Subgroups — C₂×C₁₃×#2231 #15719975

Order 3328.

526 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2231	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×#125	—	2	1	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
11	C₁₃×#499	—	2	2	no	—
12	C₁₃×#499	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×#107	—	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	C₁₃×#135	—	2	1	no	—
25	C₁₃×#135	—	2	1	no	—
26	C₁₃×#135	Q₈	1	2	yes	—
27	C₁₃×#135	C₂³	1	2	yes	—
28	C₂×C₁₃×Q₁₆	Q₈	1	1	yes	characteristic
29	C₂×C₁₃×SD₁₆	—	2	1	no	—
30	C₂×C₁₃×D₈	Q₈	1	1	yes	characteristic
31	C₁₃×#131	—	2	4	no	—
32	C₁₃×#131	—	4	2	no	—
33	C₂×C₁₃×M₁₆	—	2	1	no	—
34	C₂×C₁₃×M₁₆	—	4	1	no	—
35	C₂²×C₈×C₁₃	—	4	1	no	—
36	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#125	—	2	4	no	—
39	C₁₃×#125	—	4	2	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₁₃×M₃₂	—	2	2	no	—
44	C₁₃×M₃₂	C₂³	1	4	yes	—
45	C₂×C₁₃×C₁₆	—	2	2	no	—
46	C₁₃×#108	—	2	1	no	—
47	C₁₃×#108	C₂³	1	2	yes	—
48	C₁₃×#107	—	2	4	no	—
49	C₁₃×#106	—	2	2	no	—
50	C₁₃×#106	C₂³	1	4	yes	—
51	C₁₃×#104	—	2	4	no	—
52	C₄×C₈×C₁₃	D₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	C₂×#2231	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₁₃×#41	—	2	2	no	—
56	C₁₃×#41	—	2	2	no	—
57	C₁₃×#41	—	4	1	no	—
58	C₁₃×#41	—	4	1	no	—
59	C₂×Q₈×C₁₃	—	2	1	no	—
60	C₂×D₄×C₁₃	—	2	1	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₁₃×Q₁₆	—	2	1	no	—
67	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	—
68	C₁₃×SD₁₆	—	2	2	no	—
69	C₁₃×SD₁₆	—	2	2	no	—
70	C₁₃×D₈	—	2	1	no	—
71	C₁₃×D₈	C₂×Q₈	1	2	yes	—
72	C₁₃×M₁₆	—	2	2	no	—
73	C₁₃×M₁₆	—	4	1	no	—
74	C₁₃×M₁₆	—	4	4	no	—
75	C₂×C₈×C₁₃	—	4	1	no	—
76	C₂×C₈×C₁₃	—	4	4	no	—
77	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
78	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
79	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×Q₈	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	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	4	8	no	—
89	C₁₃×C₄⋊C₄	—	4	2	no	—
90	C₄²×C₁₃	—	2	1	no	—
91	C₄²×C₁₃	C₂×D₄	1	2	yes	—
92	C₁₃×C₁₆	—	2	4	no	—
93	C₂×#499	C₂×C₁₃	1	1	yes	characteristic, complemented
94	C₂×#443	C₂×C₁₃	1	1	yes	characteristic
95	#2231	C₂×C₁₃	1	8	yes	direct product factor
96	C₂×#365	C₂×C₁₃	1	2	yes	complemented
97	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
98	C₂×#360	C₂×C₁₃	1	2	yes	—
99	C₂³×C₁₃	—	4	1	no	—
100	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	Q₈×C₁₃	—	2	2	no	—
102	Q₈×C₁₃	—	4	1	no	—
103	D₄×C₁₃	—	2	2	no	—
104	D₄×C₁₃	—	4	2	no	—
105	D₄×C₁₃	—	4	2	no	—
106	D₄×C₁₃	—	4	1	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	8	4	no	—
114	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×SD₁₆	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	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₁₃	—	4	2	no	—
121	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
124	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
125	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
126	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
127	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
128	C₂×#131	—	2	1	no	—
129	C₂×#125	—	2	1	no	—
130	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
131	#499	—	2	2	no	—
132	#499	C₂²×C₁₃	1	4	yes	complemented
133	#443	—	2	2	no	—
134	#443	C₂²×C₁₃	1	4	yes	—
135	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×#107	—	2	1	no	—
137	C₂×#106	C₂²×C₁₃	1	2	yes	—
138	C₂×#104	—	2	1	no	—
139	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
140	#365	C₂²×C₁₃	1	8	yes	complemented
141	#364	C₂²×C₁₃	1	4	yes	complemented
142	#360	C₂²×C₁₃	1	8	yes	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	4	2	no	—
145	C₂²×C₁₃	—	8	2	no	—
146	C₂²×C₁₃	#476	1	1	yes	characteristic
147	C₂²×C₁₃	#476	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₄×C₁₃	—	4	2	no	—
150	C₄×C₁₃	—	4	2	no	—
151	C₄×C₁₃	—	8	4	no	—
152	C₄×C₁₃	#476	1	1	yes	characteristic
