Subgroups — C₁₃×#33718 #15720057

Order 3328.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 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	4	yes	maximal, complemented
2	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2251	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2248	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2247	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	—	2	1	no	—
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	C₂²×C₁₃×Q₁₆	—	2	1	no	—
11	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
12	C₁₃×#532	—	2	1	no	—
13	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
14	C₁₃×#510	—	2	16	no	—
15	C₂×C₁₃×Q₃₂	C₂²	1	8	yes	—
16	C₂×C₁₃×SD₃₂	C₂²	1	8	yes	—
17	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
18	C₁₃×#495	—	2	1	no	—
19	C₁₃×#495	C₂²	1	2	yes	—
20	C₁₃×#494	—	2	1	no	—
21	C₁₃×#494	—	2	1	no	—
22	C₁₃×#494	C₂²	1	1	yes	characteristic
23	C₁₃×#494	C₂²	1	2	yes	—
24	C₁₃×#494	C₂²	1	2	yes	—
25	C₁₃×#493	C₂²	1	1	yes	characteristic
26	C₁₃×#492	—	2	1	no	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#346	C₂²	1	2	yes	—
29	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
30	C₂×C₁₃×#41	—	2	2	no	—
31	C₂²×Q₈×C₁₃	—	2	2	no	—
32	C₁₃×#135	—	2	8	no	—
33	C₁₃×#135	—	4	4	no	—
34	C₂×C₁₃×Q₁₆	—	2	8	no	—
35	C₂×C₁₃×Q₁₆	—	2	2	no	—
36	C₂×C₁₃×Q₁₆	—	2	2	no	—
37	C₂×C₁₃×Q₁₆	—	2	2	no	—
38	C₂×C₁₃×Q₁₆	—	2	1	no	—
39	C₂×C₁₃×Q₁₆	—	4	4	no	—
40	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
41	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×SD₁₆	—	2	2	no	—
43	C₂×C₁₃×SD₁₆	—	2	2	no	—
44	C₂×C₁₃×D₈	—	2	1	no	—
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×C₄⋊Q₈	—	2	2	no	—
50	C₁₃×C₄⋊Q₈	—	2	2	no	—
51	C₁₃×C₄⋊Q₈	—	4	2	no	—
52	C₁₃×#127	—	2	1	no	—
53	C₁₃×#124	—	2	2	no	—
54	C₁₃×#121	—	4	2	no	—
55	C₄×D₄×C₁₃	—	4	2	no	—
56	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
57	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
58	C₁₃×Q₃₂	—	2	16	no	—
59	C₁₃×SD₃₂	—	2	16	no	—
60	C₁₃×M₃₂	D₄	1	8	yes	complemented
61	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#33718	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₁₃×#41	—	4	4	no	—
68	C₁₃×#41	—	4	4	no	—
69	C₂×Q₈×C₁₃	—	2	2	no	—
70	C₂×Q₈×C₁₃	—	2	4	no	—
71	C₂×Q₈×C₁₃	—	4	2	no	—
72	C₂×Q₈×C₁₃	—	4	4	no	—
73	C₂×Q₈×C₁₃	—	4	4	no	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	4	2	no	—
76	C₂×D₄×C₁₃	—	4	2	no	—
77	C₂²×C₄×C₁₃	—	4	2	no	—
78	C₂²×C₄×C₁₃	—	4	2	no	—
79	C₂²×C₄×C₁₃	D₈	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₁₃×Q₁₆	—	2	8	no	—
82	C₁₃×Q₁₆	—	2	4	no	—
83	C₁₃×Q₁₆	—	4	4	no	—
84	C₁₃×Q₁₆	—	4	4	no	—
85	C₁₃×Q₁₆	—	4	4	no	—
86	C₁₃×Q₁₆	—	4	2	no	—
87	C₁₃×SD₁₆	—	4	4	no	—
88	C₁₃×SD₁₆	—	4	4	no	—
89	C₁₃×D₈	—	2	4	no	—
90	C₁₃×D₈	—	4	2	no	—
91	C₂×C₈×C₁₃	D₈	1	4	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	4	4	no	—
98	C₁₃×C₄⋊C₄	—	4	4	no	—
99	C₁₃×C₄⋊C₄	—	4	2	no	—
100	C₁₃×C₂²⋊C₄	—	4	4	no	—
101	C₄²×C₁₃	D₈	1	2	yes	—
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₁₃×C₁₆	C₂×D₄	1	8	yes	—
105	C₂×#510	C₂×C₁₃	1	4	yes	complemented
106	#3147	C₂×C₁₃	1	1	yes	characteristic
107	C₂×#494	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#2251	C₂×C₁₃	1	4	yes	complemented
109	#2248	C₂×C₁₃	1	2	yes	complemented
110	#2247	C₂×C₁₃	1	2	yes	complemented
111	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂³×C₁₃	—	4	2	no	—
113	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₁₃	—	4	2	no	—
115	Q₈×C₁₃	—	4	4	no	—
116	Q₈×C₁₃	—	4	2	no	—
117	Q₈×C₁₃	—	4	4	no	—
118	Q₈×C₁₃	—	8	4	no	—
119	D₄×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	8	4	no	—
122	D₄×C₁₃	—	8	4	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	8	4	no	—
127	C₂×C₄×C₁₃	—	8	4	no	—
128	C₂×C₄×C₁₃	#127	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	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	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₁₃	#127	1	2	yes	—
135	C₈×C₁₃	C₂×D₈	1	2	yes	—
136	C₈×C₁₃	C₂×D₈	1	2	yes	—
137	C₈×C₁₃	C₂²×D₄	1	2	yes	—
138	C₂×#135	—	2	1	no	—
139	C₂×#135	C₂²×C₁₃	1	2	yes	—
140	C₂²×Q₁₆	—	2	1	no	—
141	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
142	#532	—	2	1	no	—
143	C₂×C₄⋊Q₈	—	2	1	no	—
144	#510	—	2	16	no	—
145	C₂×Q₃₂	C₂²×C₁₃	1	8	yes	—
146	C₂×SD₃₂	C₂²×C₁₃	1	8	yes	—
147	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
148	#495	—	2	1	no	—
