Subgroups — C₂²×C₁₃×#393 #15720871

Order 3328.

7118 conjugacy classes of subgroups.

The subgroup lattice has 186 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	C₂³×C₁₃×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₂×C₁₃×#393	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₁₃×#95	C₂	1	3	yes	maximal, complemented
5	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
6	C₂³×Q₈×C₁₃	—	2	24	no	—
7	C₂³×Q₈×C₁₃	C₂²	1	12	yes	complemented
8	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
9	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
10	C₂²×C₁₃×C₂²⋊C₄	—	2	24	no	—
11	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	36	yes	—
12	C₁₃×#393	C₂²	1	64	yes	direct product factor
13	C₂×C₁₃×#95	C₂²	1	36	yes	complemented
14	C₂⁵×C₁₃	—	2	12	no	—
15	C₂⁵×C₁₃	—	2	12	no	—
16	C₂⁵×C₁₃	Q₈	1	8	yes	complemented
17	C₂⁵×C₁₃	C₂³	1	6	yes	—
18	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×Q₈×C₁₃	—	2	128	no	—
20	C₂²×Q₈×C₁₃	—	2	144	no	—
21	C₂²×Q₈×C₁₃	C₂³	1	16	yes	complemented
22	C₂³×C₄×C₁₃	—	2	24	no	—
23	C₂³×C₄×C₁₃	—	2	12	no	—
24	C₂³×C₄×C₁₃	—	2	24	no	—
25	C₂³×C₄×C₁₃	D₄	1	24	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	18	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	18	yes	—
28	C₂×C₁₃×C₂²⋊C₄	—	2	144	no	—
29	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	—
30	C₁₃×#95	C₂³	1	48	yes	complemented
31	C₂²×#393	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₁₃	—	2	96	no	—
33	C₂⁴×C₁₃	—	2	36	no	—
34	C₂⁴×C₁₃	—	2	36	no	—
35	C₂⁴×C₁₃	—	4	16	no	—
36	C₂⁴×C₁₃	—	4	48	no	—
37	C₂⁴×C₁₃	C₂×D₄	1	12	yes	—
38	C₂⁴×C₁₃	C₂×Q₈	1	4	yes	—
39	C₂⁴×C₁₃	C₂×Q₈	1	24	yes	complemented
40	C₂⁴×C₁₃	#41	1	12	yes	—
41	C₂⁴×C₁₃	C₂⁴	1	4	yes	—
42	C₂⁴×C₁₃	C₂⁴	1	3	yes	—
43	C₂×Q₈×C₁₃	—	2	384	no	—
44	C₂×Q₈×C₁₃	—	2	96	no	—
45	C₂²×C₄×C₁₃	—	2	96	no	—
46	C₂²×C₄×C₁₃	—	2	72	no	—
47	C₂²×C₄×C₁₃	—	2	36	no	—
48	C₂²×C₄×C₁₃	—	2	72	no	—
49	C₂²×C₄×C₁₃	—	4	48	no	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	72	yes	—
51	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
52	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
53	C₁₃×C₂²⋊C₄	—	2	96	no	—
54	C₂⁴×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
55	C₂³×C₂²⋊C₄	C₂×C₁₃	1	3	yes	—
56	C₂×#393	C₂×C₁₃	1	24	yes	direct product factor
57	C₂²×#95	C₂×C₁₃	1	3	yes	complemented
58	C₂³×C₁₃	—	2	144	no	—
59	C₂³×C₁₃	—	2	12	no	—
60	C₂³×C₁₃	—	2	12	no	—
61	C₂³×C₁₃	—	4	24	no	—
62	C₂³×C₁₃	—	4	72	no	—
63	C₂³×C₁₃	—	4	128	no	—
64	C₂³×C₁₃	#120	1	16	yes	—
65	C₂³×C₁₃	C₂²⋊Q₈	1	48	yes	—
