Subgroups — C₂×C₁₃×#2251 #15720054

Order 3328.

826 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	C₂²×C₁₃×Q₃₂	C₂	1	2	yes	maximal, complemented
2	C₂²×C₁₃×SD₃₂	C₂	1	2	yes	maximal
3	C₂×C₁₃×#495	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2251	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×Q₁₆	—	2	1	no	—
8	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×SD₁₆	—	2	1	no	—
11	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₁₃×#124	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	—	2	8	no	—
15	C₂×C₁₃×Q₃₂	C₂²	1	8	yes	complemented
16	C₂×C₁₃×SD₃₂	—	2	8	no	—
17	C₂×C₁₃×SD₃₂	C₂²	1	8	yes	—
18	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
19	C₁₃×#495	—	2	2	no	—
20	C₁₃×#495	C₂²	1	4	yes	—
21	C₁₃×#494	—	2	2	no	—
22	C₁₃×#494	C₂²	1	4	yes	complemented
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#346	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₁₃	—	2	1	no	—
26	C₂²×Q₈×C₁₃	—	2	2	no	—
27	C₂²×D₄×C₁₃	—	2	1	no	—
28	C₂×C₁₃×Q₁₆	—	2	4	no	—
29	C₂×C₁₃×Q₁₆	—	2	4	no	—
30	C₂×C₁₃×Q₁₆	—	2	4	no	—
31	C₂×C₁₃×Q₁₆	—	2	2	no	—
32	C₂×C₁₃×Q₁₆	—	2	4	no	—
33	C₂×C₁₃×Q₁₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	—	4	4	no	—
35	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
36	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
37	C₂×C₁₃×SD₁₆	—	2	4	no	—
38	C₂×C₁₃×SD₁₆	—	4	4	no	—
39	C₂×C₁₃×D₈	—	2	4	no	—
40	C₂×C₁₃×D₈	—	2	1	no	—
41	C₂×C₁₃×D₈	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊Q₈	—	2	4	no	—
45	C₁₃×C₄⋊Q₈	—	4	2	no	—
46	C₁₃×#124	—	2	4	no	—
47	C₁₃×#124	—	4	2	no	—
48	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
49	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
50	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
51	C₁₃×Q₃₂	—	2	16	no	—
52	C₁₃×SD₃₂	—	2	16	no	—
53	C₂×C₁₃×C₁₆	D₄	1	8	yes	—
54	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
55	C₄×C₈×C₁₃	D₄	1	2	yes	—
56	C₄×C₈×C₁₃	C₂³	1	2	yes	—
57	C₂×#2251	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	—	4	1	no	—
59	C₂×Q₈×C₁₃	—	2	2	no	—
60	C₂×Q₈×C₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₃	—	4	1	no	—
62	C₂×Q₈×C₁₃	—	4	2	no	—
63	C₂×Q₈×C₁₃	—	4	4	no	—
64	C₂×Q₈×C₁₃	—	4	2	no	—
65	C₂×Q₈×C₁₃	—	4	2	no	—
66	C₂×Q₈×C₁₃	—	4	4	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	4	2	no	—
69	C₂×D₄×C₁₃	—	4	2	no	—
70	C₂×D₄×C₁₃	—	4	1	no	—
71	C₂²×C₄×C₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	D₈	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₁₃×Q₁₆	—	2	4	no	—
76	C₁₃×Q₁₆	—	2	4	no	—
77	C₁₃×Q₁₆	—	2	4	no	—
78	C₁₃×Q₁₆	—	4	4	no	—
79	C₁₃×Q₁₆	—	4	2	no	—
80	C₁₃×Q₁₆	—	4	8	no	—
81	C₁₃×SD₁₆	—	4	8	no	—
82	C₁₃×D₈	—	2	4	no	—
83	C₁₃×D₈	—	4	2	no	—
84	C₂×C₈×C₁₃	D₈	1	4	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	—	4	8	no	—
90	C₁₃×C₂²⋊C₄	—	4	8	no	—
91	C₄²×C₁₃	D₈	1	2	yes	—
92	C₄²×C₁₃	C₂×D₄	1	2	yes	—
93	C₁₃×C₁₆	C₂×D₄	1	8	yes	—
