Subgroups — C₁₃×#29155 #15715494

Order 3328.

608 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3162	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#444	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2031	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1976	C₂	1	2	yes	maximal
5	C₁₃×#1833	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1814	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1812	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	C₂²	1	2	yes	—
11	C₁₃×#533	C₂²	1	2	yes	—
12	C₁₃×#501	—	2	4	no	—
13	C₁₃×#444	—	2	4	no	—
14	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₂²	1	2	yes	—
18	C₁₃×#427	C₂²	1	2	yes	—
19	C₁₃×#421	C₂²	1	4	yes	—
20	C₂×C₁₃×#104	C₂²	1	2	yes	—
21	C₁₃×#410	C₂²	1	4	yes	—
22	C₁₃×#405	C₂²	1	2	yes	—
23	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
24	C₁₃×#344	C₂²	1	4	yes	—
25	C₁₃×#339	C₂²	1	2	yes	—
26	C₁₃×H(3,ℤ/4)	C₂²	1	4	yes	—
27	C₂×C₁₃×#41	C₂³	1	2	yes	—
28	C₁₃×#135	—	2	2	no	—
29	C₁₃×#135	—	2	2	no	—
30	C₂×C₁₃×Q₁₆	Q₈	1	1	yes	characteristic, complemented
31	C₂×C₁₃×SD₁₆	Q₈	1	2	yes	complemented
32	C₂×C₁₃×D₈	Q₈	1	1	yes	characteristic, complemented
33	C₁₃×#131	—	2	2	no	—
34	C₁₃×#131	—	2	2	no	—
35	C₁₃×#131	—	2	2	no	—
36	C₁₃×#131	—	2	2	no	—
37	C₂×C₁₃×M₁₆	—	2	2	no	—
38	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
39	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	—	2	2	no	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×C₄⋊Q₈	—	2	2	no	—
44	C₁₃×#125	—	2	2	no	—
45	C₁₃×C₂²⋊Q₈	—	4	2	no	—
46	C₄×Q₈×C₁₃	—	4	2	no	—
47	C₁₃×#117	C₂³	1	2	yes	—
48	C₁₃×#117	C₂³	1	4	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×#108	—	2	2	no	—
51	C₁₃×#108	—	2	2	no	—
52	C₁₃×#107	—	2	2	no	—
53	C₁₃×#107	—	2	2	no	—
54	C₁₃×#107	—	2	1	no	—
55	C₁₃×#106	—	2	2	no	—
56	C₁₃×#106	—	2	2	no	—
57	C₁₃×#106	—	2	1	no	—
58	C₁₃×#104	—	2	4	no	—
59	C₁₃×#104	—	4	4	no	—
60	C₁₃×#99	—	4	4	no	—
61	C₁₃×#97	—	2	2	no	—
62	C₁₃×#97	D₄	1	4	yes	—
63	C₄×C₈×C₁₃	—	2	2	no	—
64	#29155	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₁₃×#41	—	2	2	no	—
66	C₁₃×#41	—	2	2	no	—
67	C₁₃×#41	—	2	2	no	—
68	C₁₃×#41	—	2	2	no	—
69	C₂×Q₈×C₁₃	—	4	2	no	—
70	C₂×Q₈×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂×D₄×C₁₃	—	2	2	no	—
72	C₂×D₄×C₁₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₁₃	—	2	1	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₁₃×Q₁₆	—	2	2	no	—
77	C₁₃×SD₁₆	—	2	4	no	—
78	C₁₃×D₈	—	2	2	no	—
79	C₁₃×M₁₆	—	2	2	no	—
80	C₁₃×M₁₆	—	2	2	no	—
81	C₁₃×M₁₆	—	2	2	no	—
82	C₁₃×M₁₆	—	2	2	no	—
83	C₁₃×M₁₆	—	4	2	no	—
84	C₁₃×M₁₆	—	4	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	4	2	no	—
88	C₂×C₈×C₁₃	—	4	2	no	—
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₁₃	#41	1	2	yes	—
94	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₁₃	#41	1	2	yes	—
97	C₁₃×C₄⋊C₄	—	2	4	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	4	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₂²⋊C₄	—	2	2	no	—
