Subgroups — C₁₃×#33686 #15720025

Order 3328.

544 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3176	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2234	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2229	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2207	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2185	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2175	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×SD₃₂	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₁₃×#540	—	2	1	no	—
12	C₁₃×#519	—	2	1	no	—
13	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
15	C₁₃×#499	C₂²	1	1	yes	characteristic
16	C₁₃×#493	C₂²	1	1	yes	characteristic
17	C₁₃×#490	—	2	1	no	—
18	C₁₃×#469	—	2	1	no	—
19	C₁₃×#466	C₂²	1	2	yes	—
20	C₁₃×#442	—	2	1	no	—
21	C₁₃×#441	—	2	1	no	—
22	C₁₃×#441	—	2	1	no	—
23	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
24	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
25	C₄×C₁₃×D₈	C₂²	1	2	yes	—
26	C₁₃×#427	C₂²	1	1	yes	characteristic
27	C₁₃×#419	—	2	1	no	—
28	C₁₃×#405	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	2	yes	—
30	C₁₃×#366	C₂²	1	2	yes	—
31	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₁₃×#358	C₂²	1	2	yes	—
33	C₁₃×#358	C₂²	1	2	yes	—
34	C₁₃×#357	C₂²	1	2	yes	—
35	C₁₃×#357	C₂²	1	2	yes	—
36	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₁₃×#346	C₂²	1	1	yes	characteristic
38	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₁₃	—	2	1	no	—
40	C₂×C₁₃×Q₁₆	—	2	4	no	—
41	C₂×C₁₃×Q₁₆	—	2	1	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
43	C₂×C₁₃×D₈	—	2	1	no	—
44	C₂×C₁₃×D₈	C₂³	1	2	yes	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×#127	—	2	1	no	—
48	C₁₃×#125	—	2	1	no	—
49	C₁₃×#123	—	4	2	no	—
50	C₁₃×#121	—	2	2	no	—
51	C₁₃×#121	—	4	2	no	—
52	C₄×Q₈×C₁₃	—	2	2	no	—
53	C₄×Q₈×C₁₃	—	2	2	no	—
54	C₄×Q₈×C₁₃	—	4	2	no	—
55	C₄×D₄×C₁₃	—	2	2	no	—
56	C₁₃×#117	—	2	1	no	—
57	C₁₃×#117	—	4	2	no	—
58	C₁₃×#117	D₄	1	1	yes	characteristic
59	C₁₃×SD₃₂	—	2	4	no	—
60	C₁₃×SD₃₂	—	2	4	no	—
61	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₁₃×#107	C₂³	1	2	yes	—
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₁₃×#106	—	2	1	no	—
66	C₁₃×#106	—	2	1	no	—
67	C₁₃×#103	—	2	2	no	—
68	C₁₃×#103	—	2	2	no	—
69	C₁₃×#102	—	2	2	no	—
70	C₁₃×#102	—	2	2	no	—
71	C₁₃×#97	D₄	1	1	yes	characteristic
72	C₁₃×#97	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#33686	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×Q₈×C₁₃	—	2	4	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	4	2	no	—
79	C₂×Q₈×C₁₃	—	4	1	no	—
80	C₂×D₄×C₁₃	—	2	2	no	—
81	C₂×D₄×C₁₃	—	4	2	no	—
82	C₂×D₄×C₁₃	—	4	2	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	—	4	2	no	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₁₃×Q₁₆	—	2	4	no	—
87	C₁₃×Q₁₆	—	2	2	no	—
88	C₁₃×Q₁₆	#41	1	4	yes	—
89	C₁₃×D₈	—	2	2	no	—
90	C₁₃×D₈	—	2	2	no	—
91	C₁₃×D₈	—	4	2	no	—
92	C₁₃×M₁₆	—	4	1	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₄⋊C₄	—	4	2	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
104	C₁₃×C₂²⋊C₄	—	4	1	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
108	C₄²×C₁₃	—	4	1	no	—
109	C₄²×C₁₃	—	4	2	no	—
110	C₄²×C₁₃	—	4	2	no	—
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₁₃×C₁₆	—	2	2	no	—
114	C₁₃×C₁₆	—	2	2	no	—
115	#3176	C₂×C₁₃	1	1	yes	characteristic
116	#2950	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#2234	C₂×C₁₃	1	2	yes	complemented
118	#2229	C₂×C₁₃	1	2	yes	complemented
119	#2207	C₂×C₁₃	1	2	yes	complemented
120	#2185	C₂×C₁₃	1	2	yes	complemented
121	#2175	C₂×C₁₃	1	2	yes	complemented
122	C₄×SD₃₂	C₂×C₁₃	1	2	yes	complemented
123	#2164	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
126	Q₈×C₁₃	—	2	4	no	—
127	Q₈×C₁₃	—	4	4	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	D₄×C₁₃	—	4	2	no	—
130	D₄×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	16	2	no	—
132	D₄×C₁₃	—	16	2	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	4	4	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	16	2	no	—
140	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	C₂×#41	1	2	yes	—
148	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
