Subgroups — C₂×C₁₃×#2230 #15719974

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2230	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#365	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#360	C₂	1	2	yes	maximal
7	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#131	—	2	1	no	—
9	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
11	C₁₃×#498	—	2	2	no	—
12	C₁₃×#498	C₂²	1	4	yes	complemented
13	C₁₃×#443	—	2	2	no	—
14	C₁₃×#443	C₂²	1	4	yes	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#106	—	2	1	no	—
17	C₂×C₁₃×#106	C₂²	1	2	yes	—
18	C₂×C₁₃×#104	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#365	C₂²	1	8	yes	complemented
21	C₁₃×#364	C₂²	1	4	yes	complemented
22	C₁₃×#360	C₂²	1	8	yes	—
23	C₂×C₁₃×#41	—	2	1	no	—
24	C₂²×Q₈×C₁₃	—	4	1	no	—
25	C₁₃×#135	—	2	1	no	—
26	C₁₃×#135	—	2	1	no	—
27	C₁₃×#135	Q₈	1	2	yes	complemented
28	C₁₃×#135	C₂³	1	2	yes	—
29	C₂×C₁₃×Q₁₆	Q₈	1	1	yes	characteristic, complemented
30	C₂×C₁₃×SD₁₆	—	2	1	no	—
31	C₂×C₁₃×D₈	Q₈	1	1	yes	characteristic, complemented
32	C₁₃×#131	—	2	4	no	—
33	C₁₃×#131	—	4	2	no	—
34	C₂×C₁₃×M₁₆	—	2	1	no	—
35	C₂×C₁₃×M₁₆	—	4	1	no	—
36	C₂²×C₈×C₁₃	—	4	1	no	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊Q₈	—	2	4	no	—
40	C₁₃×C₄⋊Q₈	—	4	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₁₃×M₃₂	—	2	2	no	—
45	C₁₃×M₃₂	C₂³	1	4	yes	—
46	C₂×C₁₃×C₁₆	—	2	2	no	—
47	C₁₃×#108	—	2	1	no	—
48	C₁₃×#108	C₂³	1	2	yes	—
49	C₁₃×#106	—	2	4	no	—
50	C₁₃×#106	—	2	2	no	—
51	C₁₃×#106	C₂³	1	4	yes	—
52	C₁₃×#104	—	2	4	no	—
53	C₄×C₈×C₁₃	D₄	1	2	yes	—
54	C₄×C₈×C₁₃	C₂³	1	2	yes	—
55	C₂×#2230	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₁₃×#41	—	2	2	no	—
57	C₁₃×#41	—	2	2	no	—
58	C₁₃×#41	—	4	1	no	—
59	C₁₃×#41	—	4	1	no	—
60	C₂×Q₈×C₁₃	—	2	1	no	—
61	C₂×Q₈×C₁₃	—	4	4	no	—
62	C₂×Q₈×C₁₃	—	8	4	no	—
63	C₂×D₄×C₁₃	—	2	1	no	—
64	C₂×D₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	—	4	1	no	—
67	C₂²×C₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₁₃×Q₁₆	—	2	1	no	—
70	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	complemented
71	C₁₃×SD₁₆	—	2	2	no	—
72	C₁₃×SD₁₆	—	2	2	no	—
73	C₁₃×D₈	—	2	1	no	—
74	C₁₃×D₈	C₂×Q₈	1	2	yes	complemented
75	C₁₃×M₁₆	—	2	2	no	—
76	C₁₃×M₁₆	—	4	1	no	—
77	C₁₃×M₁₆	—	4	4	no	—
78	C₂×C₈×C₁₃	—	4	1	no	—
79	C₂×C₈×C₁₃	—	4	4	no	—
80	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
81	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
82	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	4	4	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₄²×C₁₃	—	2	1	no	—
94	C₄²×C₁₃	C₂×D₄	1	2	yes	—
95	C₁₃×C₁₆	—	2	4	no	—
96	C₂×#498	C₂×C₁₃	1	1	yes	characteristic, complemented
97	C₂×#443	C₂×C₁₃	1	1	yes	characteristic
98	#2230	C₂×C₁₃	1	8	yes	direct product factor
99	C₂×#365	C₂×C₁₃	1	2	yes	complemented
100	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
101	C₂×#360	C₂×C₁₃	1	2	yes	—
102	C₂³×C₁₃	—	4	1	no	—
103	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	Q₈×C₁₃	—	2	2	no	—
105	Q₈×C₁₃	—	4	1	no	—
106	Q₈×C₁₃	—	8	8	no	—
107	D₄×C₁₃	—	2	2	no	—
108	D₄×C₁₃	—	4	2	no	—
109	D₄×C₁₃	—	4	2	no	—
110	D₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	—	8	4	no	—
118	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₁₃	—	4	2	no	—
125	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
128	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
129	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
130	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×#131	—	2	1	no	—
133	C₂×C₄⋊Q₈	—	2	1	no	—
134	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
135	#498	—	2	2	no	—
136	#498	C₂²×C₁₃	1	4	yes	complemented
137	#443	—	2	2	no	—
138	#443	C₂²×C₁₃	1	4	yes	—
139	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×#106	—	2	1	no	—
141	C₂×#106	C₂²×C₁₃	1	2	yes	—
142	C₂×#104	—	2	1	no	—
143	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
144	#365	C₂²×C₁₃	1	8	yes	complemented
145	#364	C₂²×C₁₃	1	4	yes	complemented
146	#360	C₂²×C₁₃	1	8	yes	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	4	2	no	—
149	C₂²×C₁₃	—	8	2	no	—
150	C₂²×C₁₃	#476	1	1	yes	characteristic
151	C₂²×C₁₃	#476	1	1	yes	characteristic
152	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₄×C₁₃	—	2	4	no	—
154	C₄×C₁₃	—	4	2	no	—
155	C₄×C₁₃	—	8	4	no	—
156	C₄×C₁₃	#476	1	1	yes	characteristic
157	C₄×C₁₃	#476	1	1	yes	characteristic
158	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