94	C₂²×Q₃₂	C₂×C₁₃	1	2	yes	complemented
95	C₂²×SD₃₂	C₂×C₁₃	1	2	yes	—
96	C₂×#495	C₂×C₁₃	1	1	yes	characteristic
97	C₂×#494	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#2251	C₂×C₁₃	1	8	yes	direct product factor
99	C₂×#346	C₂×C₁₃	1	1	yes	characteristic, complemented
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	—	8	2	no	—
102	C₂³×C₁₃	—	8	4	no	—
103	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
104	Q₈×C₁₃	—	4	4	no	—
105	Q₈×C₁₃	—	4	2	no	—
106	Q₈×C₁₃	—	4	2	no	—
107	Q₈×C₁₃	—	4	4	no	—
108	Q₈×C₁₃	—	8	2	no	—
109	Q₈×C₁₃	—	8	4	no	—
110	D₄×C₁₃	—	4	2	no	—
111	D₄×C₁₃	—	4	2	no	—
112	D₄×C₁₃	—	8	2	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	8	2	no	—
117	C₂×C₄×C₁₃	—	8	4	no	—
118	C₂×C₄×C₁₃	#127	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
120	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₁₃	#127	1	2	yes	—
124	C₈×C₁₃	C₂×D₈	1	4	yes	—
125	C₈×C₁₃	C₂²×D₄	1	2	yes	—
126	C₂²×Q₁₆	—	2	1	no	—
127	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
128	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
129	C₂²×SD₁₆	—	2	1	no	—
130	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×C₄⋊Q₈	—	2	1	no	—
132	C₂×#124	—	2	1	no	—
133	C₂×Q₃₂	—	2	8	no	—
134	C₂×Q₃₂	C₂²×C₁₃	1	8	yes	complemented
135	C₂×SD₃₂	—	2	8	no	—
136	C₂×SD₃₂	C₂²×C₁₃	1	8	yes	—
137	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
138	#495	—	2	2	no	—
139	#495	C₂²×C₁₃	1	4	yes	—
140	#494	—	2	2	no	—
141	#494	C₂²×C₁₃	1	4	yes	complemented
142	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
143	#346	C₂²×C₁₃	1	4	yes	complemented
144	C₂²×C₁₃	—	8	2	no	—
145	C₂²×C₁₃	—	8	4	no	—
146	C₂²×C₁₃	—	16	4	no	—
147	C₂²×C₁₃	#493	1	2	yes	—
148	C₂²×C₁₃	#510	1	4	yes	—
149	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₁₃	—	2	4	no	—
151	C₄×C₁₃	—	8	2	no	—
152	C₄×C₁₃	—	8	4	no	—
153	C₄×C₁₃	#493	1	2	yes	—
154	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
155	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
156	C₂²×Q₈	—	2	1	no	—
157	C₂²×Q₈	—	2	2	no	—
158	C₂²×D₄	—	2	1	no	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	2	4	no	—
163	C₂×Q₁₆	—	2	2	no	—
164	C₂×Q₁₆	—	2	1	no	—
165	C₂×Q₁₆	—	4	4	no	—
166	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
167	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
168	C₂×SD₁₆	—	2	4	no	—
169	C₂×SD₁₆	—	4	4	no	—
170	C₂×D₈	—	2	4	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	C₂³×C₁₃	1	2	yes	—
173	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
175	C₄⋊Q₈	—	2	4	no	—
176	C₄⋊Q₈	—	4	2	no	—
177	#124	—	2	4	no	—
178	#124	—	4	2	no	—
179	C₂×C₄⋊C₄	—	4	2	no	—
180	C₂×C₂²⋊C₄	—	4	2	no	—
181	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
182	Q₃₂	—	2	16	no	—
183	SD₃₂	—	2	16	no	—
184	C₂×C₁₆	D₄×C₁₃	1	8	yes	—
185	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
186	C₄×C₈	D₄×C₁₃	1	2	yes	—
187	C₄×C₈	C₂³×C₁₃	1	2	yes	—
188	C₂×C₁₃	—	16	2	no	—
