Subgroups — C₂×C₁₃×#3293 #15747443

Order 3328.

2598 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#578	C₂	1	4	yes	maximal, complemented
2	C₁₃×#3293	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#473	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#456	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₃×#443	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×2₋¹⁺⁴	C₂²	1	4	yes	—
8	C₁₃×#578	—	2	16	no	—
9	C₁₃×#578	C₂²	1	32	yes	complemented
10	C₂×C₁₃×#137	—	2	8	no	—
11	C₂×C₁₃×#137	C₂²	1	8	yes	—
12	C₂×C₁₃×#135	—	2	4	no	—
13	C₂×C₁₃×#135	C₂²	1	2	yes	—
14	C₂²×C₁₃×Q₁₆	—	2	4	no	—
15	C₂²×C₁₃×Q₁₆	C₂²	1	4	yes	—
16	C₂×C₁₃×#131	C₂²	1	4	yes	—
17	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₁₃×#494	C₂²	1	8	yes	complemented
19	C₁₃×#473	C₂²	1	32	yes	complemented
20	C₁₃×#456	C₂²	1	32	yes	complemented
21	C₁₃×#443	C₂²	1	16	yes	complemented
22	C₂×C₁₃×#108	C₂²	1	2	yes	—
23	C₂×C₁₃×#104	C₂²	1	4	yes	—
24	C₂×C₁₃×#101	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×2₋¹⁺⁴	—	2	16	no	—
27	C₁₃×2₋¹⁺⁴	—	4	4	no	—
28	C₁₃×2₋¹⁺⁴	C₂³	1	16	yes	—
29	C₂×C₁₃×#41	—	2	8	no	—
30	C₂×C₁₃×#41	—	4	4	no	—
31	C₂×C₁₃×#41	C₂³	1	4	yes	—
32	C₂²×Q₈×C₁₃	—	2	8	no	—
33	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
34	C₁₃×#137	—	2	16	no	—
35	C₁₃×#137	—	2	64	no	—
36	C₁₃×#137	C₂³	1	32	yes	—
37	C₁₃×#135	—	2	32	no	—
38	C₁₃×#135	D₄	1	8	yes	—
39	C₁₃×#135	C₂³	1	8	yes	—
40	C₂×C₁₃×Q₁₆	—	2	8	no	—
41	C₂×C₁₃×Q₁₆	—	2	32	no	—
42	C₂×C₁₃×Q₁₆	—	2	4	no	—
43	C₂×C₁₃×Q₁₆	—	2	4	no	—
44	C₂×C₁₃×Q₁₆	—	2	8	no	—
45	C₂×C₁₃×Q₁₆	D₄	1	2	yes	—
46	C₂×C₁₃×Q₁₆	C₂³	1	16	yes	—
47	C₂×C₁₃×SD₁₆	—	2	8	no	—
48	C₂×C₁₃×SD₁₆	D₄	1	4	yes	—
49	C₂×C₁₃×D₈	D₄	1	2	yes	—
50	C₁₃×#131	—	2	8	no	—
51	C₁₃×#131	C₂³	1	16	yes	—
52	C₂×C₁₃×M₁₆	—	2	4	no	—
53	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
54	C₂²×C₈×C₁₃	—	2	4	no	—
55	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
56	C₁₃×C₄⋊Q₈	—	4	2	no	—
57	C₁₃×C₄⋊Q₈	C₂³	1	8	yes	—
58	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
59	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×#108	C₂³	1	8	yes	—
61	C₁₃×#104	C₂³	1	16	yes	—
62	C₁₃×#101	C₂³	1	16	yes	—
63	C₄×C₈×C₁₃	C₂³	1	4	yes	—
64	C₂×#3293	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₁₃×#41	—	2	4	no	—
66	C₁₃×#41	—	2	4	no	—
67	C₁₃×#41	—	2	32	no	—
68	C₁₃×#41	—	4	16	no	—
69	C₁₃×#41	—	4	16	no	—
70	C₁₃×#41	—	4	16	no	—
71	C₁₃×#41	C₂×D₄	1	8	yes	—
72	C₁₃×#41	C₂⁴	1	8	yes	—
73	C₂×Q₈×C₁₃	—	2	4	no	—
74	C₂×Q₈×C₁₃	—	2	32	no	—