160	C₂×#41	—	2	1	no	—
161	C₂²×Q₈	—	4	1	no	—
162	#135	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	Q₈×C₁₃	1	2	yes	complemented
165	#135	C₂³×C₁₃	1	2	yes	—
166	C₂×Q₁₆	Q₈×C₁₃	1	1	yes	characteristic, complemented
167	C₂×SD₁₆	—	2	1	no	—
168	C₂×D₈	Q₈×C₁₃	1	1	yes	characteristic, complemented
169	#131	—	2	4	no	—
170	#131	—	4	2	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	4	1	no	—
173	C₂²×C₈	—	4	1	no	—
174	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
176	C₄⋊Q₈	—	2	4	no	—
177	C₄⋊Q₈	—	4	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	4	1	no	—
180	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
181	M₃₂	—	2	2	no	—
182	M₃₂	C₂³×C₁₃	1	4	yes	—
183	C₂×C₁₆	—	2	2	no	—
184	#108	—	2	1	no	—
185	#108	C₂³×C₁₃	1	2	yes	—
186	#106	—	2	4	no	—
187	#106	—	2	2	no	—
188	#106	C₂³×C₁₃	1	4	yes	—
189	#104	—	2	4	no	—
190	C₄×C₈	D₄×C₁₃	1	2	yes	—
191	C₄×C₈	C₂³×C₁₃	1	2	yes	—
192	C₂×C₁₃	—	2	1	no	—
193	C₂×C₁₃	—	2	1	no	—
194	C₂×C₁₃	—	8	2	no	—
195	C₂×C₁₃	#2230	1	2	yes	direct product factor
196	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	#41	—	4	1	no	—
200	#41	—	4	1	no	—
201	C₂×Q₈	—	2	1	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	8	4	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
210	Q₁₆	—	2	1	no	—
211	Q₁₆	C₂×Q₈×C₁₃	1	2	yes	complemented
212	SD₁₆	—	2	2	no	—
213	SD₁₆	—	2	2	no	—
214	D₈	—	2	1	no	—
215	D₈	C₂×Q₈×C₁₃	1	2	yes	complemented
216	M₁₆	—	2	2	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	4	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
222	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
223	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
224	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
226	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
228	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
229	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
230	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	C₂×D₄×C₁₃	1	2	yes	—
236	C₁₆	—	2	4	no	—
237	C₁₃	C₂×#2230	1	1	yes	characteristic, direct product factor, Sylow
238	C₂³	—	4	1	no	—
239	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	4	1	no	—
242	Q₈	—	8	8	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	4	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
256	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
257	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
258	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
260	C₈	—	4	2	no	—
261	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
262	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
263	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
264	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
265	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
266	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	C₁₃×#476	1	1	yes	characteristic
271	C₂²	C₁₃×#476	1	1	yes	characteristic
272	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	8	4	no	—
276	C₄	C₁₃×#476	1	1	yes	characteristic
277	C₄	C₁₃×#476	1	1	yes	characteristic
278	C₄	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
279	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	8	2	no	—
283	C₂	C₁₃×#2230	1	2	yes	direct product factor
284	C₂	C₂×C₁₃×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#498	C₂	semidirect
3. C₁₃×#2230	C₂	direct (N × H)
4. C₂×C₁₃×#365	C₂	semidirect
5. C₂×C₁₃×#364	C₂	semidirect
12. C₁₃×#498	C₂²	semidirect
20. C₁₃×#365	C₂²	semidirect
21. C₁₃×#364	C₂²	semidirect
27. C₁₃×#135	Q₈	semidirect
29. C₂×C₁₃×Q₁₆	Q₈	semidirect
31. C₂×C₁₃×D₈	Q₈	semidirect
55. C₂×#2230	C₁₃	direct (N × H)
70. C₁₃×Q₁₆	C₂×Q₈	semidirect
74. C₁₃×D₈	C₂×Q₈	semidirect
96. C₂×#498	C₂×C₁₃	semidirect
98. #2230	C₂×C₁₃	direct (N × H)
99. C₂×#365	C₂×C₁₃	semidirect
100. C₂×#364	C₂×C₁₃	semidirect
136. #498	C₂²×C₁₃	semidirect
144. #365	C₂²×C₁₃	semidirect
145. #364	C₂²×C₁₃	semidirect
164. #135	Q₈×C₁₃	semidirect
166. C₂×Q₁₆	Q₈×C₁₃	semidirect
168. C₂×D₈	Q₈×C₁₃	semidirect
195. C₂×C₁₃	#2230	direct (N × H)
211. Q₁₆	C₂×Q₈×C₁₃	semidirect
215. D₈	C₂×Q₈×C₁₃	semidirect
237. C₁₃	C₂×#2230	direct (N × H)
283. C₂	C₁₃×#2230	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×#498	2
C₂×C₁₃×#443	2
C₁₃×#2230	2
C₂×C₁₃×#365	2
C₂×C₁₃×#364	2
C₂×C₁₃×#360	2
C₂×#2230	13

Subgroups — C2×C13×#2230 #15719974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2230 #15719974