Subgroups — C₁₃×#33773 #15720112

Order 3328.

636 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2210	C₂	1	4	yes	maximal, complemented
8	C₁₃×#2186	C₂	1	4	yes	maximal, complemented
9	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#135	—	2	1	no	—
11	C₂×C₁₃×#135	C₂²	1	2	yes	—
12	C₁₃×#532	—	2	1	no	—
13	C₁₃×#532	—	2	1	no	—
14	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
15	C₂×C₁₃×D₁₆	C₂²	1	4	yes	—
16	C₁₃×#500	C₂²	1	1	yes	characteristic
17	C₁₃×#495	—	2	1	no	—
18	C₁₃×#495	C₂²	1	2	yes	—
19	C₁₃×#494	C₂²	1	1	yes	characteristic
20	C₁₃×#493	C₂²	1	1	yes	characteristic
21	C₁₃×#493	C₂²	1	1	yes	characteristic
22	C₁₃×#492	—	2	1	no	—
23	C₁₃×#486	—	2	1	no	—
24	C₁₃×#466	—	2	1	no	—
25	C₁₃×#466	C₂²	1	2	yes	—
26	C₄×C₁₃×D₈	—	2	1	no	—
27	C₄×C₁₃×D₈	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₁₃×#358	C₂²	1	4	yes	—
31	C₁₃×#357	C₂²	1	4	yes	—
32	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×C₁₃×#41	—	2	2	no	—
36	C₂×C₁₃×#41	—	4	1	no	—
37	C₁₃×#135	—	4	4	no	—
38	C₁₃×#135	—	4	4	no	—
39	C₂×C₁₃×Q₁₆	—	2	1	no	—
40	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
41	C₂×C₁₃×SD₁₆	—	2	2	no	—
42	C₂×C₁₃×SD₁₆	—	2	2	no	—
43	C₂×C₁₃×D₈	—	2	1	no	—
44	C₂×C₁₃×D₈	—	2	1	no	—
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₂×C₁₃×D₈	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×C₄⋊Q₈	—	2	1	no	—
51	C₁₃×#127	—	2	1	no	—
52	C₁₃×#127	—	2	1	no	—
53	C₁₃×#124	—	2	2	no	—
54	C₁₃×#124	—	4	1	no	—
55	C₁₃×#121	—	2	2	no	—
56	C₁₃×#121	—	4	1	no	—
57	C₁₃×#121	—	4	1	no	—
58	C₁₃×#121	—	4	1	no	—
59	C₄×D₄×C₁₃	—	2	2	no	—
60	C₄×D₄×C₁₃	—	4	1	no	—
61	C₄×D₄×C₁₃	—	4	1	no	—
62	C₄×D₄×C₁₃	—	4	1	no	—
63	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
64	C₁₃×SD₃₂	—	4	4	no	—
65	C₁₃×D₁₆	—	4	4	no	—
66	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
67	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
68	C₁₃×#107	—	2	1	no	—
69	C₁₃×#107	C₂³	1	2	yes	—
70	C₁₃×#102	—	2	2	no	—
71	C₁₃×#102	—	2	2	no	—
72	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	#33773	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₁₃×#41	—	4	4	no	—
78	C₁₃×#41	—	4	4	no	—
79	C₁₃×#41	—	8	4	no	—
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	2	2	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂×D₄×C₁₃	—	4	2	no	—
86	C₂×D₄×C₁₃	—	4	1	no	—
87	C₂×D₄×C₁₃	—	4	1	no	—
88	C₂×D₄×C₁₃	—	4	1	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂²×C₄×C₁₃	—	4	2	no	—
91	C₂²×C₄×C₁₃	—	4	2	no	—
92	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₁₃×Q₁₆	—	2	4	no	—
96	C₁₃×Q₁₆	—	4	2	no	—
97	C₁₃×SD₁₆	—	4	4	no	—
98	C₁₃×SD₁₆	—	4	4	no	—
99	C₁₃×D₈	—	2	4	no	—
100	C₁₃×D₈	—	2	4	no	—
101	C₁₃×D₈	—	4	2	no	—
102	C₁₃×D₈	—	4	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	4	1	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₁₃×C₂²⋊C₄	—	4	2	no	—
115	C₁₃×C₂²⋊C₄	—	4	2	no	—
116	C₁₃×C₂²⋊C₄	—	4	2	no	—
117	C₄²×C₁₃	D₈	1	1	yes	characteristic
118	C₄²×C₁₃	D₈	1	1	yes	characteristic
119	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₁₃×C₁₆	—	2	2	no	—
122	C₁₃×C₁₆	—	2	2	no	—
123	#3169	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#3147	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2248	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2246	C₂×C₁₃	1	1	yes	characteristic, complemented
