Subgroups — C₂×C₁₃×#2399 #15720885

Order 3328.

2814 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	C₂²×C₄×Q₈×C₁₃	C₂	1	2	yes	maximal
2	C₁₃×#2399	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#398	C₂	1	4	yes	maximal
4	C₂×C₁₃×#395	C₂	1	4	yes	maximal
5	C₂×C₄×C₁₃×C₄⋊C₄	C₂	1	4	yes	maximal
6	C₂×C₁₃×#376	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₁₃	—	2	16	no	—
9	C₂×C₄×Q₈×C₁₃	C₂²	1	16	yes	—
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	8	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
14	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#398	C₂²	1	32	yes	—
17	C₁₃×#395	C₂²	1	32	yes	—
18	C₄×C₁₃×C₄⋊C₄	C₂²	1	32	yes	—
19	C₁₃×#376	C₂²	1	8	yes	—
20	C₂×C₁₃×#95	C₂²	1	4	yes	—
21	C₂×C₁₃×#95	C₂²	1	4	yes	—
22	C₂×C₁₃×#95	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₃	—	2	8	no	—
24	C₂²×Q₈×C₁₃	—	2	16	no	—
25	C₂²×Q₈×C₁₃	C₂³	1	8	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₁₃	—	2	64	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	32	no	—
35	C₂×C₁₃×C₄⋊C₄	C₂³	1	32	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
38	C₂×C₄²×C₁₃	—	2	16	no	—
39	C₂×C₄²×C₁₃	D₄	1	8	yes	—
40	C₂×C₄²×C₁₃	Q₈	1	16	yes	complemented
41	C₂×C₄²×C₁₃	C₂³	1	16	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	8	yes	—
44	C₁₃×#95	C₂³	1	16	yes	—
45	C₁₃×#95	C₂³	1	16	yes	—
46	C₁₃×#95	C₂³	1	16	yes	—
47	C₂×#2399	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
49	C₂×Q₈×C₁₃	—	2	64	no	—
50	C₂×Q₈×C₁₃	—	2	16	no	—
51	C₂×Q₈×C₁₃	—	2	32	no	—
52	C₂²×C₄×C₁₃	—	2	8	no	—
53	C₂²×C₄×C₁₃	—	2	16	no	—
54	C₂²×C₄×C₁₃	—	2	8	no	—
55	C₂²×C₄×C₁₃	—	2	8	no	—
56	C₂²×C₄×C₁₃	—	2	8	no	—
57	C₂²×C₄×C₁₃	—	2	16	no	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂×Q₈	1	16	yes	—
61	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
62	C₂²×C₄×C₁₃	#41	1	16	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
68	C₁₃×C₄⋊C₄	—	2	32	no	—
69	C₁₃×C₄⋊C₄	—	2	16	no	—
70	C₁₃×C₄⋊C₄	—	2	64	no	—
71	C₄²×C₁₃	—	2	32	no	—
72	C₄²×C₁₃	C₂×D₄	1	16	yes	—
73	C₄²×C₁₃	C₂×Q₈	1	32	yes	complemented
74	C₂²×C₄×Q₈	C₂×C₁₃	1	2	yes	—
75	#2399	C₂×C₁₃	1	16	yes	direct product factor
76	C₂×#398	C₂×C₁₃	1	4	yes	—
77	C₂×#395	C₂×C₁₃	1	4	yes	—
78	C₂×C₄×C₄⋊C₄	C₂×C₁₃	1	4	yes	—
79	C₂×#376	C₂×C₁₃	1	1	yes	characteristic
80	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
81	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
82	C₂³×C₁₃	C₂×#41	1	8	yes	—
83	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
84	Q₈×C₁₃	—	2	64	no	—
85	C₂×C₄×C₁₃	—	2	16	no	—
86	C₂×C₄×C₁₃	—	2	32	no	—
87	C₂×C₄×C₁₃	—	2	32	no	—
88	C₂×C₄×C₁₃	—	2	8	no	—
89	C₂×C₄×C₁₃	—	2	16	no	—
90	C₂×C₄×C₁₃	—	2	8	no	—
91	C₂×C₄×C₁₃	—	2	8	no	—
92	C₂×C₄×C₁₃	—	2	8	no	—
93	C₂×C₄×C₁₃	—	2	16	no	—
94	C₂×C₄×C₁₃	—	4	16	no	—
95	C₂×C₄×C₁₃	C₄⋊Q₈	1	16	yes	—
96	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
97	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
98	C₂×C₄×C₁₃	C₂²×Q₈	1	16	yes	—
99	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
100	C₂×C₄×C₁₃	C₂×#41	1	16	yes	—
101	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
102	C₂×C₄×Q₈	—	2	16	no	—
103	C₂×C₄×Q₈	C₂²×C₁₃	1	16	yes	—
104	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
105	C₂²×C₄⋊C₄	C₂²×C₁₃	1	8	yes	—
106	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
107	C₂²×C₄²	C₂²×C₁₃	1	4	yes	—
108	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
109	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
110	#398	C₂²×C₁₃	1	32	yes	—
111	#395	C₂²×C₁₃	1	32	yes	—
112	C₄×C₄⋊C₄	C₂²×C₁₃	1	32	yes	—
113	#376	C₂²×C₁₃	1	8	yes	—
114	C₂×#95	C₂²×C₁₃	1	4	yes	—
115	C₂×#95	C₂²×C₁₃	1	4	yes	—
116	C₂×#95	C₂²×C₁₃	1	4	yes	—
117	C₂²×C₁₃	#525	1	8	yes	—
118	C₂²×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
119	C₂²×C₁₃	#533	1	16	yes	—
120	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
121	C₂²×C₁₃	C₂³×Q₈	1	2	yes	—
122	C₂²×C₁₃	C₂²×#41	1	4	yes	—
123	C₄×C₁₃	—	2	8	no	—
124	C₄×C₁₃	—	2	16	no	—
