Subgroups — C₁₃×#33829 #15720168

Order 3328.

488 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×#3175	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2226	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2225	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2219	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2218	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#2217	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2211	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2210	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#542	—	2	2	no	—
11	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
13	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#500	C₂²	1	2	yes	—
17	C₁₃×#495	C₂²	1	2	yes	—
18	C₁₃×#489	—	2	2	no	—
19	C₁₃×#468	—	2	2	no	—
20	C₁₃×#467	C₂²	1	2	yes	—
21	C₁₃×#466	C₂²	1	2	yes	—
22	C₁₃×#440	—	2	2	no	—
23	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
24	C₄×C₁₃×D₈	C₂²	1	2	yes	—
25	C₁₃×#405	C₂²	1	1	yes	characteristic
26	C₁₃×#367	C₂²	1	2	yes	—
27	C₁₃×#367	C₂²	1	1	yes	characteristic
28	C₁₃×#367	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₁₃×#358	C₂²	1	2	yes	—
32	C₁₃×#358	C₂²	1	2	yes	—
33	C₁₃×#357	C₂²	1	2	yes	—
34	C₁₃×#357	C₂²	1	2	yes	—
35	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
36	C₂×C₁₃×SD₁₆	—	2	2	no	—
37	C₂×C₁₃×D₈	C₂³	1	2	yes	—
38	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊Q₈	—	2	2	no	—
41	C₁₃×#126	—	4	2	no	—
42	C₁₃×#124	—	2	2	no	—
43	C₁₃×#124	—	4	2	no	—
44	C₁₃×#123	—	4	2	no	—
45	C₁₃×C₂²⋊Q₈	—	2	2	no	—
46	C₁₃×C₂²⋊Q₈	—	4	2	no	—
47	C₁₃×#121	—	2	2	no	—
48	C₄×Q₈×C₁₃	—	2	2	no	—
49	C₄×D₄×C₁₃	—	2	2	no	—
50	C₁₃×#117	D₄	1	1	yes	characteristic
51	C₁₃×Q₃₂	—	4	1	no	—
52	C₁₃×SD₃₂	—	4	2	no	—
53	C₁₃×D₁₆	—	4	1	no	—
54	C₁₃×M₃₂	—	4	1	no	—
55	C₂×C₁₃×C₁₆	—	4	1	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₃×#107	—	2	2	no	—
60	C₁₃×#107	C₂³	1	2	yes	—
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#103	—	2	2	no	—
63	C₁₃×#103	—	2	2	no	—
64	C₁₃×#102	—	2	2	no	—
65	C₁₃×#102	—	2	2	no	—
66	C₁₃×#97	D₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	#33829	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×Q₈×C₁₃	—	2	2	no	—
70	C₂×Q₈×C₁₃	—	4	2	no	—
71	C₂×D₄×C₁₃	—	2	2	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	—	4	2	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₁₃×Q₁₆	—	2	2	no	—
76	C₁₃×Q₁₆	—	2	2	no	—
77	C₁₃×SD₁₆	—	4	4	no	—
78	C₁₃×D₈	—	2	2	no	—
79	C₁₃×D₈	—	2	2	no	—
80	C₁₃×M₁₆	—	4	1	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	—	2	2	no	—
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	D₈	1	2	yes	—
92	C₁₃×C₂²⋊C₄	—	4	2	no	—
93	C₁₃×C₂²⋊C₄	—	4	2	no	—
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	D₈	1	1	yes	characteristic
98	C₁₃×C₂²⋊C₄	D₈	1	1	yes	characteristic
99	C₄²×C₁₃	—	4	2	no	—
100	C₄²×C₁₃	C₂×D₄	1	2	yes	—
101	C₁₃×C₁₆	—	2	2	no	—
102	C₁₃×C₁₆	—	2	1	no	—
103	C₁₃×C₁₆	—	2	1	no	—
104	#3175	C₂×C₁₃	1	2	yes	complemented
105	#2226	C₂×C₁₃	1	2	yes	complemented
106	#2225	C₂×C₁₃	1	2	yes	complemented
107	#2219	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#2218	C₂×C₁₃	1	1	yes	characteristic
109	#2217	C₂×C₁₃	1	2	yes	complemented
110	#2211	C₂×C₁₃	1	2	yes	complemented
111	#2210	C₂×C₁₃	1	2	yes	complemented
112	#2151	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	—	4	2	no	—
114	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₁₃	—	4	2	no	—
116	Q₈×C₁₃	—	4	2	no	—
117	Q₈×C₁₃	—	16	2	no	—
118	D₄×C₁₃	—	4	2	no	—
119	D₄×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	16	2	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	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	—	16	2	no	—
127	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₁₃	—	2	1	no	—
132	C₈×C₁₃	—	2	2	no	—
133	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
134	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
135	#542	—	2	2	no	—
136	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
138	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
140	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
141	#500	C₂²×C₁₃	1	2	yes	—
142	#495	C₂²×C₁₃	1	2	yes	—