102	C₁₃×C₂²⋊C₄	—	2	4	no	—
103	C₄²×C₁₃	—	2	1	no	—
104	C₄²×C₁₃	—	2	2	no	—
105	C₄²×C₁₃	—	2	4	no	—
106	C₄²×C₁₃	C₂×D₄	1	2	yes	—
107	#3162	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂×#444	C₂×C₁₃	1	1	yes	characteristic
109	#2031	C₂×C₁₃	1	4	yes	complemented
110	#1976	C₂×C₁₃	1	2	yes	—
111	#1833	C₂×C₁₃	1	2	yes	complemented
112	#1814	C₂×C₁₃	1	2	yes	complemented
113	#1812	C₂×C₁₃	1	2	yes	complemented
114	#1773	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂³×C₁₃	—	2	2	no	—
116	C₂³×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
117	Q₈×C₁₃	—	2	2	no	—
118	Q₈×C₁₃	—	2	2	no	—
119	Q₈×C₁₃	—	8	4	no	—
120	D₄×C₁₃	—	2	2	no	—
121	D₄×C₁₃	—	2	2	no	—
122	D₄×C₁₃	—	4	2	no	—
123	D₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	8	1	no	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₁₃	—	4	2	no	—
138	C₈×C₁₃	—	4	2	no	—
139	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
140	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
141	C₈×C₁₃	#124	1	2	yes	—
142	C₈×C₁₃	#124	1	2	yes	—
143	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×#131	C₂²×C₁₃	1	2	yes	—
145	#533	C₂²×C₁₃	1	2	yes	—
146	#501	—	2	4	no	—
147	#444	—	2	4	no	—
148	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
150	#427	C₂²×C₁₃	1	1	yes	characteristic
151	#427	C₂²×C₁₃	1	2	yes	—
152	#427	C₂²×C₁₃	1	2	yes	—
153	#421	C₂²×C₁₃	1	4	yes	—
154	C₂×#104	C₂²×C₁₃	1	2	yes	—
155	#410	C₂²×C₁₃	1	4	yes	—
156	#405	C₂²×C₁₃	1	2	yes	—
157	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
158	#344	C₂²×C₁₃	1	4	yes	—
159	#339	C₂²×C₁₃	1	2	yes	—
160	H(3,ℤ/4)	C₂²×C₁₃	1	4	yes	—
161	C₂²×C₁₃	—	4	1	no	—
162	C₂²×C₁₃	—	4	2	no	—
163	C₂²×C₁₃	—	8	2	no	—
164	C₂²×C₁₃	#535	1	1	yes	characteristic
165	C₂²×C₁₃	#554	1	1	yes	characteristic
166	C₂²×C₁₃	#555	1	1	yes	characteristic
167	C₄×C₁₃	—	4	2	no	—
168	C₄×C₁₃	—	4	2	no	—
169	C₄×C₁₃	—	8	2	no	—
170	C₄×C₁₃	—	8	4	no	—
171	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
173	C₄×C₁₃	#554	1	1	yes	characteristic
174	C₄×C₁₃	#555	1	1	yes	characteristic
175	C₂×#41	C₂³×C₁₃	1	2	yes	—
176	#135	—	2	2	no	—
177	#135	—	2	2	no	—
178	C₂×Q₁₆	Q₈×C₁₃	1	1	yes	characteristic, complemented
179	C₂×SD₁₆	Q₈×C₁₃	1	2	yes	complemented
180	C₂×D₈	Q₈×C₁₃	1	1	yes	characteristic, complemented
181	#131	—	2	2	no	—
182	#131	—	2	2	no	—
183	#131	—	2	2	no	—
184	#131	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
187	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
191	C₄⋊Q₈	—	2	2	no	—
192	#125	—	2	2	no	—
193	C₂²⋊Q₈	—	4	2	no	—
194	C₄×Q₈	—	4	2	no	—
195	#117	C₂³×C₁₃	1	4	yes	—
196	#117	C₂³×C₁₃	1	2	yes	—
197	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
198	#108	—	2	2	no	—
199	#108	—	2	2	no	—
200	#107	—	2	2	no	—
201	#107	—	2	2	no	—
202	#107	—	2	1	no	—
203	#106	—	2	2	no	—
204	#106	—	2	2	no	—
205	#106	—	2	1	no	—
206	#104	—	2	4	no	—
207	#104	—	4	4	no	—
208	#99	—	4	4	no	—
209	#97	—	2	2	no	—
210	#97	D₄×C₁₃	1	4	yes	—
211	C₄×C₈	—	2	2	no	—
212	C₂×C₁₃	—	2	1	no	—
