Subgroups — C₁₃×#10865 #15697204

Order 3328.

766 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#108	C₂	1	2	yes	maximal
2	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#341	C₂	1	2	yes	maximal
4	C₁₃×#1728	C₂	1	2	yes	maximal
5	C₂²×C₁₃×M₁₆	C₂²	1	4	yes	—
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#108	—	2	4	no	—
10	C₂×C₁₃×#108	—	2	4	no	—
11	C₂×C₁₃×#108	C₄	1	8	yes	complemented
12	C₂×C₁₃×#97	—	2	4	no	—
13	C₂×C₁₃×#97	C₄	1	4	yes	—
14	C₁₃×#341	—	2	4	no	—
15	C₁₃×#336	—	2	4	no	—
16	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
17	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₁₃×M₁₆	—	2	4	no	—
19	C₂×C₁₃×M₁₆	—	2	4	no	—
20	C₂×C₁₃×M₁₆	—	2	4	no	—
21	C₂×C₁₃×M₁₆	—	2	4	no	—
22	C₂×C₁₃×M₁₆	—	2	4	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	4	no	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃	D₄	1	2	yes	—
31	C₂²×C₈×C₁₃	D₄	1	4	yes	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
35	C₂×C₄²×C₁₃	—	2	2	no	—
36	C₂×C₄²×C₁₃	—	2	2	no	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃	D₄	1	2	yes	—
39	C₁₃×#108	—	2	16	no	—
40	C₁₃×#97	—	2	8	no	—
41	#10865	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	—	4	4	no	—
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	#41	1	2	yes	—
55	C₂²×C₄×C₁₃	#41	1	2	yes	—
56	C₁₃×M₁₆	—	2	8	no	—
57	C₁₃×M₁₆	—	2	8	no	—
58	C₁₃×M₁₆	—	4	4	no	—
59	C₁₃×M₁₆	—	4	4	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	—	2	8	no	—
65	C₂×C₈×C₁₃	—	4	4	no	—
66	C₂×C₈×C₁₃	—	4	4	no	—
67	C₂×C₈×C₁₃	C₄²	1	4	yes	—
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
69	C₂×C₈×C₁₃	C₂²⋊C₄	1	8	yes	—
70	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
72	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
73	C₄²×C₁₃	—	2	2	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	2	4	no	—
76	C₂²×#108	C₂×C₁₃	1	2	yes	—
77	C₂²×#97	C₂×C₁₃	1	1	yes	characteristic
78	C₂×#341	C₂×C₁₃	1	2	yes	—
79	#1728	C₂×C₁₃	1	2	yes	—
80	C₂³×C₁₃	—	2	4	no	—
81	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
83	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
84	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	4	no	—
87	C₂×C₄×C₁₃	—	2	8	no	—
88	C₂×C₄×C₁₃	—	4	8	no	—
89	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₁₃	#121	1	2	yes	—
101	C₂×C₄×C₁₃	#121	1	2	yes	—
102	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₁₃	#124	1	2	yes	—
104	C₂×C₄×C₁₃	#125	1	2	yes	—
105	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
107	C₈×C₁₃	—	2	4	no	—
108	C₈×C₁₃	—	4	8	no	—
109	C₈×C₁₃	#95	1	8	yes	—
110	C₂²×M₁₆	C₂²×C₁₃	1	4	yes	—
111	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
112	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
113	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
114	C₂×#108	—	2	4	no	—
115	C₂×#108	—	2	4	no	—
116	C₂×#108	C₄×C₁₃	1	8	yes	complemented
117	C₂×#97	—	2	4	no	—
118	C₂×#97	C₄×C₁₃	1	4	yes	—
119	#341	—	2	4	no	—
120	#336	—	2	4	no	—
121	C₂²×C₁₃	—	2	4	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	—	2	2	no	—
124	C₂²×C₁₃	—	2	4	no	—
125	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₁₃	#380	1	1	yes	characteristic
127	C₂²×C₁₃	#384	1	1	yes	characteristic
128	C₂²×C₁₃	#389	1	2	yes	—
129	C₂²×C₁₃	#391	1	2	yes	—
130	C₂²×C₁₃	#430	1	4	yes	—
131	C₄×C₁₃	—	4	4	no	—
132	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
133	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₁₃	#380	1	1	yes	characteristic
135	C₄×C₁₃	#382	1	1	yes	characteristic
136	C₄×C₁₃	#389	1	2	yes	—
137	C₄×C₁₃	#390	1	2	yes	—
138	C₂³×C₄	Q₈×C₁₃	1	2	yes	—