66	C₂³×C₁₃	#124	1	48	yes	—
67	C₂³×C₁₃	C₂²×D₄	1	18	yes	—
68	C₂³×C₁₃	C₂²×Q₈	1	6	yes	—
69	C₂³×C₁₃	C₂²×Q₈	1	8	yes	complemented
70	C₂³×C₁₃	C₂×#41	1	18	yes	—
71	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
72	Q₈×C₁₃	—	2	128	no	—
73	C₂×C₄×C₁₃	—	2	144	no	—
74	C₂×C₄×C₁₃	—	2	24	no	—
75	C₂×C₄×C₁₃	—	2	12	no	—
76	C₂×C₄×C₁₃	—	2	24	no	—
77	C₂×C₄×C₁₃	—	4	72	no	—
78	C₂×C₄×C₁₃	C₂²×D₄	1	24	yes	—
79	C₂⁶	C₂²×C₁₃	1	1	yes	characteristic
80	C₂³×Q₈	—	2	24	no	—
81	C₂³×Q₈	C₂²×C₁₃	1	12	yes	complemented
82	C₂⁴×C₄	C₂²×C₁₃	1	3	yes	—
83	C₂⁴×C₄	C₂²×C₁₃	1	3	yes	—
84	C₂²×C₂²⋊C₄	—	2	24	no	—
85	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	36	yes	—
86	#393	C₂²×C₁₃	1	64	yes	direct product factor
87	C₂×#95	C₂²×C₁₃	1	36	yes	complemented
88	C₂²×C₁₃	—	2	24	no	—
89	C₂²×C₁₃	—	4	96	no	—
90	C₂²×C₁₃	—	4	4	no	—
91	C₂²×C₁₃	—	4	12	no	—
92	C₂²×C₁₃	#393	1	64	yes	direct product factor
93	C₂²×C₁₃	C₂×#120	1	12	yes	—
94	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	36	yes	—
95	C₂²×C₁₃	C₂×#124	1	36	yes	—
96	C₂²×C₁₃	C₂³×D₄	1	3	yes	—
97	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
98	C₂²×C₁₃	C₂²×#41	1	3	yes	—
99	C₄×C₁₃	—	2	24	no	—
100	C₄×C₁₃	—	4	12	no	—
101	C₂⁵	—	2	12	no	—
102	C₂⁵	—	2	12	no	—
103	C₂⁵	Q₈×C₁₃	1	8	yes	complemented
104	C₂⁵	C₂³×C₁₃	1	6	yes	—
105	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
106	C₂²×Q₈	—	2	128	no	—
107	C₂²×Q₈	—	2	144	no	—
108	C₂²×Q₈	C₂³×C₁₃	1	16	yes	complemented
109	C₂³×C₄	—	2	24	no	—
110	C₂³×C₄	—	2	12	no	—
111	C₂³×C₄	—	2	24	no	—
112	C₂³×C₄	D₄×C₁₃	1	24	yes	—
113	C₂³×C₄	C₂³×C₁₃	1	18	yes	—
114	C₂³×C₄	C₂³×C₁₃	1	18	yes	—
115	C₂×C₂²⋊C₄	—	2	144	no	—
116	C₂×C₂²⋊C₄	C₂³×C₁₃	1	48	yes	—
117	#95	C₂³×C₁₃	1	48	yes	complemented
118	C₂×C₁₃	—	4	8	no	—
119	C₂×C₁₃	C₂×#393	1	24	yes	direct product factor
120	C₂×C₁₃	C₂²×#120	1	1	yes	characteristic
121	C₂×C₁₃	C₂²×C₂²⋊Q₈	1	3	yes	—
122	C₂×C₁₃	C₂²×#124	1	3	yes	—
123	C₂⁴	—	2	96	no	—
124	C₂⁴	—	2	36	no	—
125	C₂⁴	—	2	36	no	—
126	C₂⁴	—	4	16	no	—
127	C₂⁴	—	4	48	no	—
128	C₂⁴	C₂×D₄×C₁₃	1	12	yes	—
129	C₂⁴	C₂×Q₈×C₁₃	1	4	yes	—
130	C₂⁴	C₂×Q₈×C₁₃	1	24	yes	complemented
131	C₂⁴	C₁₃×#41	1	12	yes	—
132	C₂⁴	C₂⁴×C₁₃	1	4	yes	—
133	C₂⁴	C₂⁴×C₁₃	1	3	yes	—
134	C₂×Q₈	—	2	384	no	—
135	C₂×Q₈	—	2	96	no	—
136	C₂²×C₄	—	2	96	no	—
137	C₂²×C₄	—	2	72	no	—
138	C₂²×C₄	—	2	36	no	—
139	C₂²×C₄	—	2	72	no	—
140	C₂²×C₄	—	4	48	no	—
141	C₂²×C₄	C₂×D₄×C₁₃	1	72	yes	—
142	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
143	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
144	C₂²⋊C₄	—	2	96	no	—
