Subgroups — C₁₃×#62637 #15748976

Order 3328.

2026 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#578	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#3417	C₂	1	8	yes	maximal, complemented
4	C₁₃×#3416	C₂	1	8	yes	maximal, complemented
5	C₁₃×#3414	C₂	1	6	yes	maximal, complemented
6	C₂×C₁₃×#510	C₂	1	6	yes	maximal, complemented
7	C₁₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#585	—	2	1	no	—
9	C₂×C₁₃×2₋¹⁺⁴	—	2	1	no	—
10	C₁₃×#578	—	2	4	no	—
11	C₁₃×#578	C₂²	1	8	yes	complemented
12	C₁₃×#578	C₂²	1	4	yes	complemented
13	C₁₃×#577	—	2	4	no	—
14	C₁₃×#576	C₂²	1	4	yes	complemented
15	C₁₃×#575	—	2	3	no	—
16	C₂×C₁₃×#137	—	2	3	no	—
17	C₂×C₁₃×#135	C₂²	1	6	yes	—
18	C₂×C₁₃×#135	C₂²	1	3	yes	complemented
19	C₂²×C₁₃×Q₁₆	C₂²	1	3	yes	complemented
20	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
21	C₁₃×#510	C₂²	1	12	yes	complemented
22	C₁₃×#510	C₂²	1	24	yes	complemented
23	C₁₃×#510	C₂²	1	12	yes	complemented
24	C₁₃×#509	C₂²	1	4	yes	complemented
25	C₁₃×#509	C₂²	1	12	yes	complemented
26	C₁₃×#508	C₂²	1	24	yes	complemented
27	C₂×C₁₃×Q₃₂	C₂²	1	12	yes	complemented
28	C₂×C₁₃×SD₃₂	C₂²	1	12	yes	complemented
29	C₁₃×#504	C₂²	1	8	yes	complemented
30	C₂×C₁₃×M₃₂	C₂²	1	6	yes	complemented
31	C₁₃×2₋¹⁺⁴	—	2	4	no	—
32	C₁₃×2₋¹⁺⁴	—	2	8	no	—
33	C₁₃×2₋¹⁺⁴	—	4	4	no	—
34	C₁₃×2₋¹⁺⁴	—	4	1	no	—
35	C₁₃×H(5,𝔽₂)	—	2	4	no	—
36	C₁₃×H(5,𝔽₂)	—	4	3	no	—
37	C₂×C₁₃×#41	—	2	6	no	—
38	C₂×C₁₃×#41	—	2	3	no	—
39	C₂×C₁₃×#41	—	4	4	no	—
40	C₂×C₁₃×#41	—	4	3	no	—
41	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₁₃	—	2	3	no	—
43	C₂²×Q₈×C₁₃	—	4	1	no	—
44	C₁₃×#137	—	2	24	no	—
45	C₁₃×#137	—	2	12	no	—
46	C₁₃×AGL(1,ℤ/8)	—	2	12	no	—
47	C₁₃×#135	—	2	6	no	—
48	C₁₃×#135	—	2	6	no	—
49	C₁₃×#135	—	2	6	no	—
50	C₁₃×#135	C₂³	1	12	yes	—
51	C₁₃×#135	C₂³	1	12	yes	—
52	C₁₃×#135	C₂³	1	4	yes	—
53	C₂×C₁₃×Q₁₆	—	2	3	no	—
54	C₂×C₁₃×Q₁₆	—	2	6	no	—
55	C₂×C₁₃×Q₁₆	C₂³	1	12	yes	—
56	C₂×C₁₃×Q₁₆	C₂³	1	6	yes	—
57	C₂×C₁₃×Q₁₆	C₂³	1	3	yes	—
58	C₂×C₁₃×SD₁₆	—	2	6	no	—
59	C₂×C₁₃×SD₁₆	—	2	3	no	—
60	C₂×C₁₃×D₈	C₂³	1	6	yes	—
61	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
62	C₁₃×#131	D₄	1	4	yes	—
63	C₁₃×#131	C₂³	1	4	yes	—
64	C₂×C₁₃×M₁₆	D₄	1	3	yes	—
65	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
66	C₁₃×Q₃₂	C₂³	1	24	yes	—
67	C₁₃×SD₃₂	C₂³	1	24	yes	—
68	C₁₃×SD₃₂	C₂³	1	8	yes	—
69	C₁₃×D₁₆	C₂³	1	8	yes	—
70	C₁₃×M₃₂	C₂³	1	4	yes	—
71	C₁₃×M₃₂	C₂³	1	12	yes	—