139	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	C₂×C₄×C₁₃	1	8	yes	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
150	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
151	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
152	C₂²×C₈	D₄×C₁₃	1	2	yes	—
153	C₂²×C₈	D₄×C₁₃	1	4	yes	—
154	C₂²×C₈	D₄×C₁₃	1	2	yes	—
155	C₂²×C₈	Q₈×C₁₃	1	2	yes	—
156	C₂²×C₈	Q₈×C₁₃	1	2	yes	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
160	C₂×C₄²	D₄×C₁₃	1	2	yes	—
161	#108	—	2	16	no	—
162	#97	—	2	8	no	—
163	C₂×C₁₃	—	2	2	no	—
164	C₂×C₁₃	—	2	2	no	—
165	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
166	C₂×C₁₃	#1778	1	2	yes	—
167	C₂×C₁₃	#1811	1	2	yes	—
168	C₂×C₁₃	#1849	1	2	yes	—
169	C₂⁴	C₂×Q₈×C₁₃	1	1	yes	characteristic
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	4	4	no	—
172	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
175	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
176	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₁₃×#41	1	2	yes	—
182	C₂²×C₄	C₁₃×#41	1	2	yes	—
183	M₁₆	—	2	8	no	—
184	M₁₆	—	2	8	no	—
185	M₁₆	—	4	4	no	—
186	M₁₆	—	4	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	—	4	4	no	—
194	C₂×C₈	C₄²×C₁₃	1	4	yes	—
195	C₂×C₈	C₁₃×C₂²⋊C₄	1	4	yes	—
196	C₂×C₈	C₁₃×C₂²⋊C₄	1	8	yes	—
197	C₂×C₈	C₁₃×C₄⋊C₄	1	4	yes	—
198	C₂×C₈	C₁₃×C₄⋊C₄	1	4	yes	—
199	C₂×C₈	C₁₃×C₄⋊C₄	1	8	yes	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	4	no	—
203	C₁₃	#10865	1	1	yes	characteristic, direct product factor, Sylow
204	C₂³	—	2	4	no	—
205	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
206	C₂³	C₄×Q₈×C₁₃	1	2	yes	—
207	C₂³	C₁₃×C₂²⋊Q₈	1	2	yes	—
208	C₂³	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
209	C₂³	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	4	8	no	—
213	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
216	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
218	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
219	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
220	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
221	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
222	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
223	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
224	C₂×C₄	C₁₃×#121	1	2	yes	—
225	C₂×C₄	C₁₃×#121	1	2	yes	—
226	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
227	C₂×C₄	C₁₃×#124	1	2	yes	—
228	C₂×C₄	C₁₃×#125	1	2	yes	—
229	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
230	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
231	C₈	—	2	4	no	—
232	C₈	—	4	8	no	—
233	C₈	C₁₃×#95	1	8	yes	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
239	C₂²	C₁₃×#380	1	1	yes	characteristic
240	C₂²	C₁₃×#384	1	1	yes	characteristic
241	C₂²	C₁₃×#389	1	2	yes	—
242	C₂²	C₁₃×#391	1	2	yes	—
243	C₂²	C₁₃×#430	1	4	yes	—
244	C₄	—	4	4	no	—
245	C₄	C₂×C₁₃×#95	1	1	yes	characteristic
246	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
247	C₄	C₁₃×#380	1	1	yes	characteristic
248	C₄	C₁₃×#382	1	1	yes	characteristic
249	C₄	C₁₃×#389	1	2	yes	—
250	C₄	C₁₃×#390	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	2	2	no	—
253	C₂	C₁₃×C₄⋊#95	1	1	yes	characteristic
254	C₂	C₁₃×#1778	1	2	yes	—
255	C₂	C₁₃×#1811	1	2	yes	—
256	C₂	C₁₃×#1849	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂×C₁₃×#108	C₄	semidirect
41. #10865	C₁₃	direct (N × H)
116. C₂×#108	C₄×C₁₃	semidirect
203. C₁₃	#10865	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×#108	2
C₂²×C₁₃×#97	2
C₂×C₁₃×#341	2
C₁₃×#1728	2
#10865	13

Subgroups — C13×#10865 #15697204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10865 #15697204