Subgroups — C₂²×C₁₃×#331 #15704032

Order 3328.

2022 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×#331	C₂	1	12	yes	maximal, direct product factor
4	C₂³×D₄×C₁₃	—	2	2	no	—
5	C₂³×D₄×C₁₃	C₄	1	2	yes	—
6	C₂³×C₈×C₁₃	—	2	2	no	—
7	C₂×C₁₃×#127	—	2	6	no	—
8	C₂×C₁₃×#127	C₄	1	6	yes	—
9	C₂×C₁₃×#127	C₂²	1	6	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₈	—	2	6	no	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	12	yes	complemented
13	C₁₃×#331	C₂²	1	16	yes	direct product factor
14	C₂⁵×C₁₃	—	2	2	no	—
15	C₂²×D₄×C₁₃	—	2	6	no	—
16	C₂²×D₄×C₁₃	—	2	12	no	—
17	C₂²×D₄×C₁₃	—	2	12	no	—
18	C₂²×D₄×C₁₃	—	4	8	no	—
19	C₂²×D₄×C₁₃	—	4	8	no	—
20	C₂²×D₄×C₁₃	—	4	8	no	—
21	C₂²×D₄×C₁₃	C₂×C₄	1	12	yes	—
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	D₄	1	2	yes	—
24	C₂²×C₈×C₁₃	—	2	12	no	—
25	C₂²×C₈×C₁₃	—	4	8	no	—
26	C₁₃×#127	—	2	12	no	—
27	C₁₃×#127	—	2	12	no	—
28	C₁₃×#127	C₂×C₄	1	12	yes	—
29	C₁₃×#127	C₂³	1	4	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₁₃	D₄	1	6	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
33	C₁₃×C₄⋊C₈	—	2	12	no	—
34	C₁₃×C₄⋊C₈	C₂³	1	8	yes	complemented
35	C₂²×#331	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂⁴×C₁₃	—	2	12	no	—
37	C₂⁴×C₁₃	—	4	8	no	—
38	C₂⁴×C₁₃	—	4	16	no	—
39	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×D₄×C₁₃	—	2	12	no	—
41	C₂×D₄×C₁₃	—	2	24	no	—
42	C₂×D₄×C₁₃	—	2	8	no	—
43	C₂×D₄×C₁₃	—	4	24	no	—
44	C₂×D₄×C₁₃	—	4	24	no	—
45	C₂×D₄×C₁₃	—	4	48	no	—
46	C₂×D₄×C₁₃	—	4	24	no	—
47	C₂×D₄×C₁₃	—	4	24	no	—
48	C₂×D₄×C₁₃	C₂²×C₄	1	8	yes	—
49	C₂²×C₄×C₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₃	C₂²⋊C₄	1	3	yes	—
51	C₂²×C₄×C₁₃	C₂²⋊C₄	1	6	yes	—
52	C₂²×C₄×C₁₃	D₈	1	8	yes	—
53	C₂²×C₄×C₁₃	SD₁₆	1	8	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
56	C₂×C₈×C₁₃	—	2	8	no	—
57	C₂×C₈×C₁₃	—	4	24	no	—
58	C₄²×C₁₃	C₂²⋊C₄	1	6	yes	—
59	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
60	C₄²×C₁₃	C₂×D₄	1	6	yes	—
61	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₂²×#127	C₂×C₁₃	1	1	yes	characteristic
63	C₂²×C₄⋊C₈	C₂×C₁₃	1	2	yes	complemented
64	C₂×#331	C₂×C₁₃	1	12	yes	direct product factor
65	C₂³×C₁₃	—	2	8	no	—
66	C₂³×C₁₃	—	4	24	no	—
67	C₂³×C₁₃	—	4	48	no	—
68	C₂³×C₁₃	—	8	32	no	—
69	C₂³×C₁₃	#100	1	4	yes	—
70	C₂³×C₁₃	#102	1	8	yes	—
71	C₂³×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
72	D₄×C₁₃	—	4	24	no	—