149	#540	—	2	1	no	—
150	#519	—	2	1	no	—
151	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
152	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
153	#499	C₂²×C₁₃	1	1	yes	characteristic
154	#493	C₂²×C₁₃	1	1	yes	characteristic
155	#490	—	2	1	no	—
156	#469	—	2	1	no	—
157	#466	C₂²×C₁₃	1	2	yes	—
158	#442	—	2	1	no	—
159	#441	—	2	1	no	—
160	#441	—	2	1	no	—
161	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
162	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
163	C₄×D₈	C₂²×C₁₃	1	2	yes	—
164	#427	C₂²×C₁₃	1	1	yes	characteristic
165	#419	—	2	1	no	—
166	#405	C₂²×C₁₃	1	1	yes	characteristic
167	#367	C₂²×C₁₃	1	2	yes	—
168	#366	C₂²×C₁₃	1	2	yes	—
169	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
170	#358	C₂²×C₁₃	1	2	yes	—
171	#358	C₂²×C₁₃	1	2	yes	—
172	#357	C₂²×C₁₃	1	2	yes	—
173	#357	C₂²×C₁₃	1	2	yes	—
174	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
175	#346	C₂²×C₁₃	1	1	yes	characteristic
176	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	4	4	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	#525	1	1	yes	characteristic
190	C₄×C₁₃	#575	1	1	yes	characteristic
191	C₂²×Q₈	—	2	1	no	—
192	C₂×Q₁₆	—	2	4	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	C₂³×C₁₃	1	2	yes	—
197	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
199	#127	—	2	1	no	—
200	#125	—	2	1	no	—
201	#123	—	4	2	no	—
202	#121	—	2	2	no	—
203	#121	—	4	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	4	2	no	—
207	C₄×D₄	—	2	2	no	—
208	#117	—	2	1	no	—
209	#117	—	4	2	no	—
210	#117	D₄×C₁₃	1	1	yes	characteristic
211	SD₃₂	—	2	4	no	—
212	SD₃₂	—	2	4	no	—
213	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
214	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
215	#107	C₂³×C₁₃	1	2	yes	—
216	#107	C₂³×C₁₃	1	2	yes	—
217	#106	—	2	1	no	—
218	#106	—	2	1	no	—
219	#103	—	2	2	no	—
220	#103	—	2	2	no	—
221	#102	—	2	2	no	—
222	#102	—	2	2	no	—
223	#97	D₄×C₁₃	1	1	yes	characteristic
224	#97	D₄×C₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
227	C₂×C₁₃	—	4	1	no	—
228	C₂×C₁₃	—	16	2	no	—
229	C₂×C₁₃	#3100	1	1	yes	characteristic
230	C₂×C₁₃	#3416	1	2	yes	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	C₁₃×#41	1	4	yes	—
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	D₈	—	4	2	no	—
247	M₁₆	—	4	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
252	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
263	C₄²	—	4	1	no	—
264	C₄²	—	4	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
267	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
268	C₁₆	—	2	2	no	—
269	C₁₆	—	2	2	no	—
270	C₁₃	#33686	1	1	yes	characteristic, direct product factor, Sylow
271	C₂³	—	4	2	no	—
272	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
273	Q₈	—	2	4	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	16	2	no	—
279	D₄	—	16	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	16	2	no	—
287	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
288	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
289	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	C₂×C₁₃×#41	1	2	yes	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	C₁₃×#525	1	1	yes	characteristic
308	C₄	C₁₃×#575	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	—	16	2	no	—
311	C₂	C₁₃×#3100	1	1	yes	characteristic
312	C₂	C₁₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2950	C₂	semidirect
3. C₁₃×#2234	C₂	semidirect
4. C₁₃×#2229	C₂	semidirect
5. C₁₃×#2207	C₂	semidirect
6. C₁₃×#2185	C₂	semidirect
7. C₁₃×#2175	C₂	semidirect
8. C₄×C₁₃×SD₃₂	C₂	semidirect
9. C₁₃×#2164	C₂	semidirect
75. #33686	C₁₃	direct (N × H)
116. #2950	C₂×C₁₃	semidirect
117. #2234	C₂×C₁₃	semidirect
118. #2229	C₂×C₁₃	semidirect
119. #2207	C₂×C₁₃	semidirect
120. #2185	C₂×C₁₃	semidirect
121. #2175	C₂×C₁₃	semidirect
122. C₄×SD₃₂	C₂×C₁₃	semidirect
123. #2164	C₂×C₁₃	semidirect
270. C₁₃	#33686	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×#3176	2
C₁₃×#2950	2
C₁₃×#2234	2
C₁₃×#2229	2
C₁₃×#2207	2
C₁₃×#2185	2
C₁₃×#2175	2
C₄×C₁₃×SD₃₂	2
C₁₃×#2164	2
#33686	13

Subgroups — C13×#33686 #15720025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33686 #15720025