Subgroups — C₁₃×#29233 #15715572

Order 3328.

532 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×#2102	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2050	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2031	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1995	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1932	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1906	C₂	1	2	yes	maximal
7	C₁₃×#1841	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1827	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1759	C₂	1	2	yes	maximal, complemented
10	C₁₃×#575	C₂²	1	1	yes	characteristic
11	C₁₃×#533	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×#41	C₂²	1	2	yes	—
13	C₁₃×#430	C₂²	1	1	yes	characteristic
14	C₁₃×#423	C₂²	1	1	yes	characteristic
15	C₁₃×#421	C₂²	1	2	yes	—
16	C₂×C₁₃×#104	C₂²	1	2	yes	—
17	C₂×C₁₃×#104	C₂²	1	2	yes	—
18	C₁₃×#413	C₂²	1	2	yes	—
19	C₁₃×#410	C₂²	1	2	yes	—
20	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
21	C₁₃×#376	C₂²	1	1	yes	characteristic
22	C₁₃×#344	C₂²	1	2	yes	—
23	C₁₃×#344	C₂²	1	2	yes	—
24	C₁₃×#339	C₂²	1	2	yes	—
25	C₁₃×#339	C₂²	1	2	yes	—
26	C₁₃×#339	C₂²	1	2	yes	—
27	C₁₃×#339	C₂²	1	2	yes	—
28	C₁₃×#338	C₂²	1	1	yes	characteristic
29	C₁₃×H(3,ℤ/4)	C₂²	1	2	yes	—
30	C₁₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
31	C₂×C₁₃×#41	C₂³	1	2	yes	—
32	C₁₃×#137	—	2	2	no	—
33	C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
34	C₁₃×#135	—	4	1	no	—
35	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
36	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
37	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
38	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊Q₈	—	2	1	no	—
40	C₁₃×#125	—	2	1	no	—
41	C₁₃×C₂²⋊Q₈	—	4	1	no	—
42	C₄×Q₈×C₁₃	—	2	2	no	—
43	C₄×Q₈×C₁₃	—	4	1	no	—
44	C₄×D₄×C₁₃	—	2	2	no	—
45	C₄×D₄×C₁₃	—	4	2	no	—
46	C₁₃×#117	—	2	2	no	—
47	C₁₃×#117	C₂³	1	2	yes	—
48	C₁₃×#117	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×#108	—	4	1	no	—
53	C₁₃×C₄⋊C₈	—	4	1	no	—
54	C₁₃×#104	—	2	2	no	—
55	C₁₃×#104	—	2	2	no	—
56	C₁₃×#104	—	2	2	no	—
57	C₁₃×#104	—	2	2	no	—
58	C₁₃×#104	—	4	2	no	—
59	C₁₃×#101	—	2	2	no	—
60	C₁₃×#100	—	2	2	no	—
61	C₁₃×#99	—	4	2	no	—
62	C₁₃×#97	—	2	2	no	—
63	C₄×C₈×C₁₃	—	2	2	no	—
64	C₁₃×#95	—	4	1	no	—
65	#29233	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₁₃×#41	—	2	2	no	—
67	C₁₃×#41	—	2	2	no	—
68	C₁₃×#41	—	2	2	no	—
69	C₁₃×#41	—	2	2	no	—
70	C₂×Q₈×C₁₃	—	4	1	no	—
71	C₂×Q₈×C₁₃	#41	1	2	yes	—
72	C₂×D₄×C₁₃	—	2	2	no	—
73	C₂×D₄×C₁₃	#41	1	2	yes	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	2	1	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	4	2	no	—
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×Q₁₆	—	4	1	no	—
80	C₁₃×SD₁₆	—	4	2	no	—
81	C₁₃×D₈	—	4	1	no	—
82	C₁₃×M₁₆	—	2	2	no	—
83	C₁₃×M₁₆	—	2	2	no	—
84	C₁₃×M₁₆	—	2	2	no	—
85	C₁₃×M₁₆	—	2	2	no	—