143	#489	—	2	2	no	—
144	#468	—	2	2	no	—
145	#467	C₂²×C₁₃	1	2	yes	—
146	#466	C₂²×C₁₃	1	2	yes	—
147	#440	—	2	2	no	—
148	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
149	C₄×D₈	C₂²×C₁₃	1	2	yes	—
150	#405	C₂²×C₁₃	1	1	yes	characteristic
151	#367	C₂²×C₁₃	1	2	yes	—
152	#367	C₂²×C₁₃	1	1	yes	characteristic
153	#367	C₂²×C₁₃	1	1	yes	characteristic
154	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
155	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
156	#358	C₂²×C₁₃	1	2	yes	—
157	#358	C₂²×C₁₃	1	2	yes	—
158	#357	C₂²×C₁₃	1	2	yes	—
159	#357	C₂²×C₁₃	1	2	yes	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	8	2	no	—
163	C₂²×C₁₃	—	8	2	no	—
164	C₂²×C₁₃	—	8	2	no	—
165	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	1	no	—
169	C₄×C₁₃	—	8	2	no	—
170	C₄×C₁₃	—	8	2	no	—
171	C₄×C₁₃	—	8	2	no	—
172	C₄×C₁₃	#537	1	1	yes	characteristic
173	C₄×C₁₃	#575	1	1	yes	characteristic
174	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
175	C₂×SD₁₆	—	2	2	no	—
176	C₂×D₈	C₂³×C₁₃	1	2	yes	—
177	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	2	no	—
180	#126	—	4	2	no	—
181	#124	—	2	2	no	—
182	#124	—	4	2	no	—
183	#123	—	4	2	no	—
184	C₂²⋊Q₈	—	2	2	no	—
185	C₂²⋊Q₈	—	4	2	no	—
186	#121	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×D₄	—	2	2	no	—
189	#117	D₄×C₁₃	1	1	yes	characteristic
190	Q₃₂	—	4	1	no	—
191	SD₃₂	—	4	2	no	—
192	D₁₆	—	4	1	no	—
193	M₃₂	—	4	1	no	—
194	C₂×C₁₆	—	4	1	no	—
195	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
196	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
197	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
198	#107	—	2	2	no	—
199	#107	C₂³×C₁₃	1	2	yes	—
200	#107	C₂³×C₁₃	1	2	yes	—
201	#103	—	2	2	no	—
202	#103	—	2	2	no	—
203	#102	—	2	2	no	—
204	#102	—	2	2	no	—
205	#97	D₄×C₁₃	1	2	yes	—
206	C₄×C₈	C₂³×C₁₃	1	2	yes	—
207	C₂×C₁₃	—	4	1	no	—
208	C₂×C₁₃	—	16	2	no	—
209	C₂×C₁₃	#3245	1	1	yes	characteristic
210	C₂×C₁₃	#3416	1	2	yes	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
217	Q₁₆	—	2	2	no	—
218	Q₁₆	—	2	2	no	—
219	SD₁₆	—	4	4	no	—
220	D₈	—	2	2	no	—
221	D₈	—	2	2	no	—
222	M₁₆	—	4	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
228	C₄⋊C₄	—	2	2	no	—
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	C₁₃×D₈	1	2	yes	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	C₁₃×D₈	1	1	yes	characteristic
240	C₂²⋊C₄	C₁₃×D₈	1	1	yes	characteristic
241	C₄²	—	4	2	no	—
242	C₄²	C₂×D₄×C₁₃	1	2	yes	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	2	1	no	—
245	C₁₆	—	2	1	no	—
246	C₁₃	#33829	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³	—	4	2	no	—
248	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	2	no	—
251	Q₈	—	16	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	16	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	16	2	no	—
261	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
262	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
263	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
268	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	2	no	—
281	C₄	C₁₃×#537	1	1	yes	characteristic
282	C₄	C₁₃×#575	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	16	2	no	—
285	C₂	C₁₃×#3245	1	1	yes	characteristic
286	C₂	C₁₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#2218	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3175	C₂	semidirect
2. C₁₃×#2226	C₂	semidirect
3. C₁₃×#2225	C₂	semidirect
4. C₁₃×#2219	C₂	semidirect
6. C₁₃×#2217	C₂	semidirect
7. C₁₃×#2211	C₂	semidirect
8. C₁₃×#2210	C₂	semidirect
9. C₁₃×#2151	C₂	semidirect
68. #33829	C₁₃	direct (N × H)
104. #3175	C₂×C₁₃	semidirect
105. #2226	C₂×C₁₃	semidirect
106. #2225	C₂×C₁₃	semidirect
107. #2219	C₂×C₁₃	semidirect
109. #2217	C₂×C₁₃	semidirect
110. #2211	C₂×C₁₃	semidirect
111. #2210	C₂×C₁₃	semidirect
112. #2151	C₂×C₁₃	semidirect
246. C₁₃	#33829	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×#3175	2
C₁₃×#2226	2
C₁₃×#2225	2
C₁₃×#2219	2
C₁₃×#2218	2
C₁₃×#2217	2
C₁₃×#2211	2
C₁₃×#2210	2
C₁₃×#2151	2
#33829	13

Subgroups — C13×#33829 #15720168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33829 #15720168