Subgroups — C₁₃×#33769 #15720108

Order 3328.

576 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3061	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2236	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2232	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2220	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2189	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2186	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#135	C₂²	1	2	yes	—
11	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
12	C₁₃×#537	—	2	2	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₁₃×#468	—	2	2	no	—
19	C₁₃×#468	—	2	2	no	—
20	C₁₃×#467	C₂²	1	2	yes	—
21	C₁₃×#467	C₂²	1	2	yes	—
22	C₁₃×#466	C₂²	1	2	yes	—
23	C₁₃×#466	C₂²	1	2	yes	—
24	C₂×C₁₃×#107	C₂²	1	2	yes	—
25	C₁₃×#417	—	2	2	no	—
26	C₁₃×#366	C₂²	1	1	yes	characteristic
27	C₁₃×#366	C₂²	1	1	yes	characteristic
28	C₁₃×#366	C₂²	1	2	yes	—
29	C₁₃×#358	C₂²	1	2	yes	—
30	C₁₃×#358	C₂²	1	2	yes	—
31	C₁₃×#357	C₂²	1	2	yes	—
32	C₁₃×#357	C₂²	1	2	yes	—
33	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₂×C₁₃×#41	—	2	2	no	—
36	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
37	C₁₃×#135	—	4	2	no	—
38	C₁₃×#135	—	4	2	no	—
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₁₃×SD₁₆	—	2	2	no	—
41	C₂×C₁₃×D₈	C₂³	1	2	yes	—
42	C₁₃×#131	—	4	1	no	—
43	C₁₃×#131	—	4	1	no	—
44	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₂²⋊Q₈	—	2	2	no	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₁₃×C₂²⋊Q₈	—	4	2	no	—
51	C₁₃×#121	—	2	2	no	—
52	C₁₃×#121	—	2	2	no	—
53	C₁₃×#121	—	4	2	no	—
54	C₁₃×#121	—	4	2	no	—
55	C₁₃×#121	—	4	2	no	—
56	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×Q₃₂	—	4	1	no	—
58	C₁₃×SD₃₂	—	4	2	no	—
59	C₁₃×D₁₆	—	4	1	no	—
60	C₁₃×M₃₂	—	4	1	no	—
61	C₂×C₁₃×C₁₆	—	4	1	no	—
62	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
64	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
65	C₁₃×#107	—	2	2	no	—
66	C₁₃×#107	C₂³	1	2	yes	—
67	C₁₃×#107	C₂³	1	2	yes	—
68	C₁₃×#103	—	2	2	no	—
69	C₁₃×#103	—	2	2	no	—
70	C₁₃×#102	—	2	2	no	—
71	C₁₃×#102	—	2	2	no	—
72	#33769	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₁₃×#41	—	4	1	no	—
74	C₁₃×#41	—	4	1	no	—
75	C₁₃×#41	—	4	4	no	—
76	C₁₃×#41	—	4	4	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	4	2	no	—
81	C₂×D₄×C₁₃	—	4	2	no	—
82	C₂×D₄×C₁₃	—	4	2	no	—
83	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
84	C₂×D₄×C₁₃	D₈	1	2	yes	—
85	C₂²×C₄×C₁₃	—	4	2	no	—
86	C₂²×C₄×C₁₃	—	4	2	no	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₁₃×Q₁₆	—	2	2	no	—
90	C₁₃×Q₁₆	—	2	2	no	—
91	C₁₃×SD₁₆	—	4	4	no	—
92	C₁₃×D₈	—	2	2	no	—
93	C₁₃×D₈	—	2	2	no	—
94	C₁₃×M₁₆	—	4	2	no	—
95	C₁₃×M₁₆	—	4	1	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	—	2	1	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	4	2	no	—
106	C₁₃×C₄⋊C₄	—	4	2	no	—
107	C₁₃×C₄⋊C₄	—	4	2	no	—
108	C₁₃×C₂²⋊C₄	—	4	2	no	—
109	C₁₃×C₂²⋊C₄	—	4	2	no	—
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	—	4	2	no	—
112	C₁₃×C₁₆	—	2	1	no	—
113	C₁₃×C₁₆	—	2	1	no	—
114	C₁₃×C₁₆	—	2	2	no	—
115	#3061	C₂×C₁₃	1	2	yes	complemented
116	#2236	C₂×C₁₃	1	2	yes	complemented
117	#2232	C₂×C₁₃	1	2	yes	complemented
118	#2220	C₂×C₁₃	1	2	yes	complemented
119	#2216	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#2215	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#2189	C₂×C₁₃	1	2	yes	complemented
122	#2186	C₂×C₁₃	1	2	yes	complemented
123	#2113	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
126	C₂³×C₁₃	C₂×D₈	1	2	yes	—
127	Q₈×C₁₃	—	4	1	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	4	2	no	—
130	D₄×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	4	1	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	8	4	no	—
135	D₄×C₁₃	—	16	2	no	—
136	D₄×C₁₃	—	16	2	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	8	4	no	—
145	C₂×C₄×C₁₃	—	16	2	no	—
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	2	no	—
151	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
152	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×#135	C₂²×C₁₃	1	2	yes	—
154	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
155	#537	—	2	2	no	—
156	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
158	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
160	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
161	#468	—	2	2	no	—
