Subgroups — C₂×C₁₃×#2397 #15720883

Order 3328.

5150 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₁₃	C₂	1	6	yes	maximal, complemented
2	C₁₃×#2397	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#399	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#392	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×C₁₃	C₂²	1	6	yes	—
7	C₂⁴×C₄×C₁₃	C₂²	1	4	yes	—
8	C₂×C₄×D₄×C₁₃	—	2	48	no	—
9	C₂×C₄×D₄×C₁₃	C₂²	1	48	yes	complemented
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
11	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	12	yes	—
12	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
13	C₁₃×#399	C₂²	1	32	yes	complemented
14	C₁₃×#392	C₂²	1	32	yes	complemented
15	C₁₃×#376	C₂²	1	8	yes	complemented
16	C₂×C₁₃×#95	C₂²	1	4	yes	—
17	C₂⁵×C₁₃	C₂³	1	4	yes	—
18	C₂²×D₄×C₁₃	—	2	48	no	—
19	C₂²×D₄×C₁₃	—	2	24	no	—
20	C₂²×D₄×C₁₃	C₂³	1	24	yes	—
21	C₂³×C₄×C₁₃	—	2	48	no	—
22	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	16	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₄×D₄×C₁₃	—	2	192	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	24	no	—
28	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
29	C₂×C₁₃×C₂²⋊C₄	—	2	48	no	—
30	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	—
31	C₂×C₄²×C₁₃	D₄	1	24	yes	complemented
32	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
33	C₁₃×#95	C₂³	1	16	yes	—
34	C₂×#2397	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂⁴×C₁₃	—	2	24	no	—
36	C₂⁴×C₁₃	—	2	24	no	—
37	C₂⁴×C₁₃	#41	1	16	yes	—
38	C₂⁴×C₁₃	C₂⁴	1	8	yes	—
39	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
40	C₂×D₄×C₁₃	—	2	192	no	—
41	C₂×D₄×C₁₃	—	2	96	no	—
42	C₂×D₄×C₁₃	—	2	48	no	—
43	C₂²×C₄×C₁₃	—	2	24	no	—
44	C₂²×C₄×C₁₃	—	2	24	no	—
45	C₂²×C₄×C₁₃	—	2	96	no	—
46	C₂²×C₄×C₁₃	—	4	64	no	—
47	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
49	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
50	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
51	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
52	C₁₃×C₄⋊C₄	—	2	48	no	—
53	C₁₃×C₂²⋊C₄	—	2	96	no	—
54	C₄²×C₁₃	C₂×D₄	1	48	yes	complemented
55	C₂²×C₄×D₄	C₂×C₁₃	1	6	yes	complemented
56	#2397	C₂×C₁₃	1	16	yes	direct product factor
57	C₂×#399	C₂×C₁₃	1	4	yes	complemented
58	C₂×#392	C₂×C₁₃	1	4	yes	complemented
59	C₂×#376	C₂×C₁₃	1	1	yes	characteristic, complemented
60	C₂³×C₁₃	—	2	96	no	—
61	C₂³×C₁₃	—	2	24	no	—
62	C₂³×C₁₃	—	2	24	no	—
63	C₂³×C₁₃	—	4	48	no	—
64	C₂³×C₁₃	—	4	16	no	—
65	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
66	C₂³×C₁₃	C₂×#41	1	16	yes	—
67	C₂³×C₁₃	C₂×#41	1	8	yes	—
68	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
69	D₄×C₁₃	—	2	192	no	—
70	C₂×C₄×C₁₃	—	2	48	no	—
71	C₂×C₄×C₁₃	—	2	24	no	—
72	C₂×C₄×C₁₃	—	2	24	no	—
73	C₂×C₄×C₁₃	—	4	16	no	—
74	C₂×C₄×C₁₃	—	4	64	no	—
75	C₂×C₄×C₁₃	#127	1	16	yes	—
76	C₂×C₄×C₁₃	C₂²×D₄	1	24	yes	—
77	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
78	C₂³×D₄	C₂²×C₁₃	1	6	yes	—
79	C₂⁴×C₄	C₂²×C₁₃	1	4	yes	—
80	C₂×C₄×D₄	—	2	48	no	—
81	C₂×C₄×D₄	C₂²×C₁₃	1	48	yes	complemented
82	C₂²×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
83	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	12	yes	—
84	C₂²×C₄²	C₂²×C₁₃	1	3	yes	—
85	#399	C₂²×C₁₃	1	32	yes	complemented
86	#392	C₂²×C₁₃	1	32	yes	complemented
87	#376	C₂²×C₁₃	1	8	yes	complemented
88	C₂×#95	C₂²×C₁₃	1	4	yes	—
89	C₂²×C₁₃	—	2	48	no	—
90	C₂²×C₁₃	—	4	64	no	—
91	C₂²×C₁₃	—	4	24	no	—
92	C₂²×C₁₃	—	4	8	no	—
93	C₂²×C₁₃	#525	1	24	yes	—
94	C₂²×C₁₃	C₂×#127	1	4	yes	—
95	C₂²×C₁₃	C₂³×D₄	1	3	yes	—
96	C₂²×C₁₃	C₂²×#41	1	4	yes	—
97	C₄×C₁₃	—	2	24	no	—
98	C₄×C₁₃	—	4	8	no	—
