Subgroups — C₂×C₁₃×#3255 #15747244

Order 3328.

1866 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#533	C₂	1	2	yes	maximal
2	C₁₃×#3255	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#475	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#470	C₂	1	4	yes	maximal
5	C₂×C₁₃×#462	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×Q₁₆	C₂²	1	4	yes	—
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₁₃×#533	—	2	8	no	—
10	C₁₃×#533	C₂²	1	16	yes	—
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₁₃×#125	—	2	2	no	—
14	C₂×C₁₃×C₂²⋊Q₈	—	2	2	no	—
15	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
16	C₂×C₄×Q₈×C₁₃	—	2	2	no	—
17	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
18	C₂×C₁₃×#117	—	2	2	no	—
19	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#475	C₂²	1	32	yes	complemented
21	C₁₃×#470	C₂²	1	32	yes	—
22	C₁₃×#462	C₂²	1	32	yes	complemented
23	C₂×C₁₃×#106	C₂²	1	4	yes	—
24	C₁₃×#423	C₂²	1	8	yes	—
25	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
26	C₂×C₁₃×#103	C₂²	1	8	yes	—
27	C₂²×Q₈×C₁₃	—	2	2	no	—
28	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
29	C₂³×C₄×C₁₃	D₄	1	2	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₃×Q₁₆	—	4	8	no	—
32	C₂×C₁₃×Q₁₆	C₂³	1	16	yes	—
33	C₂×C₁₃×M₁₆	—	4	4	no	—
34	C₂×C₁₃×M₁₆	C₂³	1	8	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
36	C₁₃×C₄⋊Q₈	—	2	4	no	—
37	C₁₃×C₄⋊Q₈	—	2	4	no	—
38	C₁₃×C₄⋊Q₈	C₂³	1	8	yes	—
39	C₁₃×C₄⋊Q₈	C₂³	1	8	yes	—
40	C₁₃×#125	—	2	16	no	—
41	C₁₃×C₂²⋊Q₈	—	2	16	no	—
42	C₁₃×C₂²⋊Q₈	—	2	8	no	—
43	C₁₃×C₂²⋊Q₈	C₂³	1	16	yes	—
44	C₄×Q₈×C₁₃	—	2	16	no	—
45	C₄×Q₈×C₁₃	—	2	8	no	—
46	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
47	C₁₃×#117	—	2	16	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
52	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
53	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
54	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
55	C₂×C₄²×C₁₃	—	2	4	no	—
56	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₁₃	D₄	1	4	yes	—
59	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
61	C₁₃×#106	C₂³	1	16	yes	—
62	C₁₃×C₄⋊C₈	C₂³	1	16	yes	—
63	C₁₃×#103	C₂³	1	32	yes	—
64	C₂×#3255	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×Q₈×C₁₃	—	2	4	no	—
67	C₂×Q₈×C₁₃	—	2	8	no	—
68	C₂×Q₈×C₁₃	—	2	8	no	—
69	C₂×Q₈×C₁₃	—	2	8	no	—
70	C₂×Q₈×C₁₃	—	4	8	no	—
71	C₂×Q₈×C₁₃	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₁₃	—	2	8	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	2	4	no	—
76	C₂²×C₄×C₁₃	—	2	4	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₁₃×Q₁₆	—	4	16	no	—
85	C₁₃×M₁₆	—	4	8	no	—
86	C₂×C₈×C₁₃	—	2	8	no	—
87	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
88	C₁₃×C₄⋊C₄	—	2	8	no	—
89	C₁₃×C₄⋊C₄	—	2	8	no	—
90	C₁₃×C₄⋊C₄	—	2	8	no	—
91	C₁₃×C₄⋊C₄	—	2	16	no	—
92	C₁₃×C₄⋊C₄	—	2	16	no	—
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
95	C₁₃×C₂²⋊C₄	—	2	16	no	—
96	C₄²×C₁₃	—	2	16	no	—
97	C₄²×C₁₃	C₂×D₄	1	4	yes	—
98	C₄²×C₁₃	C₂×D₄	1	4	yes	—
99	C₄²×C₁₃	C₂×D₄	1	4	yes	—
100	C₄²×C₁₃	C₂⁴	1	4	yes	—