72	C₂×C₁₃×C₁₆	C₂³	1	12	yes	—
73	#62637	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₁₃×#41	—	2	4	no	—
75	C₁₃×#41	—	2	12	no	—
76	C₁₃×#41	—	2	12	no	—
77	C₁₃×#41	—	4	4	no	—
78	C₁₃×#41	—	4	12	no	—
79	C₁₃×#41	—	4	24	no	—
80	C₁₃×#41	—	4	6	no	—
81	C₁₃×#41	—	4	6	no	—
82	C₁₃×#41	—	4	6	no	—
83	C₁₃×#41	D₈	1	4	yes	—
84	C₁₃×#41	C₂×D₄	1	4	yes	—
85	C₂×Q₈×C₁₃	—	2	3	no	—
86	C₂×Q₈×C₁₃	—	2	6	no	—
87	C₂×Q₈×C₁₃	—	2	12	no	—
88	C₂×Q₈×C₁₃	—	4	8	no	—
89	C₂×Q₈×C₁₃	—	4	4	no	—
90	C₂×Q₈×C₁₃	—	4	3	no	—
91	C₂×Q₈×C₁₃	—	4	6	no	—
92	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
93	C₂×D₄×C₁₃	—	2	1	no	—
94	C₂×D₄×C₁₃	—	2	6	no	—
95	C₂×D₄×C₁₃	—	4	3	no	—
96	C₂×D₄×C₁₃	—	4	6	no	—
97	C₂×D₄×C₁₃	—	4	12	no	—
98	C₂×D₄×C₁₃	D₈	1	3	yes	—
99	C₂²×C₄×C₁₃	—	4	3	no	—
100	C₂²×C₄×C₁₃	—	4	3	no	—
101	C₂²×C₄×C₁₃	—	4	6	no	—
102	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
103	C₁₃×Q₁₆	—	2	3	no	—
104	C₁₃×Q₁₆	—	2	12	no	—
105	C₁₃×Q₁₆	—	2	6	no	—
106	C₁₃×Q₁₆	C₂⁴	1	6	yes	—
107	C₁₃×Q₁₆	C₂⁴	1	4	yes	—
108	C₁₃×SD₁₆	—	2	4	no	—
109	C₁₃×SD₁₆	—	2	12	no	—
110	C₁₃×SD₁₆	—	2	12	no	—
111	C₁₃×D₈	—	2	1	no	—
112	C₁₃×D₈	—	2	6	no	—
113	C₁₃×D₈	C₂⁴	1	2	yes	—
114	C₁₃×D₈	C₂⁴	1	4	yes	—
115	C₁₃×M₁₆	C₂×D₄	1	12	yes	—
116	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
117	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
118	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
119	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
120	C₁₃×C₁₆	C₂⁴	1	8	yes	—
121	#3584	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂×#578	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#3417	C₂×C₁₃	1	8	yes	complemented
124	#3416	C₂×C₁₃	1	8	yes	complemented
125	#3414	C₂×C₁₃	1	6	yes	complemented
126	C₂×#510	C₂×C₁₃	1	6	yes	complemented
127	#3407	C₂×C₁₃	1	1	yes	characteristic, complemented
128	C₂³×C₁₃	—	4	1	no	—
129	C₂³×C₁₃	—	4	6	no	—
130	C₂³×C₁₃	C₂×D₈	1	3	yes	—
131	Q₈×C₁₃	—	2	4	no	—
132	Q₈×C₁₃	—	2	6	no	—
133	Q₈×C₁₃	—	4	3	no	—
134	Q₈×C₁₃	—	4	12	no	—
135	Q₈×C₁₃	—	4	6	no	—
136	Q₈×C₁₃	C₂×D₈	1	4	yes	—
137	D₄×C₁₃	—	2	4	no	—
138	D₄×C₁₃	—	2	2	no	—
139	D₄×C₁₃	—	4	4	no	—
140	D₄×C₁₃	—	4	12	no	—
141	D₄×C₁₃	—	4	12	no	—
142	D₄×C₁₃	—	4	6	no	—
143	D₄×C₁₃	—	4	1	no	—
144	D₄×C₁₃	C₂×D₈	1	12	yes	—
145	C₂×C₄×C₁₃	—	4	3	no	—
146	C₂×C₄×C₁₃	—	4	6	no	—
147	C₂×C₄×C₁₃	—	4	12	no	—
148	C₂×C₄×C₁₃	—	4	4	no	—
149	C₂×C₄×C₁₃	—	4	12	no	—
150	C₂×C₄×C₁₃	—	4	12	no	—
