Subgroups — C₁₃×#33647 #15719986

Order 3328.

584 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2237	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2234	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2159	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2139	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2112	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#525	—	2	1	no	—
11	C₂×C₁₃×#117	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₁₃×#467	C₂²	1	2	yes	—
14	C₁₃×#466	C₂²	1	2	yes	—
15	C₁₃×#465	—	2	2	no	—
16	C₁₃×#427	—	2	2	no	—
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₁₃×#427	C₂²	1	4	yes	—
19	C₂×C₁₃×#107	C₂²	1	2	yes	—
20	C₂×C₁₃×#106	—	2	1	no	—
21	C₁₃×#416	—	2	1	no	—
22	C₁₃×#367	C₂²	1	4	yes	complemented
23	C₁₃×#366	C₂²	1	4	yes	complemented
24	C₁₃×#358	C₂²	1	4	yes	complemented
25	C₁₃×#357	C₂²	1	4	yes	complemented
26	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
29	C₁₃×#135	—	2	4	no	—
30	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
31	C₂×C₁₃×SD₁₆	—	2	1	no	—
32	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	—	2	4	no	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	D₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#123	—	4	1	no	—
39	C₁₃×C₂²⋊Q₈	—	2	2	no	—
40	C₁₃×#121	—	2	2	no	—
41	C₁₃×#120	—	4	1	no	—
42	C₄×D₄×C₁₃	—	4	2	no	—
43	C₁₃×#117	—	2	1	no	—
44	C₁₃×#117	—	2	4	no	—
45	C₁₃×#117	—	4	2	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
48	C₂×C₄²×C₁₃	—	4	1	no	—
49	C₁₃×M₃₂	—	2	4	no	—
50	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
51	C₁₃×#107	—	2	2	no	—
52	C₁₃×#107	C₂³	1	4	yes	—
53	C₁₃×#107	C₂³	1	2	yes	—
54	C₁₃×#106	—	2	4	no	—
55	C₁₃×#106	—	2	1	no	—
56	C₁₃×#103	—	2	2	no	—
57	C₁₃×#102	—	2	2	no	—
58	#33647	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
60	C₁₃×#41	—	4	2	no	—
61	C₁₃×#41	—	4	2	no	—
62	C₂×Q₈×C₁₃	—	2	1	no	—
63	C₂×D₄×C₁₃	—	2	1	no	—
64	C₂×D₄×C₁₃	—	4	2	no	—
65	C₂×D₄×C₁₃	—	4	1	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	4	2	no	—
69	C₂²×C₄×C₁₃	—	4	1	no	—
70	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	D₈	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₁₃×Q₁₆	—	2	2	no	—
75	C₁₃×SD₁₆	—	4	2	no	—
76	C₁₃×D₈	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	4	yes	—
83	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	4	2	no	—
87	C₁₃×C₄⋊C₄	—	4	2	no	—
88	C₁₃×C₂²⋊C₄	—	4	4	no	—
89	C₁₃×C₂²⋊C₄	—	4	1	no	—
90	C₁₃×C₂²⋊C₄	—	4	2	no	—
91	C₁₃×C₂²⋊C₄	—	4	2	no	—
92	C₄²×C₁₃	—	4	4	no	—
93	C₄²×C₁₃	—	4	1	no	—
94	C₁₃×C₁₆	—	2	4	no	—
95	#3050	C₂×C₁₃	1	1	yes	characteristic
96	C₂×#427	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#2237	C₂×C₁₃	1	4	yes	complemented
98	#2234	C₂×C₁₃	1	4	yes	complemented
99	#2159	C₂×C₁₃	1	2	yes	complemented
100	#2139	C₂×C₁₃	1	2	yes	complemented
101	#2112	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂³×C₁₃	—	2	2	no	—
103	C₂³×C₁₃	—	4	2	no	—
104	C₂³×C₁₃	—	4	1	no	—
105	C₂³×C₁₃	C₂×D₈	1	2	yes	—
106	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
107	Q₈×C₁₃	—	4	1	no	—
108	Q₈×C₁₃	—	4	1	no	—
109	D₄×C₁₃	—	4	1	no	—
110	D₄×C₁₃	—	4	1	no	—
111	D₄×C₁₃	—	8	4	no	—
112	D₄×C₁₃	—	8	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	8	4	no	—
121	C₂×C₄×C₁₃	—	8	2	no	—
122	C₂×C₄×C₁₃	#123	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#137	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₁₃	—	2	2	no	—
128	C₈×C₁₃	#123	1	2	yes	—
129	C₈×C₁₃	C₂×#41	1	2	yes	—
130	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
131	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
132	#525	—	2	1	no	—
133	C₂×#117	—	2	1	no	—
134	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
135	#467	C₂²×C₁₃	1	2	yes	—
136	#466	C₂²×C₁₃	1	2	yes	—
137	#465	—	2	2	no	—
138	#427	—	2	2	no	—
139	#427	C₂²×C₁₃	1	4	yes	—
140	#427	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×#107	C₂²×C₁₃	1	2	yes	—
142	C₂×#106	—	2	1	no	—
143	#416	—	2	1	no	—
144	#367	C₂²×C₁₃	1	4	yes	complemented
145	#366	C₂²×C₁₃	1	4	yes	complemented
146	#358	C₂²×C₁₃	1	4	yes	complemented
147	#357	C₂²×C₁₃	1	4	yes	complemented