75	C₂×Q₈×C₁₃	—	2	8	no	—
76	C₂×Q₈×C₁₃	—	2	8	no	—
77	C₂×Q₈×C₁₃	—	4	4	no	—
78	C₂×Q₈×C₁₃	—	4	16	no	—
79	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
80	C₂×Q₈×C₁₃	C₂⁴	1	8	yes	—
81	C₂×D₄×C₁₃	—	2	4	no	—
82	C₂×D₄×C₁₃	—	4	4	no	—
83	C₂×D₄×C₁₃	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₁₃	—	2	1	no	—
85	C₂²×C₄×C₁₃	—	2	4	no	—
86	C₂²×C₄×C₁₃	—	2	4	no	—
87	C₂²×C₄×C₁₃	—	4	8	no	—
88	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₁₃×Q₁₆	—	2	32	no	—
90	C₁₃×Q₁₆	—	2	16	no	—
91	C₁₃×Q₁₆	—	2	16	no	—
92	C₁₃×Q₁₆	C₂×D₄	1	8	yes	—
93	C₁₃×SD₁₆	—	2	32	no	—
94	C₁₃×SD₁₆	C₂×D₄	1	16	yes	—
95	C₁₃×D₈	C₂×D₄	1	8	yes	—
96	C₁₃×M₁₆	—	2	4	no	—
97	C₁₃×M₁₆	—	2	16	no	—
98	C₁₃×M₁₆	C₂⁴	1	8	yes	—
99	C₂×C₈×C₁₃	—	2	4	no	—
100	C₂×C₈×C₁₃	—	2	16	no	—
101	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
104	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
105	C₁₃×C₄⋊C₄	—	4	8	no	—
106	C₄²×C₁₃	—	2	1	no	—
107	C₄²×C₁₃	C₂⁴	1	2	yes	—
108	C₂×#578	C₂×C₁₃	1	4	yes	complemented
109	#3293	C₂×C₁₃	1	16	yes	direct product factor
110	C₂×#494	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂×#473	C₂×C₁₃	1	4	yes	complemented
112	C₂×#456	C₂×C₁₃	1	4	yes	complemented
113	C₂×#443	C₂×C₁₃	1	2	yes	complemented
114	C₂³×C₁₃	—	2	4	no	—
115	C₂³×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
116	Q₈×C₁₃	—	2	4	no	—
117	Q₈×C₁₃	—	2	16	no	—
118	Q₈×C₁₃	—	4	16	no	—
119	Q₈×C₁₃	—	4	8	no	—
120	Q₈×C₁₃	C₂²×D₄	1	8	yes	—
121	D₄×C₁₃	—	2	8	no	—
122	D₄×C₁₃	—	2	8	no	—
123	D₄×C₁₃	—	2	4	no	—
124	D₄×C₁₃	—	4	16	no	—
125	D₄×C₁₃	C₂²×D₄	1	8	yes	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	8	no	—
130	C₂×C₄×C₁₃	—	2	8	no	—
131	C₂×C₄×C₁₃	—	2	8	no	—
132	C₂×C₄×C₁₃	—	4	32	no	—
133	C₂×C₄×C₁₃	—	4	4	no	—
134	C₂×C₄×C₁₃	—	4	4	no	—
135	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
137	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
139	C₈×C₁₃	—	2	8	no	—
140	C₈×C₁₃	C₂²×D₄	1	4	yes	—
141	C₈×C₁₃	C₂²×D₄	1	4	yes	—
142	C₂×2₋¹⁺⁴	C₂²×C₁₃	1	4	yes	—
143	#578	—	2	16	no	—
144	#578	C₂²×C₁₃	1	32	yes	complemented
145	C₂×#137	—	2	8	no	—
146	C₂×#137	C₂²×C₁₃	1	8	yes	—
147	C₂×#135	—	2	4	no	—
148	C₂×#135	C₂²×C₁₃	1	2	yes	—
149	C₂²×Q₁₆	—	2	4	no	—
150	C₂²×Q₁₆	C₂²×C₁₃	1	4	yes	—
151	C₂×#131	C₂²×C₁₃	1	4	yes	—
152	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
153	#494	C₂²×C₁₃	1	8	yes	complemented
154	#473	C₂²×C₁₃	1	32	yes	complemented
155	#456	C₂²×C₁₃	1	32	yes	complemented
156	#443	C₂²×C₁₃	1	16	yes	complemented