153	C₄×C₁₃	#476	1	1	yes	characteristic
154	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
155	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
156	C₂×#41	—	2	1	no	—
157	#135	—	2	1	no	—
158	#135	—	2	1	no	—
159	#135	Q₈×C₁₃	1	2	yes	—
160	#135	C₂³×C₁₃	1	2	yes	—
161	C₂×Q₁₆	Q₈×C₁₃	1	1	yes	characteristic
162	C₂×SD₁₆	—	2	1	no	—
163	C₂×D₈	Q₈×C₁₃	1	1	yes	characteristic
164	#131	—	2	4	no	—
165	#131	—	4	2	no	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	—	4	1	no	—
168	C₂²×C₈	—	4	1	no	—
169	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
171	#125	—	2	4	no	—
172	#125	—	4	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	4	2	no	—
175	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
176	M₃₂	—	2	2	no	—
177	M₃₂	C₂³×C₁₃	1	4	yes	—
178	C₂×C₁₆	—	2	2	no	—
179	#108	—	2	1	no	—
180	#108	C₂³×C₁₃	1	2	yes	—
181	#107	—	2	4	no	—
182	#106	—	2	2	no	—
183	#106	C₂³×C₁₃	1	4	yes	—
184	#104	—	2	4	no	—
185	C₄×C₈	D₄×C₁₃	1	2	yes	—
186	C₄×C₈	C₂³×C₁₃	1	2	yes	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	—	8	2	no	—
190	C₂×C₁₃	#2231	1	2	yes	direct product factor
191	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	—	4	1	no	—
195	#41	—	4	1	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
203	Q₁₆	—	2	1	no	—
204	Q₁₆	C₂×Q₈×C₁₃	1	2	yes	—
205	SD₁₆	—	2	2	no	—
206	SD₁₆	—	2	2	no	—
207	D₈	—	2	1	no	—
208	D₈	C₂×Q₈×C₁₃	1	2	yes	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	4	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
215	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
216	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
219	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
220	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
221	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
222	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
223	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	8	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	C₂×D₄×C₁₃	1	2	yes	—
229	C₁₆	—	2	4	no	—
230	C₁₃	C₂×#2231	1	1	yes	characteristic, direct product factor, Sylow
231	C₂³	—	4	1	no	—
232	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
233	Q₈	—	2	2	no	—
234	Q₈	—	4	1	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
248	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
249	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
250	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
254	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
255	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
256	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
257	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
258	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	C₁₃×#476	1	1	yes	characteristic
263	C₂²	C₁₃×#476	1	1	yes	characteristic
264	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	8	4	no	—
268	C₄	C₁₃×#476	1	1	yes	characteristic
269	C₄	C₁₃×#476	1	1	yes	characteristic
270	C₄	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
271	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	8	2	no	—
275	C₂	C₁₃×#2231	1	2	yes	direct product factor
276	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×#499	C₂	semidirect
3. C₁₃×#2231	C₂	direct (N × H)
4. C₂×C₁₃×#365	C₂	semidirect
5. C₂×C₁₃×#364	C₂	semidirect
12. C₁₃×#499	C₂²	semidirect
20. C₁₃×#365	C₂²	semidirect
21. C₁₃×#364	C₂²	semidirect
54. C₂×#2231	C₁₃	direct (N × H)
93. C₂×#499	C₂×C₁₃	semidirect
95. #2231	C₂×C₁₃	direct (N × H)
96. C₂×#365	C₂×C₁₃	semidirect
97. C₂×#364	C₂×C₁₃	semidirect
132. #499	C₂²×C₁₃	semidirect
140. #365	C₂²×C₁₃	semidirect
141. #364	C₂²×C₁₃	semidirect
190. C₂×C₁₃	#2231	direct (N × H)
230. C₁₃	C₂×#2231	direct (N × H)
275. C₂	C₁₃×#2231	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×#499	2
C₂×C₁₃×#443	2
C₁₃×#2231	2
C₂×C₁₃×#365	2
C₂×C₁₃×#364	2
C₂×C₁₃×#360	2
C₂×#2231	13

Subgroups — C2×C13×#2231 #15719975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2231 #15719975