Subgroups — C₂×C₁₃×#2090 #15710012

Order 3328.

810 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#428	C₂	1	2	yes	maximal
2	C₂²×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#424	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2090	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#343	C₂	1	2	yes	maximal
6	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₁₃×#428	—	2	4	no	—
13	C₁₃×#428	C₂²	1	8	yes	—
14	C₂×C₁₃×#107	—	2	2	no	—
15	C₂×C₁₃×#106	—	2	2	no	—
16	C₂×C₁₃×#106	—	2	2	no	—
17	C₂×C₁₃×#106	—	2	2	no	—
18	C₂×C₁₃×#106	C₂²	1	4	yes	—
19	C₁₃×#424	—	2	2	no	—
20	C₁₃×#424	C₂²	1	4	yes	—
21	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
22	C₁₃×#343	C₂²	1	8	yes	—
23	C₁₃×#341	C₂²	1	4	yes	—
24	C₂³×C₄×C₁₃	—	2	2	no	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	4	no	—
28	C₂×C₁₃×M₁₆	—	2	1	no	—
29	C₂×C₁₃×M₁₆	—	2	4	no	—
30	C₂×C₁₃×M₁₆	Q₈	1	8	yes	—
31	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
32	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₃	—	2	2	no	—
34	C₂²×C₈×C₁₃	—	2	4	no	—
35	C₂²×C₈×C₁₃	—	2	1	no	—
36	C₂²×C₈×C₁₃	D₄	1	2	yes	—
37	C₂²×C₈×C₁₃	D₄	1	2	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₁₃×#117	—	2	1	no	—
40	C₁₃×#117	—	2	1	no	—
41	C₁₃×#117	D₄	1	2	yes	—
42	C₁₃×#117	C₂³	1	2	yes	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
50	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
51	C₂×C₄²×C₁₃	—	2	1	no	—
52	C₁₃×#107	—	2	8	no	—
53	C₁₃×#106	—	2	8	no	—
54	C₁₃×#106	—	2	8	no	—
55	C₁₃×C₄⋊C₈	—	2	8	no	—
56	C₂×#2090	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	—	8	4	no	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	4	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₁₃×M₁₆	—	2	8	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₁₃×M₁₆	—	2	4	no	—
74	C₁₃×M₁₆	—	4	2	no	—
75	C₁₃×M₁₆	C₂×Q₈	1	8	yes	—
76	C₂×C₈×C₁₃	—	2	8	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	4	2	no	—
88	C₁₃×C₄⋊C₄	—	4	2	no	—
89	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
90	C₁₃×C₂²⋊C₄	—	2	2	no	—
91	C₁₃×C₂²⋊C₄	—	2	2	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	—	2	2	no	—
94	C₂×#428	C₂×C₁₃	1	2	yes	—
95	C₂²×#106	C₂×C₁₃	1	1	yes	characteristic
96	C₂×#424	C₂×C₁₃	1	1	yes	characteristic
97	#2090	C₂×C₁₃	1	8	yes	direct product factor
98	C₂×#343	C₂×C₁₃	1	2	yes	—
99	C₂×#341	C₂×C₁₃	1	1	yes	characteristic
100	C₂³×C₁₃	—	2	4	no	—
101	C₂³×C₁₃	#121	1	1	yes	characteristic
102	C₂³×C₁₃	#121	1	1	yes	characteristic
103	C₂³×C₁₃	#125	1	4	yes	—
104	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	8	4	no	—
109	C₂×C₄×C₁₃	—	8	4	no	—
110	C₂×C₄×C₁₃	—	8	4	no	—
111	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
114	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
115	C₂×C₄×C₁₃	#123	1	2	yes	—
116	C₂×C₄×C₁₃	#123	1	2	yes	—
117	C₂×C₄×C₁₃	#125	1	4	yes	—
118	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
124	C₈×C₁₃	—	2	8	no	—
125	C₈×C₁₃	—	2	4	no	—
126	C₈×C₁₃	—	4	2	no	—
127	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
128	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
129	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
131	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
132	#428	—	2	4	no	—
133	#428	C₂²×C₁₃	1	8	yes	—
134	C₂×#107	—	2	2	no	—
135	C₂×#106	—	2	2	no	—
136	C₂×#106	—	2	2	no	—
137	C₂×#106	—	2	2	no	—
138	C₂×#106	C₂²×C₁₃	1	4	yes	—
139	#424	—	2	2	no	—
140	#424	C₂²×C₁₃	1	4	yes	—
141	C₂×C₄⋊C₈	—	2	2	no	—
142	#343	C₂²×C₁₃	1	8	yes	—
143	#341	C₂²×C₁₃	1	4	yes	—
144	C₂²×C₁₃	—	2	8	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	#398	1	2	yes	—
