Subgroups — C₂×C₁₃×#2170 #15719778

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#504	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2170	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#350	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#135	C₄	1	1	yes	characteristic
8	C₂×C₁₃×#131	C₂²	1	2	yes	—
9	C₁₃×#504	—	2	4	no	—
10	C₁₃×#504	C₂²	1	8	yes	complemented
11	C₂×C₁₃×M₃₂	—	2	2	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₂²×C₁₃×C₁₆	—	2	2	no	—
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#443	C₄	1	4	yes	—
17	C₁₃×#443	C₂²	1	4	yes	—
18	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
19	C₂×C₁₃×#104	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#364	C₂²	1	4	yes	complemented
22	C₁₃×#350	C₂²	1	8	yes	complemented
23	C₄×C₁₃×C₁₆	C₂²	1	4	yes	complemented
24	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
25	C₁₃×#135	C₈	1	4	yes	—
26	C₁₃×#135	C₂×C₄	1	4	yes	—
27	C₂×C₁₃×Q₁₆	C₈	1	1	yes	characteristic
28	C₂×C₁₃×SD₁₆	C₈	1	2	yes	—
29	C₂×C₁₃×D₈	C₈	1	1	yes	characteristic
30	C₁₃×#131	—	2	2	no	—
31	C₁₃×#131	—	2	2	no	—
32	C₁₃×#131	C₂×C₄	1	4	yes	—
33	C₁₃×#131	C₂³	1	4	yes	—
34	C₂×C₁₃×M₁₆	—	2	2	no	—
35	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	—	2	2	no	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₁₃×M₃₂	—	2	2	no	—
41	C₁₃×M₃₂	—	2	8	no	—
42	C₁₃×M₃₂	C₂³	1	4	yes	—
43	C₂×C₁₃×C₁₆	—	2	2	no	—
44	C₂×C₁₃×C₁₆	—	2	1	no	—
45	C₂×C₁₃×C₁₆	—	2	1	no	—
46	C₂×C₁₃×C₁₆	—	2	8	no	—
47	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
48	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
49	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
50	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
51	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
52	C₁₃×#108	C₂×C₄	1	4	yes	—
53	C₁₃×#104	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₁₃	C₂³	1	2	yes	—
56	C₂×#2170	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₁₃×#41	—	2	2	no	—
58	C₁₃×#41	—	2	2	no	—
59	C₁₃×#41	C₂×C₈	1	4	yes	—
60	C₁₃×#41	C₂²×C₄	1	4	yes	—
61	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₁₃×Q₁₆	C₂×C₈	1	4	yes	—
68	C₁₃×SD₁₆	C₂×C₈	1	8	yes	—
69	C₁₃×D₈	C₂×C₈	1	4	yes	—
70	C₁₃×M₁₆	—	2	2	no	—
71	C₁₃×M₁₆	—	2	4	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₁₃×M₁₆	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₄²×C₁₃	—	2	1	no	—
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₁₃×C₁₆	—	2	4	no	—
90	C₁₃×C₁₆	—	2	2	no	—
91	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
92	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
93	C₂×#504	C₂×C₁₃	1	2	yes	complemented
94	C₂×#443	C₂×C₁₃	1	1	yes	characteristic
95	#2170	C₂×C₁₃	1	8	yes	direct product factor
96	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
97	C₂×#350	C₂×C₁₃	1	2	yes	complemented
98	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂³×C₁₃	—	2	2	no	—
100	C₂³×C₁₃	#131	1	1	yes	characteristic
101	Q₈×C₁₃	—	2	2	no	—
102	Q₈×C₁₃	C₂²×C₈	1	4	yes	—
103	D₄×C₁₃	—	2	4	no	—
104	D₄×C₁₃	—	2	4	no	—
105	D₄×C₁₃	—	2	2	no	—
106	D₄×C₁₃	C₂²×C₈	1	4	yes	—
107	C₂×C₄×C₁₃	—	2	4	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	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₁₃	—	2	4	no	—
119	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₈×C₁₃	C₂²×C₈	1	2	yes	—
122	C₈×C₁₃	C₂²×C₈	1	2	yes	—
123	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×#135	C₄×C₁₃	1	1	yes	characteristic
126	C₂×#131	C₂²×C₁₃	1	2	yes	—
127	#504	—	2	4	no	—
128	#504	C₂²×C₁₃	1	8	yes	complemented
129	C₂×M₃₂	—	2	2	no	—
130	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
131	C₂²×C₁₆	—	2	2	no	—
132	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
133	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
134	#443	C₄×C₁₃	1	4	yes	—
135	#443	C₂²×C₁₃	1	4	yes	—
136	C₂×#108	C₄×C₁₃	1	1	yes	characteristic
137	C₂×#104	C₄×C₁₃	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
139	#364	C₂²×C₁₃	1	4	yes	complemented
140	#350	C₂²×C₁₃	1	8	yes	complemented
141	C₄×C₁₆	C₂²×C₁₃	1	4	yes	complemented
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	2	4	no	—
144	C₂²×C₁₃	—	4	4	no	—
145	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
146	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	—	2	4	no	—
150	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
151	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
152	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
