Subgroups — C₁₃×#33774 #15720113

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2248	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2240	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2207	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2187	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
10	C₁₃×#533	—	2	1	no	—
11	C₂×C₁₃×#127	—	2	1	no	—
12	C₂×C₁₃×SD₃₂	C₂²	1	8	yes	complemented
13	C₁₃×#500	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#493	—	2	1	no	—
16	C₁₃×#493	—	2	1	no	—
17	C₁₃×#493	C₂²	1	2	yes	—
18	C₁₃×#493	C₂²	1	2	yes	—
19	C₁₃×#487	—	2	1	no	—
20	C₁₃×#467	—	2	1	no	—
21	C₁₃×#467	C₂²	1	2	yes	—
22	C₄×C₁₃×Q₁₆	—	2	1	no	—
23	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	complemented
24	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
26	C₁₃×#357	C₂²	1	8	yes	complemented
27	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
29	C₂²×D₄×C₁₃	—	2	2	no	—
30	C₂²×D₄×C₁₃	—	4	1	no	—
31	C₂²×D₄×C₁₃	—	4	1	no	—
32	C₂×C₁₃×Q₁₆	—	2	1	no	—
33	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
34	C₂×C₁₃×D₈	—	2	2	no	—
35	C₂×C₁₃×D₈	—	2	2	no	—
36	C₂×C₁₃×D₈	—	2	2	no	—
37	C₂×C₁₃×D₈	—	4	4	no	—
38	C₂×C₁₃×D₈	—	4	4	no	—
39	C₂×C₁₃×D₈	C₂³	1	4	yes	—
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×#127	—	2	2	no	—
45	C₁₃×#127	—	2	2	no	—
46	C₁₃×#127	—	4	1	no	—
47	C₁₃×#127	—	4	1	no	—
48	C₁₃×#125	—	4	1	no	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₁₃×C₂²⋊Q₈	—	4	1	no	—
51	C₄×Q₈×C₁₃	—	2	2	no	—
52	C₄×Q₈×C₁₃	—	4	1	no	—
53	C₁₃×#117	—	4	1	no	—
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₁₃×SD₃₂	—	4	8	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
57	C₁₃×#107	—	2	1	no	—
58	C₁₃×#107	C₂³	1	2	yes	—
59	C₁₃×#103	—	2	2	no	—
60	C₁₃×#103	—	2	2	no	—
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	#33774	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂⁴×C₁₃	—	4	2	no	—
67	C₂×Q₈×C₁₃	—	2	2	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	2	4	no	—
70	C₂×D₄×C₁₃	—	4	4	no	—
71	C₂×D₄×C₁₃	—	4	4	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂×D₄×C₁₃	—	4	2	no	—
74	C₂×D₄×C₁₃	—	4	2	no	—
75	C₂×D₄×C₁₃	—	4	2	no	—
76	C₂×D₄×C₁₃	—	4	2	no	—
77	C₂×D₄×C₁₃	—	8	4	no	—
78	C₂×D₄×C₁₃	—	8	4	no	—
79	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×Q₁₆	—	2	4	no	—
83	C₁₃×Q₁₆	—	4	2	no	—
84	C₁₃×D₈	—	2	8	no	—
85	C₁₃×D₈	—	4	4	no	—
86	C₁₃×D₈	—	4	4	no	—
87	C₁₃×D₈	—	4	4	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₂²⋊C₄	—	4	2	no	—
101	C₄²×C₁₃	—	4	2	no	—
102	C₄²×C₁₃	D₈	1	1	yes	characteristic
103	C₄²×C₁₃	D₈	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₁₃×C₁₆	—	2	4	no	—
107	#3170	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂×#493	C₂×C₁₃	1	1	yes	characteristic
109	#2248	C₂×C₁₃	1	2	yes	complemented
110	#2240	C₂×C₁₃	1	2	yes	complemented
111	#2207	C₂×C₁₃	1	4	yes	complemented
112	#2187	C₂×C₁₃	1	4	yes	complemented
113	#2162	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂³×C₁₃	—	4	4	no	—
115	C₂³×C₁₃	—	8	4	no	—
116	C₂³×C₁₃	—	8	4	no	—
117	C₂³×C₁₃	—	8	4	no	—
118	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₁₃	—	4	2	no	—
120	Q₈×C₁₃	—	4	2	no	—
121	Q₈×C₁₃	—	8	2	no	—
122	D₄×C₁₃	—	4	4	no	—
123	D₄×C₁₃	—	4	4	no	—
124	D₄×C₁₃	—	8	4	no	—
125	D₄×C₁₃	—	8	4	no	—
126	D₄×C₁₃	—	8	4	no	—
127	D₄×C₁₃	—	8	4	no	—
128	D₄×C₁₃	—	8	4	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₁₃	—	2	1	no	—
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
144	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
145	C₂²×D₈	—	2	1	no	—
146	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
147	#533	—	2	1	no	—
148	C₂×#127	—	2	1	no	—
149	C₂×SD₃₂	C₂²×C₁₃	1	8	yes	complemented
150	#500	C₂²×C₁₃	1	1	yes	characteristic, complemented
151	#494	C₂²×C₁₃	1	1	yes	characteristic, complemented
152	#493	—	2	1	no	—
153	#493	—	2	1	no	—
154	#493	C₂²×C₁₃	1	2	yes	—
155	#493	C₂²×C₁₃	1	2	yes	—
156	#487	—	2	1	no	—
157	#467	—	2	1	no	—
158	#467	C₂²×C₁₃	1	2	yes	—
159	C₄×Q₁₆	—	2	1	no	—
160	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	complemented
161	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	complemented
163	#357	C₂²×C₁₃	1	8	yes	complemented
164	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
165	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
166	C₂²×C₁₃	—	2	1	no	—
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	—	8	4	no	—
