Subgroups — C₁₃×#33970 #15720309

Order 3328.

528 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3396	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2250	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2211	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2210	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2206	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2175	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#559	—	2	2	no	—
11	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
13	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
17	C₁₃×#495	C₂²	1	2	yes	—
18	C₁₃×#495	C₂²	1	2	yes	—
19	C₁₃×#493	C₂²	1	2	yes	—
20	C₁₃×#463	—	2	2	no	—
21	C₁₃×#463	—	2	2	no	—
22	C₁₃×#460	—	2	2	no	—
23	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
25	C₄×C₁₃×D₈	C₂²	1	2	yes	—
26	C₄×C₁₃×D₈	C₂²	1	2	yes	—
27	C₁₃×#366	C₂²	1	1	yes	characteristic
28	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₃×#358	C₂²	1	2	yes	—
31	C₁₃×#357	C₂²	1	2	yes	—
32	C₁₃×#357	C₂²	1	2	yes	—
33	C₁₃×#346	C₂²	1	2	yes	—
34	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
36	C₂×C₁₃×SD₁₆	—	2	2	no	—
37	C₂×C₁₃×SD₁₆	—	2	2	no	—
38	C₂×C₁₃×D₈	—	2	2	no	—
39	C₂×C₁₃×D₈	C₂³	1	2	yes	—
40	C₂×C₁₃×D₈	C₂³	1	2	yes	—
41	C₁₃×#127	—	2	2	no	—
42	C₁₃×#124	—	2	2	no	—
43	C₁₃×#124	—	2	2	no	—
44	C₁₃×#124	—	4	2	no	—
45	C₁₃×#123	—	4	2	no	—
46	C₁₃×#123	—	4	2	no	—
47	C₄×Q₈×C₁₃	—	2	2	no	—
48	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
49	C₄×D₄×C₁₃	—	2	2	no	—
50	C₄×D₄×C₁₃	—	2	2	no	—
51	C₄×D₄×C₁₃	—	4	2	no	—
52	C₁₃×Q₃₂	—	4	1	no	—
53	C₁₃×SD₃₂	—	2	4	no	—
54	C₁₃×SD₃₂	—	4	2	no	—
55	C₁₃×D₁₆	—	2	2	no	—
56	C₁₃×D₁₆	—	4	2	no	—
57	C₁₃×D₁₆	—	4	1	no	—
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
63	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₁₃×#103	—	2	2	no	—
65	C₁₃×#102	—	2	2	no	—
66	C₁₃×#102	—	2	2	no	—
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	2	yes	—
69	#33970	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
72	C₂×D₄×C₁₃	—	2	2	no	—
73	C₂×D₄×C₁₃	—	2	2	no	—
74	C₂×D₄×C₁₃	—	4	2	no	—
75	C₂×D₄×C₁₃	—	4	2	no	—
76	C₂²×C₄×C₁₃	—	4	2	no	—
77	C₂²×C₄×C₁₃	—	4	2	no	—
78	C₁₃×Q₁₆	—	2	2	no	—
79	C₁₃×Q₁₆	—	2	2	no	—
80	C₁₃×SD₁₆	—	8	2	no	—
81	C₁₃×SD₁₆	—	8	2	no	—
82	C₁₃×D₈	—	2	2	no	—
83	C₁₃×D₈	—	2	2	no	—
84	C₁₃×D₈	—	2	2	no	—
85	C₁₃×D₈	—	2	2	no	—
86	C₁₃×D₈	—	8	2	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	4	2	no	—
92	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
93	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
94	C₁₃×C₂²⋊C₄	—	4	2	no	—
95	C₁₃×C₂²⋊C₄	—	4	2	no	—
96	C₁₃×C₂²⋊C₄	—	4	2	no	—
97	C₁₃×C₂²⋊C₄	—	4	2	no	—
98	C₁₃×C₂²⋊C₄	—	4	2	no	—
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₄²×C₁₃	—	4	2	no	—
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	2	yes	—
103	C₁₃×C₁₆	—	2	2	no	—
104	C₁₃×C₁₆	—	2	1	no	—
105	C₁₃×C₁₆	#41	1	2	yes	—
106	#3396	C₂×C₁₃	1	2	yes	complemented
107	#2250	C₂×C₁₃	1	2	yes	complemented
108	#2249	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#2211	C₂×C₁₃	1	2	yes	complemented
110	#2210	C₂×C₁₃	1	2	yes	complemented
111	#2206	C₂×C₁₃	1	2	yes	complemented
112	#2183	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2175	C₂×C₁₃	1	2	yes	complemented
114	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂³×C₁₃	—	4	2	no	—
116	C₂³×C₁₃	—	4	2	no	—
117	Q₈×C₁₃	—	4	1	no	—
118	Q₈×C₁₃	—	4	2	no	—
119	Q₈×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	4	2	no	—
122	D₄×C₁₃	—	4	2	no	—
123	D₄×C₁₃	—	4	2	no	—
124	D₄×C₁₃	—	8	4	no	—
125	D₄×C₁₃	—	16	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	16	2	no	—
129	C₂×C₄×C₁₃	—	16	2	no	—
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	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
139	#559	—	2	2	no	—
140	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
142	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
143	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
146	#495	C₂²×C₁₃	1	2	yes	—
147	#495	C₂²×C₁₃	1	2	yes	—
148	#493	C₂²×C₁₃	1	2	yes	—
149	#463	—	2	2	no	—
150	#463	—	2	2	no	—
151	#460	—	2	2	no	—
152	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
153	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
154	C₄×D₈	C₂²×C₁₃	1	2	yes	—
155	C₄×D₈	C₂²×C₁₃	1	2	yes	—
156	#366	C₂²×C₁₃	1	1	yes	characteristic