154	C₂×#41	C₂×C₄×C₁₃	1	2	yes	—
155	#135	C₈×C₁₃	1	4	yes	—
156	#135	C₂×C₄×C₁₃	1	4	yes	—
157	C₂×Q₁₆	C₈×C₁₃	1	1	yes	characteristic
158	C₂×SD₁₆	C₈×C₁₃	1	2	yes	—
159	C₂×D₈	C₈×C₁₃	1	1	yes	characteristic
160	#131	—	2	2	no	—
161	#131	—	2	2	no	—
162	#131	C₂×C₄×C₁₃	1	4	yes	—
163	#131	C₂³×C₁₃	1	4	yes	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	8	no	—
172	M₃₂	C₂³×C₁₃	1	4	yes	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	2	8	no	—
177	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
178	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
179	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
180	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
181	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
182	#108	C₂×C₄×C₁₃	1	4	yes	—
183	#104	C₂×C₄×C₁₃	1	8	yes	—
184	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
185	C₄×C₈	C₂³×C₁₃	1	2	yes	—
186	C₂×C₁₃	—	2	1	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	4	4	no	—
189	C₂×C₁₃	#2170	1	2	yes	direct product factor
190	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	#41	C₂×C₈×C₁₃	1	4	yes	—
194	#41	C₂²×C₄×C₁₃	1	4	yes	—
195	C₂×Q₈	C₂×C₈×C₁₃	1	2	yes	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
201	Q₁₆	C₂×C₈×C₁₃	1	4	yes	—
202	SD₁₆	C₂×C₈×C₁₃	1	8	yes	—
203	D₈	C₂×C₈×C₁₃	1	4	yes	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	C₂²×C₄×C₁₃	1	4	yes	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
216	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
217	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
219	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
220	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
221	C₄²	—	2	1	no	—
222	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
223	C₁₆	—	2	4	no	—
224	C₁₆	—	2	2	no	—
225	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
226	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
227	C₁₃	C₂×#2170	1	1	yes	characteristic, direct product factor, Sylow
228	C₂³	—	2	2	no	—
229	C₂³	C₁₃×#131	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	C₂²×C₈×C₁₃	1	4	yes	—
232	D₄	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	2	2	no	—
235	D₄	C₂²×C₈×C₁₃	1	4	yes	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
242	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
245	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
246	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
249	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
250	C₈	C₂²×C₈×C₁₃	1	2	yes	—
251	C₈	C₂²×C₈×C₁₃	1	2	yes	—
252	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
253	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₈×D₄×C₁₃	1	1	yes	characteristic
258	C₂²	C₈×D₄×C₁₃	1	1	yes	characteristic
259	C₂²	C₂×C₁₃×#131	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
263	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
264	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
265	C₄	C₂³×C₈×C₁₃	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	4	4	no	—
269	C₂	C₁₃×#2170	1	2	yes	direct product factor
270	C₂	C₂×C₈×D₄×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₃×C₁₆	D₄
commutator	C₄	C₂³×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂×C₁₃×#135	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃×C₁₆ ◁ C₂²×C₁₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#504	C₂	semidirect
3. C₁₃×#2170	C₂	direct (N × H)
4. C₂×C₁₃×#364	C₂	semidirect
5. C₂×C₁₃×#350	C₂	semidirect
6. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
10. C₁₃×#504	C₂²	semidirect
21. C₁₃×#364	C₂²	semidirect
22. C₁₃×#350	C₂²	semidirect
23. C₄×C₁₃×C₁₆	C₂²	semidirect
56. C₂×#2170	C₁₃	direct (N × H)
93. C₂×#504	C₂×C₁₃	semidirect
95. #2170	C₂×C₁₃	direct (N × H)
96. C₂×#364	C₂×C₁₃	semidirect
97. C₂×#350	C₂×C₁₃	semidirect
98. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
128. #504	C₂²×C₁₃	semidirect
139. #364	C₂²×C₁₃	semidirect
140. #350	C₂²×C₁₃	semidirect
141. C₄×C₁₆	C₂²×C₁₃	semidirect
189. C₂×C₁₃	#2170	direct (N × H)
227. C₁₃	C₂×#2170	direct (N × H)
269. C₂	C₁₃×#2170	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×#504	2
C₂×C₁₃×#443	2
C₁₃×#2170	2
C₂×C₁₃×#364	2
C₂×C₁₃×#350	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#2170	13

Subgroups — C2×C13×#2170 #15719778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2170 #15719778