Subgroups — C₁₃×#33583 #15719922

Order 3328.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#510	C₂	1	2	yes	maximal
2	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2209	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2208	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2141	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×Q₁₆	—	2	1	no	—
10	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
12	C₄×C₁₃×#41	—	2	1	no	—
13	C₁₃×#510	—	2	8	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	—
15	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
16	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
17	C₁₃×#494	—	2	2	no	—
18	C₁₃×#494	C₂²	1	4	yes	—
19	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
20	C₄×C₁₃×SD₁₆	—	2	2	no	—
21	C₄×C₁₃×D₈	C₂²	1	2	yes	—
22	C₁₃×#427	C₂²	1	1	yes	characteristic
23	C₁₃×#416	—	2	1	no	—
24	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
26	C₁₃×#358	C₂²	1	8	yes	—
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂²×Q₈×C₁₃	—	2	2	no	—
29	C₁₃×#135	—	2	2	no	—
30	C₁₃×#135	D₄	1	4	yes	—
31	C₂×C₁₃×Q₁₆	—	2	4	no	—
32	C₂×C₁₃×Q₁₆	—	2	2	no	—
33	C₂×C₁₃×Q₁₆	—	2	4	no	—
34	C₂×C₁₃×Q₁₆	—	2	4	no	—
35	C₂×C₁₃×Q₁₆	—	4	4	no	—
36	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
37	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
38	C₂×C₁₃×SD₁₆	—	2	1	no	—
39	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊Q₈	—	2	4	no	—
43	C₁₃×C₄⋊Q₈	—	4	1	no	—
44	C₁₃×C₄⋊Q₈	—	4	1	no	—
45	C₄×Q₈×C₁₃	—	2	2	no	—
46	C₄×D₄×C₁₃	—	2	2	no	—
47	C₄×D₄×C₁₃	—	4	2	no	—
48	C₁₃×#117	—	2	1	no	—
49	C₁₃×#117	—	4	1	no	—
50	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
51	C₂×C₄²×C₁₃	—	4	1	no	—
52	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
53	C₁₃×Q₃₂	—	2	8	no	—
54	C₁₃×SD₃₂	—	2	8	no	—
55	C₁₃×M₃₂	—	2	4	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
57	C₁₃×#107	C₂³	1	2	yes	—
58	C₁₃×#106	—	2	1	no	—
59	C₁₃×#103	—	2	2	no	—
60	C₁₃×#102	—	2	2	no	—
61	C₄×C₈×C₁₃	D₄	1	2	yes	—
62	C₄×C₈×C₁₃	C₂³	1	2	yes	—
63	#33583	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₁₃×#41	—	2	4	no	—
65	C₁₃×#41	—	4	2	no	—
66	C₂×Q₈×C₁₃	—	2	1	no	—
67	C₂×Q₈×C₁₃	—	2	4	no	—
68	C₂×Q₈×C₁₃	—	4	2	no	—
69	C₂×Q₈×C₁₃	—	4	4	no	—
70	C₂×Q₈×C₁₃	—	4	4	no	—
71	C₂×D₄×C₁₃	—	2	1	no	—
72	C₂×D₄×C₁₃	—	4	1	no	—
73	C₂²×C₄×C₁₃	—	4	1	no	—
74	C₂²×C₄×C₁₃	—	4	2	no	—
75	C₂²×C₄×C₁₃	—	4	2	no	—
76	C₂²×C₄×C₁₃	D₈	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₁₃×Q₁₆	—	2	4	no	—
79	C₁₃×Q₁₆	—	2	1	no	—
80	C₁₃×Q₁₆	—	2	4	no	—
81	C₁₃×Q₁₆	—	4	8	no	—
82	C₁₃×Q₁₆	—	4	4	no	—
83	C₁₃×Q₁₆	C₂×D₄	1	2	yes	—
84	C₁₃×SD₁₆	—	2	4	no	—
85	C₁₃×D₈	—	2	1	no	—
86	C₁₃×D₈	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃	#41	1	2	yes	—
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	4	4	no	—
95	C₁₃×C₄⋊C₄	—	4	4	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₂²⋊C₄	—	4	1	no	—
98	C₁₃×C₂²⋊C₄	—	4	2	no	—
99	C₄²×C₁₃	—	4	1	no	—
100	C₄²×C₁₃	—	4	2	no	—
101	C₄²×C₁₃	D₈	1	1	yes	characteristic
102	C₄²×C₁₃	D₈	1	1	yes	characteristic
103	C₄²×C₁₃	C₂×D₄	1	2	yes	—
104	C₁₃×C₁₆	—	2	4	no	—
105	C₂×#510	C₂×C₁₃	1	2	yes	—
106	C₂×#494	C₂×C₁₃	1	1	yes	characteristic, complemented
107	C₄×#135	C₂×C₁₃	1	1	yes	characteristic
108	#2209	C₂×C₁₃	1	4	yes	complemented
109	#2208	C₂×C₁₃	1	4	yes	complemented
110	#2150	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#2141	C₂×C₁₃	1	2	yes	complemented
112	C₂³×C₁₃	—	4	1	no	—
113	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₁₃	—	2	2	no	—
115	Q₈×C₁₃	—	4	1	no	—
116	Q₈×C₁₃	—	4	4	no	—
117	Q₈×C₁₃	—	4	4	no	—
118	Q₈×C₁₃	—	8	4	no	—
119	D₄×C₁₃	—	2	2	no	—
120	D₄×C₁₃	—	4	4	no	—
121	D₄×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	—	8	4	no	—
128	C₂×C₄×C₁₃	—	8	4	no	—
129	C₂×C₄×C₁₃	#121	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	#121	1	2	yes	—
137	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
140	C₂²×Q₁₆	—	2	1	no	—
141	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
142	C₂×C₄⋊Q₈	—	2	1	no	—
143	C₄×#41	—	2	1	no	—
144	#510	—	2	8	no	—
145	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	—
146	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
147	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
148	#494	—	2	2	no	—
149	#494	C₂²×C₁₃	1	4	yes	—
150	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
