Subgroups — C₁₃×#33687 #15720026

Order 3328.

520 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3177	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2950	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2237	C₂	1	2	yes	maximal
4	C₁₃×#2229	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2209	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2188	C₂	1	2	yes	maximal
7	C₁₃×#2176	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×Q₃₂	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2164	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₁₃×#541	—	2	1	no	—
12	C₁₃×#519	—	2	1	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
14	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
15	C₁₃×#499	C₂²	1	1	yes	characteristic
16	C₁₃×#494	C₂²	1	1	yes	characteristic
17	C₁₃×#491	—	2	1	no	—
18	C₁₃×#470	—	2	1	no	—
19	C₁₃×#467	C₂²	1	2	yes	—
20	C₁₃×#441	—	2	1	no	—
21	C₁₃×#441	—	2	1	no	—
22	C₁₃×#441	—	2	1	no	—
23	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
24	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
25	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
26	C₁₃×#427	C₂²	1	1	yes	characteristic
27	C₁₃×#419	—	2	1	no	—
28	C₁₃×#405	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	2	yes	—
30	C₁₃×#366	C₂²	1	2	yes	—
31	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₁₃×#358	C₂²	1	2	yes	—
33	C₁₃×#358	C₂²	1	2	yes	—
34	C₁₃×#358	C₂²	1	2	yes	—
35	C₁₃×#358	C₂²	1	2	yes	—
36	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₁₃×#346	C₂²	1	1	yes	characteristic
38	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₁₃	—	2	1	no	—
40	C₂×C₁₃×Q₁₆	—	2	4	no	—
41	C₂×C₁₃×Q₁₆	—	2	1	no	—
42	C₂×C₁₃×Q₁₆	—	2	1	no	—
43	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊Q₈	—	2	1	no	—
48	C₁₃×#126	—	4	2	no	—
49	C₁₃×#125	—	2	1	no	—
50	C₁₃×#125	—	4	2	no	—
51	C₁₃×C₂²⋊Q₈	—	2	2	no	—
52	C₄×Q₈×C₁₃	—	2	2	no	—
53	C₄×Q₈×C₁₃	—	2	2	no	—
54	C₄×Q₈×C₁₃	—	2	2	no	—
55	C₄×Q₈×C₁₃	—	4	2	no	—
56	C₁₃×#117	—	2	1	no	—
57	C₁₃×#117	—	4	2	no	—
58	C₁₃×#117	D₄	1	1	yes	characteristic
59	C₁₃×Q₃₂	—	2	4	no	—
60	C₁₃×Q₃₂	—	2	4	no	—
61	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₁₃×#107	C₂³	1	2	yes	—
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₁₃×#106	—	2	1	no	—
66	C₁₃×#106	—	2	1	no	—
67	C₁₃×#103	—	2	2	no	—
68	C₁₃×#103	—	2	2	no	—
69	C₁₃×#103	—	2	2	no	—
70	C₁₃×#103	—	2	2	no	—
71	C₁₃×#97	D₄	1	1	yes	characteristic
72	C₁₃×#97	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#33687	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×Q₈×C₁₃	—	2	4	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	2	2	no	—
79	C₂×Q₈×C₁₃	—	4	2	no	—
80	C₂×Q₈×C₁₃	—	4	1	no	—
81	C₂²×C₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₁₃×Q₁₆	—	2	4	no	—
84	C₁₃×Q₁₆	—	2	2	no	—
85	C₁₃×Q₁₆	—	2	2	no	—
86	C₁₃×Q₁₆	—	2	2	no	—
87	C₁₃×Q₁₆	—	4	2	no	—
88	C₁₃×Q₁₆	#41	1	4	yes	—
89	C₁₃×M₁₆	—	4	1	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	2	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₄⋊C₄	—	4	2	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
104	C₁₃×C₂²⋊C₄	—	4	1	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
107	C₄²×C₁₃	—	4	1	no	—
108	C₄²×C₁₃	—	4	2	no	—
109	C₄²×C₁₃	—	4	2	no	—
110	C₄²×C₁₃	—	4	2	no	—
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₁₃×C₁₆	—	2	2	no	—
114	C₁₃×C₁₆	—	2	2	no	—
115	#3177	C₂×C₁₃	1	1	yes	characteristic
116	#2950	C₂×C₁₃	1	1	yes	characteristic
117	#2237	C₂×C₁₃	1	2	yes	—
118	#2229	C₂×C₁₃	1	2	yes	complemented
119	#2209	C₂×C₁₃	1	2	yes	complemented
120	#2188	C₂×C₁₃	1	2	yes	—
121	#2176	C₂×C₁₃	1	2	yes	complemented
122	C₄×Q₃₂	C₂×C₁₃	1	2	yes	complemented
123	#2164	C₂×C₁₃	1	1	yes	characteristic
124	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
125	Q₈×C₁₃	—	2	4	no	—
126	Q₈×C₁₃	—	4	4	no	—
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	4	4	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₁₃	—	2	1	no	—
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	C₂×#41	1	2	yes	—
145	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
