Subgroups — C₁₃×#3103 #13112161

Order 1664.

428 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#481	C₂	1	2	yes	maximal, complemented
4	C₁₃×#475	C₂	1	2	yes	maximal, complemented
5	C₁₃×#460	C₂	1	2	yes	maximal, complemented
6	C₁₃×#448	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×SD₁₆	C₂	1	4	yes	maximal, complemented
8	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#41	—	2	1	no	—
10	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×SD₁₆	C₂²	1	4	yes	—
12	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₁₃×#127	C₂²	1	1	yes	characteristic
14	C₁₃×#124	—	2	1	no	—
15	C₁₃×#121	—	2	1	no	—
16	C₁₃×#121	C₂²	1	2	yes	—
17	C₄×Q₈×C₁₃	—	2	1	no	—
18	C₄×Q₈×C₁₃	—	2	1	no	—
19	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
20	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
21	C₄×D₄×C₁₃	—	2	1	no	—
22	C₄×D₄×C₁₃	C₂²	1	2	yes	—
23	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
24	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	—	2	1	no	—
26	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#106	C₂²	1	4	yes	—
28	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₁₃×#103	C₂²	1	4	yes	—
30	C₁₃×#102	C₂²	1	4	yes	—
31	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₄×C₈×C₁₃	C₂²	1	2	yes	—
33	C₁₃×#41	—	4	4	no	—
34	C₂×Q₈×C₁₃	—	2	4	no	—
35	C₂×Q₈×C₁₃	—	2	1	no	—
36	C₂×Q₈×C₁₃	—	2	1	no	—
37	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
38	C₂×D₄×C₁₃	—	2	2	no	—
39	C₂×D₄×C₁₃	—	2	1	no	—
40	C₂×D₄×C₁₃	—	2	1	no	—
41	C₂×D₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	—	2	1	no	—
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×SD₁₆	—	2	8	no	—
49	C₂×C₈×C₁₃	C₂³	1	4	yes	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₁₃×C₂²⋊C₄	—	2	2	no	—
57	C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	2	2	no	—
60	C₄²×C₁₃	D₄	1	1	yes	characteristic
61	C₄²×C₁₃	D₄	1	1	yes	characteristic
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₄²×C₁₃	C₂³	1	1	yes	characteristic
64	#3103	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₁₃	—	2	2	no	—
66	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₁₃	—	2	4	no	—
68	Q₈×C₁₃	—	2	2	no	—
69	Q₈×C₁₃	—	4	2	no	—
70	Q₈×C₁₃	#41	1	4	yes	complemented
71	D₄×C₁₃	—	2	4	no	—
72	D₄×C₁₃	—	4	2	no	—
73	D₄×C₁₃	—	4	4	no	—
74	D₄×C₁₃	—	4	2	no	—
75	C₂×C₄×C₁₃	—	2	4	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	4	1	no	—
81	C₂×C₄×C₁₃	—	4	2	no	—
82	C₂×C₄×C₁₃	—	4	4	no	—
83	C₂×C₄×C₁₃	SD₁₆	1	4	yes	complemented
84	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₈×C₁₃	—	2	4	no	—
91	#532	C₂×C₁₃	1	1	yes	characteristic
92	C₂×C₄×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
93	#481	C₂×C₁₃	1	2	yes	complemented
94	#475	C₂×C₁₃	1	2	yes	complemented
95	#460	C₂×C₁₃	1	2	yes	complemented
96	#448	C₂×C₁₃	1	2	yes	complemented
97	C₄×SD₁₆	C₂×C₁₃	1	4	yes	complemented
98	#431	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂²×C₁₃	—	2	1	no	—
100	C₂²×C₁₃	—	2	1	no	—
101	C₂²×C₁₃	—	4	2	no	—
102	C₂²×C₁₃	—	4	2	no	—
103	C₂²×C₁₃	—	4	2	no	—
104	C₂²×C₁₃	C₂×SD₁₆	1	2	yes	—
105	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₁₃	—	2	4	no	—
107	C₄×C₁₃	—	2	1	no	—
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	—	2	1	no	—
110	C₄×C₁₃	—	4	2	no	—
111	C₄×C₁₃	—	4	2	no	—
112	C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
113	C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
114	C₄×C₁₃	C₂×#41	1	2	yes	—
115	C₂×#41	—	2	1	no	—
116	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
117	C₂×SD₁₆	C₂²×C₁₃	1	4	yes	—