157	C₂×#108	C₂²×C₁₃	1	2	yes	—
158	C₂×#104	C₂²×C₁₃	1	4	yes	—
159	C₂×#101	C₂²×C₁₃	1	4	yes	—
160	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	8	no	—
163	C₂²×C₁₃	—	4	8	no	—
164	C₂²×C₁₃	D₄²	1	1	yes	characteristic
165	C₂²×C₁₃	D₄²	1	1	yes	characteristic
166	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
167	C₄×C₁₃	—	2	4	no	—
168	C₄×C₁₃	—	2	8	no	—
169	C₄×C₁₃	—	4	8	no	—
170	C₄×C₁₃	D₄²	1	2	yes	—
171	C₄×C₁₃	C₂³×D₄	1	2	yes	—
172	2₋¹⁺⁴	—	2	16	no	—
173	2₋¹⁺⁴	—	4	4	no	—
174	2₋¹⁺⁴	C₂³×C₁₃	1	16	yes	—
175	C₂×#41	—	2	8	no	—
176	C₂×#41	—	4	4	no	—
177	C₂×#41	C₂³×C₁₃	1	4	yes	—
178	C₂²×Q₈	—	2	8	no	—
179	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
180	#137	—	2	16	no	—
181	#137	—	2	64	no	—
182	#137	C₂³×C₁₃	1	32	yes	—
183	#135	—	2	32	no	—
184	#135	D₄×C₁₃	1	8	yes	—
185	#135	C₂³×C₁₃	1	8	yes	—
186	C₂×Q₁₆	—	2	8	no	—
187	C₂×Q₁₆	—	2	32	no	—
188	C₂×Q₁₆	—	2	4	no	—
189	C₂×Q₁₆	—	2	4	no	—
190	C₂×Q₁₆	—	2	8	no	—
191	C₂×Q₁₆	D₄×C₁₃	1	2	yes	—
192	C₂×Q₁₆	C₂³×C₁₃	1	16	yes	—
193	C₂×SD₁₆	—	2	8	no	—
194	C₂×SD₁₆	D₄×C₁₃	1	4	yes	—
195	C₂×D₈	D₄×C₁₃	1	2	yes	—
196	#131	—	2	8	no	—
197	#131	C₂³×C₁₃	1	16	yes	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
202	C₄⋊Q₈	—	4	2	no	—
203	C₄⋊Q₈	C₂³×C₁₃	1	8	yes	—
204	C₂×C₄⋊C₄	—	4	2	no	—
205	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
206	#108	C₂³×C₁₃	1	8	yes	—
207	#104	C₂³×C₁₃	1	16	yes	—
208	#101	C₂³×C₁₃	1	16	yes	—
209	C₄×C₈	C₂³×C₁₃	1	4	yes	—
210	C₂×C₁₃	—	2	1	no	—
211	C₂×C₁₃	—	2	1	no	—
212	C₂×C₁₃	—	4	8	no	—
213	C₂×C₁₃	#3293	1	2	yes	direct product factor
214	C₂×C₁₃	C₂×D₄²	1	1	yes	characteristic
215	#41	—	2	4	no	—
216	#41	—	2	4	no	—
217	#41	—	2	32	no	—
218	#41	—	4	16	no	—
219	#41	—	4	16	no	—
220	#41	—	4	16	no	—
221	#41	C₂×D₄×C₁₃	1	8	yes	—
222	#41	C₂⁴×C₁₃	1	8	yes	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	2	32	no	—
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	2	8	no	—
227	C₂×Q₈	—	4	4	no	—
228	C₂×Q₈	—	4	16	no	—
229	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
230	C₂×Q₈	C₂⁴×C₁₃	1	8	yes	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	C₂×D₄×C₁₃	1	4	yes	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	4	8	no	—
238	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
239	Q₁₆	—	2	32	no	—
240	Q₁₆	—	2	16	no	—
241	Q₁₆	—	2	16	no	—
242	Q₁₆	C₂×D₄×C₁₃	1	8	yes	—
243	SD₁₆	—	2	32	no	—
244	SD₁₆	C₂×D₄×C₁₃	1	16	yes	—
245	D₈	C₂×D₄×C₁₃	1	8	yes	—
246	M₁₆	—	2	4	no	—
247	M₁₆	—	2	16	no	—
