Subgroups — C₁₃×#12817 #15699156

Order 3328.

430 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#360	C₂	1	2	yes	maximal
4	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1386	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
7	C₄×C₁₃×#41	—	2	1	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
9	C₄×C₁₃×Q₁₆	C₄	1	2	yes	complemented
10	C₄×C₁₃×SD₁₆	—	2	2	no	—
11	C₄×C₁₃×D₈	C₄	1	2	yes	complemented
12	C₁₃×#427	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	—	2	1	no	—
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₁₃×#416	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#382	—	2	1	no	—
18	C₁₃×C₄⋊C₁₆	—	2	2	no	—
19	C₁₃×#360	—	2	4	no	—
20	C₂×C₁₃×#41	—	2	1	no	—
21	C₁₃×#135	—	2	1	no	—
22	C₁₃×#135	—	2	1	no	—
23	C₁₃×#135	D₄	1	2	yes	—
24	C₁₃×#135	Q₈	1	2	yes	—
25	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₃×SD₁₆	—	2	1	no	—
27	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₄×Q₈×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₄×D₄×C₁₃	—	4	2	no	—
33	C₁₃×#117	—	2	1	no	—
34	C₁₃×#117	—	4	1	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₁₃	—	4	1	no	—
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₁₃×M₃₂	—	2	2	no	—
39	C₁₃×M₃₂	—	2	2	no	—
40	C₂×C₁₃×C₁₆	—	2	2	no	—
41	C₁₃×#107	—	4	2	no	—
42	C₁₃×#107	C₂×C₄	1	2	yes	—
43	C₁₃×#106	—	2	1	no	—
44	C₁₃×#106	—	2	2	no	—
45	C₁₃×#106	—	2	2	no	—
46	C₁₃×#103	—	2	2	no	—
47	C₁₃×#102	—	2	2	no	—
48	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	D₄	1	2	yes	—
50	C₁₃×#95	—	4	2	no	—
51	#12817	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₁₃×#41	—	2	2	no	—
53	C₁₃×#41	—	2	2	no	—
54	C₁₃×#41	—	4	1	no	—
55	C₁₃×#41	—	4	1	no	—
56	C₂×Q₈×C₁₃	—	2	1	no	—
57	C₂×D₄×C₁₃	—	2	1	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂²×C₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	D₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×Q₁₆	—	2	1	no	—
65	C₁₃×Q₁₆	C₄⋊C₄	1	2	yes	—
66	C₁₃×SD₁₆	—	2	2	no	—
67	C₁₃×SD₁₆	—	2	2	no	—
68	C₁₃×D₈	—	2	1	no	—
69	C₁₃×D₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	4	2	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₂²⋊C₄	—	4	1	no	—
84	C₁₃×C₂²⋊C₄	—	4	2	no	—
85	C₄²×C₁₃	—	4	2	no	—
86	C₄²×C₁₃	—	4	1	no	—
87	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
89	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
90	C₁₃×C₁₆	—	4	2	no	—
91	C₄×#135	C₂×C₁₃	1	1	yes	characteristic
92	#2150	C₂×C₁₃	1	1	yes	characteristic, complemented
93	C₂×#360	C₂×C₁₃	1	2	yes	—
94	#1846	C₂×C₁₃	1	1	yes	characteristic, complemented
95	#1386	C₂×C₁₃	1	2	yes	complemented
96	C₂³×C₁₃	—	4	1	no	—
97	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
98	Q₈×C₁₃	—	2	2	no	—
99	Q₈×C₁₃	—	4	1	no	—
100	D₄×C₁₃	—	2	2	no	—
101	D₄×C₁₃	—	4	2	no	—
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	4	1	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	1	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	8	4	no	—
110	C₂×C₄×C₁₃	—	8	4	no	—
111	C₂×C₄×C₁₃	—	8	2	no	—
112	C₂×C₄×C₁₃	#102	1	2	yes	—
113	C₂×C₄×C₁₃	#102	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₈×C₁₃	#117	1	1	yes	characteristic
123	C₈×C₁₃	#121	1	1	yes	characteristic
124	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
126	C₄×#41	—	2	1	no	—
127	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
128	C₄×Q₁₆	C₄×C₁₃	1	2	yes	complemented
129	C₄×SD₁₆	—	2	2	no	—
130	C₄×D₈	C₄×C₁₃	1	2	yes	complemented
131	#427	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×#107	—	2	1	no	—
133	C₂×#106	C₂²×C₁₃	1	2	yes	—
134	#416	—	2	1	no	—
135	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
136	#382	—	2	1	no	—
137	C₄⋊C₁₆	—	2	2	no	—
138	#360	—	2	4	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	4	2	no	—
141	C₂²×C₁₃	—	8	2	no	—
142	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