125	C₄×C₁₃	—	2	16	no	—
126	C₄×C₁₃	—	4	8	no	—
127	C₄×C₁₃	C₂×C₄⋊Q₈	1	8	yes	—
128	C₂²×Q₈	—	2	8	no	—
129	C₂²×Q₈	—	2	16	no	—
130	C₂²×Q₈	C₂³×C₁₃	1	8	yes	—
131	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
132	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
133	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
134	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
135	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
136	C₄×Q₈	—	2	64	no	—
137	C₂×C₄⋊C₄	—	2	16	no	—
138	C₂×C₄⋊C₄	—	2	8	no	—
139	C₂×C₄⋊C₄	—	2	32	no	—
140	C₂×C₄⋊C₄	C₂³×C₁₃	1	16	yes	—
141	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
142	C₂×C₄⋊C₄	C₂³×C₁₃	1	32	yes	—
143	C₂×C₄²	—	2	16	no	—
144	C₂×C₄²	D₄×C₁₃	1	8	yes	—
145	C₂×C₄²	Q₈×C₁₃	1	16	yes	complemented
146	C₂×C₄²	C₂³×C₁₃	1	16	yes	—
147	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
148	C₂×C₄²	C₂³×C₁₃	1	8	yes	—
149	#95	C₂³×C₁₃	1	16	yes	—
150	#95	C₂³×C₁₃	1	16	yes	—
151	#95	C₂³×C₁₃	1	16	yes	—
152	C₂×C₁₃	#2399	1	8	yes	direct product factor
153	C₂×C₁₃	C₂×#525	1	2	yes	—
154	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
155	C₂×C₁₃	C₂×#533	1	4	yes	—
156	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
157	C₂×Q₈	—	2	64	no	—
158	C₂×Q₈	—	2	16	no	—
159	C₂×Q₈	—	2	32	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	16	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	2	16	no	—
166	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
167	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
168	C₂²×C₄	C₂×Q₈×C₁₃	1	16	yes	—
169	C₂²×C₄	C₂×Q₈×C₁₃	1	8	yes	—
170	C₂²×C₄	C₁₃×#41	1	16	yes	—
171	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
172	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
173	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
174	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
175	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
176	C₄⋊C₄	—	2	32	no	—
177	C₄⋊C₄	—	2	16	no	—
178	C₄⋊C₄	—	2	64	no	—
179	C₄²	—	2	32	no	—
180	C₄²	C₂×D₄×C₁₃	1	16	yes	—
181	C₄²	C₂×Q₈×C₁₃	1	32	yes	complemented
182	C₁₃	C₂×#2399	1	1	yes	characteristic, direct product factor, Sylow
183	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
184	C₂³	C₂²×Q₈×C₁₃	1	4	yes	—
185	C₂³	C₂×C₁₃×#41	1	8	yes	—
186	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
187	Q₈	—	2	64	no	—
188	C₂×C₄	—	2	16	no	—
189	C₂×C₄	—	2	32	no	—
190	C₂×C₄	—	2	32	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	16	no	—
197	C₂×C₄	—	4	16	no	—
198	C₂×C₄	C₁₃×C₄⋊Q₈	1	16	yes	—
199	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
200	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
201	C₂×C₄	C₂²×Q₈×C₁₃	1	16	yes	—
202	C₂×C₄	C₂²×Q₈×C₁₃	1	8	yes	—
203	C₂×C₄	C₂×C₁₃×#41	1	16	yes	—
204	C₂²	C₁₃×#525	1	8	yes	—
205	C₂²	C₂×C₁₃×C₄⋊Q₈	1	4	yes	—
206	C₂²	C₁₃×#533	1	16	yes	—
207	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
208	C₂²	C₂³×Q₈×C₁₃	1	2	yes	—
209	C₂²	C₂²×C₁₃×#41	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	—	2	16	no	—
212	C₄	—	2	16	no	—
213	C₄	—	4	8	no	—
214	C₄	C₂×C₁₃×C₄⋊Q₈	1	8	yes	—
215	C₂	C₁₃×#2399	1	8	yes	direct product factor
216	C₂	C₂×C₁₃×#525	1	2	yes	—
217	C₂	C₂²×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
218	C₂	C₂×C₁₃×#533	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
socle, prime radical	C₂⁴×C₁₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2399	C₂	direct (N × H)
40. C₂×C₄²×C₁₃	Q₈	semidirect
47. C₂×#2399	C₁₃	direct (N × H)
73. C₄²×C₁₃	C₂×Q₈	semidirect
75. #2399	C₂×C₁₃	direct (N × H)
145. C₂×C₄²	Q₈×C₁₃	semidirect
152. C₂×C₁₃	#2399	direct (N × H)
181. C₄²	C₂×Q₈×C₁₃	semidirect
182. C₁₃	C₂×#2399	direct (N × H)
215. C₂	C₁₃×#2399	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×C₄×Q₈×C₁₃	2
C₁₃×#2399	2
C₂×C₁₃×#398	2
C₂×C₁₃×#395	2
C₂×C₄×C₁₃×C₄⋊C₄	2
C₂×C₁₃×#376	2
C₂×#2399	13

Subgroups — C2×C13×#2399 #15720885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2399 #15720885