Subgroups — C₁₃×#29232 #15715571

Order 3328.

552 conjugacy classes of subgroups.

The subgroup lattice has 316 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×#1994	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	C₄×C₁₃×#104	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	C₄³×C₁₃	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	—	4	1	no	—
51	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×#108	—	4	1	no	—
54	C₁₃×C₄⋊C₈	—	4	1	no	—
55	C₁₃×#104	—	2	2	no	—
56	C₁₃×#104	—	2	2	no	—
57	C₁₃×#104	—	2	2	no	—
58	C₁₃×#104	—	2	2	no	—
59	C₁₃×#104	—	4	2	no	—
60	C₁₃×#101	—	2	2	no	—
61	C₁₃×#100	—	2	2	no	—
62	C₁₃×#99	—	4	2	no	—
63	C₁₃×#97	—	2	2	no	—
64	C₄×C₈×C₁₃	—	2	2	no	—
65	#29232	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	—	4	4	no	—
106	C₄²×C₁₃	#41	1	2	yes	—
107	C₄²×C₁₃	#41	1	2	yes	—
108	C₄²×C₁₃	#41	1	1	yes	characteristic
109	C₄²×C₁₃	#41	1	1	yes	characteristic
110	#2102	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#2050	C₂×C₁₃	1	1	yes	characteristic
112	#2031	C₂×C₁₃	1	2	yes	complemented
113	#1994	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#1932	C₂×C₁₃	1	2	yes	complemented
115	#1906	C₂×C₁₃	1	2	yes	—
116	#1841	C₂×C₁₃	1	2	yes	complemented
117	#1827	C₂×C₁₃	1	2	yes	complemented
118	C₄×#104	C₂×C₁₃	1	2	yes	complemented
119	C₂³×C₁₃	—	2	2	no	—
120	C₂³×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	2	no	—
122	Q₈×C₁₃	—	2	2	no	—
123	Q₈×C₁₃	—	8	2	no	—
124	D₄×C₁₃	—	2	2	no	—
125	D₄×C₁₃	—	2	2	no	—
126	D₄×C₁₃	—	4	2	no	—
127	D₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	4	4	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	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	—	8	4	no	—
140	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
143	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₁₃	—	4	2	no	—
145	C₈×C₁₃	—	4	1	no	—
146	C₈×C₁₃	—	4	1	no	—
147	C₈×C₁₃	—	4	2	no	—
148	#575	C₂²×C₁₃	1	1	yes	characteristic
149	#533	C₂²×C₁₃	1	1	yes	characteristic
150	C₄×#41	C₂²×C₁₃	1	2	yes	—
151	#430	C₂²×C₁₃	1	1	yes	characteristic
152	#423	C₂²×C₁₃	1	1	yes	characteristic
153	#421	C₂²×C₁₃	1	2	yes	—
154	C₂×#104	C₂²×C₁₃	1	2	yes	—
155	C₂×#104	C₂²×C₁₃	1	2	yes	—
156	#413	C₂²×C₁₃	1	2	yes	—
157	#410	C₂²×C₁₃	1	2	yes	—
158	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
159	C₄³	C₂²×C₁₃	1	1	yes	characteristic
160	#344	C₂²×C₁₃	1	2	yes	—
161	#344	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	#339	C₂²×C₁₃	1	2	yes	—
166	#338	C₂²×C₁₃	1	1	yes	characteristic
167	H(3,ℤ/4)	C₂²×C₁₃	1	1	yes	characteristic
168	H(3,ℤ/4)	C₂²×C₁₃	1	2	yes	—
169	C₂²×C₁₃	—	4	1	no	—
170	C₂²×C₁₃	—	4	2	no	—
171	C₂²×C₁₃	—	8	2	no	—
172	C₂²×C₁₃	#538	1	1	yes	characteristic
173	C₂²×C₁₃	#551	1	2	yes	—
174	C₄×C₁₃	—	2	4	no	—
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	—	4	1	no	—
177	C₄×C₁₃	—	4	2	no	—
178	C₄×C₁₃	—	4	2	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	#529	1	2	yes	—
182	C₄×C₁₃	#551	1	1	yes	characteristic
183	C₄×C₁₃	#555	1	1	yes	characteristic
184	C₂×#41	C₂³×C₁₃	1	2	yes	—
185	#137	—	2	2	no	—
186	AGL(1,ℤ/8)	—	2	2	no	—
187	#135	—	4	1	no	—