118	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
119	#127	C₂²×C₁₃	1	1	yes	characteristic
120	#124	—	2	1	no	—
121	#121	—	2	1	no	—
122	#121	C₂²×C₁₃	1	2	yes	—
123	C₄×Q₈	—	2	1	no	—
124	C₄×Q₈	—	2	1	no	—
125	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
126	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
127	C₄×D₄	—	2	1	no	—
128	C₄×D₄	C₂²×C₁₃	1	2	yes	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
133	#106	C₂²×C₁₃	1	4	yes	—
134	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
135	#103	C₂²×C₁₃	1	4	yes	—
136	#102	C₂²×C₁₃	1	4	yes	—
137	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
138	C₄×C₈	C₂²×C₁₃	1	2	yes	—
139	C₂×C₁₃	—	2	2	no	—
140	C₂×C₁₃	—	8	2	no	—
141	C₂×C₁₃	#525	1	1	yes	characteristic
142	C₂×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
143	C₂×C₁₃	#575	1	1	yes	characteristic
144	#41	—	4	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	2	1	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	C₂³×C₁₃	1	2	yes	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
157	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
159	SD₁₆	—	2	8	no	—
160	C₂×C₈	C₂³×C₁₃	1	4	yes	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
166	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	D₄×C₁₃	1	1	yes	characteristic
172	C₄²	D₄×C₁₃	1	1	yes	characteristic
173	C₄²	C₂³×C₁₃	1	1	yes	characteristic
174	C₄²	C₂³×C₁₃	1	1	yes	characteristic
175	C₁₃	#3103	1	1	yes	characteristic, direct product factor, Sylow
176	C₂³	—	2	2	no	—
177	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
178	Q₈	—	2	4	no	—
179	Q₈	—	2	2	no	—
180	Q₈	—	4	2	no	—
181	Q₈	C₁₃×#41	1	4	yes	complemented
182	D₄	—	2	4	no	—
183	D₄	—	4	2	no	—
184	D₄	—	4	4	no	—
185	D₄	—	4	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	C₁₃×SD₁₆	1	4	yes	complemented
195	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
196	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
197	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
198	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
201	C₈	—	2	4	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	C₂×C₁₃×SD₁₆	1	2	yes	—
208	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
209	C₄	—	2	4	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	2	no	—
215	C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
216	C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
217	C₄	C₂×C₁₃×#41	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	8	2	no	—
220	C₂	C₁₃×#525	1	1	yes	characteristic
221	C₂	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
222	C₂	C₁₃×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₁₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₁₃	C₂	semidirect
3. C₁₃×#481	C₂	semidirect
4. C₁₃×#475	C₂	semidirect
5. C₁₃×#460	C₂	semidirect
6. C₁₃×#448	C₂	semidirect
7. C₄×C₁₃×SD₁₆	C₂	semidirect
8. C₁₃×#431	C₂	semidirect
64. #3103	C₁₃	direct (N × H)
70. Q₈×C₁₃	#41	semidirect
83. C₂×C₄×C₁₃	SD₁₆	semidirect
92. C₂×C₄×Q₈	C₂×C₁₃	semidirect
93. #481	C₂×C₁₃	semidirect
94. #475	C₂×C₁₃	semidirect
95. #460	C₂×C₁₃	semidirect
96. #448	C₂×C₁₃	semidirect
97. C₄×SD₁₆	C₂×C₁₃	semidirect
98. #431	C₂×C₁₃	semidirect
175. C₁₃	#3103	direct (N × H)
181. Q₈	C₁₃×#41	semidirect
194. C₂×C₄	C₁₃×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#532	2
C₂×C₄×Q₈×C₁₃	2
C₁₃×#481	2
C₁₃×#475	2
C₁₃×#460	2
C₁₃×#448	2
C₄×C₁₃×SD₁₆	2
C₁₃×#431	2
#3103	13

Subgroups — C13×#3103 #13112161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3103 #13112161