Subgroups — C₁₃×#3384 #13112442

Order 1664.

374 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#501	C₂	1	2	yes	maximal, complemented
4	C₁₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#478	C₂	1	2	yes	maximal, complemented
6	C₁₃×#476	C₂	1	2	yes	maximal, complemented
7	C₁₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#442	C₂	1	2	yes	maximal
11	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₁₃	—	2	1	no	—
13	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₁₃×#125	C₂²	1	2	yes	—
18	C₁₃×C₂²⋊Q₈	—	2	1	no	—
19	C₁₃×C₂²⋊Q₈	—	2	1	no	—
20	C₁₃×C₂²⋊Q₈	—	2	2	no	—
21	C₁₃×C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₃	C₂²	1	2	yes	—
26	C₄×D₄×C₁₃	C₂²	1	2	yes	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₁₃×#107	C₂²	1	1	yes	characteristic
29	C₁₃×#107	C₂²	1	2	yes	—
30	C₁₃×#106	C₂²	1	2	yes	—
31	C₁₃×#106	C₂²	1	2	yes	—
32	C₁₃×#106	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
35	C₁₃×#102	C₂²	1	1	yes	characteristic
36	C₁₃×#102	C₂²	1	1	yes	characteristic
37	C₁₃×#102	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₁₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	—	2	2	no	—
42	C₂×Q₈×C₁₃	—	2	1	no	—
43	C₂×Q₈×C₁₃	—	2	1	no	—
44	C₂×Q₈×C₁₃	—	4	4	no	—
45	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
46	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₁₃×D₈	Q₈	1	4	yes	complemented
53	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂³	1	2	yes	—
55	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
63	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
66	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₁₃×C₂²⋊C₄	—	2	1	no	—
68	C₁₃×C₂²⋊C₄	—	2	2	no	—
69	C₁₃×C₂²⋊C₄	—	2	1	no	—
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₄²×C₁₃	C₂³	1	1	yes	characteristic
72	C₄²×C₁₃	C₂³	1	2	yes	—
73	#3384	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₁₃	—	2	1	no	—
75	C₂³×C₁₃	—	2	1	no	—
76	Q₈×C₁₃	—	2	2	no	—
77	Q₈×C₁₃	—	4	4	no	—
78	Q₈×C₁₃	—	4	2	no	—
79	Q₈×C₁₃	—	8	1	no	—
80	D₄×C₁₃	—	2	1	no	—
81	D₄×C₁₃	—	2	1	no	—
82	D₄×C₁₃	C₂×Q₈	1	2	yes	—
83	D₄×C₁₃	C₂×Q₈	1	2	yes	complemented
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	4	4	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	—	8	2	no	—
92	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₈×C₁₃	—	2	1	no	—
98	C₈×C₁₃	—	2	2	no	—
99	C₈×C₁₃	C₂×Q₈	1	2	yes	complemented
100	#554	C₂×C₁₃	1	1	yes	characteristic, complemented
101	D₄×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#501	C₂×C₁₃	1	2	yes	complemented
103	#498	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#478	C₂×C₁₃	1	2	yes	complemented
105	#476	C₂×C₁₃	1	2	yes	complemented
106	#476	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#474	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#442	C₂×C₁₃	1	2	yes	—
110	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	4	1	no	—
114	C₂²×C₁₃	—	4	1	no	—
115	C₂²×C₁₃	—	4	1	no	—
116	C₂²×C₁₃	—	4	1	no	—
117	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₄×C₁₃	—	2	2	no	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	4	1	no	—
122	C₄×C₁₃	—	4	2	no	—
123	C₄×C₁₃	—	4	1	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
126	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
127	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂²×Q₈	—	2	1	no	—
129	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
130	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
131	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
132	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
133	#125	C₂²×C₁₃	1	2	yes	—