127	#2241	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#2240	C₂×C₁₃	1	1	yes	characteristic, complemented
129	#2210	C₂×C₁₃	1	4	yes	complemented
130	#2186	C₂×C₁₃	1	4	yes	complemented
131	#2162	C₂×C₁₃	1	1	yes	characteristic, complemented
132	C₂³×C₁₃	—	4	2	no	—
133	C₂³×C₁₃	—	4	2	no	—
134	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
135	Q₈×C₁₃	—	4	2	no	—
136	Q₈×C₁₃	—	4	2	no	—
137	Q₈×C₁₃	—	8	2	no	—
138	D₄×C₁₃	—	4	2	no	—
139	D₄×C₁₃	—	4	2	no	—
140	D₄×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	4	2	no	—
142	D₄×C₁₃	—	8	4	no	—
143	D₄×C₁₃	—	8	4	no	—
144	D₄×C₁₃	—	8	2	no	—
145	D₄×C₁₃	—	8	2	no	—
146	D₄×C₁₃	—	8	2	no	—
147	D₄×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	—	2	1	no	—
149	C₂×C₄×C₁₃	—	2	1	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	8	4	no	—
155	C₂×C₄×C₁₃	—	8	4	no	—
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
165	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
166	C₂×#135	—	2	1	no	—
167	C₂×#135	C₂²×C₁₃	1	2	yes	—
168	#532	—	2	1	no	—
169	#532	—	2	1	no	—
170	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
171	C₂×D₁₆	C₂²×C₁₃	1	4	yes	—
172	#500	C₂²×C₁₃	1	1	yes	characteristic
173	#495	—	2	1	no	—
174	#495	C₂²×C₁₃	1	2	yes	—
175	#494	C₂²×C₁₃	1	1	yes	characteristic
176	#493	C₂²×C₁₃	1	1	yes	characteristic
177	#493	C₂²×C₁₃	1	1	yes	characteristic
178	#492	—	2	1	no	—
179	#486	—	2	1	no	—
180	#466	—	2	1	no	—
181	#466	C₂²×C₁₃	1	2	yes	—
182	C₄×D₈	—	2	1	no	—
183	C₄×D₈	C₂²×C₁₃	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
185	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
186	#358	C₂²×C₁₃	1	4	yes	—
187	#357	C₂²×C₁₃	1	4	yes	—
188	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
189	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
190	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	—	8	2	no	—
196	C₂²×C₁₃	—	8	2	no	—
197	C₂²×C₁₃	—	8	2	no	—
198	C₂²×C₁₃	—	8	2	no	—
199	C₂²×C₁₃	—	8	2	no	—
200	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	—	2	1	no	—
203	C₄×C₁₃	—	2	1	no	—
204	C₄×C₁₃	—	8	2	no	—
205	C₄×C₁₃	—	8	2	no	—
206	C₄×C₁₃	#508	1	2	yes	—
207	C₄×C₁₃	#509	1	2	yes	—
208	C₄×C₁₃	#535	1	1	yes	characteristic
209	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₂×#41	—	2	2	no	—
211	C₂×#41	—	4	1	no	—
212	#135	—	4	4	no	—
213	#135	—	4	4	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×D₈	—	2	1	no	—
219	C₂×D₈	—	2	1	no	—
220	C₂×D₈	C₂³×C₁₃	1	2	yes	—
221	C₂×D₈	C₂³×C₁₃	1	2	yes	—
222	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
223	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	#127	—	2	1	no	—
227	#127	—	2	1	no	—
228	#124	—	2	2	no	—
229	#124	—	4	1	no	—
230	#121	—	2	2	no	—
231	#121	—	4	1	no	—
232	#121	—	4	1	no	—
233	#121	—	4	1	no	—
234	C₄×D₄	—	2	2	no	—
235	C₄×D₄	—	4	1	no	—
236	C₄×D₄	—	4	1	no	—
237	C₄×D₄	—	4	1	no	—
238	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
239	SD₃₂	—	4	4	no	—
240	D₁₆	—	4	4	no	—
241	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
242	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
243	#107	—	2	1	no	—
