	6	no	—
154	C₂⁴	—	2	8	no	—
155	C₂⁴	C₂×D₄×C₁₃	1	12	yes	—
156	C₂⁴	C₂⁴×C₁₃	1	2	yes	—
157	C₂⁴	C₂⁴×C₁₃	1	2	yes	—
158	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
159	C₂×Q₈	—	2	128	no	—
160	C₂×Q₈	—	2	256	no	—
161	C₂×Q₈	—	2	96	no	—
162	C₂×D₄	—	2	24	no	—
163	C₂×D₄	—	4	16	no	—
164	C₂²×C₄	—	2	96	no	—
165	C₂²×C₄	—	2	48	no	—
166	C₂²×C₄	—	2	24	no	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	2	6	no	—
169	C₂²×C₄	—	2	24	no	—
170	C₂²×C₄	C₂×D₄×C₁₃	1	24	yes	—
171	C₂²×C₄	C₂×D₄×C₁₃	1	48	yes	—
172	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
173	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
174	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
175	C₄⋊C₄	—	2	48	no	—
176	C₂²⋊C₄	—	2	24	no	—
177	C₂²⋊C₄	—	2	48	no	—
178	C₁₃	C₂×#2614	1	1	yes	characteristic, direct product factor, Sylow
179	C₂³	—	2	48	no	—
180	C₂³	—	2	8	no	—
181	C₂³	—	2	6	no	—
182	C₂³	—	2	8	no	—
183	C₂³	—	4	4	no	—
184	C₂³	—	4	24	no	—
185	C₂³	C₁₃×#120	1	16	yes	—
186	C₂³	C₂²×D₄×C₁₃	1	12	yes	—
187	C₂³	C₂²×D₄×C₁₃	1	6	yes	—
188	C₂³	C₁₃×2₋¹⁺⁴	1	8	yes	—
189	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
190	Q₈	—	2	128	no	—
191	D₄	—	4	16	no	—
192	C₂×C₄	—	2	96	no	—
193	C₂×C₄	—	2	48	no	—
194	C₂×C₄	—	2	24	no	—
195	C₂×C₄	—	2	12	no	—
196	C₂×C₄	—	2	6	no	—
197	C₂×C₄	—	2	24	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	24	no	—
200	C₂×C₄	C₂²×D₄×C₁₃	1	24	yes	—
201	C₂²	—	2	32	no	—
202	C₂²	—	2	24	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	12	no	—
205	C₂²	—	8	8	no	—
206	C₂²	C₂×C₁₃×#120	1	8	yes	—
207	C₂²	C₂×C₁₃×#120	1	4	yes	—
208	C₂²	C₁₃×#536	1	24	yes	—
209	C₂²	C₂³×D₄×C₁₃	1	3	yes	—
210	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	4	yes	—
211	C₄	—	2	24	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	12	no	—
214	C₂	—	2	8	no	—
215	C₂	—	8	2	no	—
216	C₂	C₁₃×#2614	1	8	yes	direct product factor
217	C₂	C₂²×C₁₃×#120	1	1	yes	characteristic
218	C₂	C₂×C₁₃×#536	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
prime radical	C₂³×D₄×C₁₃	C₂²
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. C₂⁴×Q₈×C₁₃	C₂	semidirect
3. C₁₃×#2614	C₂	direct (N × H)
4. C₂×C₁₃×#395	C₂	semidirect
5. C₂×C₁₃×#393	C₂	semidirect
6. C₂×C₁₃×#381	C₂	semidirect
9. C₂³×Q₈×C₁₃	C₂²	semidirect
17. C₁₃×#395	C₂²	semidirect
18. C₁₃×#393	C₂²	semidirect
19. C₁₃×#381	C₂²	semidirect
43. C₂×#2614	C₁₃	direct (N × H)
70. C₂⁴×Q₈	C₂×C₁₃	semidirect
72. #2614	C₂×C₁₃	direct (N × H)
73. C₂×#395	C₂×C₁₃	semidirect
74. C₂×#393	C₂×C₁₃	semidirect
75. C₂×#381	C₂×C₁₃	semidirect
100. C₂³×Q₈	C₂²×C₁₃	semidirect
108. #395	C₂²×C₁₃	semidirect
109. #393	C₂²×C₁₃	semidirect
110. #381	C₂²×C₁₃	semidirect
149. C₂×C₁₃	#2614	direct (N × H)
178. C₁₃	C₂×#2614	direct (N × H)
216. C₂	C₁₃×#2614	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₁₃	2
C₂²×C₁₃×#123	2
C₁₃×#2614	2
C₂×C₁₃×#395	2
C₂×C₁₃×#393	2
C₂×C₁₃×#381	2
C₂×#2614	13

Subgroups — C2×C13×#2614 #15720937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2614 #15720937