148	C₂²×C₁₃	#398	1	2	yes	—
149	C₂²×C₁₃	#471	1	4	yes	—
150	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
151	C₂²×C₁₃	C₂×#125	1	2	yes	—
152	C₄×C₁₃	—	4	4	no	—
153	C₄×C₁₃	—	8	4	no	—
154	C₄×C₁₃	#398	1	2	yes	—
155	C₄×C₁₃	#398	1	2	yes	—
156	C₄×C₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
157	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
158	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	C₂³×C₄	—	2	2	no	—
160	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	—	2	1	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	Q₈×C₁₃	1	8	yes	—
166	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
167	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	D₄×C₁₃	1	2	yes	—
172	C₂²×C₈	D₄×C₁₃	1	2	yes	—
173	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
174	#117	—	2	1	no	—
175	#117	—	2	1	no	—
176	#117	D₄×C₁₃	1	2	yes	—
177	#117	C₂³×C₁₃	1	2	yes	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₄²	—	2	1	no	—
187	#107	—	2	8	no	—
188	#106	—	2	8	no	—
189	#106	—	2	8	no	—
190	C₄⋊C₈	—	2	8	no	—
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	—	2	2	no	—
193	C₂×C₁₃	#2090	1	4	yes	direct product factor
194	C₂×C₁₃	C₂×#398	1	1	yes	characteristic
195	C₂×C₁₃	C₂×#471	1	2	yes	—
196	C₂⁴	C₁₃×#41	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	8	4	no	—
202	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
207	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
208	C₂²×C₄	C₁₃×#41	1	4	yes	—
209	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
210	M₁₆	—	2	8	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	C₂×Q₈×C₁₃	1	8	yes	—
215	C₂×C₈	—	2	8	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₁₃	C₂×#2090	1	1	yes	characteristic, direct product factor, Sylow
234	C₂³	—	2	4	no	—
235	C₂³	C₁₃×#121	1	1	yes	characteristic
236	C₂³	C₁₃×#121	1	1	yes	characteristic
237	C₂³	C₁₃×#125	1	4	yes	—
238	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
246	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
247	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
248	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
249	C₂×C₄	C₁₃×#123	1	2	yes	—
250	C₂×C₄	C₁₃×#123	1	2	yes	—
251	C₂×C₄	C₁₃×#125	1	4	yes	—
252	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
253	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
254	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×Q₈×C₁₃	1	2	yes	—
257	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
258	C₈	—	2	8	no	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	8	no	—
264	C₂²	C₁₃×#398	1	2	yes	—
265	C₂²	C₁₃×#398	1	2	yes	—
266	C₂²	C₁₃×#471	1	4	yes	—
267	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
268	C₂²	C₂×C₁₃×#125	1	2	yes	—
269	C₄	—	4	4	no	—
270	C₄	—	8	4	no	—
271	C₄	C₁₃×#398	1	2	yes	—
272	C₄	C₁₃×#398	1	2	yes	—
273	C₄	C₂×C₁₃×C₂²⋊Q₈	1	2	yes	—
274	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
275	C₄	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	2	2	no	—
278	C₂	C₁₃×#2090	1	4	yes	direct product factor
279	C₂	C₂×C₁₃×#398	1	1	yes	characteristic
280	C₂	C₂×C₁₃×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#2090	C₂	direct (N × H)
56. C₂×#2090	C₁₃	direct (N × H)
97. #2090	C₂×C₁₃	direct (N × H)
193. C₂×C₁₃	#2090	direct (N × H)
233. C₁₃	C₂×#2090	direct (N × H)
278. C₂	C₁₃×#2090	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#428	2
C₂²×C₁₃×#106	2
C₂×C₁₃×#424	2
C₁₃×#2090	2
C₂×C₁₃×#343	2
C₂×C₁₃×#341	2
C₂×#2090	13

Subgroups — C2×C13×#2090 #15710012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2090 #15710012