151	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
152	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
155	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
156	C₈×C₁₃	C₂²×D₄	1	6	yes	—
157	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
158	#585	—	2	1	no	—
159	C₂×2₋¹⁺⁴	—	2	1	no	—
160	#578	—	2	4	no	—
161	#578	C₂²×C₁₃	1	4	yes	complemented
162	#578	C₂²×C₁₃	1	8	yes	complemented
163	#577	—	2	4	no	—
164	#576	C₂²×C₁₃	1	4	yes	complemented
165	#575	—	2	3	no	—
166	C₂×#137	—	2	3	no	—
167	C₂×#135	C₂²×C₁₃	1	6	yes	—
168	C₂×#135	C₂²×C₁₃	1	3	yes	complemented
169	C₂²×Q₁₆	C₂²×C₁₃	1	3	yes	complemented
170	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
171	#510	C₂²×C₁₃	1	12	yes	complemented
172	#510	C₂²×C₁₃	1	24	yes	complemented
173	#510	C₂²×C₁₃	1	12	yes	complemented
174	#509	C₂²×C₁₃	1	4	yes	complemented
175	#509	C₂²×C₁₃	1	12	yes	complemented
176	#508	C₂²×C₁₃	1	24	yes	complemented
177	C₂×Q₃₂	C₂²×C₁₃	1	12	yes	complemented
178	C₂×SD₃₂	C₂²×C₁₃	1	12	yes	complemented
179	#504	C₂²×C₁₃	1	8	yes	complemented
180	C₂×M₃₂	C₂²×C₁₃	1	6	yes	complemented
181	C₂²×C₁₃	—	4	3	no	—
182	C₂²×C₁₃	—	4	2	no	—
183	C₂²×C₁₃	—	4	4	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	—	16	6	no	—
186	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
187	C₂²×C₁₃	C₂²×D₈	1	6	yes	—
188	C₄×C₁₃	—	4	4	no	—
189	C₄×C₁₃	—	4	6	no	—
190	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
191	C₄×C₁₃	C₂²×D₈	1	6	yes	—
192	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
193	2₋¹⁺⁴	—	2	4	no	—
194	2₋¹⁺⁴	—	2	8	no	—
195	2₋¹⁺⁴	—	4	4	no	—
196	2₋¹⁺⁴	—	4	1	no	—
197	H(5,𝔽₂)	—	2	4	no	—
198	H(5,𝔽₂)	—	4	3	no	—
199	C₂×#41	—	2	6	no	—
200	C₂×#41	—	2	3	no	—
201	C₂×#41	—	4	4	no	—
202	C₂×#41	—	4	3	no	—
203	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
204	C₂²×Q₈	—	2	3	no	—
205	C₂²×Q₈	—	4	1	no	—
206	#137	—	2	24	no	—
207	#137	—	2	12	no	—
208	AGL(1,ℤ/8)	—	2	12	no	—
209	#135	—	2	6	no	—
210	#135	—	2	6	no	—
211	#135	—	2	6	no	—
212	#135	C₂³×C₁₃	1	4	yes	—
213	#135	C₂³×C₁₃	1	12	yes	—
214	#135	C₂³×C₁₃	1	12	yes	—
215	C₂×Q₁₆	—	2	3	no	—
216	C₂×Q₁₆	—	2	6	no	—
217	C₂×Q₁₆	C₂³×C₁₃	1	3	yes	—
218	C₂×Q₁₆	C₂³×C₁₃	1	12	yes	—
219	C₂×Q₁₆	C₂³×C₁₃	1	6	yes	—
220	C₂×SD₁₆	—	2	6	no	—
221	C₂×SD₁₆	—	2	3	no	—
222	C₂×D₈	C₂³×C₁₃	1	6	yes	—
223	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
224	#131	D₄×C₁₃	1	4	yes	—
225	#131	C₂³×C₁₃	1	4	yes	—
