Subgroups — C₁₃×#33707 #15720046

Order 3328.

622 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2243	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2241	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2240	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2139	C₂	1	4	yes	maximal, complemented
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	—	2	1	no	—
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	C₁₃×#532	—	2	1	no	—
11	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₁₃×#500	—	2	1	no	—
14	C₁₃×#500	—	2	1	no	—
15	C₁₃×#500	C₂²	1	2	yes	—
16	C₁₃×#500	C₂²	1	2	yes	—
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×#492	—	2	1	no	—
22	C₂×C₁₃×#107	—	2	1	no	—
23	C₂×C₁₃×#107	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×#358	C₂²	1	8	yes	—
26	C₁₃×#357	C₂²	1	8	yes	—
27	C₁₃×#346	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
29	C₂×C₁₃×#41	—	2	2	no	—
30	C₂²×Q₈×C₁₃	—	4	1	no	—
31	C₁₃×#135	—	2	8	no	—
32	C₁₃×#135	—	4	4	no	—
33	C₂×C₁₃×Q₁₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
35	C₂×C₁₃×SD₁₆	—	2	2	no	—
36	C₂×C₁₃×SD₁₆	—	2	2	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊Q₈	—	2	2	no	—
42	C₁₃×C₄⋊Q₈	—	2	2	no	—
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×C₄⋊Q₈	—	4	1	no	—
45	C₁₃×C₄⋊Q₈	—	4	1	no	—
46	C₁₃×#127	—	2	1	no	—
47	C₁₃×#124	—	2	2	no	—
48	C₁₃×#121	—	4	1	no	—
49	C₁₃×#121	—	4	1	no	—
50	C₄×D₄×C₁₃	—	4	1	no	—
51	C₄×D₄×C₁₃	—	4	1	no	—
52	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₁₃×M₃₂	—	2	4	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
57	C₁₃×#107	—	2	2	no	—
58	C₁₃×#107	—	2	2	no	—
59	C₁₃×#107	—	2	2	no	—
60	C₁₃×#107	C₂³	1	4	yes	—
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	#33707	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₁₃×#41	—	4	4	no	—
67	C₁₃×#41	—	4	4	no	—
68	C₂×Q₈×C₁₃	—	2	2	no	—
69	C₂×Q₈×C₁₃	—	4	2	no	—
70	C₂×Q₈×C₁₃	—	4	2	no	—
71	C₂×Q₈×C₁₃	—	8	4	no	—
72	C₂×D₄×C₁₃	—	2	2	no	—
73	C₂×D₄×C₁₃	—	4	2	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	—	4	2	no	—
77	C₂²×C₄×C₁₃	—	4	2	no	—
78	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₁₃×Q₁₆	—	2	4	no	—
82	C₁₃×Q₁₆	—	4	2	no	—
83	C₁₃×SD₁₆	—	4	4	no	—
84	C₁₃×SD₁₆	—	4	4	no	—
85	C₁₃×D₈	—	2	4	no	—
86	C₁₃×D₈	—	4	2	no	—
87	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	#41	1	4	yes	—
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₁₃×C₄⋊C₄	—	4	1	no	—
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₁₃×C₂²⋊C₄	—	4	2	no	—
101	C₄²×C₁₃	D₈	1	1	yes	characteristic
102	C₄²×C₁₃	D₈	1	1	yes	characteristic
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₁₃×C₁₆	—	2	4	no	—
106	C₂×#500	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#3147	C₂×C₁₃	1	1	yes	characteristic
108	#2243	C₂×C₁₃	1	4	yes	complemented
109	#2241	C₂×C₁₃	1	2	yes	complemented
110	#2240	C₂×C₁₃	1	2	yes	complemented
111	#2139	C₂×C₁₃	1	4	yes	complemented
112	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	—	4	2	no	—
114	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₁₃	—	4	2	no	—
116	Q₈×C₁₃	—	4	2	no	—
117	Q₈×C₁₃	—	8	4	no	—
118	Q₈×C₁₃	—	8	4	no	—
119	D₄×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	8	4	no	—
122	D₄×C₁₃	—	8	2	no	—
123	D₄×C₁₃	—	8	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	8	4	no	—
129	C₂×C₄×C₁₃	—	8	4	no	—
130	C₂×C₄×C₁₃	#124	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₁₃	#124	1	2	yes	—
139	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
140	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
141	C₈×C₁₃	C₂×#41	1	2	yes	—
142	C₂×#135	—	2	1	no	—
143	C₂×#135	C₂²×C₁₃	1	2	yes	—
144	#532	—	2	1	no	—
145	C₂×C₄⋊Q₈	—	2	1	no	—
146	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
147	#500	—	2	1	no	—
148	#500	—	2	1	no	—
149	#500	C₂²×C₁₃	1	2	yes	—
150	#500	C₂²×C₁₃	1	2	yes	—
151	#495	—	2	1	no	—
152	#495	C₂²×C₁₃	1	2	yes	—
153	#494	C₂²×C₁₃	1	1	yes	characteristic
154	#493	C₂²×C₁₃	1	1	yes	characteristic
155	#492	—	2	1	no	—
156	C₂×#107	—	2	1	no	—
157	C₂×#107	C₂²×C₁₃	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
159	#358	C₂²×C₁₃	1	8	yes	—
160	#357	C₂²×C₁₃	1	8	yes	—
161	#346	C₂²×C₁₃	1	2	yes	—
162	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
163	C₂²×C₁₃	—	2	1	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	—	8	2	no	—