188	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
189	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
190	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
191	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
192	C₄⋊Q₈	—	2	1	no	—
193	#125	—	2	1	no	—
194	C₂²⋊Q₈	—	4	1	no	—
195	C₄×Q₈	—	2	2	no	—
196	C₄×Q₈	—	4	1	no	—
197	C₄×D₄	—	2	2	no	—
198	C₄×D₄	—	4	2	no	—
199	#117	—	2	2	no	—
200	#117	C₂³×C₁₃	1	2	yes	—
201	#117	C₂³×C₁₃	1	2	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	4	1	no	—
204	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
205	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
206	#108	—	4	1	no	—
207	C₄⋊C₈	—	4	1	no	—
208	#104	—	2	2	no	—
209	#104	—	2	2	no	—
210	#104	—	2	2	no	—
211	#104	—	2	2	no	—
212	#104	—	4	2	no	—
213	#101	—	2	2	no	—
214	#100	—	2	2	no	—
215	#99	—	4	2	no	—
216	#97	—	2	2	no	—
217	C₄×C₈	—	2	2	no	—
218	C₂×C₁₃	—	2	1	no	—
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	—	8	2	no	—
221	C₂×C₁₃	#2572	1	1	yes	characteristic
222	#41	—	2	2	no	—
223	#41	—	2	2	no	—
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	C₁₃×#41	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₁₃×#41	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
235	Q₁₆	—	4	1	no	—
236	SD₁₆	—	4	2	no	—
237	D₈	—	4	1	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	M₁₆	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	—	4	4	no	—
262	C₄²	C₁₃×#41	1	2	yes	—
263	C₄²	C₁₃×#41	1	2	yes	—
264	C₄²	C₁₃×#41	1	1	yes	characteristic
265	C₄²	C₁₃×#41	1	1	yes	characteristic
266	C₁₃	#29232	1	1	yes	characteristic, direct product factor, Sylow
267	C₂³	—	2	2	no	—
268	C₂³	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
269	Q₈	—	2	2	no	—
270	Q₈	—	2	2	no	—
271	Q₈	—	8	2	no	—
272	D₄	—	2	2	no	—
273	D₄	—	2	2	no	—
274	D₄	—	4	2	no	—
275	D₄	—	4	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
289	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
290	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
291	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
292	C₈	—	4	2	no	—
293	C₈	—	4	1	no	—
294	C₈	—	4	1	no	—
295	C₈	—	4	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	C₁₃×#538	1	1	yes	characteristic
300	C₂²	C₁₃×#551	1	2	yes	—
301	C₄	—	2	4	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	C₁₃×#529	1	2	yes	—
309	C₄	C₁₃×#551	1	1	yes	characteristic
310	C₄	C₁₃×#555	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	8	2	no	—
314	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×#1994	C₂	semidirect
5. C₁₃×#1932	C₂	semidirect
7. C₁₃×#1841	C₂	semidirect
8. C₁₃×#1827	C₂	semidirect
9. C₄×C₁₃×#104	C₂	semidirect
65. #29232	C₁₃	direct (N × H)
110. #2102	C₂×C₁₃	semidirect
112. #2031	C₂×C₁₃	semidirect
113. #1994	C₂×C₁₃	semidirect
114. #1932	C₂×C₁₃	semidirect
116. #1841	C₂×C₁₃	semidirect
117. #1827	C₂×C₁₃	semidirect
118. C₄×#104	C₂×C₁₃	semidirect
266. C₁₃	#29232	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×#1994	2
C₁₃×#1932	2
C₁₃×#1906	2
C₁₃×#1841	2
C₁₃×#1827	2
C₄×C₁₃×#104	2
#29232	13

Subgroups — C13×#29232 #15715571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29232 #15715571