86	C₁₃×M₁₆	—	2	2	no	—
87	C₁₃×M₁₆	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₁₃×C₂²⋊C₄	—	2	2	no	—
98	C₁₃×C₂²⋊C₄	—	2	2	no	—
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₄²×C₁₃	—	2	2	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	4	2	no	—
105	C₄²×C₁₃	#41	1	2	yes	—
106	C₄²×C₁₃	#41	1	2	yes	—
107	C₄²×C₁₃	#41	1	1	yes	characteristic
108	C₄²×C₁₃	#41	1	1	yes	characteristic
109	#2102	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#2050	C₂×C₁₃	1	1	yes	characteristic
111	#2031	C₂×C₁₃	1	2	yes	complemented
112	#1995	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#1932	C₂×C₁₃	1	2	yes	complemented
114	#1906	C₂×C₁₃	1	2	yes	—
115	#1841	C₂×C₁₃	1	2	yes	complemented
116	#1827	C₂×C₁₃	1	2	yes	complemented
117	#1759	C₂×C₁₃	1	2	yes	complemented
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
120	Q₈×C₁₃	—	2	2	no	—
121	Q₈×C₁₃	—	2	2	no	—
122	Q₈×C₁₃	—	8	2	no	—
123	D₄×C₁₃	—	2	2	no	—
124	D₄×C₁₃	—	2	2	no	—
125	D₄×C₁₃	—	4	2	no	—
126	D₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	8	2	no	—
137	C₂×C₄×C₁₃	—	8	1	no	—
138	C₂×C₄×C₁₃	—	8	4	no	—
139	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
142	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
143	C₈×C₁₃	—	4	2	no	—
144	C₈×C₁₃	—	4	1	no	—
145	C₈×C₁₃	—	4	1	no	—
146	C₈×C₁₃	—	4	2	no	—
147	#575	C₂²×C₁₃	1	1	yes	characteristic
148	#533	C₂²×C₁₃	1	1	yes	characteristic
149	C₄×#41	C₂²×C₁₃	1	2	yes	—
150	#430	C₂²×C₁₃	1	1	yes	characteristic
151	#423	C₂²×C₁₃	1	1	yes	characteristic
152	#421	C₂²×C₁₃	1	2	yes	—
153	C₂×#104	C₂²×C₁₃	1	2	yes	—
154	C₂×#104	C₂²×C₁₃	1	2	yes	—
155	#413	C₂²×C₁₃	1	2	yes	—
156	#410	C₂²×C₁₃	1	2	yes	—
157	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
158	#376	C₂²×C₁₃	1	1	yes	characteristic
159	#344	C₂²×C₁₃	1	2	yes	—
160	#344	C₂²×C₁₃	1	2	yes	—
161	#339	C₂²×C₁₃	1	2	yes	—
162	#339	C₂²×C₁₃	1	2	yes	—
163	#339	C₂²×C₁₃	1	2	yes	—
164	#339	C₂²×C₁₃	1	2	yes	—
165	#338	C₂²×C₁₃	1	1	yes	characteristic
166	H(3,ℤ/4)	C₂²×C₁₃	1	1	yes	characteristic
167	H(3,ℤ/4)	C₂²×C₁₃	1	2	yes	—
168	C₂²×C₁₃	—	4	1	no	—
169	C₂²×C₁₃	—	4	2	no	—
170	C₂²×C₁₃	—	8	2	no	—
171	C₂²×C₁₃	#538	1	1	yes	characteristic
172	C₂²×C₁₃	#551	1	2	yes	—
173	C₄×C₁₃	—	4	1	no	—
174	C₄×C₁₃	—	4	2	no	—
175	C₄×C₁₃	—	4	2	no	—
176	C₄×C₁₃	—	4	1	no	—
177	C₄×C₁₃	—	4	2	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	#529	1	2	yes	—
181	C₄×C₁₃	#551	1	1	yes	characteristic
182	C₄×C₁₃	#555	1	1	yes	characteristic
183	C₂×#41	C₂³×C₁₃	1	2	yes	—
184	#137	—	2	2	no	—
185	AGL(1,ℤ/8)	—	2	2	no	—
186	#135	—	4	1	no	—
187	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
188	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
189	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
190	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
191	C₄⋊Q₈	—	2	1	no	—
192	#125	—	2	1	no	—
193	C₂²⋊Q₈	—	4	1	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×Q₈	—	4	1	no	—
