Subgroups — C₁₃×#33658 #15719997

Order 3328.

534 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3059	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2916	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2239	C₂	1	8	yes	maximal, complemented
4	C₁₃×#2155	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#362	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2115	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
8	C₁₃×#568	—	2	1	no	—
9	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
10	C₁₃×#536	—	2	1	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₁₃×#470	C₂²	1	4	yes	—
13	C₁₃×#467	—	2	2	no	—
14	C₁₃×#430	—	2	1	no	—
15	C₁₃×#430	C₂²	1	4	yes	—
16	C₂×C₁₃×#108	—	2	2	no	—
17	C₂×C₁₃×#108	C₂²	1	2	yes	—
18	C₁₃×#427	C₂²	1	1	yes	characteristic
19	C₁₃×#419	—	2	1	no	—
20	C₁₃×#365	C₂²	1	8	yes	—
21	C₁₃×#362	C₂²	1	8	yes	—
22	C₁₃×#349	C₂²	1	4	yes	—
23	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
24	C₂²×Q₈×C₁₃	—	2	1	no	—
25	C₂×C₁₃×Q₁₆	—	2	1	no	—
26	C₂×C₁₃×Q₁₆	—	2	2	no	—
27	C₂×C₁₃×Q₁₆	—	2	2	no	—
28	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
29	C₁₃×#131	—	2	2	no	—
30	C₁₃×#131	—	2	2	no	—
31	C₁₃×#131	—	4	4	no	—
32	C₂×C₁₃×M₁₆	—	2	4	no	—
33	C₂×C₁₃×M₁₆	—	2	2	no	—
34	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
35	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	D₄	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×C₄⋊Q₈	—	4	1	no	—
39	C₁₃×#124	—	4	1	no	—
40	C₁₃×#123	—	4	2	no	—
41	C₁₃×C₂²⋊Q₈	—	2	4	no	—
42	C₁₃×#117	—	2	1	no	—
43	C₁₃×M₃₂	—	2	2	no	—
44	C₁₃×M₃₂	C₂³	1	4	yes	—
45	C₂×C₁₃×C₁₆	—	2	2	no	—
46	C₁₃×#108	—	2	2	no	—
47	C₁₃×#108	—	2	4	no	—
48	C₁₃×#108	C₂³	1	4	yes	—
49	C₁₃×#107	—	2	1	no	—
50	C₁₃×#106	C₂³	1	2	yes	—
51	C₁₃×#103	—	2	4	no	—
52	#33658	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₁₃×#41	—	2	2	no	—
54	C₁₃×#41	—	2	2	no	—
55	C₂×Q₈×C₁₃	—	2	2	no	—
56	C₂×Q₈×C₁₃	—	4	1	no	—
57	C₂×Q₈×C₁₃	—	4	2	no	—
58	C₂×Q₈×C₁₃	—	4	2	no	—
59	C₂×Q₈×C₁₃	SD₁₆	1	1	yes	characteristic
60	C₂×D₄×C₁₃	SD₁₆	1	2	yes	—
61	C₂×D₄×C₁₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₁₃×Q₁₆	—	2	4	no	—
66	C₁₃×Q₁₆	—	4	1	no	—
67	C₁₃×Q₁₆	—	4	1	no	—
68	C₁₃×M₁₆	—	2	2	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	4	4	no	—
72	C₁₃×M₁₆	—	4	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	4	4	no	—
76	C₂×C₈×C₁₃	—	4	2	no	—
77	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	#41	1	2	yes	—
80	C₂×C₈×C₁₃	#41	1	2	yes	—
81	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	4	4	no	—
84	C₁₃×C₂²⋊C₄	—	4	1	no	—
85	C₁₃×C₂²⋊C₄	—	4	4	no	—
86	C₄²×C₁₃	—	4	1	no	—
87	C₁₃×C₁₆	—	2	4	no	—
88	#3059	C₂×C₁₃	1	1	yes	characteristic
89	#2916	C₂×C₁₃	1	1	yes	characteristic
90	#2239	C₂×C₁₃	1	8	yes	complemented
91	#2155	C₂×C₁₃	1	2	yes	complemented
92	C₂×#362	C₂×C₁₃	1	2	yes	complemented
93	#2115	C₂×C₁₃	1	1	yes	characteristic
94	C₂³×C₁₃	C₂×SD₁₆	1	2	yes	—
95	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
96	Q₈×C₁₃	—	2	2	no	—
97	Q₈×C₁₃	—	4	2	no	—
98	Q₈×C₁₃	—	4	2	no	—
99	Q₈×C₁₃	—	8	2	no	—
100	D₄×C₁₃	—	2	4	no	—
101	D₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	8	2	no	—
107	C₂×C₄×C₁₃	#123	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	—	4	4	no	—
114	C₈×C₁₃	#123	1	1	yes	characteristic
115	C₈×C₁₃	#123	1	1	yes	characteristic
116	C₈×C₁₃	C₂×#41	1	2	yes	—
117	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
118	#568	—	2	1	no	—
119	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
120	#536	—	2	1	no	—
121	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
122	#470	C₂²×C₁₃	1	4	yes	—
123	#467	—	2	2	no	—
124	#430	—	2	1	no	—
125	#430	C₂²×C₁₃	1	4	yes	—
126	C₂×#108	—	2	2	no	—
127	C₂×#108	C₂²×C₁₃	1	2	yes	—
