Subgroups — C₂²×C₁₃×#357 #15719713

Order 3328.

1470 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#357	C₂	1	12	yes	maximal, direct product factor
5	C₂³×D₄×C₁₃	—	2	1	no	—
6	C₂²×C₁₃×D₈	—	2	3	no	—
7	C₂²×C₁₃×D₈	—	2	4	no	—
8	C₂²×C₁₃×D₈	C₄	1	8	yes	complemented
9	C₂²×C₁₃×D₈	C₂²	1	6	yes	—
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
12	C₂²×C₁₃×C₁₆	—	2	4	no	—
13	C₂²×C₁₃×C₁₆	C₂²	1	6	yes	complemented
14	C₂×C₁₃×#107	—	2	3	no	—
15	C₂×C₁₃×#107	C₂²	1	6	yes	complemented
16	C₁₃×#357	C₂²	1	16	yes	direct product factor
17	C₂⁵×C₁₃	—	4	1	no	—
18	C₂²×D₄×C₁₃	—	2	8	no	—
19	C₂²×D₄×C₁₃	—	2	6	no	—
20	C₂²×D₄×C₁₃	—	4	3	no	—
21	C₂²×D₄×C₁₃	—	4	4	no	—
22	C₂³×C₄×C₁₃	—	4	1	no	—
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₁₃×D₈	—	2	24	no	—
25	C₂×C₁₃×D₈	—	2	6	no	—
26	C₂×C₁₃×D₈	—	2	12	no	—
27	C₂×C₁₃×D₈	C₂×C₄	1	24	yes	complemented
28	C₂×C₁₃×D₈	C₂³	1	4	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃	D₄	1	3	yes	—
31	C₂²×C₈×C₁₃	D₄	1	4	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
33	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
34	C₂×C₁₃×C₄⋊C₄	—	4	3	no	—
35	C₂×C₁₃×C₁₆	—	2	12	no	—
36	C₂×C₁₃×C₁₆	C₂³	1	4	yes	complemented
37	C₁₃×#107	—	2	6	no	—
38	C₁₃×#107	C₂³	1	4	yes	complemented
39	C₂²×#357	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂⁴×C₁₃	—	4	8	no	—
41	C₂⁴×C₁₃	—	4	6	no	—
42	C₂⁴×C₁₃	—	8	4	no	—
43	C₂⁴×C₁₃	—	8	4	no	—
44	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
45	C₂×D₄×C₁₃	—	2	24	no	—
46	C₂×D₄×C₁₃	—	2	4	no	—
47	C₂×D₄×C₁₃	—	4	24	no	—
48	C₂×D₄×C₁₃	—	4	6	no	—
49	C₂×D₄×C₁₃	—	4	12	no	—
50	C₂²×C₄×C₁₃	—	4	6	no	—
51	C₂²×C₄×C₁₃	—	8	4	no	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₃	D₈	1	3	yes	—
54	C₂²×C₄×C₁₃	SD₁₆	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
56	C₁₃×D₈	—	2	24	no	—
57	C₁₃×D₈	—	2	4	no	—
58	C₁₃×D₈	C₂²×C₄	1	8	yes	complemented
59	C₂×C₈×C₁₃	C₂²⋊C₄	1	12	yes	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
61	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
62	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
63	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	—	2	4	no	—
65	C₁₃×C₄⋊C₄	—	4	6	no	—
66	C₁₃×C₁₆	—	2	4	no	—
67	C₂³×D₈	C₂×C₁₃	1	1	yes	characteristic
68	C₂³×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
69	C₂²×#107	C₂×C₁₃	1	1	yes	characteristic, complemented
70	C₂×#357	C₂×C₁₃	1	12	yes	direct product factor
71	C₂³×C₁₃	—	4	4	no	—
72	C₂³×C₁₃	—	4	24	no	—
73	C₂³×C₁₃	—	8	32	no	—
74	C₂³×C₁₃	—	8	12	no	—
75	C₂³×C₁₃	—	8	12	no	—
76	C₂³×C₁₃	#102	1	4	yes	—
77	C₂³×C₁₃	D₁₆	1	4	yes	—
78	C₂³×C₁₃	SD₃₂	1	4	yes	—
79	C₂³×C₁₃	C₂×D₈	1	3	yes	—
80	D₄×C₁₃	—	2	8	no	—
81	D₄×C₁₃	—	4	4	no	—
82	D₄×C₁₃	—	4	24	no	—
83	C₂×C₄×C₁₃	—	4	4	no	—
84	C₂×C₄×C₁₃	—	8	12	no	—
85	C₂×C₄×C₁₃	#102	1	12	yes	—
86	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