149	#495	C₂²×C₁₃	1	2	yes	—
150	#494	—	2	1	no	—
151	#494	—	2	1	no	—
152	#494	C₂²×C₁₃	1	2	yes	—
153	#494	C₂²×C₁₃	1	2	yes	—
154	#494	C₂²×C₁₃	1	1	yes	characteristic
155	#493	C₂²×C₁₃	1	1	yes	characteristic
156	#492	—	2	1	no	—
157	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
158	#346	C₂²×C₁₃	1	2	yes	—
159	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	8	2	no	—
162	C₂²×C₁₃	—	8	4	no	—
163	C₂²×C₁₃	#493	1	2	yes	—
164	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	8	2	no	—
167	C₄×C₁₃	—	8	4	no	—
168	C₄×C₁₃	#493	1	2	yes	—
169	C₄×C₁₃	#510	1	4	yes	complemented
170	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
171	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
172	C₂×#41	—	2	2	no	—
173	C₂²×Q₈	—	2	2	no	—
174	#135	—	2	8	no	—
175	#135	—	4	4	no	—
176	C₂×Q₁₆	—	2	8	no	—
177	C₂×Q₁₆	—	2	2	no	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	—	2	1	no	—
181	C₂×Q₁₆	—	4	4	no	—
182	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
183	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
184	C₂×SD₁₆	—	2	2	no	—
185	C₂×SD₁₆	—	2	2	no	—
186	C₂×D₈	—	2	1	no	—
187	C₂×D₈	C₂³×C₁₃	1	2	yes	—
188	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	4	2	no	—
194	#127	—	2	1	no	—
195	#124	—	2	2	no	—
196	#121	—	4	2	no	—
197	C₄×D₄	—	4	2	no	—
198	C₂×C₄⋊C₄	—	4	2	no	—
199	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
200	Q₃₂	—	2	16	no	—
201	SD₃₂	—	2	16	no	—
202	M₃₂	D₄×C₁₃	1	8	yes	complemented
203	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
204	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
205	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
208	C₂×C₁₃	—	2	2	no	—
209	C₂×C₁₃	—	16	2	no	—
210	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
211	C₂×C₁₃	C₂×#510	1	2	yes	—
212	#41	—	4	4	no	—
213	#41	—	4	4	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₁₃×D₈	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
226	Q₁₆	—	2	8	no	—
227	Q₁₆	—	2	4	no	—
228	Q₁₆	—	4	4	no	—
229	Q₁₆	—	4	4	no	—
230	Q₁₆	—	4	4	no	—
231	Q₁₆	—	4	2	no	—
232	SD₁₆	—	4	4	no	—
233	SD₁₆	—	4	4	no	—
234	D₈	—	2	4	no	—
235	D₈	—	4	2	no	—
236	C₂×C₈	C₁₃×D₈	1	4	yes	—
237	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
239	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₄²	C₁₃×D₈	1	2	yes	—
247	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
249	C₁₆	C₂×D₄×C₁₃	1	8	yes	—
250	C₁₃	#33718	1	1	yes	characteristic, direct product factor, Sylow
251	C₂³	—	4	2	no	—
252	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	4	no	—
255	Q₈	—	4	2	no	—
256	Q₈	—	4	4	no	—
257	Q₈	—	8	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	4	no	—
261	D₄	—	8	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	C₁₃×#127	1	2	yes	—
268	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
269	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
270	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
271	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
273	C₈	C₁₃×#127	1	2	yes	—
274	C₈	C₂×C₁₃×D₈	1	2	yes	—
275	C₈	C₂×C₁₃×D₈	1	2	yes	—
276	C₈	C₂²×D₄×C₁₃	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	4	no	—
280	C₂²	C₁₃×#493	1	2	yes	—
281	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	C₁₃×#493	1	2	yes	—
286	C₄	C₁₃×#510	1	4	yes	complemented
287	C₄	C₂×C₁₃×#127	1	1	yes	characteristic
288	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	16	2	no	—
291	C₂	C₂×C₁₃×#493	1	1	yes	characteristic
292	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	C₁₃×#3147	C₂
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
1. C₂×C₁₃×#510	C₂	semidirect
3. C₂×C₁₃×#494	C₂	semidirect
4. C₁₃×#2251	C₂	semidirect
5. C₁₃×#2248	C₂	semidirect
6. C₁₃×#2247	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
60. C₁₃×M₃₂	D₄	semidirect
66. #33718	C₁₃	direct (N × H)
105. C₂×#510	C₂×C₁₃	semidirect
107. C₂×#494	C₂×C₁₃	semidirect
108. #2251	C₂×C₁₃	semidirect
109. #2248	C₂×C₁₃	semidirect
110. #2247	C₂×C₁₃	semidirect
111. C₄×M₃₂	C₂×C₁₃	semidirect
169. C₄×C₁₃	#510	semidirect
202. M₃₂	D₄×C₁₃	semidirect
250. C₁₃	#33718	direct (N × H)
286. 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
C₁₃×#3147	2
C₂×C₁₃×#494	2
C₁₃×#2251	2
C₁₃×#2248	2
C₁₃×#2247	2
C₄×C₁₃×M₃₂	2
#33718	13

Subgroups — C13×#33718 #15720057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33718 #15720057