151	C₄×SD₁₆	—	2	2	no	—
152	C₄×D₈	C₂²×C₁₃	1	2	yes	—
153	#427	C₂²×C₁₃	1	1	yes	characteristic
154	#416	—	2	1	no	—
155	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
156	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	—
157	#358	C₂²×C₁₃	1	8	yes	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	—	4	2	no	—
160	C₂²×C₁₃	—	8	2	no	—
161	C₂²×C₁₃	#459	1	2	yes	—
162	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	4	2	no	—
165	C₄×C₁₃	—	8	2	no	—
166	C₄×C₁₃	—	8	4	no	—
167	C₄×C₁₃	#459	1	2	yes	—
168	C₄×C₁₃	#510	1	4	yes	complemented
169	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
170	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂×#41	—	2	1	no	—
172	C₂²×Q₈	—	2	2	no	—
173	#135	—	2	2	no	—
174	#135	D₄×C₁₃	1	4	yes	—
175	C₂×Q₁₆	—	2	4	no	—
176	C₂×Q₁₆	—	2	4	no	—
177	C₂×Q₁₆	—	2	2	no	—
178	C₂×Q₁₆	—	2	4	no	—
179	C₂×Q₁₆	—	4	4	no	—
180	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
181	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
182	C₂×SD₁₆	—	2	1	no	—
183	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
184	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
186	C₄⋊Q₈	—	2	4	no	—
187	C₄⋊Q₈	—	4	1	no	—
188	C₄⋊Q₈	—	4	1	no	—
189	C₄×Q₈	—	2	2	no	—
190	C₄×D₄	—	2	2	no	—
191	C₄×D₄	—	4	2	no	—
192	#117	—	2	1	no	—
193	#117	—	4	1	no	—
194	C₂×C₄⋊C₄	—	4	2	no	—
195	C₂×C₄²	—	4	1	no	—
196	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
197	Q₃₂	—	2	8	no	—
198	SD₃₂	—	2	8	no	—
199	M₃₂	—	2	4	no	—
200	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
201	#107	C₂³×C₁₃	1	2	yes	—
202	#106	—	2	1	no	—
203	#103	—	2	2	no	—
204	#102	—	2	2	no	—
205	C₄×C₈	D₄×C₁₃	1	2	yes	—
206	C₄×C₈	C₂³×C₁₃	1	2	yes	—
207	C₂×C₁₃	—	2	2	no	—
208	C₂×C₁₃	—	8	2	no	—
209	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
210	C₂×C₁₃	C₂×#510	1	2	yes	—
211	#41	—	2	4	no	—
212	#41	—	4	2	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₁₃×D₈	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
225	Q₁₆	—	2	4	no	—
226	Q₁₆	—	2	1	no	—
227	Q₁₆	—	2	4	no	—
228	Q₁₆	—	4	8	no	—
229	Q₁₆	—	4	4	no	—
230	Q₁₆	C₂×D₄×C₁₃	1	2	yes	—
231	SD₁₆	—	2	4	no	—
232	D₈	—	2	1	no	—
233	D₈	C₂×D₄×C₁₃	1	2	yes	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
236	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
238	C₂×C₈	C₁₃×#41	1	2	yes	—
239	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₁₃×D₈	1	1	yes	characteristic
249	C₄²	C₁₃×D₈	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₁₃	1	2	yes	—
251	C₁₆	—	2	4	no	—
252	C₁₃	#33583	1	1	yes	characteristic, direct product factor, Sylow
253	C₂³	—	4	1	no	—
254	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	4	1	no	—
257	Q₈	—	4	4	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	8	4	no	—
260	D₄	—	2	2	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	C₁₃×#121	1	2	yes	—
271	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
272	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
273	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
274	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
275	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	C₁₃×#121	1	2	yes	—
278	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
279	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	C₁₃×#459	1	2	yes	—
284	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	8	4	no	—
288	C₄	—	8	2	no	—
289	C₄	C₁₃×#459	1	2	yes	—
290	C₄	C₁₃×#510	1	4	yes	complemented
291	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
292	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	8	2	no	—
295	C₂	C₂×C₁₃×#459	1	1	yes	characteristic
296	C₂	C₂×C₁₃×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#494	C₂	semidirect
4. C₁₃×#2209	C₂	semidirect
5. C₁₃×#2208	C₂	semidirect
6. C₁₃×#2150	C₂	semidirect
7. C₁₃×#2141	C₂	semidirect
63. #33583	C₁₃	direct (N × H)
106. C₂×#494	C₂×C₁₃	semidirect
108. #2209	C₂×C₁₃	semidirect
109. #2208	C₂×C₁₃	semidirect
110. #2150	C₂×C₁₃	semidirect
111. #2141	C₂×C₁₃	semidirect
168. C₄×C₁₃	#510	semidirect
252. C₁₃	#33583	direct (N × H)
290. C₄	C₁₃×#510	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#510	2
C₂×C₁₃×#494	2
C₄×C₁₃×#135	2
C₁₃×#2209	2
C₁₃×#2208	2
C₁₃×#2150	2
C₁₃×#2141	2
#33583	13

Subgroups — C13×#33583 #15719922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33583 #15719922