169	C₂²×C₁₃	—	8	4	no	—
170	C₂²×C₁₃	—	8	4	no	—
171	C₂²×C₁₃	—	16	8	no	—
172	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
173	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	8	2	no	—
177	C₄×C₁₃	—	8	2	no	—
178	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
179	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
180	C₄×C₁₃	#509	1	2	yes	—
181	C₄×C₁₃	#535	1	1	yes	characteristic
182	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂²×D₄	—	2	2	no	—
184	C₂²×D₄	—	4	1	no	—
185	C₂²×D₄	—	4	1	no	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	4	4	no	—
192	C₂×D₈	—	4	4	no	—
193	C₂×D₈	C₂³×C₁₃	1	4	yes	—
194	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	—	2	1	no	—
198	#127	—	2	2	no	—
199	#127	—	2	2	no	—
200	#127	—	4	1	no	—
201	#127	—	4	1	no	—
202	#125	—	4	1	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	4	1	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	4	1	no	—
207	#117	—	4	1	no	—
208	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
209	SD₃₂	—	4	8	no	—
210	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
211	#107	—	2	1	no	—
212	#107	C₂³×C₁₃	1	2	yes	—
213	#103	—	2	2	no	—
214	#103	—	2	2	no	—
215	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
216	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	—	16	4	no	—
221	C₂×C₁₃	#3205	1	1	yes	characteristic
222	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
224	C₂⁴	—	4	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	8	4	no	—
236	C₂×D₄	—	8	4	no	—
237	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
238	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	Q₁₆	—	2	4	no	—
241	Q₁₆	—	4	2	no	—
242	D₈	—	2	8	no	—
243	D₈	—	4	4	no	—
244	D₈	—	4	4	no	—
245	D₈	—	4	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	C₁₃×D₈	1	1	yes	characteristic
261	C₄²	C₁₃×D₈	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
264	C₁₆	—	2	4	no	—
265	C₁₃	#33774	1	1	yes	characteristic, direct product factor, Sylow
266	C₂³	—	4	4	no	—
267	C₂³	—	8	4	no	—
268	C₂³	—	8	4	no	—
269	C₂³	—	8	4	no	—
270	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	Q₈	—	8	2	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	8	4	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	4	no	—
280	D₄	—	8	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₁₃×SD₃₂	1	4	yes	complemented
287	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
288	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
289	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
290	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
291	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
296	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	16	8	no	—
303	C₂²	C₂×C₁₃×SD₃₂	1	2	yes	—
304	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₂×C₁₃×SD₃₂	1	2	yes	—
310	C₄	C₂×C₁₃×SD₃₂	1	2	yes	—
311	C₄	C₁₃×#509	1	2	yes	—
312	C₄	C₁₃×#535	1	1	yes	characteristic
313	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	16	4	no	—
316	C₂	C₁₃×#3205	1	1	yes	characteristic
317	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic
318	C₂	C₂×C₁₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. C₁₃×#3170	C₂	semidirect
3. C₁₃×#2248	C₂	semidirect
4. C₁₃×#2240	C₂	semidirect
5. C₁₃×#2207	C₂	semidirect
6. C₁₃×#2187	C₂	semidirect
7. C₁₃×#2162	C₂	semidirect
12. C₂×C₁₃×SD₃₂	C₂²	semidirect
13. C₁₃×#500	C₂²	semidirect
14. C₁₃×#494	C₂²	semidirect
23. C₄×C₁₃×Q₁₆	C₂²	semidirect
25. C₁₃×C₄⋊C₁₆	C₂²	semidirect
26. C₁₃×#357	C₂²	semidirect
28. C₄×C₁₃×C₁₆	C₂²	semidirect
65. #33774	C₁₃	direct (N × H)
107. #3170	C₂×C₁₃	semidirect
109. #2248	C₂×C₁₃	semidirect
110. #2240	C₂×C₁₃	semidirect
111. #2207	C₂×C₁₃	semidirect
112. #2187	C₂×C₁₃	semidirect
113. #2162	C₂×C₁₃	semidirect
134. C₂×C₄×C₁₃	SD₃₂	semidirect
149. C₂×SD₃₂	C₂²×C₁₃	semidirect
150. #500	C₂²×C₁₃	semidirect
151. #494	C₂²×C₁₃	semidirect
160. C₄×Q₁₆	C₂²×C₁₃	semidirect
162. C₄⋊C₁₆	C₂²×C₁₃	semidirect
163. #357	C₂²×C₁₃	semidirect
165. C₄×C₁₆	C₂²×C₁₃	semidirect
265. C₁₃	#33774	direct (N × H)
286. C₂×C₄	C₁₃×SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#3170	2
C₂×C₁₃×#493	2
C₁₃×#2248	2
C₁₃×#2240	2
C₁₃×#2207	2
C₁₃×#2187	2
C₁₃×#2162	2
#33774	13

Subgroups — C13×#33774 #15720113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33774 #15720113