146	#541	—	2	1	no	—
147	#519	—	2	1	no	—
148	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
149	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
150	#499	C₂²×C₁₃	1	1	yes	characteristic
151	#494	C₂²×C₁₃	1	1	yes	characteristic
152	#491	—	2	1	no	—
153	#470	—	2	1	no	—
154	#467	C₂²×C₁₃	1	2	yes	—
155	#441	—	2	1	no	—
156	#441	—	2	1	no	—
157	#441	—	2	1	no	—
158	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
159	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
160	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
161	#427	C₂²×C₁₃	1	1	yes	characteristic
162	#419	—	2	1	no	—
163	#405	C₂²×C₁₃	1	1	yes	characteristic
164	#367	C₂²×C₁₃	1	2	yes	—
165	#366	C₂²×C₁₃	1	2	yes	—
166	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
167	#358	C₂²×C₁₃	1	2	yes	—
168	#358	C₂²×C₁₃	1	2	yes	—
169	#358	C₂²×C₁₃	1	2	yes	—
170	#358	C₂²×C₁₃	1	2	yes	—
171	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
172	#346	C₂²×C₁₃	1	1	yes	characteristic
173	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	2	no	—
176	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	2	2	no	—
180	C₄×C₁₃	—	4	4	no	—
181	C₄×C₁₃	—	8	2	no	—
182	C₄×C₁₃	—	8	2	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	#525	1	1	yes	characteristic
185	C₄×C₁₃	#575	1	1	yes	characteristic
186	C₂²×Q₈	—	2	1	no	—
187	C₂×Q₁₆	—	2	4	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
191	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
192	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	1	no	—
195	#126	—	4	2	no	—
196	#125	—	2	1	no	—
197	#125	—	4	2	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×Q₈	—	2	2	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×Q₈	—	4	2	no	—
203	#117	—	2	1	no	—
204	#117	—	4	2	no	—
205	#117	D₄×C₁₃	1	1	yes	characteristic
206	Q₃₂	—	2	4	no	—
207	Q₃₂	—	2	4	no	—
208	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
209	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
210	#107	C₂³×C₁₃	1	2	yes	—
211	#107	C₂³×C₁₃	1	2	yes	—
212	#106	—	2	1	no	—
213	#106	—	2	1	no	—
214	#103	—	2	2	no	—
215	#103	—	2	2	no	—
216	#103	—	2	2	no	—
217	#103	—	2	2	no	—
218	#97	D₄×C₁₃	1	1	yes	characteristic
219	#97	D₄×C₁₃	1	1	yes	characteristic
220	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
221	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
222	C₂×C₁₃	—	4	1	no	—
223	C₂×C₁₃	#3100	1	1	yes	characteristic
224	C₂×C₁₃	#3417	1	2	yes	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
232	Q₁₆	—	2	4	no	—
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	4	2	no	—
237	Q₁₆	C₁₃×#41	1	4	yes	—
238	M₁₆	—	4	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
253	C₂²⋊C₄	—	4	1	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
256	C₄²	—	4	1	no	—
257	C₄²	—	4	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₁₆	—	2	2	no	—
263	C₁₆	—	2	2	no	—
264	C₁₃	#33687	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
266	Q₈	—	2	4	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
279	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
280	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	C₂×C₁₃×#41	1	2	yes	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	4	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	C₁₃×#525	1	1	yes	characteristic
297	C₄	C₁₃×#575	1	1	yes	characteristic
298	C₂	—	4	1	no	—
299	C₂	C₁₃×#3100	1	1	yes	characteristic
300	C₂	C₁₃×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂³×C₁₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#2229	C₂	semidirect
5. C₁₃×#2209	C₂	semidirect
7. C₁₃×#2176	C₂	semidirect
8. C₄×C₁₃×Q₃₂	C₂	semidirect
75. #33687	C₁₃	direct (N × H)
118. #2229	C₂×C₁₃	semidirect
119. #2209	C₂×C₁₃	semidirect
121. #2176	C₂×C₁₃	semidirect
122. C₄×Q₃₂	C₂×C₁₃	semidirect
264. C₁₃	#33687	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×#3177	2
C₁₃×#2950	2
C₁₃×#2237	2
C₁₃×#2229	2
C₁₃×#2209	2
C₁₃×#2188	2
C₁₃×#2176	2
C₄×C₁₃×Q₃₂	2
C₁₃×#2164	2
#33687	13

Subgroups — C13×#33687 #15720026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33687 #15720026