196	C₄×D₄	—	2	2	no	—
197	C₄×D₄	—	4	2	no	—
198	#117	—	2	2	no	—
199	#117	C₂³×C₁₃	1	2	yes	—
200	#117	C₂³×C₁₃	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
203	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
204	#108	—	4	1	no	—
205	C₄⋊C₈	—	4	1	no	—
206	#104	—	2	2	no	—
207	#104	—	2	2	no	—
208	#104	—	2	2	no	—
209	#104	—	2	2	no	—
210	#104	—	4	2	no	—
211	#101	—	2	2	no	—
212	#100	—	2	2	no	—
213	#99	—	4	2	no	—
214	#97	—	2	2	no	—
215	C₄×C₈	—	2	2	no	—
216	#95	—	4	1	no	—
217	C₂×C₁₃	—	2	1	no	—
218	C₂×C₁₃	—	2	2	no	—
219	C₂×C₁₃	—	8	2	no	—
220	C₂×C₁₃	#2572	1	1	yes	characteristic
221	#41	—	2	2	no	—
222	#41	—	2	2	no	—
223	#41	—	2	2	no	—
224	#41	—	2	2	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×Q₈	C₁₃×#41	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₁₃×#41	1	2	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
234	Q₁₆	—	4	1	no	—
235	SD₁₆	—	4	2	no	—
236	D₈	—	4	1	no	—
237	M₁₆	—	2	2	no	—
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	C₁₃×#41	1	2	yes	—
261	C₄²	C₁₃×#41	1	2	yes	—
262	C₄²	C₁₃×#41	1	1	yes	characteristic
263	C₄²	C₁₃×#41	1	1	yes	characteristic
264	C₁₃	#29233	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³	—	2	2	no	—
266	C₂³	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
267	Q₈	—	2	2	no	—
268	Q₈	—	2	2	no	—
269	Q₈	—	8	2	no	—
270	D₄	—	2	2	no	—
271	D₄	—	2	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
287	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
288	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
289	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
290	C₈	—	4	2	no	—
291	C₈	—	4	1	no	—
292	C₈	—	4	1	no	—
293	C₈	—	4	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	C₁₃×#538	1	1	yes	characteristic
298	C₂²	C₁₃×#551	1	2	yes	—
299	C₄	—	4	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	C₁₃×#529	1	2	yes	—
307	C₄	C₁₃×#551	1	1	yes	characteristic
308	C₄	C₁₃×#555	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	2	no	—
311	C₂	—	8	2	no	—
312	C₂	C₁₃×#2572	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂×C₁₃	#2572
prime radical	C₁₃×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2102	C₂	semidirect
3. C₁₃×#2031	C₂	semidirect
4. C₁₃×#1995	C₂	semidirect
5. C₁₃×#1932	C₂	semidirect
7. C₁₃×#1841	C₂	semidirect
8. C₁₃×#1827	C₂	semidirect
9. C₁₃×#1759	C₂	semidirect
65. #29233	C₁₃	direct (N × H)
109. #2102	C₂×C₁₃	semidirect
111. #2031	C₂×C₁₃	semidirect
112. #1995	C₂×C₁₃	semidirect
113. #1932	C₂×C₁₃	semidirect
115. #1841	C₂×C₁₃	semidirect
116. #1827	C₂×C₁₃	semidirect
117. #1759	C₂×C₁₃	semidirect
264. C₁₃	#29233	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×#2102	2
C₁₃×#2050	2
C₁₃×#2031	2
C₁₃×#1995	2
C₁₃×#1932	2
C₁₃×#1906	2
C₁₃×#1841	2
C₁₃×#1827	2
C₁₃×#1759	2
#29233	13

Subgroups — C13×#29233 #15715572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29233 #15715572