Subgroups — C₁₃×#32815 #15719154

Order 3328.

746 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2642	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2641	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2055	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2026	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2025	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2014	C₂	1	2	yes	maximal
7	C₁₃×#2013	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1832	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×SD₁₆	C₂²	1	4	yes	—
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×#103	C₂²	1	4	yes	—
16	C₂×C₁₃×#102	C₂²	1	4	yes	—
17	C₁₃×#400	—	2	4	no	—
18	C₁₃×#394	—	2	4	no	—
19	C₁₃×#391	C₂²	1	2	yes	—
20	C₁₃×#391	C₂²	1	2	yes	—
21	C₁₃×#390	C₂²	1	2	yes	—
22	C₁₃×#390	C₂²	1	2	yes	—
23	C₁₃×#376	C₂²	1	1	yes	characteristic
24	C₁₃×#340	C₂²	1	4	yes	—
25	C₁₃×#336	C₂²	1	2	yes	—
26	C₁₃×#336	C₂²	1	2	yes	—
27	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
28	C₂²×D₄×C₁₃	—	2	2	no	—
29	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
30	C₂×C₁₃×SD₁₆	—	4	4	no	—
31	C₂×C₁₃×SD₁₆	—	4	4	no	—
32	C₂×C₁₃×SD₁₆	—	4	4	no	—
33	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
34	C₁₃×C₄⋊Q₈	—	4	2	no	—
35	C₁₃×#127	—	4	2	no	—
36	C₁₃×#125	—	4	2	no	—
37	C₁₃×#125	—	4	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
44	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊C₈	—	4	2	no	—
49	C₁₃×#103	—	4	4	no	—
50	C₁₃×#102	—	4	4	no	—
51	C₁₃×#95	—	2	4	no	—
52	C₁₃×#95	—	2	4	no	—
53	C₁₃×#95	D₄	1	4	yes	—
54	#32815	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₃	—	2	2	no	—
56	C₂×Q₈×C₁₃	—	2	4	no	—
57	C₂×Q₈×C₁₃	—	2	4	no	—
58	C₂×Q₈×C₁₃	—	2	2	no	—
59	C₂×Q₈×C₁₃	—	2	2	no	—
60	C₂×D₄×C₁₃	—	2	4	no	—
61	C₂×D₄×C₁₃	—	2	4	no	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	2	2	no	—
64	C₂×D₄×C₁₃	—	4	4	no	—
65	C₂×D₄×C₁₃	—	8	4	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×SD₁₆	—	4	16	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₄⋊C₄	—	2	4	no	—
79	C₁₃×C₄⋊C₄	—	4	4	no	—
80	C₁₃×C₄⋊C₄	—	4	4	no	—
81	C₁₃×C₄⋊C₄	—	4	4	no	—
82	C₁₃×C₂²⋊C₄	—	8	4	no	—
83	C₁₃×C₂²⋊C₄	—	8	4	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	2	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	1	no	—
88	#2642	C₂×C₁₃	1	1	yes	characteristic, complemented
89	#2641	C₂×C₁₃	1	1	yes	characteristic
90	#2055	C₂×C₁₃	1	4	yes	complemented
91	#2026	C₂×C₁₃	1	2	yes	complemented
92	#2025	C₂×C₁₃	1	2	yes	complemented
93	#2014	C₂×C₁₃	1	2	yes	—
94	#2013	C₂×C₁₃	1	2	yes	complemented
95	#1832	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₂³×C₁₃	—	4	2	no	—
97	C₂³×C₁₃	—	4	2	no	—
98	C₂³×C₁₃	—	4	2	no	—
99	C₂³×C₁₃	—	4	4	no	—
100	C₂³×C₁₃	—	4	4	no	—
101	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
102	Q₈×C₁₃	—	4	4	no	—
103	Q₈×C₁₃	—	4	4	no	—
104	D₄×C₁₃	—	4	4	no	—
105	D₄×C₁₃	—	4	4	no	—
106	D₄×C₁₃	—	8	8	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
122	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
123	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
124	C₈×C₁₃	—	4	4	no	—
125	C₂²×SD₁₆	C₂²×C₁₃	1	4	yes	—
126	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
128	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
131	C₂×#103	C₂²×C₁₃	1	4	yes	—
132	C₂×#102	C₂²×C₁₃	1	4	yes	—
133	#400	—	2	4	no	—
134	#394	—	2	4	no	—
135	#391	C₂²×C₁₃	1	2	yes	—
136	#391	C₂²×C₁₃	1	2	yes	—
137	#390	C₂²×C₁₃	1	2	yes	—
138	#390	C₂²×C₁₃	1	2	yes	—
139	#376	C₂²×C₁₃	1	1	yes	characteristic