189	C₂×C₁₃	#2251	1	4	yes	direct product factor
190	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
191	C₂×C₁₃	C₂×#510	1	2	yes	—
192	C₂⁴	—	4	1	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂×Q₈	—	4	4	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₁₃×D₈	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
209	Q₁₆	—	2	4	no	—
210	Q₁₆	—	2	4	no	—
211	Q₁₆	—	2	4	no	—
212	Q₁₆	—	4	4	no	—
213	Q₁₆	—	4	2	no	—
214	Q₁₆	—	4	8	no	—
215	SD₁₆	—	4	8	no	—
216	D₈	—	2	4	no	—
217	D₈	—	4	2	no	—
218	C₂×C₈	C₁₃×D₈	1	4	yes	—
219	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
221	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
222	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
223	C₄⋊C₄	—	4	8	no	—
224	C₂²⋊C₄	—	4	8	no	—
225	C₄²	C₁₃×D₈	1	2	yes	—
226	C₄²	C₂×D₄×C₁₃	1	2	yes	—
227	C₁₆	C₂×D₄×C₁₃	1	8	yes	—
228	C₁₃	C₂×#2251	1	1	yes	characteristic, direct product factor, Sylow
229	C₂³	—	4	2	no	—
230	C₂³	—	8	2	no	—
231	C₂³	—	8	4	no	—
232	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
233	Q₈	—	4	4	no	—
234	Q₈	—	4	2	no	—
235	Q₈	—	4	2	no	—
236	Q₈	—	4	4	no	—
237	Q₈	—	8	2	no	—
238	Q₈	—	8	4	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	8	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	C₁₃×#127	1	2	yes	—
248	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
249	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
250	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
251	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
252	C₈	C₁₃×#127	1	2	yes	—
253	C₈	C₂×C₁₃×D₈	1	4	yes	—
254	C₈	C₂²×D₄×C₁₃	1	2	yes	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	16	4	no	—
258	C₂²	C₁₃×#493	1	2	yes	—
259	C₂²	C₁₃×#510	1	4	yes	—
260	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	8	2	no	—
263	C₄	—	8	4	no	—
264	C₄	C₁₃×#493	1	2	yes	—
265	C₄	C₂×C₁₃×#127	1	1	yes	characteristic
266	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
267	C₂	—	16	2	no	—
268	C₂	C₁₃×#2251	1	4	yes	direct product factor
269	C₂	C₂×C₁₃×#493	1	1	yes	characteristic
270	C₂	C₂×C₁₃×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×Q₃₂	C₂	semidirect
4. C₂×C₁₃×#494	C₂	semidirect
5. C₁₃×#2251	C₂	direct (N × H)
6. C₂×C₁₃×#346	C₂	semidirect
15. C₂×C₁₃×Q₃₂	C₂²	semidirect
22. C₁₃×#494	C₂²	semidirect
24. C₁₃×#346	C₂²	semidirect
57. C₂×#2251	C₁₃	direct (N × H)
94. C₂²×Q₃₂	C₂×C₁₃	semidirect
97. C₂×#494	C₂×C₁₃	semidirect
98. #2251	C₂×C₁₃	direct (N × H)
99. C₂×#346	C₂×C₁₃	semidirect
134. C₂×Q₃₂	C₂²×C₁₃	semidirect
141. #494	C₂²×C₁₃	semidirect
143. #346	C₂²×C₁₃	semidirect
189. C₂×C₁₃	#2251	direct (N × H)
228. C₁₃	C₂×#2251	direct (N × H)
268. C₂	C₁₃×#2251	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×Q₃₂	2
C₂²×C₁₃×SD₃₂	2
C₂×C₁₃×#495	2
C₂×C₁₃×#494	2
C₁₃×#2251	2
C₂×C₁₃×#346	2
C₂×#2251	13

Subgroups — C2×C13×#2251 #15720054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2251 #15720054