99	C₄×C₁₃	C₂×#127	1	8	yes	—
100	C₂⁵	C₂³×C₁₃	1	4	yes	—
101	C₂²×D₄	—	2	48	no	—
102	C₂²×D₄	—	2	24	no	—
103	C₂²×D₄	C₂³×C₁₃	1	24	yes	—
104	C₂³×C₄	—	2	48	no	—
105	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
106	C₂³×C₄	C₂³×C₁₃	1	16	yes	—
107	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
108	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
109	C₄×D₄	—	2	192	no	—
110	C₂×C₄⋊C₄	—	2	24	no	—
111	C₂×C₄⋊C₄	C₂³×C₁₃	1	24	yes	—
112	C₂×C₂²⋊C₄	—	2	48	no	—
113	C₂×C₂²⋊C₄	C₂³×C₁₃	1	48	yes	—
114	C₂×C₄²	D₄×C₁₃	1	24	yes	complemented
115	C₂×C₄²	C₂³×C₁₃	1	12	yes	—
116	#95	C₂³×C₁₃	1	16	yes	—
117	C₂×C₁₃	—	4	16	no	—
118	C₂×C₁₃	#2397	1	8	yes	direct product factor
119	C₂×C₁₃	C₂×#525	1	6	yes	—
120	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
121	C₂⁴	—	2	24	no	—
122	C₂⁴	—	2	24	no	—
123	C₂⁴	C₁₃×#41	1	16	yes	—
124	C₂⁴	C₂⁴×C₁₃	1	8	yes	—
125	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
126	C₂×D₄	—	2	192	no	—
127	C₂×D₄	—	2	96	no	—
128	C₂×D₄	—	2	48	no	—
129	C₂²×C₄	—	2	24	no	—
130	C₂²×C₄	—	2	24	no	—
131	C₂²×C₄	—	2	96	no	—
132	C₂²×C₄	—	4	64	no	—
133	C₂²×C₄	C₂×D₄×C₁₃	1	24	yes	—
134	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
135	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
136	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
137	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
138	C₄⋊C₄	—	2	48	no	—
139	C₂²⋊C₄	—	2	96	no	—
140	C₄²	C₂×D₄×C₁₃	1	48	yes	complemented
141	C₁₃	C₂×#2397	1	1	yes	characteristic, direct product factor, Sylow
142	C₂³	—	2	96	no	—
143	C₂³	—	2	24	no	—
144	C₂³	—	2	24	no	—
145	C₂³	—	4	48	no	—
146	C₂³	—	4	16	no	—
147	C₂³	C₂²×D₄×C₁₃	1	6	yes	—
148	C₂³	C₂×C₁₃×#41	1	16	yes	—
149	C₂³	C₂×C₁₃×#41	1	8	yes	—
150	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
151	D₄	—	2	192	no	—
152	C₂×C₄	—	2	48	no	—
153	C₂×C₄	—	2	24	no	—
154	C₂×C₄	—	2	24	no	—
155	C₂×C₄	—	4	16	no	—
156	C₂×C₄	—	4	64	no	—
157	C₂×C₄	C₁₃×#127	1	16	yes	—
158	C₂×C₄	C₂²×D₄×C₁₃	1	24	yes	—
159	C₂×C₄	C₂²×D₄×C₁₃	1	12	yes	—
160	C₂²	—	2	48	no	—
161	C₂²	—	4	64	no	—
162	C₂²	—	4	24	no	—
163	C₂²	—	4	8	no	—
164	C₂²	C₁₃×#525	1	24	yes	—
165	C₂²	C₂×C₁₃×#127	1	4	yes	—
166	C₂²	C₂³×D₄×C₁₃	1	3	yes	—
167	C₂²	C₂²×C₁₃×#41	1	4	yes	—
168	C₄	—	2	24	no	—
169	C₄	—	4	8	no	—
170	C₄	C₂×C₁₃×#127	1	8	yes	—
171	C₂	—	4	16	no	—
172	C₂	C₁₃×#2397	1	8	yes	direct product factor
173	C₂	C₂×C₁₃×#525	1	6	yes	—
174	C₂	C₂²×C₁₃×#127	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₁₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
socle	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
1. C₂²×C₄×D₄×C₁₃	C₂	semidirect
2. C₁₃×#2397	C₂	direct (N × H)
3. C₂×C₁₃×#399	C₂	semidirect
4. C₂×C₁₃×#392	C₂	semidirect
5. C₂×C₁₃×#376	C₂	semidirect
9. C₂×C₄×D₄×C₁₃	C₂²	semidirect
13. C₁₃×#399	C₂²	semidirect
14. C₁₃×#392	C₂²	semidirect
15. C₁₃×#376	C₂²	semidirect
31. C₂×C₄²×C₁₃	D₄	semidirect
34. C₂×#2397	C₁₃	direct (N × H)
54. C₄²×C₁₃	C₂×D₄	semidirect
55. C₂²×C₄×D₄	C₂×C₁₃	semidirect
56. #2397	C₂×C₁₃	direct (N × H)
57. C₂×#399	C₂×C₁₃	semidirect
58. C₂×#392	C₂×C₁₃	semidirect
59. C₂×#376	C₂×C₁₃	semidirect
81. C₂×C₄×D₄	C₂²×C₁₃	semidirect
85. #399	C₂²×C₁₃	semidirect
86. #392	C₂²×C₁₃	semidirect
87. #376	C₂²×C₁₃	semidirect
114. C₂×C₄²	D₄×C₁₃	semidirect
118. C₂×C₁₃	#2397	direct (N × H)
140. C₄²	C₂×D₄×C₁₃	semidirect
141. C₁₃	C₂×#2397	direct (N × H)
172. C₂	C₁₃×#2397	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×D₄×C₁₃	2
C₁₃×#2397	2
C₂×C₁₃×#399	2
C₂×C₁₃×#392	2
C₂×C₁₃×#376	2
C₂×#2397	13

Subgroups — C2×C13×#2397 #15720883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2397 #15720883