73	D₄×C₁₃	—	4	8	no	—
74	D₄×C₁₃	—	4	8	no	—
75	D₄×C₁₃	—	4	8	no	—
76	D₄×C₁₃	—	4	48	no	—
77	C₂×C₄×C₁₃	—	2	6	no	—
78	C₂×C₄×C₁₃	—	2	6	no	—
79	C₂×C₄×C₁₃	#102	1	24	yes	—
80	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
81	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
82	C₂×C₄×C₁₃	C₂×D₈	1	12	yes	—
83	C₂×C₄×C₁₃	C₂×SD₁₆	1	12	yes	—
84	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
86	C₈×C₁₃	—	4	8	no	—
87	C₂³×D₄	—	2	2	no	—
88	C₂³×D₄	C₄×C₁₃	1	2	yes	—
89	C₂³×C₈	—	2	2	no	—
90	C₂×#127	—	2	6	no	—
91	C₂×#127	C₄×C₁₃	1	6	yes	—
92	C₂×#127	C₂²×C₁₃	1	6	yes	—
93	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
94	C₂×C₄⋊C₈	—	2	6	no	—
95	C₂×C₄⋊C₈	C₂²×C₁₃	1	12	yes	complemented
96	#331	C₂²×C₁₃	1	16	yes	direct product factor
97	C₂²×C₁₃	—	4	8	no	—
98	C₂²×C₁₃	—	4	16	no	—
99	C₂²×C₁₃	—	8	48	no	—
100	C₂²×C₁₃	#331	1	16	yes	direct product factor
101	C₂²×C₁₃	C₂×#100	1	6	yes	—
102	C₂²×C₁₃	C₂×#102	1	12	yes	—
103	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
104	C₄×C₁₃	—	2	3	no	—
105	C₄×C₁₃	—	2	1	no	—
106	C₄×C₁₃	C₂×#102	1	12	yes	—
107	C₄×C₁₃	C₂²×D₈	1	2	yes	—
108	C₄×C₁₃	C₂²×SD₁₆	1	2	yes	—
109	C₂⁵	—	2	2	no	—
110	C₂²×D₄	—	2	6	no	—
111	C₂²×D₄	—	2	12	no	—
112	C₂²×D₄	—	2	12	no	—
113	C₂²×D₄	—	4	8	no	—
114	C₂²×D₄	—	4	8	no	—
115	C₂²×D₄	—	4	8	no	—
116	C₂²×D₄	C₂×C₄×C₁₃	1	12	yes	—
117	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
118	C₂³×C₄	D₄×C₁₃	1	2	yes	—
119	C₂²×C₈	—	2	12	no	—
120	C₂²×C₈	—	4	8	no	—
121	#127	—	2	12	no	—
122	#127	—	2	12	no	—
123	#127	C₂×C₄×C₁₃	1	12	yes	—
124	#127	C₂³×C₁₃	1	4	yes	—
125	C₂×C₄²	C₂×C₄×C₁₃	1	3	yes	—
126	C₂×C₄²	D₄×C₁₃	1	6	yes	—
127	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
128	C₄⋊C₈	—	2	12	no	—
129	C₄⋊C₈	C₂³×C₁₃	1	8	yes	complemented
130	C₂×C₁₃	—	8	8	no	—
131	C₂×C₁₃	C₂×#331	1	12	yes	direct product factor
132	C₂×C₁₃	C₂²×#100	1	1	yes	characteristic
133	C₂×C₁₃	C₂²×#102	1	2	yes	—
134	C₂⁴	—	2	12	no	—
135	C₂⁴	—	4	8	no	—
136	C₂⁴	—	4	16	no	—
137	C₂⁴	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×D₄	—	2	12	no	—
139	C₂×D₄	—	2	24	no	—
140	C₂×D₄	—	2	8	no	—
141	C₂×D₄	—	4	24	no	—
142	C₂×D₄	—	4	24	no	—
143	C₂×D₄	—	4	48	no	—
144	C₂×D₄	—	4	24	no	—
145	C₂×D₄	—	4	24	no	—
146	C₂×D₄	C₂²×C₄×C₁₃	1	8	yes	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	C₁₃×C₂²⋊C₄	1	3	yes	—
149	C₂²×C₄	C₁₃×C₂²⋊C₄	1	6	yes	—
150	C₂²×C₄	C₁₃×D₈	1	8	yes	—