140	#340	C₂²×C₁₃	1	4	yes	—
141	#336	C₂²×C₁₃	1	2	yes	—
142	#336	C₂²×C₁₃	1	2	yes	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	8	4	no	—
145	C₂²×C₁₃	—	8	2	no	—
146	C₂²×C₁₃	—	8	2	no	—
147	C₂²×C₁₃	—	8	4	no	—
148	C₂²×C₁₃	#535	1	1	yes	characteristic
149	C₂²×C₁₃	#535	1	1	yes	characteristic
150	C₂²×C₁₃	#537	1	1	yes	characteristic
151	C₂²×C₁₃	#577	1	4	yes	—
152	C₄×C₁₃	—	4	2	no	—
153	C₄×C₁₃	—	4	2	no	—
154	C₄×C₁₃	—	4	2	no	—
155	C₄×C₁₃	—	8	2	no	—
156	C₄×C₁₃	—	8	2	no	—
157	C₄×C₁₃	—	8	2	no	—
158	C₄×C₁₃	H(3,𝔽₄)	1	2	yes	—
159	C₄×C₁₃	#562	1	2	yes	—
160	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
161	C₂²×D₄	—	2	2	no	—
162	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
163	C₂×SD₁₆	—	4	4	no	—
164	C₂×SD₁₆	—	4	4	no	—
165	C₂×SD₁₆	—	4	4	no	—
166	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
167	C₄⋊Q₈	—	4	2	no	—
168	#127	—	4	2	no	—
169	#125	—	4	2	no	—
170	#125	—	4	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
175	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
176	C₂×C₂²⋊C₄	—	2	4	no	—
177	C₂×C₂²⋊C₄	—	2	4	no	—
178	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
181	C₄⋊C₈	—	4	2	no	—
182	#103	—	4	4	no	—
183	#102	—	4	4	no	—
184	#95	—	2	4	no	—
185	#95	—	2	4	no	—
186	#95	D₄×C₁₃	1	4	yes	—
187	C₂×C₁₃	—	16	2	no	—
188	C₂×C₁₃	#2819	1	1	yes	characteristic
189	C₂×C₁₃	#3221	1	2	yes	—
190	C₂×C₁₃	#3222	1	2	yes	—
191	C₂×C₁₃	#3260	1	2	yes	—
192	C₂⁴	—	2	2	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	—	8	4	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
209	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
210	SD₁₆	—	4	16	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₂²⋊C₄	—	8	4	no	—
220	C₂²⋊C₄	—	8	4	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	1	no	—
225	C₁₃	#32815	1	1	yes	characteristic, direct product factor, Sylow
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	4	no	—
231	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
232	Q₈	—	4	4	no	—
233	Q₈	—	4	4	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	8	8	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
251	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
252	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
253	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
254	C₈	—	4	4	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	4	no	—
260	C₂²	C₁₃×#535	1	1	yes	characteristic
261	C₂²	C₁₃×#535	1	1	yes	characteristic
262	C₂²	C₁₃×#537	1	1	yes	characteristic
263	C₂²	C₁₃×#577	1	4	yes	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	8	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	2	no	—
270	C₄	C₁₃×H(3,𝔽₄)	1	2	yes	—
271	C₄	C₁₃×#562	1	2	yes	—
272	C₂	—	16	2	no	—
273	C₂	C₁₃×#2819	1	1	yes	characteristic
274	C₂	C₁₃×#3221	1	2	yes	—
275	C₂	C₁₃×#3222	1	2	yes	—
276	C₂	C₁₃×#3260	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2642	C₂	semidirect
3. C₁₃×#2055	C₂	semidirect
4. C₁₃×#2026	C₂	semidirect
5. C₁₃×#2025	C₂	semidirect
7. C₁₃×#2013	C₂	semidirect
8. C₁₃×#1832	C₂	semidirect
54. #32815	C₁₃	direct (N × H)
88. #2642	C₂×C₁₃	semidirect
90. #2055	C₂×C₁₃	semidirect
91. #2026	C₂×C₁₃	semidirect
92. #2025	C₂×C₁₃	semidirect
94. #2013	C₂×C₁₃	semidirect
95. #1832	C₂×C₁₃	semidirect
225. C₁₃	#32815	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×#2642	2
C₁₃×#2641	2
C₁₃×#2055	2
C₁₃×#2026	2
C₁₃×#2025	2
C₁₃×#2014	2
C₁₃×#2013	2
C₁₃×#1832	2
#32815	13

Subgroups — C13×#32815 #15719154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32815 #15719154