157	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
158	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
159	#358	C₂²×C₁₃	1	2	yes	—
160	#357	C₂²×C₁₃	1	2	yes	—
161	#357	C₂²×C₁₃	1	2	yes	—
162	#346	C₂²×C₁₃	1	2	yes	—
163	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
164	C₂²×C₁₃	—	8	2	no	—
165	C₂²×C₁₃	—	8	2	no	—
166	C₂²×C₁₃	—	8	2	no	—
167	C₂²×C₁₃	—	8	2	no	—
168	C₂²×C₁₃	—	8	2	no	—
169	C₂²×C₁₃	—	8	2	no	—
170	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
171	C₄×C₁₃	—	2	2	no	—
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	8	2	no	—
176	C₄×C₁₃	—	8	2	no	—
177	C₄×C₁₃	#550	1	1	yes	characteristic
178	C₄×C₁₃	#576	1	1	yes	characteristic
179	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
180	C₂×SD₁₆	—	2	2	no	—
181	C₂×SD₁₆	—	2	2	no	—
182	C₂×D₈	—	2	2	no	—
183	C₂×D₈	C₂³×C₁₃	1	2	yes	—
184	C₂×D₈	C₂³×C₁₃	1	2	yes	—
185	#127	—	2	2	no	—
186	#124	—	2	2	no	—
187	#124	—	2	2	no	—
188	#124	—	4	2	no	—
189	#123	—	4	2	no	—
190	#123	—	4	2	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
193	C₄×D₄	—	2	2	no	—
194	C₄×D₄	—	2	2	no	—
195	C₄×D₄	—	4	2	no	—
196	Q₃₂	—	4	1	no	—
197	SD₃₂	—	2	4	no	—
198	SD₃₂	—	4	2	no	—
199	D₁₆	—	2	2	no	—
200	D₁₆	—	4	2	no	—
201	D₁₆	—	4	1	no	—
202	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
203	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
204	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
205	#107	C₂³×C₁₃	1	2	yes	—
206	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
207	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
208	#103	—	2	2	no	—
209	#102	—	2	2	no	—
210	#102	—	2	2	no	—
211	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₁₃	1	2	yes	—
213	C₂×C₁₃	—	16	2	no	—
214	C₂×C₁₃	—	16	2	no	—
215	C₂×C₁₃	#3391	1	1	yes	characteristic
216	C₂×C₁₃	#3414	1	1	yes	characteristic
217	C₂×C₁₃	#3415	1	1	yes	characteristic
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	Q₁₆	—	2	2	no	—
227	Q₁₆	—	2	2	no	—
228	SD₁₆	—	8	2	no	—
229	SD₁₆	—	8	2	no	—
230	D₈	—	2	2	no	—
231	D₈	—	2	2	no	—
232	D₈	—	2	2	no	—
233	D₈	—	2	2	no	—
234	D₈	—	8	2	no	—
235	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
237	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
241	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₁₃	1	2	yes	—
251	C₁₆	—	2	2	no	—
252	C₁₆	—	2	1	no	—
253	C₁₆	C₁₃×#41	1	2	yes	—
254	C₁₃	#33970	1	1	yes	characteristic, direct product factor, Sylow
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	Q₈	—	4	1	no	—
258	Q₈	—	4	2	no	—
259	Q₈	—	4	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	D₄	—	8	4	no	—
265	D₄	—	16	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	16	2	no	—
269	C₂×C₄	—	16	2	no	—
270	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
271	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
272	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
273	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
274	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	2	1	no	—
277	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
278	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	C₁₃×#550	1	1	yes	characteristic
293	C₄	C₁₃×#576	1	1	yes	characteristic
294	C₂	—	16	2	no	—
295	C₂	—	16	2	no	—
296	C₂	C₁₃×#3391	1	1	yes	characteristic
297	C₂	C₁₃×#3414	1	1	yes	characteristic
298	C₂	C₁₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ C₈×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3396	C₂	semidirect
2. C₁₃×#2250	C₂	semidirect
3. C₁₃×#2249	C₂	semidirect
4. C₁₃×#2211	C₂	semidirect
5. C₁₃×#2210	C₂	semidirect
6. C₁₃×#2206	C₂	semidirect
7. C₁₃×#2183	C₂	semidirect
8. C₁₃×#2175	C₂	semidirect
9. C₄×C₁₃×D₁₆	C₂	semidirect
69. #33970	C₁₃	direct (N × H)
106. #3396	C₂×C₁₃	semidirect
107. #2250	C₂×C₁₃	semidirect
108. #2249	C₂×C₁₃	semidirect
109. #2211	C₂×C₁₃	semidirect
110. #2210	C₂×C₁₃	semidirect
111. #2206	C₂×C₁₃	semidirect
112. #2183	C₂×C₁₃	semidirect
113. #2175	C₂×C₁₃	semidirect
114. C₄×D₁₆	C₂×C₁₃	semidirect
254. C₁₃	#33970	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×#3396	2
C₁₃×#2250	2
C₁₃×#2249	2
C₁₃×#2211	2
C₁₃×#2210	2
C₁₃×#2206	2
C₁₃×#2183	2
C₁₃×#2175	2
C₄×C₁₃×D₁₆	2
#33970	13

Subgroups — C13×#33970 #15720309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33970 #15720309