Subgroups — C₁₃×#33397 #15719736

Order 3328.

706 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×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2151	C₂	1	8	yes	maximal, complemented
4	C₁₃×#2150	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
8	C₂×C₁₃×M₃₂	—	2	8	no	—
9	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
10	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₃×C₁₆	C₂²	1	4	yes	—
12	C₁₃×#405	C₄	1	4	yes	—
13	C₁₃×#405	C₂²	1	4	yes	—
14	C₂×C₁₃×#97	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	C₁₃×C₄⋊C₁₆	C₂²	1	16	yes	complemented
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₃×M₁₆	—	2	4	no	—
19	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	D₄	1	4	yes	—
24	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
27	C₁₃×#117	C₈	1	4	yes	—
28	C₁₃×#117	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
30	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₁₃×M₃₂	—	2	8	no	—
33	C₁₃×M₃₂	—	2	8	no	—
34	C₂×C₁₃×C₁₆	—	2	8	no	—
35	C₂×C₁₃×C₁₆	C₂³	1	8	yes	—
36	C₁₃×#97	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₁₃	C₂³	1	4	yes	—
39	#33397	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
48	C₁₃×M₁₆	—	2	8	no	—
49	C₂×C₈×C₁₃	—	2	4	no	—
50	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
51	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
56	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
61	C₁₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
62	C₄²×C₁₃	C₂×C₈	1	4	yes	—
63	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
64	C₁₃×C₁₆	—	2	8	no	—
65	C₂²×M₃₂	C₂×C₁₃	1	2	yes	—
66	C₂×#405	C₂×C₁₃	1	1	yes	characteristic
67	#2151	C₂×C₁₃	1	8	yes	complemented
68	#2150	C₂×C₁₃	1	4	yes	complemented
69	C₂³×C₁₃	—	4	2	no	—
70	C₂³×C₁₃	C₄⋊C₈	1	4	yes	—
71	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
72	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	—	4	4	no	—
74	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
75	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
79	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
83	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
84	C₈×C₁₃	—	2	4	no	—
85	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
86	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
87	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
89	C₂²×M₁₆	C₄×C₁₃	1	1	yes	characteristic
90	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
91	C₂×#117	C₄×C₁₃	1	1	yes	characteristic
92	C₂×M₃₂	—	2	8	no	—
93	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
94	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
95	C₂²×C₁₆	C₂²×C₁₃	1	4	yes	—
96	#405	C₄×C₁₃	1	4	yes	—
97	#405	C₂²×C₁₃	1	4	yes	—
98	C₂×#97	C₄×C₁₃	1	2	yes	—
99	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
100	C₄⋊C₁₆	C₂²×C₁₃	1	16	yes	complemented
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	—	2	4	no	—
103	C₂²×C₁₃	—	4	4	no	—
104	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
105	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
106	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
107	C₄×C₁₃	—	2	8	no	—
108	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
109	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
110	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
112	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
113	C₂×M₁₆	—	2	4	no	—
114	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
115	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
116	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
118	C₂²×C₈	D₄×C₁₃	1	4	yes	—
119	C₂²×C₈	Q₈×C₁₃	1	4	yes	—
120	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
122	#117	C₈×C₁₃	1	4	yes	—
123	#117	C₂×C₄×C₁₃	1	4	yes	—
124	C₂×C₄⋊C₄	C₈×C₁₃	1	2	yes	—
125	C₂×C₂²⋊C₄	C₈×C₁₃	1	2	yes	—
126	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
127	M₃₂	—	2	8	no	—
128	M₃₂	—	2	8	no	—
129	C₂×C₁₆	—	2	8	no	—
130	C₂×C₁₆	C₂³×C₁₃	1	8	yes	—
131	#97	C₂×C₄×C₁₃	1	8	yes	—
132	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
133	C₄×C₈	C₂³×C₁₃	1	4	yes	—
134	C₂×C₁₃	—	2	2	no	—
135	C₂×C₁₃	—	2	4	no	—
136	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
137	C₂×C₁₃	#3407	1	2	yes	—
138	C₂⁴	C₂×C₈×C₁₃	1	1	yes	characteristic
139	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
140	C₂²×C₄	C₂×C₈×C₁₃	1	4	yes	—
141	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
143	C₂²×C₄	C₁₃×M₁₆	1	4	yes	—
144	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
145	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
146	M₁₆	—	2	8	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	C₁₃×C₄⋊C₄	1	4	yes	—
149	C₂×C₈	C₁₃×C₄⋊C₄	1	8	yes	—
150	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
151	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
152	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
153	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
154	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
155	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
156	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
157	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
158	C₄⋊C₄	C₂×C₈×C₁₃	1	8	yes	—
159	C₂²⋊C₄	C₂×C₈×C₁₃	1	8	yes	—
160	C₄²	C₂×C₈×C₁₃	1	4	yes	—
161	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
162	C₁₆	—	2	8	no	—
163	C₁₃	#33397	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³	—	4	2	no	—
165	C₂³	C₁₃×C₄⋊C₈	1	4	yes	—
166	C₂³	C₂²×C₈×C₁₃	1	2	yes	—
167	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₁₃×C₄⋊C₈	1	4	yes	—
170	C₂×C₄	C₁₃×C₄⋊C₈	1	8	yes	—
171	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
172	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
173	C₂×C₄	C₂²×C₈×C₁₃	1	8	yes	—
174	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
176	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
177	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
178	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
179	C₈	—	2	4	no	—
180	C₈	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
181	C₈	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
182	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
183	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₂×C₁₃×C₄⋊C₈	1	2	yes	—
188	C₂²	C₂×C₁₃×C₄⋊C₈	1	4	yes	—
189	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	C₂×C₁₃×C₄⋊C₈	1	2	yes	—
192	C₄	C₂×C₁₃×C₄⋊C₈	1	4	yes	—
193	C₄	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
194	C₄	C₂²×C₁₃×M₁₆	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂²×C₁₃×C₄⋊C₈	1	1	yes	characteristic
198	C₂	C₁₃×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2151	C₂	semidirect
4. C₁₃×#2150	C₂	semidirect
16. C₁₃×C₄⋊C₁₆	C₂²	semidirect
39. #33397	C₁₃	direct (N × H)
67. #2151	C₂×C₁₃	semidirect
68. #2150	C₂×C₁₃	semidirect
100. C₄⋊C₁₆	C₂²×C₁₃	semidirect
163. C₁₃	#33397	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×M₃₂	2
C₂×C₁₃×#405	2
C₁₃×#2151	2
C₁₃×#2150	2
#33397	13

Subgroups — C13×#33397 #15719736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33397 #15719736