162	#468	—	2	2	no	—
163	#467	C₂²×C₁₃	1	2	yes	—
164	#467	C₂²×C₁₃	1	2	yes	—
165	#466	C₂²×C₁₃	1	2	yes	—
166	#466	C₂²×C₁₃	1	2	yes	—
167	C₂×#107	C₂²×C₁₃	1	2	yes	—
168	#417	—	2	2	no	—
169	#366	C₂²×C₁₃	1	1	yes	characteristic
170	#366	C₂²×C₁₃	1	1	yes	characteristic
171	#366	C₂²×C₁₃	1	2	yes	—
172	#358	C₂²×C₁₃	1	2	yes	—
173	#358	C₂²×C₁₃	1	2	yes	—
174	#357	C₂²×C₁₃	1	2	yes	—
175	#357	C₂²×C₁₃	1	2	yes	—
176	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
177	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	2	2	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	—	8	2	no	—
186	C₂²×C₁₃	—	8	2	no	—
187	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	#534	1	1	yes	characteristic
194	C₄×C₁₃	#575	1	1	yes	characteristic
195	C₂×#41	—	2	2	no	—
196	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
197	#135	—	4	2	no	—
198	#135	—	4	2	no	—
199	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
200	C₂×SD₁₆	—	2	2	no	—
201	C₂×D₈	C₂³×C₁₃	1	2	yes	—
202	#131	—	4	1	no	—
203	#131	—	4	1	no	—
204	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
205	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
207	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
208	C₂²⋊Q₈	—	2	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	4	2	no	—
211	#121	—	2	2	no	—
212	#121	—	2	2	no	—
213	#121	—	4	2	no	—
214	#121	—	4	2	no	—
215	#121	—	4	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	Q₃₂	—	4	1	no	—
218	SD₃₂	—	4	2	no	—
219	D₁₆	—	4	1	no	—
220	M₃₂	—	4	1	no	—
221	C₂×C₁₆	—	4	1	no	—
222	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
223	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
224	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
225	#107	—	2	2	no	—
226	#107	C₂³×C₁₃	1	2	yes	—
227	#107	C₂³×C₁₃	1	2	yes	—
228	#103	—	2	2	no	—
229	#103	—	2	2	no	—
230	#102	—	2	2	no	—
231	#102	—	2	2	no	—
232	C₂×C₁₃	—	4	2	no	—
233	C₂×C₁₃	—	4	1	no	—
234	C₂×C₁₃	—	16	2	no	—
235	C₂×C₁₃	#3186	1	1	yes	characteristic
236	C₂×C₁₃	#3415	1	1	yes	characteristic
237	C₂×C₁₃	#3417	1	1	yes	characteristic
238	#41	—	4	1	no	—
239	#41	—	4	1	no	—
240	#41	—	4	4	no	—
241	#41	—	4	4	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
249	C₂×D₄	C₁₃×D₈	1	2	yes	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	2	2	no	—
256	SD₁₆	—	4	4	no	—
257	D₈	—	2	2	no	—
258	D₈	—	2	2	no	—
259	M₁₆	—	4	2	no	—
260	M₁₆	—	4	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
267	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₁₆	—	2	1	no	—
278	C₁₆	—	2	1	no	—
279	C₁₆	—	2	2	no	—
280	C₁₃	#33769	1	1	yes	characteristic, direct product factor, Sylow
281	C₂³	—	4	2	no	—
282	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
283	C₂³	C₂×C₁₃×D₈	1	2	yes	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	1	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	4	no	—
292	D₄	—	16	2	no	—
293	D₄	—	16	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	16	2	no	—
303	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
304	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	2	no	—
308	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
309	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	C₁₃×#534	1	1	yes	characteristic
326	C₄	C₁₃×#575	1	1	yes	characteristic
327	C₂	—	4	2	no	—
328	C₂	—	4	1	no	—
329	C₂	—	16	2	no	—
330	C₂	C₁₃×#3186	1	1	yes	characteristic
331	C₂	C₁₃×#3415	1	1	yes	characteristic
332	C₂	C₁₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₁₃×#41 ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3061	C₂	semidirect
2. C₁₃×#2236	C₂	semidirect
3. C₁₃×#2232	C₂	semidirect
4. C₁₃×#2220	C₂	semidirect
5. C₁₃×#2216	C₂	semidirect
6. C₁₃×#2215	C₂	semidirect
7. C₁₃×#2189	C₂	semidirect
8. C₁₃×#2186	C₂	semidirect
9. C₁₃×#2113	C₂	semidirect
72. #33769	C₁₃	direct (N × H)
115. #3061	C₂×C₁₃	semidirect
116. #2236	C₂×C₁₃	semidirect
117. #2232	C₂×C₁₃	semidirect
118. #2220	C₂×C₁₃	semidirect
119. #2216	C₂×C₁₃	semidirect
120. #2215	C₂×C₁₃	semidirect
121. #2189	C₂×C₁₃	semidirect
122. #2186	C₂×C₁₃	semidirect
123. #2113	C₂×C₁₃	semidirect
280. C₁₃	#33769	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#3061	2
C₁₃×#2236	2
C₁₃×#2232	2
C₁₃×#2220	2
C₁₃×#2216	2
C₁₃×#2215	2
C₁₃×#2189	2
C₁₃×#2186	2
C₁₃×#2113	2
#33769	13

Subgroups — C13×#33769 #15720108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33769 #15720108