151	C₂²×C₄	C₁₃×SD₁₆	1	8	yes	—
152	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
153	C₂²×C₄	C₂×D₄×C₁₃	1	6	yes	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	—	4	24	no	—
156	C₄²	C₁₃×C₂²⋊C₄	1	6	yes	—
157	C₄²	C₂²×C₄×C₁₃	1	3	yes	—
158	C₄²	C₂×D₄×C₁₃	1	6	yes	—
159	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
160	C₁₃	C₂²×#331	1	1	yes	characteristic, direct product factor, Sylow
161	C₂³	—	2	8	no	—
162	C₂³	—	4	24	no	—
163	C₂³	—	4	48	no	—
164	C₂³	—	8	32	no	—
165	C₂³	C₁₃×#100	1	4	yes	—
166	C₂³	C₁₃×#102	1	8	yes	—
167	C₂³	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
168	D₄	—	4	8	no	—
169	D₄	—	4	24	no	—
170	D₄	—	4	48	no	—
171	D₄	—	4	8	no	—
172	D₄	—	4	8	no	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	2	6	no	—
175	C₂×C₄	C₁₃×#102	1	24	yes	—
176	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
177	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	6	yes	—
178	C₂×C₄	C₂×C₁₃×D₈	1	12	yes	—
179	C₂×C₄	C₂×C₁₃×SD₁₆	1	12	yes	—
180	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
182	C₈	—	4	8	no	—
183	C₂²	—	4	8	no	—
184	C₂²	—	4	16	no	—
185	C₂²	—	8	48	no	—
186	C₂²	C₁₃×#331	1	16	yes	direct product factor
187	C₂²	C₂×C₁₃×#100	1	6	yes	—
188	C₂²	C₂×C₁₃×#102	1	12	yes	—
189	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
190	C₄	—	2	3	no	—
191	C₄	—	2	1	no	—
192	C₄	C₂×C₁₃×#102	1	12	yes	—
193	C₄	C₂²×C₁₃×D₈	1	2	yes	—
194	C₄	C₂²×C₁₃×SD₁₆	1	2	yes	—
195	C₂	—	8	8	no	—
196	C₂	C₂×C₁₃×#331	1	12	yes	direct product factor
197	C₂	C₂²×C₁₃×#100	1	1	yes	characteristic
198	C₂	C₂²×C₁₃×#102	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₃×C₄⋊C₈	C₂	semidirect
3. C₂×C₁₃×#331	C₂	direct (N × H)
12. C₂×C₁₃×C₄⋊C₈	C₂²	semidirect
13. C₁₃×#331	C₂²	direct (N × H)
34. C₁₃×C₄⋊C₈	C₂³	semidirect
35. C₂²×#331	C₁₃	direct (N × H)
63. C₂²×C₄⋊C₈	C₂×C₁₃	semidirect
64. C₂×#331	C₂×C₁₃	direct (N × H)
95. C₂×C₄⋊C₈	C₂²×C₁₃	semidirect
96. #331	C₂²×C₁₃	direct (N × H)
100. C₂²×C₁₃	#331	direct (N × H)
129. C₄⋊C₈	C₂³×C₁₃	semidirect
131. C₂×C₁₃	C₂×#331	direct (N × H)
160. C₁₃	C₂²×#331	direct (N × H)
186. C₂²	C₁₃×#331	direct (N × H)
196. C₂	C₂×C₁₃×#331	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×C₁₃×#127	2
C₂²×C₁₃×C₄⋊C₈	2
C₂×C₁₃×#331	2
C₂²×#331	13

Subgroups — C22×C13×#331 #15704032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#331 #15704032