Subgroups — C₁₃×#13528 #15699867

Order 3328.

378 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1970	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	—	2	1	no	—
15	C₂×C₁₃×#102	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#390	—	2	1	no	—
18	C₁₃×#382	—	2	1	no	—
19	C₁₃×#367	—	2	2	no	—
20	C₁₃×#366	—	2	2	no	—
21	C₁₃×#357	—	2	2	no	—
22	C₁₃×#357	—	2	2	no	—
23	C₂²×D₄×C₁₃	—	2	1	no	—
24	C₂×C₁₃×D₈	—	2	1	no	—
25	C₂×C₁₃×D₈	—	2	1	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	1	no	—
29	C₂×C₁₃×D₈	Q₈	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₁₃×C₁₆	—	2	1	no	—
38	C₂×C₁₃×C₁₆	—	2	1	no	—
39	C₂×C₁₃×C₁₆	—	2	1	no	—
40	C₂×C₁₃×C₁₆	—	2	1	no	—
41	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
42	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
43	C₁₃×#107	—	2	1	no	—
44	C₁₃×#107	—	2	1	no	—
45	C₁₃×#107	—	2	1	no	—
46	C₁₃×#107	—	2	1	no	—
47	C₁₃×#107	—	2	1	no	—
48	C₁₃×#107	—	2	1	no	—
49	C₁₃×#106	—	4	2	no	—
50	C₁₃×#102	—	4	2	no	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	C₁₃×#95	—	4	2	no	—
54	#13528	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₃	—	4	1	no	—
56	C₂×D₄×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂×D₄×C₁₃	—	4	1	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	D₈	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₁₃×D₈	—	2	2	no	—
68	C₁₃×D₈	—	2	2	no	—
69	C₁₃×D₈	—	4	1	no	—
70	C₁₃×D₈	—	4	1	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	D₈	1	2	yes	—
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	—	4	1	no	—
81	C₁₃×C₄⋊C₄	—	4	1	no	—
82	C₁₃×C₄⋊C₄	—	4	1	no	—
83	C₁₃×C₄⋊C₄	—	4	1	no	—
84	C₁₃×C₄⋊C₄	—	4	1	no	—
85	C₁₃×C₄⋊C₄	—	4	1	no	—
86	C₁₃×C₂²⋊C₄	—	8	4	no	—
87	C₄²×C₁₃	—	2	1	no	—
88	C₄²×C₁₃	—	2	1	no	—
89	C₁₃×C₁₆	—	2	2	no	—
90	C₁₃×C₁₆	—	2	2	no	—
91	C₂×#367	C₂×C₁₃	1	1	yes	characteristic, complemented
92	C₂×#366	C₂×C₁₃	1	1	yes	characteristic, complemented
93	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
94	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
95	#1970	C₂×C₁₃	1	1	yes	characteristic
96	#1846	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#1374	C₂×C₁₃	1	1	yes	characteristic, complemented
98	C₂³×C₁₃	—	4	2	no	—
99	C₂³×C₁₃	—	8	1	no	—
100	C₂³×C₁₃	—	8	1	no	—
101	C₂³×C₁₃	—	8	2	no	—
102	C₂³×C₁₃	—	8	2	no	—
103	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
104	D₄×C₁₃	—	4	2	no	—
105	D₄×C₁₃	—	4	2	no	—
106	D₄×C₁₃	—	8	1	no	—
107	D₄×C₁₃	—	8	1	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	8	1	no	—
112	C₂×C₄×C₁₃	—	8	1	no	—
113	C₂×C₄×C₁₃	—	8	1	no	—
114	C₂×C₄×C₁₃	—	8	2	no	—
115	C₂×C₄×C₁₃	—	8	2	no	—
116	C₂×C₄×C₁₃	—	8	2	no	—
117	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
123	C₈×C₁₃	—	2	2	no	—
124	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₈×C₁₃	#123	1	1	yes	characteristic
127	C₈×C₁₃	#123	1	1	yes	characteristic
128	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
130	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#106	—	2	1	no	—
135	C₂×#102	—	2	1	no	—
136	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
137	#390	—	2	1	no	—
138	#382	—	2	1	no	—
139	#367	—	2	2	no	—
140	#366	—	2	2	no	—
141	#357	—	2	2	no	—
142	#357	—	2	2	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	8	4	no	—
145	C₂²×C₁₃	—	16	2	no	—
146	C₂²×C₁₃	—	16	1	no	—
147	C₂²×C₁₃	—	16	1	no	—
148	C₂²×C₁₃	#474	1	1	yes	characteristic
149	C₂²×C₁₃	#480	1	1	yes	characteristic
150	C₂²×C₁₃	#493	1	1	yes	characteristic
151	C₂²×C₁₃	#508	1	2	yes	—
152	C₂²×C₁₃	#509	1	2	yes	—
153	C₄×C₁₃	—	4	2	no	—
154	C₄×C₁₃	—	16	1	no	—
155	C₄×C₁₃	—	16	1	no	—
156	C₄×C₁₃	—	16	1	no	—
157	C₄×C₁₃	#399	1	1	yes	characteristic