134	C₂²⋊Q₈	—	2	1	no	—
135	C₂²⋊Q₈	—	2	1	no	—
136	C₂²⋊Q₈	—	2	2	no	—
137	C₂²⋊Q₈	—	2	2	no	—
138	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
139	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
140	C₄×D₄	C₂²×C₁₃	1	2	yes	—
141	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
142	C₄×D₄	C₂²×C₁₃	1	2	yes	—
143	C₂×C₄⋊C₄	—	2	1	no	—
144	#107	C₂²×C₁₃	1	1	yes	characteristic
145	#107	C₂²×C₁₃	1	2	yes	—
146	#106	C₂²×C₁₃	1	2	yes	—
147	#106	C₂²×C₁₃	1	2	yes	—
148	#106	C₂²×C₁₃	1	2	yes	—
149	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
151	#102	C₂²×C₁₃	1	1	yes	characteristic
152	#102	C₂²×C₁₃	1	1	yes	characteristic
153	#102	C₂²×C₁₃	1	2	yes	—
154	#102	C₂²×C₁₃	1	2	yes	—
155	#97	C₂²×C₁₃	1	2	yes	—
156	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₁₃	—	4	2	no	—
158	C₂×C₁₃	—	4	2	no	—
159	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
160	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×C₁₃	#577	1	1	yes	characteristic
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	4	4	no	—
166	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
167	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
168	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	D₈	Q₈×C₁₃	1	4	yes	complemented
174	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
175	C₂×C₈	C₂³×C₁₃	1	2	yes	—
176	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
183	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
184	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
185	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
187	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
188	C₂²⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₄²	C₂³×C₁₃	1	1	yes	characteristic
193	C₄²	C₂³×C₁₃	1	2	yes	—
194	C₁₃	#3384	1	1	yes	characteristic, direct product factor, Sylow
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	4	4	no	—
199	Q₈	—	4	2	no	—
200	Q₈	—	8	1	no	—
201	D₄	—	2	1	no	—
202	D₄	—	2	1	no	—
203	D₄	C₂×Q₈×C₁₃	1	2	yes	—
204	D₄	C₂×Q₈×C₁₃	1	2	yes	complemented
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
217	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	2	no	—
220	C₈	C₂×Q₈×C₁₃	1	2	yes	complemented
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
236	C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
237	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
238	C₂	—	4	2	no	—
239	C₂	—	4	2	no	—
240	C₂	D₄×Q₈×C₁₃	1	1	yes	characteristic
241	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂	C₁₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#554	C₂	semidirect
2. D₄×Q₈×C₁₃	C₂	semidirect
3. C₁₃×#501	C₂	semidirect
4. C₁₃×#498	C₂	semidirect
5. C₁₃×#478	C₂	semidirect
6. C₁₃×#476	C₂	semidirect
7. C₁₃×#476	C₂	semidirect
8. C₁₃×#474	C₂	semidirect
9. C₁₃×#446	C₂	semidirect
52. C₁₃×D₈	Q₈	semidirect
73. #3384	C₁₃	direct (N × H)
83. D₄×C₁₃	C₂×Q₈	semidirect
99. C₈×C₁₃	C₂×Q₈	semidirect
100. #554	C₂×C₁₃	semidirect
101. D₄×Q₈	C₂×C₁₃	semidirect
102. #501	C₂×C₁₃	semidirect
103. #498	C₂×C₁₃	semidirect
104. #478	C₂×C₁₃	semidirect
105. #476	C₂×C₁₃	semidirect
106. #476	C₂×C₁₃	semidirect
107. #474	C₂×C₁₃	semidirect
108. #446	C₂×C₁₃	semidirect
173. D₈	Q₈×C₁₃	semidirect
194. C₁₃	#3384	direct (N × H)
204. D₄	C₂×Q₈×C₁₃	semidirect
220. C₈	C₂×Q₈×C₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#554	2
D₄×Q₈×C₁₃	2
C₁₃×#501	2
C₁₃×#498	2
C₁₃×#478	2
C₁₃×#476	2
C₁₃×#474	2
C₁₃×#446	2
C₁₃×#442	2
C₄×C₁₃×D₈	2
#3384	13

Subgroups — C13×#3384 #13112442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3384 #13112442