148	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	4	4	no	—
154	C₂²×C₁₃	—	8	1	no	—
155	C₂²×C₁₃	—	8	2	no	—
156	C₂²×C₁₃	#480	1	2	yes	—
157	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	—	8	1	no	—
160	C₄×C₁₃	—	8	4	no	—
161	C₄×C₁₃	—	8	2	no	—
162	C₄×C₁₃	#480	1	2	yes	—
163	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
164	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
165	C₂×#41	—	2	1	no	—
166	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
167	#135	—	2	4	no	—
168	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
169	C₂×SD₁₆	—	2	1	no	—
170	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
173	C₂²×C₈	D₄×C₁₃	1	2	yes	—
174	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
175	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
176	#123	—	4	1	no	—
177	C₂²⋊Q₈	—	2	2	no	—
178	#121	—	2	2	no	—
179	#120	—	4	1	no	—
180	C₄×D₄	—	4	2	no	—
181	#117	—	2	4	no	—
182	#117	—	2	1	no	—
183	#117	—	4	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	4	1	no	—
186	C₂×C₄²	—	4	1	no	—
187	M₃₂	—	2	4	no	—
188	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
189	#107	—	2	2	no	—
190	#107	C₂³×C₁₃	1	4	yes	—
191	#107	C₂³×C₁₃	1	2	yes	—
192	#106	—	2	4	no	—
193	#106	—	2	1	no	—
194	#103	—	2	2	no	—
195	#102	—	2	2	no	—
196	C₂×C₁₃	—	2	2	no	—
197	C₂×C₁₃	—	4	2	no	—
198	C₂×C₁₃	—	16	1	no	—
199	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
200	C₂×C₁₃	#3414	1	2	yes	—
201	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
202	#41	—	4	2	no	—
203	#41	—	4	2	no	—
204	C₂×Q₈	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
213	C₂²×C₄	C₁₃×D₈	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	SD₁₆	—	4	2	no	—
218	D₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
223	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
224	C₂×C₈	C₁₃×#41	1	4	yes	—
225	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	4	1	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₂²⋊C₄	—	4	2	no	—
234	C₄²	—	4	4	no	—
235	C₄²	—	4	1	no	—
236	C₁₆	—	2	4	no	—
237	C₁₃	#33647	1	1	yes	characteristic, direct product factor, Sylow
238	C₂³	—	2	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	4	1	no	—
241	C₂³	C₂×C₁₃×D₈	1	2	yes	—
242	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
243	Q₈	—	4	1	no	—
244	Q₈	—	4	1	no	—
245	D₄	—	4	1	no	—
246	D₄	—	4	1	no	—
247	D₄	—	8	4	no	—
248	D₄	—	8	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	C₁₃×#123	1	2	yes	—
259	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
260	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
261	C₂×C₄	C₁₃×#137	1	2	yes	—
262	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	C₁₃×#123	1	2	yes	—
265	C₈	C₂×C₁₃×#41	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	8	1	no	—
272	C₂²	—	8	2	no	—
273	C₂²	C₁₃×#480	1	2	yes	—
274	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	8	1	no	—
277	C₄	—	8	4	no	—
278	C₄	—	8	2	no	—
279	C₄	C₁₃×#480	1	2	yes	—
280	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
281	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	4	2	no	—
284	C₂	—	16	1	no	—
285	C₂	C₂×C₁₃×#480	1	1	yes	characteristic
286	C₂	C₁₃×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#427	C₂	semidirect
3. C₁₃×#2237	C₂	semidirect
4. C₁₃×#2234	C₂	semidirect
5. C₁₃×#2159	C₂	semidirect
6. C₁₃×#2139	C₂	semidirect
7. C₁₃×#2112	C₂	semidirect
22. C₁₃×#367	C₂²	semidirect
23. C₁₃×#366	C₂²	semidirect
24. C₁₃×#358	C₂²	semidirect
25. C₁₃×#357	C₂²	semidirect
58. #33647	C₁₃	direct (N × H)
96. C₂×#427	C₂×C₁₃	semidirect
97. #2237	C₂×C₁₃	semidirect
98. #2234	C₂×C₁₃	semidirect
99. #2159	C₂×C₁₃	semidirect
100. #2139	C₂×C₁₃	semidirect
101. #2112	C₂×C₁₃	semidirect
144. #367	C₂²×C₁₃	semidirect
145. #366	C₂²×C₁₃	semidirect
146. #358	C₂²×C₁₃	semidirect
147. #357	C₂²×C₁₃	semidirect
237. C₁₃	#33647	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×#3050	2
C₂×C₁₃×#427	2
C₁₃×#2237	2
C₁₃×#2234	2
C₁₃×#2159	2
C₁₃×#2139	2
C₁₃×#2112	2
#33647	13

Subgroups — C13×#33647 #15719986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33647 #15719986