87	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
88	C₂×C₄×C₁₃	C₂×SD₁₆	1	6	yes	—
89	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
90	C₈×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
91	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	C₂³×D₄	—	2	1	no	—
94	C₂²×D₈	—	2	3	no	—
95	C₂²×D₈	—	2	4	no	—
96	C₂²×D₈	C₄×C₁₃	1	8	yes	complemented
97	C₂²×D₈	C₂²×C₁₃	1	6	yes	—
98	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
99	C₂²×C₄⋊C₄	—	2	1	no	—
100	C₂²×C₁₆	—	2	4	no	—
101	C₂²×C₁₆	C₂²×C₁₃	1	6	yes	complemented
102	C₂×#107	—	2	3	no	—
103	C₂×#107	C₂²×C₁₃	1	6	yes	complemented
104	#357	C₂²×C₁₃	1	16	yes	direct product factor
105	C₂²×C₁₃	—	4	8	no	—
106	C₂²×C₁₃	—	8	4	no	—
107	C₂²×C₁₃	—	8	4	no	—
108	C₂²×C₁₃	—	8	48	no	—
109	C₂²×C₁₃	#357	1	16	yes	direct product factor
110	C₂²×C₁₃	C₂×#102	1	6	yes	—
111	C₂²×C₁₃	C₂×D₁₆	1	6	yes	—
112	C₂²×C₁₃	C₂×SD₃₂	1	6	yes	—
113	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
114	C₄×C₁₃	—	8	4	no	—
115	C₄×C₁₃	C₂×#102	1	6	yes	—
116	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
117	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
118	C₂⁵	—	4	1	no	—
119	C₂²×D₄	—	2	8	no	—
120	C₂²×D₄	—	2	6	no	—
121	C₂²×D₄	—	4	3	no	—
122	C₂²×D₄	—	4	4	no	—
123	C₂³×C₄	—	4	1	no	—
124	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
125	C₂×D₈	—	2	24	no	—
126	C₂×D₈	—	2	6	no	—
127	C₂×D₈	—	2	12	no	—
128	C₂×D₈	C₂×C₄×C₁₃	1	24	yes	complemented
129	C₂×D₈	C₂³×C₁₃	1	4	yes	—
130	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
131	C₂²×C₈	D₄×C₁₃	1	3	yes	—
132	C₂²×C₈	D₄×C₁₃	1	4	yes	—
133	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
134	C₂×C₄⋊C₄	—	2	6	no	—
135	C₂×C₄⋊C₄	—	4	3	no	—
136	C₂×C₁₆	—	2	12	no	—
137	C₂×C₁₆	C₂³×C₁₃	1	4	yes	complemented
138	#107	—	2	6	no	—
139	#107	C₂³×C₁₃	1	4	yes	complemented
140	C₂×C₁₃	—	8	8	no	—
141	C₂×C₁₃	C₂×#357	1	12	yes	direct product factor
142	C₂×C₁₃	C₂²×#102	1	1	yes	characteristic
143	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
144	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
145	C₂⁴	—	4	8	no	—
146	C₂⁴	—	4	6	no	—
147	C₂⁴	—	8	4	no	—
148	C₂⁴	—	8	4	no	—
149	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
150	C₂×D₄	—	2	24	no	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	4	24	no	—
153	C₂×D₄	—	4	6	no	—
154	C₂×D₄	—	4	12	no	—
155	C₂²×C₄	—	4	6	no	—
156	C₂²×C₄	—	8	4	no	—
157	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
158	C₂²×C₄	C₁₃×D₈	1	3	yes	—
159	C₂²×C₄	C₁₃×SD₁₆	1	4	yes	—
160	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
161	D₈	—	2	24	no	—
162	D₈	—	2	4	no	—
163	D₈	C₂²×C₄×C₁₃	1	8	yes	complemented
164	C₂×C₈	C₁₃×C₂²⋊C₄	1	12	yes	—
165	C₂×C₈	C₂²×C₄×C₁₃	1	6	yes	—
166	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
167	C₂×C₈	C₂×D₄×C₁₃	1	6	yes	—