226	C₂×M₁₆	D₄×C₁₃	1	3	yes	—
227	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
228	Q₃₂	C₂³×C₁₃	1	24	yes	—
229	SD₃₂	C₂³×C₁₃	1	24	yes	—
230	SD₃₂	C₂³×C₁₃	1	8	yes	—
231	D₁₆	C₂³×C₁₃	1	8	yes	—
232	M₃₂	C₂³×C₁₃	1	4	yes	—
233	M₃₂	C₂³×C₁₃	1	12	yes	—
234	C₂×C₁₆	C₂³×C₁₃	1	12	yes	—
235	C₂×C₁₃	—	2	1	no	—
236	C₂×C₁₃	—	2	6	no	—
237	C₂×C₁₃	—	8	2	no	—
238	C₂×C₁₃	—	8	4	no	—
239	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
240	#41	—	2	4	no	—
241	#41	—	2	12	no	—
242	#41	—	2	12	no	—
243	#41	—	4	4	no	—
244	#41	—	4	12	no	—
245	#41	—	4	24	no	—
246	#41	—	4	6	no	—
247	#41	—	4	6	no	—
248	#41	—	4	6	no	—
249	#41	C₁₃×D₈	1	4	yes	—
250	#41	C₂×D₄×C₁₃	1	4	yes	—
251	C₂×Q₈	—	2	3	no	—
252	C₂×Q₈	—	2	6	no	—
253	C₂×Q₈	—	2	12	no	—
254	C₂×Q₈	—	4	8	no	—
255	C₂×Q₈	—	4	4	no	—
256	C₂×Q₈	—	4	3	no	—
257	C₂×Q₈	—	4	6	no	—
258	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	2	6	no	—
261	C₂×D₄	—	4	3	no	—
262	C₂×D₄	—	4	6	no	—
263	C₂×D₄	—	4	12	no	—
264	C₂×D₄	C₁₃×D₈	1	3	yes	—
265	C₂²×C₄	—	4	3	no	—
266	C₂²×C₄	—	4	3	no	—
267	C₂²×C₄	—	4	6	no	—
268	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
269	Q₁₆	—	2	3	no	—
270	Q₁₆	—	2	12	no	—
271	Q₁₆	—	2	6	no	—
272	Q₁₆	C₂⁴×C₁₃	1	6	yes	—
273	Q₁₆	C₂⁴×C₁₃	1	4	yes	—
274	SD₁₆	—	2	4	no	—
275	SD₁₆	—	2	12	no	—
276	SD₁₆	—	2	12	no	—
277	D₈	—	2	1	no	—
278	D₈	—	2	6	no	—
279	D₈	C₂⁴×C₁₃	1	2	yes	—
280	D₈	C₂⁴×C₁₃	1	4	yes	—
281	M₁₆	C₂×D₄×C₁₃	1	12	yes	—
282	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
283	C₂×C₈	C₂×D₄×C₁₃	1	6	yes	—
284	C₂×C₈	C₂⁴×C₁₃	1	6	yes	—
285	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
286	C₁₆	C₂⁴×C₁₃	1	8	yes	—
287	C₁₃	#62637	1	1	yes	characteristic, direct product factor, Sylow
288	C₂³	—	4	1	no	—
289	C₂³	—	4	6	no	—
290	C₂³	C₂×C₁₃×D₈	1	3	yes	—
291	Q₈	—	2	4	no	—
292	Q₈	—	2	6	no	—
293	Q₈	—	4	3	no	—
294	Q₈	—	4	12	no	—
295	Q₈	—	4	6	no	—
296	Q₈	C₂×C₁₃×D₈	1	4	yes	—
297	D₄	—	2	4	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	12	no	—
301	D₄	—	4	12	no	—
302	D₄	—	4	6	no	—
303	D₄	—	4	1	no	—
304	D₄	C₂×C₁₃×D₈	1	12	yes	—
305	C₂×C₄	—	4	3	no	—
306	C₂×C₄	—	4	6	no	—
307	C₂×C₄	—	4	12	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	12	no	—
310	C₂×C₄	—	4	12	no	—
311	C₂×C₄	C₂×C₁₃×D₈	1	3	yes	—
312	C₂×C₄	C₂×C₁₃×D₈	1	6	yes	—
313	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
315	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