101	C₂×#533	C₂×C₁₃	1	2	yes	—
102	#3255	C₂×C₁₃	1	16	yes	direct product factor
103	C₂×#475	C₂×C₁₃	1	4	yes	complemented
104	C₂×#470	C₂×C₁₃	1	4	yes	—
105	C₂×#462	C₂×C₁₃	1	4	yes	complemented
106	C₂×#423	C₂×C₁₃	1	1	yes	characteristic
107	C₂³×C₁₃	—	2	2	no	—
108	C₂³×C₁₃	—	2	4	no	—
109	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
110	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₁₃	—	2	8	no	—
112	Q₈×C₁₃	—	2	8	no	—
113	Q₈×C₁₃	—	4	8	no	—
114	Q₈×C₁₃	—	4	16	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	16	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	8	no	—
122	C₂×C₄×C₁₃	—	2	8	no	—
123	C₂×C₄×C₁₃	—	4	8	no	—
124	C₂×C₄×C₁₃	—	4	8	no	—
125	C₂×C₄×C₁₃	#137	1	8	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
130	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
131	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
132	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
133	C₈×C₁₃	—	2	8	no	—
134	C₂²×Q₁₆	C₂²×C₁₃	1	4	yes	—
135	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
136	#533	—	2	8	no	—
137	#533	C₂²×C₁₃	1	16	yes	—
138	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
139	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
140	C₂×#125	—	2	2	no	—
141	C₂×C₂²⋊Q₈	—	2	2	no	—
142	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
143	C₂×C₄×Q₈	—	2	2	no	—
144	C₂×C₄×Q₈	C₂²×C₁₃	1	4	yes	—
145	C₂×#117	—	2	2	no	—
146	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
147	#475	C₂²×C₁₃	1	32	yes	complemented
148	#470	C₂²×C₁₃	1	32	yes	—
149	#462	C₂²×C₁₃	1	32	yes	complemented
150	C₂×#106	C₂²×C₁₃	1	4	yes	—
151	#423	C₂²×C₁₃	1	8	yes	—
152	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
153	C₂×#103	C₂²×C₁₃	1	8	yes	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	4	no	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	#537	1	2	yes	—
158	C₂²×C₁₃	C₂×#137	1	4	yes	—
159	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	2	4	no	—
162	C₄×C₁₃	—	4	8	no	—
163	C₄×C₁₃	—	4	8	no	—
164	C₄×C₁₃	#537	1	2	yes	—
165	C₄×C₁₃	C₂×#137	1	8	yes	—
166	C₄×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
167	C₄×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
168	C₂²×Q₈	—	2	2	no	—
169	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
170	C₂³×C₄	D₄×C₁₃	1	2	yes	—
171	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
172	C₂×Q₁₆	—	4	8	no	—
173	C₂×Q₁₆	C₂³×C₁₃	1	16	yes	—
174	C₂×M₁₆	—	4	4	no	—
175	C₂×M₁₆	C₂³×C₁₃	1	8	yes	—
176	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
177	C₄⋊Q₈	—	2	4	no	—
178	C₄⋊Q₈	—	2	4	no	—
179	C₄⋊Q₈	C₂³×C₁₃	1	8	yes	—
180	C₄⋊Q₈	C₂³×C₁₃	1	8	yes	—
181	#125	—	2	16	no	—
182	C₂²⋊Q₈	—	2	16	no	—
183	C₂²⋊Q₈	—	2	8	no	—
184	C₂²⋊Q₈	C₂³×C₁₃	1	16	yes	—
185	C₄×Q₈	—	2	16	no	—
186	C₄×Q₈	—	2	8	no	—
187	C₄×Q₈	C₂³×C₁₃	1	16	yes	—
188	#117	—	2	16	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
195	C₂×C₂²⋊C₄	—	2	4	no	—
196	C₂×C₄²	—	2	4	no	—
197	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₁₃	1	4	yes	—
200	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