166	C₂²×C₁₃	—	8	2	no	—
167	C₂²×C₁₃	—	8	2	no	—
168	C₂²×C₁₃	#486	1	2	yes	—
169	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	8	2	no	—
173	C₄×C₁₃	—	8	4	no	—
174	C₄×C₁₃	#486	1	2	yes	—
175	C₄×C₁₃	#509	1	2	yes	—
176	C₄×C₁₃	#510	1	2	yes	—
177	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
178	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
179	C₂×#41	—	2	2	no	—
180	C₂²×Q₈	—	4	1	no	—
181	#135	—	2	8	no	—
182	#135	—	4	4	no	—
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
185	C₂×SD₁₆	—	2	2	no	—
186	C₂×SD₁₆	—	2	2	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	C₂³×C₁₃	1	2	yes	—
189	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	4	1	no	—
195	C₄⋊Q₈	—	4	1	no	—
196	#127	—	2	1	no	—
197	#124	—	2	2	no	—
198	#121	—	4	1	no	—
199	#121	—	4	1	no	—
200	C₄×D₄	—	4	1	no	—
201	C₄×D₄	—	4	1	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	4	1	no	—
204	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
205	M₃₂	—	2	4	no	—
206	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
207	#107	—	2	2	no	—
208	#107	—	2	2	no	—
209	#107	—	2	2	no	—
210	#107	C₂³×C₁₃	1	4	yes	—
211	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
212	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
213	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
215	C₂×C₁₃	—	2	2	no	—
216	C₂×C₁₃	—	16	2	no	—
217	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
218	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
219	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
220	#41	—	4	4	no	—
221	#41	—	4	4	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×Q₈	—	8	4	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
233	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
235	Q₁₆	—	2	4	no	—
236	Q₁₆	—	4	2	no	—
237	SD₁₆	—	4	4	no	—
238	SD₁₆	—	4	4	no	—
239	D₈	—	2	4	no	—
240	D₈	—	4	2	no	—
241	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
242	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₂×C₈	C₁₃×#41	1	4	yes	—
246	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₄²	C₁₃×D₈	1	1	yes	characteristic
256	C₄²	C₁₃×D₈	1	1	yes	characteristic
257	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₁₆	—	2	4	no	—
260	C₁₃	#33707	1	1	yes	characteristic, direct product factor, Sylow
261	C₂³	—	4	2	no	—
262	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	8	4	no	—
266	Q₈	—	8	4	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	8	4	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	C₁₃×#124	1	2	yes	—
279	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
280	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
281	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
282	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
284	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
285	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
286	C₈	C₁₃×#124	1	2	yes	—
287	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
288	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
289	C₈	C₂×C₁₃×#41	1	2	yes	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	C₁₃×#486	1	2	yes	—
296	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	4	no	—
301	C₄	C₁₃×#486	1	2	yes	—
302	C₄	C₁₃×#509	1	2	yes	—
303	C₄	C₁₃×#510	1	2	yes	—
304	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
305	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	16	2	no	—
308	C₂	C₂×C₁₃×#486	1	1	yes	characteristic
309	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
310	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3147	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. C₂×C₁₃×#500	C₂	semidirect
3. C₁₃×#2243	C₂	semidirect
4. C₁₃×#2241	C₂	semidirect
5. C₁₃×#2240	C₂	semidirect
6. C₁₃×#2139	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
65. #33707	C₁₃	direct (N × H)
106. C₂×#500	C₂×C₁₃	semidirect
108. #2243	C₂×C₁₃	semidirect
109. #2241	C₂×C₁₃	semidirect
110. #2240	C₂×C₁₃	semidirect
111. #2139	C₂×C₁₃	semidirect
112. C₄×M₃₂	C₂×C₁₃	semidirect
260. C₁₃	#33707	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×#500	2
C₁₃×#3147	2
C₁₃×#2243	2
C₁₃×#2241	2
C₁₃×#2240	2
C₁₃×#2139	2
C₄×C₁₃×M₃₂	2
#33707	13

Subgroups — C13×#33707 #15720046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33707 #15720046