Subgroups — C₁₃×#33775 #15720114

Order 3328.

582 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2247	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2241	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2209	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2188	C₂	1	4	yes	maximal
7	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×Q₁₆	—	2	1	no	—
9	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
10	C₁₃×#533	—	2	1	no	—
11	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₁₃×Q₃₂	C₂²	1	8	yes	—
13	C₁₃×#500	C₂²	1	1	yes	characteristic
14	C₁₃×#494	—	2	1	no	—
15	C₁₃×#494	—	2	1	no	—
16	C₁₃×#494	C₂²	1	2	yes	—
17	C₁₃×#494	C₂²	1	1	yes	characteristic
18	C₁₃×#494	C₂²	1	2	yes	—
19	C₁₃×#487	—	2	1	no	—
20	C₁₃×#467	—	2	1	no	—
21	C₁₃×#467	C₂²	1	2	yes	—
22	C₄×C₁₃×Q₁₆	—	2	1	no	—
23	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₁₃×#358	C₂²	1	8	yes	—
27	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃	—	2	2	no	—
30	C₂×C₁₃×Q₁₆	—	2	1	no	—
31	C₂×C₁₃×Q₁₆	—	2	2	no	—
32	C₂×C₁₃×Q₁₆	—	2	2	no	—
33	C₂×C₁₃×Q₁₆	—	2	2	no	—
34	C₂×C₁₃×Q₁₆	—	4	4	no	—
35	C₂×C₁₃×Q₁₆	—	4	4	no	—
36	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
37	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	—	2	2	no	—
43	C₁₃×C₄⋊Q₈	—	2	2	no	—
44	C₁₃×C₄⋊Q₈	—	4	1	no	—
45	C₁₃×C₄⋊Q₈	—	4	1	no	—
46	C₁₃×#125	—	4	1	no	—
47	C₁₃×C₂²⋊Q₈	—	2	2	no	—
48	C₁₃×C₂²⋊Q₈	—	4	1	no	—
49	C₄×Q₈×C₁₃	—	2	2	no	—
50	C₄×Q₈×C₁₃	—	4	1	no	—
51	C₁₃×#117	—	4	1	no	—
52	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
53	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₁₃×Q₃₂	—	4	8	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
57	C₁₃×#107	—	2	1	no	—
58	C₁₃×#107	C₂³	1	2	yes	—
59	C₁₃×#103	—	2	2	no	—
60	C₁₃×#103	—	2	2	no	—
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	#33775	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×Q₈×C₁₃	—	2	2	no	—
67	C₂×Q₈×C₁₃	—	2	4	no	—
68	C₂×Q₈×C₁₃	—	4	4	no	—
69	C₂×Q₈×C₁₃	—	4	4	no	—
70	C₂×Q₈×C₁₃	—	4	1	no	—
71	C₂×Q₈×C₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₁₃×Q₁₆	—	2	4	no	—
77	C₁₃×Q₁₆	—	2	8	no	—
78	C₁₃×Q₁₆	—	4	4	no	—
79	C₁₃×Q₁₆	—	4	4	no	—
80	C₁₃×Q₁₆	—	4	2	no	—
81	C₁₃×Q₁₆	—	4	4	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	4	2	no	—
90	C₁₃×C₄⋊C₄	—	4	2	no	—
91	C₁₃×C₄⋊C₄	—	4	2	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₁₃×C₄⋊C₄	—	4	1	no	—
95	C₁₃×C₄⋊C₄	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₁₃×C₂²⋊C₄	—	4	2	no	—
99	C₄²×C₁₃	—	4	2	no	—
100	C₄²×C₁₃	D₈	1	1	yes	characteristic
101	C₄²×C₁₃	D₈	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₁₃×C₁₆	—	2	4	no	—
105	#3170	C₂×C₁₃	1	1	yes	characteristic
106	C₂×#494	C₂×C₁₃	1	1	yes	characteristic
107	#2247	C₂×C₁₃	1	2	yes	complemented
108	#2241	C₂×C₁₃	1	2	yes	complemented
109	#2209	C₂×C₁₃	1	4	yes	complemented
110	#2188	C₂×C₁₃	1	4	yes	—
111	#2162	C₂×C₁₃	1	1	yes	characteristic
112	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₁₃	—	4	4	no	—
114	Q₈×C₁₃	—	4	2	no	—
115	Q₈×C₁₃	—	4	4	no	—
116	Q₈×C₁₃	—	4	2	no	—
117	Q₈×C₁₃	—	8	4	no	—
118	Q₈×C₁₃	—	8	2	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	8	4	no	—
126	C₂×C₄×C₁₃	Q₃₂	1	4	yes	complemented
127	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₁₃	—	2	1	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
136	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
137	C₂²×Q₁₆	—	2	1	no	—
138	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
139	#533	—	2	1	no	—
140	C₂×C₄⋊Q₈	—	2	1	no	—
141	C₂×Q₃₂	C₂²×C₁₃	1	8	yes	—
142	#500	C₂²×C₁₃	1	1	yes	characteristic
143	#494	—	2	1	no	—
144	#494	—	2	1	no	—
145	#494	C₂²×C₁₃	1	2	yes	—
146	#494	C₂²×C₁₃	1	2	yes	—
147	#494	C₂²×C₁₃	1	1	yes	characteristic
148	#487	—	2	1	no	—
149	#467	—	2	1	no	—
150	#467	C₂²×C₁₃	1	2	yes	—
151	C₄×Q₁₆	—	2	1	no	—
152	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
153	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