316	C₈	C₂²×D₄×C₁₃	1	6	yes	—
317	C₈	C₂⁵×C₁₃	1	1	yes	characteristic
318	C₂²	—	4	3	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	4	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	16	6	no	—
323	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
324	C₂²	C₂²×C₁₃×D₈	1	6	yes	—
325	C₄	—	4	4	no	—
326	C₄	—	4	6	no	—
327	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
328	C₄	C₂²×C₁₃×D₈	1	6	yes	—
329	C₄	C₂³×D₄×C₁₃	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	6	no	—
332	C₂	—	8	2	no	—
333	C₂	—	8	4	no	—
334	C₂	C₂³×C₁₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	C₂³×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₁₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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×#3584	C₂	semidirect
2. C₂×C₁₃×#578	C₂	semidirect
3. C₁₃×#3417	C₂	semidirect
4. C₁₃×#3416	C₂	semidirect
5. C₁₃×#3414	C₂	semidirect
6. C₂×C₁₃×#510	C₂	semidirect
7. C₁₃×#3407	C₂	semidirect
11. C₁₃×#578	C₂²	semidirect
12. C₁₃×#578	C₂²	semidirect
14. C₁₃×#576	C₂²	semidirect
18. C₂×C₁₃×#135	C₂²	semidirect
19. C₂²×C₁₃×Q₁₆	C₂²	semidirect
21. C₁₃×#510	C₂²	semidirect
22. C₁₃×#510	C₂²	semidirect
23. C₁₃×#510	C₂²	semidirect
24. C₁₃×#509	C₂²	semidirect
25. C₁₃×#509	C₂²	semidirect
26. C₁₃×#508	C₂²	semidirect
27. C₂×C₁₃×Q₃₂	C₂²	semidirect
28. C₂×C₁₃×SD₃₂	C₂²	semidirect
29. C₁₃×#504	C₂²	semidirect
30. C₂×C₁₃×M₃₂	C₂²	semidirect
73. #62637	C₁₃	direct (N × H)
121. #3584	C₂×C₁₃	semidirect
122. C₂×#578	C₂×C₁₃	semidirect
123. #3417	C₂×C₁₃	semidirect
124. #3416	C₂×C₁₃	semidirect
125. #3414	C₂×C₁₃	semidirect
126. C₂×#510	C₂×C₁₃	semidirect
127. #3407	C₂×C₁₃	semidirect
161. #578	C₂²×C₁₃	semidirect
162. #578	C₂²×C₁₃	semidirect
164. #576	C₂²×C₁₃	semidirect
168. C₂×#135	C₂²×C₁₃	semidirect
169. C₂²×Q₁₆	C₂²×C₁₃	semidirect
171. #510	C₂²×C₁₃	semidirect
172. #510	C₂²×C₁₃	semidirect
173. #510	C₂²×C₁₃	semidirect
174. #509	C₂²×C₁₃	semidirect
175. #509	C₂²×C₁₃	semidirect
176. #508	C₂²×C₁₃	semidirect
177. C₂×Q₃₂	C₂²×C₁₃	semidirect
178. C₂×SD₃₂	C₂²×C₁₃	semidirect
179. #504	C₂²×C₁₃	semidirect
180. C₂×M₃₂	C₂²×C₁₃	semidirect
287. C₁₃	#62637	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×#3584	2
C₂×C₁₃×#578	2
C₁₃×#3417	2
C₁₃×#3416	2
C₁₃×#3414	2
C₂×C₁₃×#510	2
C₁₃×#3407	2
#62637	13

Subgroups — C13×#62637 #15748976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#62637 #15748976