213	C₂×C₁₃	—	2	1	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	8	2	no	—
216	C₂×C₁₃	#2563	1	1	yes	characteristic
217	#41	—	2	2	no	—
218	#41	—	2	2	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	C₂×Q₈	—	4	2	no	—
222	C₂×Q₈	C₂×Q₈×C₁₃	1	2	yes	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	C₂×Q₈×C₁₃	1	2	yes	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
228	Q₁₆	—	2	2	no	—
229	SD₁₆	—	2	4	no	—
230	D₈	—	2	2	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
242	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
243	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
244	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
245	C₂×C₈	C₁₃×#41	1	2	yes	—
246	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
247	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
248	C₂×C₈	C₁₃×#41	1	2	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	4	no	—
258	C₄²	C₂×D₄×C₁₃	1	2	yes	—
259	C₁₃	#29155	1	1	yes	characteristic, direct product factor, Sylow
260	C₂³	—	2	2	no	—
261	C₂³	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	—	2	2	no	—
264	Q₈	—	8	4	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
279	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
280	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
281	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
285	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
286	C₈	C₁₃×#124	1	2	yes	—
287	C₈	C₁₃×#124	1	2	yes	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	C₁₃×#535	1	1	yes	characteristic
292	C₂²	C₁₃×#554	1	1	yes	characteristic
293	C₂²	C₁₃×#555	1	1	yes	characteristic
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	8	4	no	—
298	C₄	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
299	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
300	C₄	C₁₃×#554	1	1	yes	characteristic
301	C₄	C₁₃×#555	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	8	2	no	—
306	C₂	C₁₃×#2563	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂×C₁₃	#2563
prime radical	C₂×C₁₃×#135	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×#3162	C₂	semidirect
3. C₁₃×#2031	C₂	semidirect
5. C₁₃×#1833	C₂	semidirect
6. C₁₃×#1814	C₂	semidirect
7. C₁₃×#1812	C₂	semidirect
8. C₁₃×#1773	C₂	semidirect
30. C₂×C₁₃×Q₁₆	Q₈	semidirect
31. C₂×C₁₃×SD₁₆	Q₈	semidirect
32. C₂×C₁₃×D₈	Q₈	semidirect
64. #29155	C₁₃	direct (N × H)
107. #3162	C₂×C₁₃	semidirect
109. #2031	C₂×C₁₃	semidirect
111. #1833	C₂×C₁₃	semidirect
112. #1814	C₂×C₁₃	semidirect
113. #1812	C₂×C₁₃	semidirect
114. #1773	C₂×C₁₃	semidirect
178. C₂×Q₁₆	Q₈×C₁₃	semidirect
179. C₂×SD₁₆	Q₈×C₁₃	semidirect
180. C₂×D₈	Q₈×C₁₃	semidirect
259. C₁₃	#29155	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×#3162	2
C₂×C₁₃×#444	2
C₁₃×#2031	2
C₁₃×#1976	2
C₁₃×#1833	2
C₁₃×#1814	2
C₁₃×#1812	2
C₁₃×#1773	2
#29155	13

Subgroups — C13×#29155 #15715494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29155 #15715494