Subgroups — C₁₃×#33495 #15719834

Order 3328.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2169	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2168	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₃×C₁₆	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2113	C₂	1	4	yes	maximal, complemented
8	C₂×C₁₃×#131	C₂²	1	2	yes	—
9	C₁₃×#532	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₃×C₁₆	C₂²	1	4	yes	—
12	C₁₃×#436	C₄	1	2	yes	—
13	C₁₃×#436	C₄	1	2	yes	—
14	C₈×D₄×C₁₃	C₂²	1	2	yes	—
15	C₈×D₄×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#423	C₄	1	1	yes	characteristic
17	C₁₃×#408	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
20	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
21	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
24	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
25	C₁₃×#346	C₂²	1	2	yes	—
26	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
29	C₁₃×#131	—	4	4	no	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
35	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
36	C₁₃×#127	C₈	1	1	yes	characteristic
37	C₁₃×#124	C₈	1	2	yes	—
38	C₁₃×#121	C₈	1	2	yes	—
39	C₁₃×#121	C₈	1	2	yes	—
40	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₁₃×M₃₂	—	4	4	no	—
44	C₂×C₁₃×C₁₆	—	4	4	no	—
45	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
46	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
47	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
48	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
51	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	#33495	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₁₃×#41	—	4	2	no	—
58	C₁₃×#41	—	4	2	no	—
59	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
61	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
62	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
63	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₁₃×M₁₆	—	4	2	no	—
69	C₁₃×M₁₆	—	4	4	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	4	1	no	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
81	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
82	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
83	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₁₃×C₁₆	—	2	2	no	—
88	C₁₃×C₁₆	—	2	4	no	—
89	C₁₃×C₁₆	—	2	2	no	—
90	#2931	C₂×C₁₃	1	1	yes	characteristic
91	#2169	C₂×C₁₃	1	2	yes	complemented
92	#2168	C₂×C₁₃	1	4	yes	complemented
93	D₄×C₁₆	C₂×C₁₃	1	2	yes	complemented
94	#2163	C₂×C₁₃	1	1	yes	characteristic, complemented
95	#2162	C₂×C₁₃	1	1	yes	characteristic, complemented
96	#2113	C₂×C₁₃	1	4	yes	complemented
97	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
98	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
99	Q₈×C₁₃	—	4	2	no	—
100	D₄×C₁₃	—	2	4	no	—
101	D₄×C₁₃	—	4	2	no	—
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
115	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₈×C₁₃	—	2	4	no	—
118	C₈×C₁₃	—	2	1	no	—
119	C₈×C₁₃	—	2	1	no	—
120	C₈×C₁₃	—	2	1	no	—
121	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
122	C₂×#131	C₂²×C₁₃	1	2	yes	—
123	#532	C₄×C₁₃	1	1	yes	characteristic
124	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
125	C₂²×C₁₆	C₂²×C₁₃	1	4	yes	—
126	#436	C₄×C₁₃	1	2	yes	—
127	#436	C₄×C₁₃	1	2	yes	—
128	C₈×D₄	C₂²×C₁₃	1	2	yes	—
129	C₈×D₄	C₂²×C₁₃	1	2	yes	—
130	#423	C₄×C₁₃	1	1	yes	characteristic
131	#408	C₄×C₁₃	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
133	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
134	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
135	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
136	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
137	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
138	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
139	#346	C₂²×C₁₃	1	2	yes	—
140	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
141	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	—	2	4	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	4	no	—
146	C₂²×C₁₃	—	2	1	no	—
147	C₂²×C₁₃	—	2	4	no	—
148	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	2	1	no	—
153	C₄×C₁₃	—	2	4	no	—
154	C₄×C₁₃	#504	1	2	yes	—
155	C₄×C₁₃	#518	1	1	yes	characteristic
156	C₄×C₁₃	#568	1	1	yes	characteristic
157	C₂×#41	C₂×C₄×C₁₃	1	2	yes	—
158	#131	—	4	4	no	—
159	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
160	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
161	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
164	C₄⋊Q₈	C₈×C₁₃	1	1	yes	characteristic
165	#127	C₈×C₁₃	1	1	yes	characteristic
166	#124	C₈×C₁₃	1	2	yes	—
167	#121	C₈×C₁₃	1	2	yes	—
168	#121	C₈×C₁₃	1	2	yes	—
169	C₄×D₄	C₂×C₄×C₁₃	1	2	yes	—
170	C₄×D₄	C₂×C₄×C₁₃	1	2	yes	—
171	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
172	M₃₂	—	4	4	no	—
173	C₂×C₁₆	—	4	4	no	—
174	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
175	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
176	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
177	C₄⋊C₈	C₂×C₄×C₁₃	1	2	yes	—
178	C₄⋊C₈	C₂×C₄×C₁₃	1	2	yes	—
179	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
180	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
181	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
182	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
183	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
184	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
185	C₂×C₁₃	—	4	1	no	—
186	C₂×C₁₃	—	4	4	no	—
187	C₂×C₁₃	#2972	1	1	yes	characteristic
188	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
189	C₂×C₁₃	#3407	1	1	yes	characteristic
190	#41	—	4	2	no	—
191	#41	—	4	2	no	—
192	C₂×Q₈	C₂×C₈×C₁₃	1	2	yes	—
193	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
194	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
195	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
196	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
197	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
200	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
209	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
211	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
212	C₄⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
213	C₄⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
214	C₂²⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
215	C₂²⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
216	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
217	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
218	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
219	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
220	C₁₆	—	2	2	no	—
221	C₁₆	—	2	4	no	—
222	C₁₆	—	2	2	no	—
223	C₁₃	#33495	1	1	yes	characteristic, direct product factor, Sylow
224	C₂³	C₂²×C₈×C₁₃	1	4	yes	—
225	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
226	Q₈	—	4	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
242	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
244	C₈	—	2	4	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	C₁₃×#504	1	2	yes	—
262	C₄	C₁₃×#518	1	1	yes	characteristic
263	C₄	C₁₃×#568	1	1	yes	characteristic
264	C₂	—	4	1	no	—
265	C₂	—	4	4	no	—
266	C₂	C₁₃×#2972	1	1	yes	characteristic
267	C₂	C₂×C₁₃×#504	1	1	yes	characteristic
268	C₂	C₁₃×#3407	1	1	yes	characteristic

	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	C₁₃×#532	C₄
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
2. C₁₃×#2169	C₂	semidirect
3. C₁₃×#2168	C₂	semidirect
4. D₄×C₁₃×C₁₆	C₂	semidirect
5. C₁₃×#2163	C₂	semidirect
6. C₁₃×#2162	C₂	semidirect
7. C₁₃×#2113	C₂	semidirect
56. #33495	C₁₃	direct (N × H)
91. #2169	C₂×C₁₃	semidirect
92. #2168	C₂×C₁₃	semidirect
93. D₄×C₁₆	C₂×C₁₃	semidirect
94. #2163	C₂×C₁₃	semidirect
95. #2162	C₂×C₁₃	semidirect
96. #2113	C₂×C₁₃	semidirect
223. C₁₃	#33495	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×#2931	2
C₁₃×#2169	2
C₁₃×#2168	2
D₄×C₁₃×C₁₆	2
C₁₃×#2163	2
C₁₃×#2162	2
C₁₃×#2113	2
#33495	13

Subgroups — C13×#33495 #15719834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33495 #15719834