154	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
155	#358	C₂²×C₁₃	1	8	yes	—
156	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
157	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
161	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	8	2	no	—
165	C₄×C₁₃	—	8	4	no	—
166	C₄×C₁₃	—	8	2	no	—
167	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
168	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
169	C₄×C₁₃	#510	1	2	yes	—
170	C₄×C₁₃	#535	1	1	yes	characteristic
171	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂²×Q₈	—	2	2	no	—
173	C₂×Q₁₆	—	2	2	no	—
174	C₂×Q₁₆	—	2	2	no	—
175	C₂×Q₁₆	—	2	2	no	—
176	C₂×Q₁₆	—	2	1	no	—
177	C₂×Q₁₆	—	4	4	no	—
178	C₂×Q₁₆	—	4	4	no	—
179	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
180	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
181	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	—	4	1	no	—
188	C₄⋊Q₈	—	4	1	no	—
189	#125	—	4	1	no	—
190	C₂²⋊Q₈	—	2	2	no	—
191	C₂²⋊Q₈	—	4	1	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	4	1	no	—
194	#117	—	4	1	no	—
195	C₂×C₄⋊C₄	—	4	1	no	—
196	C₂×C₄⋊C₄	—	4	1	no	—
197	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
198	Q₃₂	—	4	8	no	—
199	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
200	#107	—	2	1	no	—
201	#107	C₂³×C₁₃	1	2	yes	—
202	#103	—	2	2	no	—
203	#103	—	2	2	no	—
204	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
205	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
208	C₂×C₁₃	—	2	2	no	—
209	C₂×C₁₃	#3205	1	1	yes	characteristic
210	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
211	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	4	4	no	—
215	C₂×Q₈	—	4	4	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
220	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
222	Q₁₆	—	2	4	no	—
223	Q₁₆	—	2	8	no	—
224	Q₁₆	—	4	4	no	—
225	Q₁₆	—	4	4	no	—
226	Q₁₆	—	4	4	no	—
227	Q₁₆	—	4	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
232	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
233	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	C₁₃×D₈	1	1	yes	characteristic
247	C₄²	C₁₃×D₈	1	1	yes	characteristic
248	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₁₆	—	2	4	no	—
251	C₁₃	#33775	1	1	yes	characteristic, direct product factor, Sylow
252	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
253	Q₈	—	4	4	no	—
254	Q₈	—	4	2	no	—
255	Q₈	—	4	4	no	—
256	Q₈	—	4	2	no	—
257	Q₈	—	8	4	no	—
258	Q₈	—	8	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	C₁₃×Q₃₂	1	4	yes	complemented
267	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
268	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
269	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
270	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
271	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
276	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	C₂×C₁₃×Q₃₂	1	2	yes	—
280	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	—	8	2	no	—
286	C₄	C₂×C₁₃×Q₃₂	1	2	yes	—
287	C₄	C₂×C₁₃×Q₃₂	1	2	yes	—
288	C₄	C₁₃×#510	1	2	yes	—
289	C₄	C₁₃×#535	1	1	yes	characteristic
290	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	C₁₃×#3205	1	1	yes	characteristic
293	C₂	C₂²×C₁₃×Q₃₂	1	1	yes	characteristic
294	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	C₂³×C₁₃	C₂×D₈
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
3. C₁₃×#2247	C₂	semidirect
4. C₁₃×#2241	C₂	semidirect
5. C₁₃×#2209	C₂	semidirect
65. #33775	C₁₃	direct (N × H)
107. #2247	C₂×C₁₃	semidirect
108. #2241	C₂×C₁₃	semidirect
109. #2209	C₂×C₁₃	semidirect
126. C₂×C₄×C₁₃	Q₃₂	semidirect
251. C₁₃	#33775	direct (N × H)
266. C₂×C₄	C₁₃×Q₃₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#3170	2
C₂×C₁₃×#494	2
C₁₃×#2247	2
C₁₃×#2241	2
C₁₃×#2209	2
C₁₃×#2188	2
C₁₃×#2162	2
#33775	13

Subgroups — C13×#33775 #15720114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33775 #15720114