Subgroups — C₁₃×#33969 #15720308

Order 3328.

504 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3393	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2251	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2249	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2211	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2210	C₂	1	2	yes	maximal
6	C₁₃×#2209	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2175	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#556	—	2	2	no	—
11	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
14	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
17	C₁₃×#495	C₂²	1	2	yes	—
18	C₁₃×#495	C₂²	1	2	yes	—
19	C₁₃×#494	C₂²	1	2	yes	—
20	C₁₃×#464	—	2	2	no	—
21	C₁₃×#464	—	2	2	no	—
22	C₁₃×#461	—	2	2	no	—
23	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
25	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
26	C₄×C₁₃×D₈	C₂²	1	2	yes	—
27	C₁₃×#366	C₂²	1	1	yes	characteristic
28	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₃×#358	C₂²	1	2	yes	—
31	C₁₃×#358	C₂²	1	2	yes	—
32	C₁₃×#357	C₂²	1	2	yes	—
33	C₁₃×#346	C₂²	1	2	yes	—
34	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×C₁₃×Q₁₆	—	2	2	no	—
36	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
37	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
38	C₂×C₁₃×SD₁₆	—	2	2	no	—
39	C₂×C₁₃×SD₁₆	—	2	2	no	—
40	C₂×C₁₃×D₈	C₂³	1	2	yes	—
41	C₁₃×C₄⋊Q₈	—	2	2	no	—
42	C₁₃×#126	—	4	2	no	—
43	C₁₃×#126	—	4	2	no	—
44	C₁₃×#124	—	2	2	no	—
45	C₁₃×#124	—	2	2	no	—
46	C₁₃×C₂²⋊Q₈	—	4	2	no	—
47	C₄×Q₈×C₁₃	—	2	2	no	—
48	C₄×Q₈×C₁₃	—	2	2	no	—
49	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
50	C₄×D₄×C₁₃	—	2	2	no	—
51	C₁₃×#117	—	4	2	no	—
52	C₁₃×Q₃₂	—	2	2	no	—
53	C₁₃×Q₃₂	—	4	1	no	—
54	C₁₃×Q₃₂	—	4	2	no	—
55	C₁₃×SD₃₂	—	2	4	no	—
56	C₁₃×SD₃₂	—	4	2	no	—
57	C₁₃×D₁₆	—	4	1	no	—
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
63	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₁₃×#103	—	2	2	no	—
65	C₁₃×#103	—	2	2	no	—
66	C₁₃×#102	—	2	2	no	—
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	2	yes	—
69	#33969	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
73	C₂×D₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	4	2	no	—
75	C₁₃×Q₁₆	—	2	2	no	—
76	C₁₃×Q₁₆	—	2	2	no	—
77	C₁₃×Q₁₆	—	2	2	no	—
78	C₁₃×Q₁₆	—	2	2	no	—
79	C₁₃×Q₁₆	—	8	2	no	—
80	C₁₃×SD₁₆	—	8	2	no	—
81	C₁₃×SD₁₆	—	8	2	no	—
82	C₁₃×D₈	—	2	2	no	—
83	C₁₃×D₈	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	4	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	D₈	1	2	yes	—
93	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
94	C₁₃×C₂²⋊C₄	—	4	2	no	—
95	C₁₃×C₂²⋊C₄	—	4	2	no	—
96	C₁₃×C₂²⋊C₄	—	4	2	no	—
97	C₁₃×C₂²⋊C₄	—	4	2	no	—
98	C₁₃×C₂²⋊C₄	—	4	2	no	—
99	C₄²×C₁₃	—	4	2	no	—
100	C₄²×C₁₃	—	4	2	no	—
101	C₄²×C₁₃	C₂×D₄	1	2	yes	—
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₁₃×C₁₆	—	2	1	no	—
104	C₁₃×C₁₆	—	2	2	no	—
105	C₁₃×C₁₆	#41	1	2	yes	—
106	#3393	C₂×C₁₃	1	2	yes	complemented
107	#2251	C₂×C₁₃	1	2	yes	complemented
108	#2249	C₂×C₁₃	1	1	yes	characteristic
109	#2211	C₂×C₁₃	1	2	yes	complemented
110	#2210	C₂×C₁₃	1	2	yes	—
111	#2209	C₂×C₁₃	1	2	yes	complemented
112	#2183	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2175	C₂×C₁₃	1	2	yes	complemented
114	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂³×C₁₃	—	4	2	no	—
116	Q₈×C₁₃	—	4	1	no	—
117	Q₈×C₁₃	—	4	2	no	—
118	Q₈×C₁₃	—	4	2	no	—
119	Q₈×C₁₃	—	4	2	no	—
120	Q₈×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	4	2	no	—
122	D₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	8	4	no	—
127	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
136	#556	—	2	2	no	—
137	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
140	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
141	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
142	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
143	#495	C₂²×C₁₃	1	2	yes	—
144	#495	C₂²×C₁₃	1	2	yes	—
145	#494	C₂²×C₁₃	1	2	yes	—
146	#464	—	2	2	no	—
147	#464	—	2	2	no	—
148	#461	—	2	2	no	—