168	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	6	no	—
171	C₁₆	—	2	4	no	—
172	C₁₃	C₂²×#357	1	1	yes	characteristic, direct product factor, Sylow
173	C₂³	—	4	24	no	—
174	C₂³	—	4	4	no	—
175	C₂³	—	8	32	no	—
176	C₂³	—	8	12	no	—
177	C₂³	—	8	12	no	—
178	C₂³	C₁₃×#102	1	4	yes	—
179	C₂³	C₁₃×D₁₆	1	4	yes	—
180	C₂³	C₁₃×SD₃₂	1	4	yes	—
181	C₂³	C₂×C₁₃×D₈	1	3	yes	—
182	D₄	—	2	8	no	—
183	D₄	—	4	24	no	—
184	D₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	12	no	—
187	C₂×C₄	C₁₃×#102	1	12	yes	—
188	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	6	yes	—
189	C₂×C₄	C₂×C₁₃×D₈	1	3	yes	—
190	C₂×C₄	C₂×C₁₃×SD₁₆	1	6	yes	—
191	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
192	C₈	C₂×C₁₃×C₂²⋊C₄	1	6	yes	—
193	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
194	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
195	C₂²	—	4	8	no	—
196	C₂²	—	8	4	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	48	no	—
199	C₂²	C₁₃×#357	1	16	yes	direct product factor
200	C₂²	C₂×C₁₃×#102	1	6	yes	—
201	C₂²	C₂×C₁₃×D₁₆	1	6	yes	—
202	C₂²	C₂×C₁₃×SD₃₂	1	6	yes	—
203	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
204	C₄	—	8	4	no	—
205	C₄	C₂×C₁₃×#102	1	6	yes	—
206	C₄	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
207	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
208	C₂	—	8	8	no	—
209	C₂	C₂×C₁₃×#357	1	12	yes	direct product factor
210	C₂	C₂²×C₁₃×#102	1	1	yes	characteristic
211	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic
212	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₃	D₈
commutator	C₈	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂³×C₁₃×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₁₃×C₁₆	C₂	semidirect
3. C₂²×C₁₃×#107	C₂	semidirect
4. C₂×C₁₃×#357	C₂	direct (N × H)
8. C₂²×C₁₃×D₈	C₄	semidirect
13. C₂²×C₁₃×C₁₆	C₂²	semidirect
15. C₂×C₁₃×#107	C₂²	semidirect
16. C₁₃×#357	C₂²	direct (N × H)
27. C₂×C₁₃×D₈	C₂×C₄	semidirect
36. C₂×C₁₃×C₁₆	C₂³	semidirect
38. C₁₃×#107	C₂³	semidirect
39. C₂²×#357	C₁₃	direct (N × H)
58. C₁₃×D₈	C₂²×C₄	semidirect
68. C₂³×C₁₆	C₂×C₁₃	semidirect
69. C₂²×#107	C₂×C₁₃	semidirect
70. C₂×#357	C₂×C₁₃	direct (N × H)
96. C₂²×D₈	C₄×C₁₃	semidirect
101. C₂²×C₁₆	C₂²×C₁₃	semidirect
103. C₂×#107	C₂²×C₁₃	semidirect
104. #357	C₂²×C₁₃	direct (N × H)
109. C₂²×C₁₃	#357	direct (N × H)
128. C₂×D₈	C₂×C₄×C₁₃	semidirect
137. C₂×C₁₆	C₂³×C₁₃	semidirect
139. #107	C₂³×C₁₃	semidirect
141. C₂×C₁₃	C₂×#357	direct (N × H)
163. D₈	C₂²×C₄×C₁₃	semidirect
172. C₁₃	C₂²×#357	direct (N × H)
199. C₂²	C₁₃×#357	direct (N × H)
209. C₂	C₂×C₁₃×#357	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×D₈	2
C₂³×C₁₃×C₁₆	2
C₂²×C₁₃×#107	2
C₂×C₁₃×#357	2
C₂²×#357	13

Subgroups — C22×C13×#357 #15719713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#357 #15719713