143	C₂²×C₁₃	#459	1	1	yes	characteristic
144	C₂²×C₁₃	#474	1	1	yes	characteristic
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	4	2	no	—
148	C₄×C₁₃	—	8	2	no	—
149	C₄×C₁₃	—	16	2	no	—
150	C₄×C₁₃	#380	1	1	yes	characteristic
151	C₄×C₁₃	#417	1	1	yes	characteristic
152	C₄×C₁₃	#461	1	1	yes	characteristic
153	C₄×C₁₃	#476	1	1	yes	characteristic
154	C₂×#41	—	2	1	no	—
155	#135	—	2	1	no	—
156	#135	—	2	1	no	—
157	#135	D₄×C₁₃	1	2	yes	—
158	#135	Q₈×C₁₃	1	2	yes	—
159	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
160	C₂×SD₁₆	—	2	1	no	—
161	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
164	C₄×Q₈	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	—	4	2	no	—
167	#117	—	2	1	no	—
168	#117	—	4	1	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄²	—	4	1	no	—
171	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
172	M₃₂	—	2	2	no	—
173	M₃₂	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	#107	—	4	2	no	—
176	#107	C₂×C₄×C₁₃	1	2	yes	—
177	#106	—	2	1	no	—
178	#106	—	2	2	no	—
179	#106	—	2	2	no	—
180	#103	—	2	2	no	—
181	#102	—	2	2	no	—
182	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
183	C₄×C₈	D₄×C₁₃	1	2	yes	—
184	#95	—	4	2	no	—
185	C₂×C₁₃	—	2	1	no	—
186	C₂×C₁₃	—	2	1	no	—
187	C₂×C₁₃	—	8	2	no	—
188	C₂×C₁₃	#1802	1	1	yes	characteristic
189	C₂×C₁₃	#2145	1	2	yes	—
190	#41	—	2	2	no	—
191	#41	—	2	2	no	—
192	#41	—	4	1	no	—
193	#41	—	4	1	no	—
194	C₂×Q₈	—	2	1	no	—
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₁₃×D₈	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
202	Q₁₆	—	2	1	no	—
203	Q₁₆	C₁₃×C₄⋊C₄	1	2	yes	—
204	SD₁₆	—	2	2	no	—
205	SD₁₆	—	2	2	no	—
206	D₈	—	2	1	no	—
207	D₈	C₁₃×C₄⋊C₄	1	2	yes	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
211	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
214	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
215	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
216	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	1	no	—
222	C₂²⋊C₄	—	4	2	no	—
223	C₄²	—	4	2	no	—
224	C₄²	—	4	1	no	—
225	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
226	C₄²	C₁₃×SD₁₆	1	1	yes	characteristic
227	C₄²	C₁₃×SD₁₆	1	1	yes	characteristic
228	C₁₆	—	4	2	no	—
229	C₁₃	#12817	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³	—	4	1	no	—
231	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	Q₈	—	4	1	no	—
234	D₄	—	2	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	C₁₃×#102	1	2	yes	—
247	C₂×C₄	C₁₃×#102	1	2	yes	—
248	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
250	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
252	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
256	C₈	C₁₃×#117	1	1	yes	characteristic
257	C₈	C₁₃×#121	1	1	yes	characteristic
258	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
263	C₂²	C₁₃×#459	1	1	yes	characteristic
264	C₂²	C₁₃×#474	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	16	2	no	—
270	C₄	C₁₃×#380	1	1	yes	characteristic
271	C₄	C₁₃×#417	1	1	yes	characteristic
272	C₄	C₁₃×#461	1	1	yes	characteristic
273	C₄	C₁₃×#476	1	1	yes	characteristic
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	8	2	no	—
277	C₂	C₁₃×#1802	1	1	yes	characteristic
278	C₂	C₁₃×#2145	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2150	C₂	semidirect
4. C₁₃×#1846	C₂	semidirect
5. C₁₃×#1386	C₂	semidirect
9. C₄×C₁₃×Q₁₆	C₄	semidirect
11. C₄×C₁₃×D₈	C₄	semidirect
51. #12817	C₁₃	direct (N × H)
92. #2150	C₂×C₁₃	semidirect
94. #1846	C₂×C₁₃	semidirect
95. #1386	C₂×C₁₃	semidirect
128. C₄×Q₁₆	C₄×C₁₃	semidirect
130. C₄×D₈	C₄×C₁₃	semidirect
229. C₁₃	#12817	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#135	2
C₁₃×#2150	2
C₂×C₁₃×#360	2
C₁₃×#1846	2
C₁₃×#1386	2
#12817	13

Subgroups — C13×#12817 #15699156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12817 #15699156