128	#427	C₂²×C₁₃	1	1	yes	characteristic
129	#419	—	2	1	no	—
130	#365	C₂²×C₁₃	1	8	yes	—
131	#362	C₂²×C₁₃	1	8	yes	—
132	#349	C₂²×C₁₃	1	4	yes	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	4	1	no	—
135	C₂²×C₁₃	—	4	2	no	—
136	C₂²×C₁₃	#481	1	2	yes	—
137	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	8	2	no	—
140	C₄×C₁₃	—	8	2	no	—
141	C₄×C₁₃	#481	1	2	yes	—
142	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
143	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
145	C₂²×Q₈	—	2	1	no	—
146	C₂×Q₁₆	—	2	1	no	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
150	#131	—	2	2	no	—
151	#131	—	2	2	no	—
152	#131	—	4	4	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
156	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
157	C₂²×C₈	D₄×C₁₃	1	2	yes	—
158	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
159	C₄⋊Q₈	—	4	1	no	—
160	#124	—	4	1	no	—
161	#123	—	4	2	no	—
162	C₂²⋊Q₈	—	2	4	no	—
163	#117	—	2	1	no	—
164	M₃₂	—	2	2	no	—
165	M₃₂	C₂³×C₁₃	1	4	yes	—
166	C₂×C₁₆	—	2	2	no	—
167	#108	—	2	2	no	—
168	#108	—	2	4	no	—
169	#108	C₂³×C₁₃	1	4	yes	—
170	#107	—	2	1	no	—
171	#106	C₂³×C₁₃	1	2	yes	—
172	#103	—	2	4	no	—
173	C₂×C₁₃	—	2	1	no	—
174	C₂×C₁₃	—	2	2	no	—
175	C₂×C₁₃	—	4	2	no	—
176	C₂×C₁₃	C₂×#481	1	1	yes	characteristic
177	#41	—	2	2	no	—
178	#41	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	4	1	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	C₁₃×SD₁₆	1	1	yes	characteristic
184	C₂×D₄	C₁₃×SD₁₆	1	2	yes	—
185	C₂×D₄	C₁₃×SD₁₆	1	1	yes	characteristic
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
188	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
189	Q₁₆	—	2	4	no	—
190	Q₁₆	—	4	1	no	—
191	Q₁₆	—	4	1	no	—
192	M₁₆	—	2	2	no	—
193	M₁₆	—	2	4	no	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	4	4	no	—
196	M₁₆	—	4	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	4	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
203	C₂×C₈	C₁₃×#41	1	2	yes	—
204	C₂×C₈	C₁₃×#41	1	2	yes	—
205	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₂²⋊C₄	—	4	1	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₄²	—	4	1	no	—
211	C₁₆	—	2	4	no	—
212	C₁₃	#33658	1	1	yes	characteristic, direct product factor, Sylow
213	C₂³	C₂×C₁₃×SD₁₆	1	2	yes	—
214	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
215	Q₈	—	2	2	no	—
216	Q₈	—	4	2	no	—
217	Q₈	—	4	2	no	—
218	Q₈	—	8	2	no	—
219	D₄	—	2	4	no	—
220	D₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₁₃×#123	1	2	yes	—
227	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
228	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
229	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
230	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	4	4	no	—
233	C₈	C₁₃×#123	1	1	yes	characteristic
234	C₈	C₁₃×#123	1	1	yes	characteristic
235	C₈	C₂×C₁₃×#41	1	2	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	C₁₃×#481	1	2	yes	—
240	C₂²	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	2	no	—
244	C₄	C₁₃×#481	1	2	yes	—
245	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
246	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	2	no	—
250	C₂	C₂×C₁₃×#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	C₂×#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2239	C₂	semidirect
4. C₁₃×#2155	C₂	semidirect
5. C₂×C₁₃×#362	C₂	semidirect
52. #33658	C₁₃	direct (N × H)
90. #2239	C₂×C₁₃	semidirect
91. #2155	C₂×C₁₃	semidirect
92. C₂×#362	C₂×C₁₃	semidirect
212. C₁₃	#33658	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#3059	2
C₁₃×#2916	2
C₁₃×#2239	2
C₁₃×#2155	2
C₂×C₁₃×#362	2
C₁₃×#2115	2
#33658	13

Subgroups — C13×#33658 #15719997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33658 #15719997