149	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
151	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
152	C₄×D₈	C₂²×C₁₃	1	2	yes	—
153	#366	C₂²×C₁₃	1	1	yes	characteristic
154	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
155	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
156	#358	C₂²×C₁₃	1	2	yes	—
157	#358	C₂²×C₁₃	1	2	yes	—
158	#357	C₂²×C₁₃	1	2	yes	—
159	#346	C₂²×C₁₃	1	2	yes	—
160	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
161	C₂²×C₁₃	—	8	2	no	—
162	C₂²×C₁₃	—	8	2	no	—
163	C₂²×C₁₃	—	8	2	no	—
164	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	8	2	no	—
170	C₄×C₁₃	—	8	2	no	—
171	C₄×C₁₃	—	8	2	no	—
172	C₄×C₁₃	#550	1	1	yes	characteristic
173	C₄×C₁₃	#576	1	1	yes	characteristic
174	C₂×Q₁₆	—	2	2	no	—
175	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
176	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
177	C₂×SD₁₆	—	2	2	no	—
178	C₂×SD₁₆	—	2	2	no	—
179	C₂×D₈	C₂³×C₁₃	1	2	yes	—
180	C₄⋊Q₈	—	2	2	no	—
181	#126	—	4	2	no	—
182	#126	—	4	2	no	—
183	#124	—	2	2	no	—
184	#124	—	2	2	no	—
185	C₂²⋊Q₈	—	4	2	no	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
189	C₄×D₄	—	2	2	no	—
190	#117	—	4	2	no	—
191	Q₃₂	—	2	2	no	—
192	Q₃₂	—	4	1	no	—
193	Q₃₂	—	4	2	no	—
194	SD₃₂	—	2	4	no	—
195	SD₃₂	—	4	2	no	—
196	D₁₆	—	4	1	no	—
197	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
198	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
199	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
200	#107	C₂³×C₁₃	1	2	yes	—
201	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
202	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
203	#103	—	2	2	no	—
204	#103	—	2	2	no	—
205	#102	—	2	2	no	—
206	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₁₃	1	2	yes	—
208	C₂×C₁₃	—	16	2	no	—
209	C₂×C₁₃	#3391	1	1	yes	characteristic
210	C₂×C₁₃	#3414	1	1	yes	characteristic
211	C₂×C₁₃	#3417	1	1	yes	characteristic
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
215	C₂×D₄	—	2	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	Q₁₆	—	2	2	no	—
218	Q₁₆	—	2	2	no	—
219	Q₁₆	—	2	2	no	—
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	8	2	no	—
222	SD₁₆	—	8	2	no	—
223	SD₁₆	—	8	2	no	—
224	D₈	—	2	2	no	—
225	D₈	—	2	2	no	—
226	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
227	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
235	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	C₂×D₄×C₁₃	1	2	yes	—
244	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₁₆	—	2	1	no	—
246	C₁₆	—	2	2	no	—
247	C₁₆	C₁₃×#41	1	2	yes	—
248	C₁₃	#33969	1	1	yes	characteristic, direct product factor, Sylow
249	C₂³	—	4	2	no	—
250	Q₈	—	4	1	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
262	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
263	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
264	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	2	no	—
268	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
269	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	2	no	—
281	C₄	C₁₃×#550	1	1	yes	characteristic
282	C₄	C₁₃×#576	1	1	yes	characteristic
283	C₂	—	16	2	no	—
284	C₂	C₁₃×#3391	1	1	yes	characteristic
285	C₂	C₁₃×#3414	1	1	yes	characteristic
286	C₂	C₁₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ C₈×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3393	C₂	semidirect
2. C₁₃×#2251	C₂	semidirect
4. C₁₃×#2211	C₂	semidirect
6. C₁₃×#2209	C₂	semidirect
7. C₁₃×#2183	C₂	semidirect
8. C₁₃×#2175	C₂	semidirect
9. C₄×C₁₃×Q₃₂	C₂	semidirect
69. #33969	C₁₃	direct (N × H)
106. #3393	C₂×C₁₃	semidirect
107. #2251	C₂×C₁₃	semidirect
109. #2211	C₂×C₁₃	semidirect
111. #2209	C₂×C₁₃	semidirect
112. #2183	C₂×C₁₃	semidirect
113. #2175	C₂×C₁₃	semidirect
114. C₄×Q₃₂	C₂×C₁₃	semidirect
248. C₁₃	#33969	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×#3393	2
C₁₃×#2251	2
C₁₃×#2249	2
C₁₃×#2211	2
C₁₃×#2210	2
C₁₃×#2209	2
C₁₃×#2183	2
C₁₃×#2175	2
C₄×C₁₃×Q₃₂	2
#33969	13

Subgroups — C13×#33969 #15720308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33969 #15720308