Subgroups — C₁₃×#17897 #15704236

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2983	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1593	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1591	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1479	C₂	1	2	yes	maximal
7	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
10	C₁₃×#446	—	2	4	no	—
11	C₁₃×#446	C₂²	1	8	yes	—
12	C₄×C₁₃×Q₁₆	C₄	1	2	yes	—
13	C₄×C₁₃×Q₁₆	C₄	1	2	yes	—
14	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
15	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₁₃×#432	—	2	4	no	—
17	C₁₃×#431	C₂²	1	2	yes	—
18	C₁₃×#431	C₂²	1	1	yes	characteristic
19	C₁₃×#431	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
21	C₂×C₁₃×#103	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#335	C₂²	1	2	yes	—
24	C₁₃×#334	C₂²	1	2	yes	—
25	C₁₃×Q₈⋊C₈	C₂²	1	8	yes	—
26	C₁₃×#322	C₂²	1	2	yes	—
27	C₈²×C₁₃	C₂²	1	2	yes	—
28	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×Q₁₆	—	2	4	no	—
30	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₁₃	—	2	2	no	—
35	C₄×Q₈×C₁₃	—	2	2	no	—
36	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	—	2	2	no	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#107	C₂×C₄	1	2	yes	—
43	C₁₃×#107	C₂×C₄	1	2	yes	—
44	C₁₃×C₄⋊C₈	—	2	8	no	—
45	C₁₃×C₄⋊C₈	—	2	1	no	—
46	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
47	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
48	C₁₃×#103	C₂×C₄	1	4	yes	—
49	C₁₃×#103	C₂×C₄	1	4	yes	—
50	C₁₃×C₂²⋊C₈	—	2	2	no	—
51	C₁₃×C₂²⋊C₈	—	2	1	no	—
52	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
53	C₁₃×#97	—	2	8	no	—
54	C₄×C₈×C₁₃	—	2	2	no	—
55	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	2	yes	—
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	2	yes	—
61	#17897	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	C₂×Q₈×C₁₃	—	2	4	no	—
64	C₂×Q₈×C₁₃	—	2	4	no	—
65	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₁₃×Q₁₆	—	2	4	no	—
72	C₁₃×Q₁₆	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
86	C₄²×C₁₃	—	2	4	no	—
87	C₄²×C₁₃	—	2	4	no	—
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	#41	1	1	yes	characteristic
91	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
92	#2983	C₂×C₁₃	1	2	yes	—
93	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
94	#1593	C₂×C₁₃	1	4	yes	complemented
95	#1591	C₂×C₁₃	1	4	yes	complemented
96	#1564	C₂×C₁₃	1	1	yes	characteristic
97	#1479	C₂×C₁₃	1	2	yes	—
98	#1450	C₂×C₁₃	1	1	yes	characteristic
99	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	4	no	—
101	Q₈×C₁₃	—	2	4	no	—
102	Q₈×C₁₃	—	4	4	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	4	1	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	8	2	no	—
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
113	C₂×C₄×C₁₃	#131	1	2	yes	—
114	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₈×C₁₃	—	2	1	no	—
118	C₈×C₁₃	—	2	1	no	—
119	C₈×C₁₃	—	2	4	no	—
120	C₈×C₁₃	—	4	2	no	—
121	C₈×C₁₃	—	4	4	no	—
122	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
123	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
124	C₂²×Q₁₆	C₄×C₁₃	1	1	yes	characteristic
125	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
126	#446	—	2	4	no	—
127	#446	C₂²×C₁₃	1	8	yes	—
128	C₄×Q₁₆	C₄×C₁₃	1	2	yes	—
129	C₄×Q₁₆	C₄×C₁₃	1	2	yes	—
130	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
131	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
132	#432	—	2	4	no	—
133	#431	C₂²×C₁₃	1	2	yes	—
134	#431	C₂²×C₁₃	1	1	yes	characteristic
135	#431	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×#107	C₄×C₁₃	1	1	yes	characteristic
137	C₂×#103	C₄×C₁₃	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
139	#335	C₂²×C₁₃	1	2	yes	—
140	#334	C₂²×C₁₃	1	2	yes	—
141	Q₈⋊C₈	C₂²×C₁₃	1	8	yes	—
142	#322	C₂²×C₁₃	1	2	yes	—
143	C₈²	C₂²×C₁₃	1	2	yes	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	#436	1	2	yes	—
147	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	4	4	no	—
152	C₄×C₁₃	—	4	4	no	—
153	C₄×C₁₃	#436	1	2	yes	—
154	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
155	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
156	C₄×C₁₃	#578	1	2	yes	—
157	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
158	C₂×Q₁₆	—	2	4	no	—
159	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
160	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
161	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
163	C₄×Q₈	—	2	2	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₄×Q₈	C₂×C₄×C₁₃	1	4	yes	—
166	C₄×Q₈	C₂³×C₁₃	1	4	yes	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
171	#107	C₂×C₄×C₁₃	1	2	yes	—
172	#107	C₂×C₄×C₁₃	1	2	yes	—
173	C₄⋊C₈	—	2	8	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
176	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
177	#103	C₂×C₄×C₁₃	1	4	yes	—
178	#103	C₂×C₄×C₁₃	1	4	yes	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₂²⋊C₈	—	2	1	no	—
181	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
182	#97	—	2	8	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
185	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
186	C₄×C₈	C₂³×C₁₃	1	2	yes	—
187	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₁₃	1	2	yes	—
190	C₂×C₁₃	—	2	2	no	—
191	C₂×C₁₃	C₂×#436	1	1	yes	characteristic
192	C₂×C₁₃	#2950	1	1	yes	characteristic
193	C₂×C₁₃	#2957	1	1	yes	characteristic
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
203	Q₁₆	—	2	4	no	—
204	Q₁₆	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₁₃×M₁₆	1	4	yes	—
209	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	4	no	—
220	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
221	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₄²	C₁₃×#41	1	1	yes	characteristic
223	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
224	C₁₃	#17897	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	2	4	no	—
228	Q₈	—	4	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
239	C₂×C₄	C₁₃×#131	1	2	yes	—
240	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
242	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	4	no	—
246	C₈	—	4	2	no	—
247	C₈	—	4	4	no	—
248	C₈	C₂×C₁₃×M₁₆	1	2	yes	—
249	C₈	C₂×C₁₃×M₁₆	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₁₃×#436	1	2	yes	—
253	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	4	no	—
259	C₄	C₁₃×#436	1	2	yes	—
260	C₄	C₂²×C₁₃×M₁₆	1	1	yes	characteristic
261	C₄	C₂×C₁₃×#131	1	1	yes	characteristic
262	C₄	C₁₃×#578	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×C₁₃×#436	1	1	yes	characteristic
265	C₂	C₁₃×#2950	1	1	yes	characteristic
266	C₂	C₁₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₁₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1593	C₂	semidirect
4. C₁₃×#1591	C₂	semidirect
61. #17897	C₁₃	direct (N × H)
94. #1593	C₂×C₁₃	semidirect
95. #1591	C₂×C₁₃	semidirect
224. C₁₃	#17897	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×#2983	2
C₂×C₄×C₁₃×Q₁₆	2
C₁₃×#1593	2
C₁₃×#1591	2
C₁₃×#1564	2
C₁₃×#1479	2
C₁₃×#1450	2
#17897	13

Subgroups — C13×#17897 #15704236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17897 #15704236