244	#107	C₂³×C₁₃	1	2	yes	—
245	#102	—	2	2	no	—
246	#102	—	2	2	no	—
247	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
248	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
251	C₂×C₁₃	—	4	1	no	—
252	C₂×C₁₃	—	16	2	no	—
253	C₂×C₁₃	—	16	2	no	—
254	C₂×C₁₃	#3205	1	1	yes	characteristic
255	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
256	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
257	#41	—	4	4	no	—
258	#41	—	4	4	no	—
259	#41	—	8	4	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
273	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
275	Q₁₆	—	2	4	no	—
276	Q₁₆	—	4	2	no	—
277	SD₁₆	—	4	4	no	—
278	SD₁₆	—	4	4	no	—
279	D₈	—	2	4	no	—
280	D₈	—	2	4	no	—
281	D₈	—	4	2	no	—
282	D₈	—	4	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
286	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
287	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
288	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	C₁₃×D₈	1	1	yes	characteristic
298	C₄²	C₁₃×D₈	1	1	yes	characteristic
299	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
301	C₁₆	—	2	2	no	—
302	C₁₆	—	2	2	no	—
303	C₁₃	#33773	1	1	yes	characteristic, direct product factor, Sylow
304	C₂³	—	4	2	no	—
305	C₂³	—	4	2	no	—
306	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	8	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	8	4	no	—
315	D₄	—	8	4	no	—
316	D₄	—	8	2	no	—
317	D₄	—	8	2	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	4	no	—
327	C₂×C₄	—	8	4	no	—
328	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
329	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
330	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
331	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
332	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
337	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	8	2	no	—
352	C₄	—	8	2	no	—
353	C₄	C₁₃×#508	1	2	yes	—
354	C₄	C₁₃×#509	1	2	yes	—
355	C₄	C₁₃×#535	1	1	yes	characteristic
356	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
357	C₂	—	4	1	no	—
358	C₂	—	16	2	no	—
359	C₂	—	16	2	no	—
360	C₂	C₁₃×#3205	1	1	yes	characteristic
361	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
362	C₂	C₂×C₁₃×#509	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×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×#3169	C₂	semidirect
2. C₁₃×#3147	C₂	semidirect
3. C₁₃×#2248	C₂	semidirect
4. C₁₃×#2246	C₂	semidirect
5. C₁₃×#2241	C₂	semidirect
6. C₁₃×#2240	C₂	semidirect
7. C₁₃×#2210	C₂	semidirect
8. C₁₃×#2186	C₂	semidirect
9. C₁₃×#2162	C₂	semidirect
76. #33773	C₁₃	direct (N × H)
123. #3169	C₂×C₁₃	semidirect
124. #3147	C₂×C₁₃	semidirect
125. #2248	C₂×C₁₃	semidirect
126. #2246	C₂×C₁₃	semidirect
127. #2241	C₂×C₁₃	semidirect
128. #2240	C₂×C₁₃	semidirect
129. #2210	C₂×C₁₃	semidirect
130. #2186	C₂×C₁₃	semidirect
131. #2162	C₂×C₁₃	semidirect
303. C₁₃	#33773	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×#3169	2
C₁₃×#3147	2
C₁₃×#2248	2
C₁₃×#2246	2
C₁₃×#2241	2
C₁₃×#2240	2
C₁₃×#2210	2
C₁₃×#2186	2
C₁₃×#2162	2
#33773	13

Subgroups — C13×#33773 #15720112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33773 #15720112