248	M₁₆	C₂⁴×C₁₃	1	8	yes	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	16	no	—
251	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
253	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
254	C₂×C₈	C₂⁴×C₁₃	1	4	yes	—
255	C₄⋊C₄	—	4	8	no	—
256	C₄²	—	2	1	no	—
257	C₄²	C₂⁴×C₁₃	1	2	yes	—
258	C₁₃	C₂×#3293	1	1	yes	characteristic, direct product factor, Sylow
259	C₂³	—	2	4	no	—
260	C₂³	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
261	Q₈	—	2	4	no	—
262	Q₈	—	2	16	no	—
263	Q₈	—	4	16	no	—
264	Q₈	—	4	8	no	—
265	Q₈	C₂²×D₄×C₁₃	1	8	yes	—
266	D₄	—	2	8	no	—
267	D₄	—	2	8	no	—
268	D₄	—	2	4	no	—
269	D₄	—	4	16	no	—
270	D₄	C₂²×D₄×C₁₃	1	8	yes	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	4	32	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
281	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
282	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
284	C₈	—	2	8	no	—
285	C₈	C₂²×D₄×C₁₃	1	4	yes	—
286	C₈	C₂²×D₄×C₁₃	1	4	yes	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	4	8	no	—
290	C₂²	D₄²×C₁₃	1	1	yes	characteristic
291	C₂²	D₄²×C₁₃	1	1	yes	characteristic
292	C₂²	C₂×C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
293	C₄	—	2	4	no	—
294	C₄	—	2	8	no	—
295	C₄	—	4	8	no	—
296	C₄	D₄²×C₁₃	1	2	yes	—
297	C₄	C₂³×D₄×C₁₃	1	2	yes	—
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	4	8	no	—
301	C₂	C₁₃×#3293	1	2	yes	direct product factor
302	C₂	C₂×D₄²×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#578	C₂	semidirect
2. C₁₃×#3293	C₂	direct (N × H)
3. C₂×C₁₃×#494	C₂	semidirect
4. C₂×C₁₃×#473	C₂	semidirect
5. C₂×C₁₃×#456	C₂	semidirect
6. C₂×C₁₃×#443	C₂	semidirect
9. C₁₃×#578	C₂²	semidirect
18. C₁₃×#494	C₂²	semidirect
19. C₁₃×#473	C₂²	semidirect
20. C₁₃×#456	C₂²	semidirect
21. C₁₃×#443	C₂²	semidirect
64. C₂×#3293	C₁₃	direct (N × H)
108. C₂×#578	C₂×C₁₃	semidirect
109. #3293	C₂×C₁₃	direct (N × H)
110. C₂×#494	C₂×C₁₃	semidirect
111. C₂×#473	C₂×C₁₃	semidirect
112. C₂×#456	C₂×C₁₃	semidirect
113. C₂×#443	C₂×C₁₃	semidirect
144. #578	C₂²×C₁₃	semidirect
153. #494	C₂²×C₁₃	semidirect
154. #473	C₂²×C₁₃	semidirect
155. #456	C₂²×C₁₃	semidirect
156. #443	C₂²×C₁₃	semidirect
213. C₂×C₁₃	#3293	direct (N × H)
258. C₁₃	C₂×#3293	direct (N × H)
301. C₂	C₁₃×#3293	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×#578	2
C₁₃×#3293	2
C₂×C₁₃×#494	2
C₂×C₁₃×#473	2
C₂×C₁₃×#456	2
C₂×C₁₃×#443	2
C₂×#3293	13

Subgroups — C2×C13×#3293 #15747443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#3293 #15747443