145	C₁₃	C₂²×#393	1	1	yes	characteristic, direct product factor, Sylow
146	C₂³	—	2	144	no	—
147	C₂³	—	2	12	no	—
148	C₂³	—	2	12	no	—
149	C₂³	—	4	128	no	—
150	C₂³	—	4	24	no	—
151	C₂³	—	4	72	no	—
152	C₂³	C₁₃×#120	1	16	yes	—
153	C₂³	C₁₃×C₂²⋊Q₈	1	48	yes	—
154	C₂³	C₁₃×#124	1	48	yes	—
155	C₂³	C₂²×D₄×C₁₃	1	18	yes	—
156	C₂³	C₂²×Q₈×C₁₃	1	6	yes	—
157	C₂³	C₂²×Q₈×C₁₃	1	8	yes	complemented
158	C₂³	C₂×C₁₃×#41	1	18	yes	—
159	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
160	Q₈	—	2	128	no	—
161	C₂×C₄	—	2	144	no	—
162	C₂×C₄	—	2	24	no	—
163	C₂×C₄	—	2	12	no	—
164	C₂×C₄	—	2	24	no	—
165	C₂×C₄	—	4	72	no	—
166	C₂×C₄	C₂²×D₄×C₁₃	1	24	yes	—
167	C₂²	—	2	24	no	—
168	C₂²	—	4	96	no	—
169	C₂²	—	4	4	no	—
170	C₂²	—	4	12	no	—
171	C₂²	C₁₃×#393	1	64	yes	direct product factor
172	C₂²	C₂×C₁₃×#120	1	12	yes	—
173	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	36	yes	—
174	C₂²	C₂×C₁₃×#124	1	36	yes	—
175	C₂²	C₂³×D₄×C₁₃	1	3	yes	—
176	C₂²	C₂³×Q₈×C₁₃	1	1	yes	characteristic
177	C₂²	C₂²×C₁₃×#41	1	3	yes	—
178	C₄	—	2	24	no	—
179	C₄	—	4	12	no	—
180	C₂	—	4	8	no	—
181	C₂	C₂×C₁₃×#393	1	24	yes	direct product factor
182	C₂	C₂²×C₁₃×#120	1	1	yes	characteristic
183	C₂	C₂²×C₁₃×C₂²⋊Q₈	1	3	yes	—
184	C₂	C₂²×C₁₃×#124	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₁₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
prime radical	C₂⁶×C₁₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×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. C₂×C₁₃×#393	C₂	direct (N × H)
4. C₂²×C₁₃×#95	C₂	semidirect
7. C₂³×Q₈×C₁₃	C₂²	semidirect
12. C₁₃×#393	C₂²	direct (N × H)
13. C₂×C₁₃×#95	C₂²	semidirect
16. C₂⁵×C₁₃	Q₈	semidirect
21. C₂²×Q₈×C₁₃	C₂³	semidirect
30. C₁₃×#95	C₂³	semidirect
31. C₂²×#393	C₁₃	direct (N × H)
39. C₂⁴×C₁₃	C₂×Q₈	semidirect
54. C₂⁴×Q₈	C₂×C₁₃	semidirect
56. C₂×#393	C₂×C₁₃	direct (N × H)
57. C₂²×#95	C₂×C₁₃	semidirect
69. C₂³×C₁₃	C₂²×Q₈	semidirect
81. C₂³×Q₈	C₂²×C₁₃	semidirect
86. #393	C₂²×C₁₃	direct (N × H)
87. C₂×#95	C₂²×C₁₃	semidirect
92. C₂²×C₁₃	#393	direct (N × H)
103. C₂⁵	Q₈×C₁₃	semidirect
108. C₂²×Q₈	C₂³×C₁₃	semidirect
117. #95	C₂³×C₁₃	semidirect
119. C₂×C₁₃	C₂×#393	direct (N × H)
130. C₂⁴	C₂×Q₈×C₁₃	semidirect
145. C₁₃	C₂²×#393	direct (N × H)
157. C₂³	C₂²×Q₈×C₁₃	semidirect
171. C₂²	C₁₃×#393	direct (N × H)
181. C₂	C₂×C₁₃×#393	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
C₂³×C₁₃×C₂²⋊C₄	2
C₂×C₁₃×#393	2
C₂²×C₁₃×#95	2
C₂²×#393	13

Subgroups — C22×C13×#393 #15720871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#393 #15720871