201	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
202	#106	C₂³×C₁₃	1	16	yes	—
203	C₄⋊C₈	C₂³×C₁₃	1	16	yes	—
204	#103	C₂³×C₁₃	1	32	yes	—
205	C₂×C₁₃	—	4	2	no	—
206	C₂×C₁₃	#3255	1	4	yes	direct product factor
207	C₂×C₁₃	C₂×#537	1	1	yes	characteristic
208	C₂×C₁₃	C₂²×#137	1	2	yes	—
209	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	2	8	no	—
212	C₂×Q₈	—	2	8	no	—
213	C₂×Q₈	—	2	8	no	—
214	C₂×Q₈	—	4	8	no	—
215	C₂×Q₈	C₂⁴×C₁₃	1	8	yes	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
225	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
226	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
227	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
228	Q₁₆	—	4	16	no	—
229	M₁₆	—	4	8	no	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	C₂⁴×C₁₃	1	8	yes	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	16	no	—
236	C₄⋊C₄	—	2	16	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	C₂⁴×C₁₃	1	8	yes	—
239	C₂²⋊C₄	—	2	16	no	—
240	C₄²	—	2	16	no	—
241	C₄²	C₂×D₄×C₁₃	1	4	yes	—
242	C₄²	C₂×D₄×C₁₃	1	4	yes	—
243	C₄²	C₂×D₄×C₁₃	1	4	yes	—
244	C₄²	C₂⁴×C₁₃	1	4	yes	—
245	C₁₃	C₂×#3255	1	1	yes	characteristic, direct product factor, Sylow
246	C₂³	—	2	2	no	—
247	C₂³	—	2	4	no	—
248	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
249	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
250	Q₈	—	2	8	no	—
251	Q₈	—	2	8	no	—
252	Q₈	—	4	8	no	—
253	Q₈	—	4	16	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	C₁₃×#137	1	8	yes	—
265	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
266	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
267	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
269	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
270	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
271	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₁₃×#537	1	2	yes	—
277	C₂²	C₂×C₁₃×#137	1	4	yes	—
278	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
279	C₄	—	2	4	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	8	no	—
282	C₄	—	4	8	no	—
283	C₄	C₁₃×#537	1	2	yes	—
284	C₄	C₂×C₁₃×#137	1	8	yes	—
285	C₄	C₂×C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
286	C₄	C₂×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
287	C₂	—	4	2	no	—
288	C₂	C₁₃×#3255	1	4	yes	direct product factor
289	C₂	C₂×C₁₃×#537	1	1	yes	characteristic
290	C₂	C₂²×C₁₃×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#3255	C₂	direct (N × H)
3. C₂×C₁₃×#475	C₂	semidirect
5. C₂×C₁₃×#462	C₂	semidirect
20. C₁₃×#475	C₂²	semidirect
22. C₁₃×#462	C₂²	semidirect
64. C₂×#3255	C₁₃	direct (N × H)
102. #3255	C₂×C₁₃	direct (N × H)
103. C₂×#475	C₂×C₁₃	semidirect
105. C₂×#462	C₂×C₁₃	semidirect
147. #475	C₂²×C₁₃	semidirect
149. #462	C₂²×C₁₃	semidirect
206. C₂×C₁₃	#3255	direct (N × H)
245. C₁₃	C₂×#3255	direct (N × H)
288. C₂	C₁₃×#3255	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×#533	2
C₁₃×#3255	2
C₂×C₁₃×#475	2
C₂×C₁₃×#470	2
C₂×C₁₃×#462	2
C₂×C₁₃×#423	2
C₂×#3255	13

Subgroups — C2×C13×#3255 #15747244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#3255 #15747244