158	C₄×C₁₃	#474	1	1	yes	characteristic
159	C₄×C₁₃	#480	1	1	yes	characteristic
160	C₄×C₁₃	#497	1	1	yes	characteristic
161	C₂²×D₄	—	2	1	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	Q₈×C₁₃	1	2	yes	—
168	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₂²⋊C₄	—	4	2	no	—
174	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
180	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
181	#107	—	2	1	no	—
182	#107	—	2	1	no	—
183	#107	—	2	1	no	—
184	#107	—	2	1	no	—
185	#107	—	2	1	no	—
186	#107	—	2	1	no	—
187	#106	—	4	2	no	—
188	#102	—	4	2	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	#95	—	4	2	no	—
192	C₂×C₁₃	—	16	2	no	—
193	C₂×C₁₃	#2093	1	1	yes	characteristic
194	C₂×C₁₃	#2228	1	2	yes	—
195	C₂×C₁₃	#2234	1	2	yes	—
196	C₂×C₁₃	#2249	1	1	yes	characteristic
197	C₂×C₁₃	#2250	1	1	yes	characteristic
198	C₂⁴	—	4	1	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	C₁₃×D₈	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
210	D₈	—	2	2	no	—
211	D₈	—	2	2	no	—
212	D₈	—	4	1	no	—
213	D₈	—	4	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	C₁₃×D₈	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
220	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
221	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
222	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₂²⋊C₄	—	8	4	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	2	1	no	—
232	C₁₆	—	2	2	no	—
233	C₁₆	—	2	2	no	—
234	C₁₃	#13528	1	1	yes	characteristic, direct product factor, Sylow
235	C₂³	—	4	2	no	—
236	C₂³	—	8	1	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	2	no	—
239	C₂³	—	8	2	no	—
240	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
241	D₄	—	4	2	no	—
242	D₄	—	4	2	no	—
243	D₄	—	8	1	no	—
244	D₄	—	8	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	8	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	—	8	1	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
256	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
257	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
258	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
259	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
262	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
263	C₈	C₁₃×#123	1	1	yes	characteristic
264	C₈	C₁₃×#123	1	1	yes	characteristic
265	C₂²	—	8	2	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	16	1	no	—
268	C₂²	—	16	1	no	—
269	C₂²	—	16	2	no	—
270	C₂²	C₁₃×#474	1	1	yes	characteristic
271	C₂²	C₁₃×#480	1	1	yes	characteristic
272	C₂²	C₁₃×#493	1	1	yes	characteristic
273	C₂²	C₁₃×#508	1	2	yes	—
274	C₂²	C₁₃×#509	1	2	yes	—
275	C₄	—	4	2	no	—
276	C₄	—	16	1	no	—
277	C₄	—	16	1	no	—
278	C₄	—	16	1	no	—
279	C₄	C₁₃×#399	1	1	yes	characteristic
280	C₄	C₁₃×#474	1	1	yes	characteristic
281	C₄	C₁₃×#480	1	1	yes	characteristic
282	C₄	C₁₃×#497	1	1	yes	characteristic
283	C₂	—	16	2	no	—
284	C₂	C₁₃×#2093	1	1	yes	characteristic
285	C₂	C₁₃×#2228	1	2	yes	—
286	C₂	C₁₃×#2234	1	2	yes	—
287	C₂	C₁₃×#2249	1	1	yes	characteristic
288	C₂	C₁₃×#2250	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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×#367	C₂	semidirect
2. C₂×C₁₃×#366	C₂	semidirect
6. C₁₃×#1846	C₂	semidirect
7. C₁₃×#1374	C₂	semidirect
54. #13528	C₁₃	direct (N × H)
91. C₂×#367	C₂×C₁₃	semidirect
92. C₂×#366	C₂×C₁₃	semidirect
96. #1846	C₂×C₁₃	semidirect
97. #1374	C₂×C₁₃	semidirect
234. C₁₃	#13528	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×#367	2
C₂×C₁₃×#366	2
C₂×C₁₃×#357	2
C₁₃×#1970	2
C₁₃×#1846	2
C₁₃×#1374	2
#13528	13

Subgroups — C13×#13528 #15699867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13528 #15699867