Subgroups — C₁₃×#33860 #15720199

Order 3328.

636 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3417	C₂	1	2	yes	maximal, complemented
2	C₁₃×#3293	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2222	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2214	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2195	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2178	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#578	—	2	2	no	—
10	C₁₃×#578	C₂²	1	4	yes	—
11	C₁₃×#510	—	2	4	no	—
12	C₁₃×#510	C₂²	1	4	yes	—
13	C₁₃×#508	—	2	2	no	—
14	C₁₃×#508	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×Q₃₂	—	2	2	no	—
16	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
17	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
18	C₁₃×#504	C₂²	1	2	yes	—
19	C₁₃×#494	C₂²	1	2	yes	—
20	C₁₃×#473	—	2	2	no	—
21	C₁₃×#473	C₂²	1	4	yes	—
22	C₁₃×#456	—	2	4	no	—
23	C₁₃×#443	C₂²	1	2	yes	—
24	C₁₃×#443	C₂²	1	1	yes	characteristic
25	C₁₃×#368	C₂²	1	1	yes	characteristic
26	C₁₃×#364	C₂²	1	1	yes	characteristic
27	C₁₃×#362	C₂²	1	4	yes	—
28	C₁₃×#359	C₂²	1	2	yes	—
29	C₁₃×#350	C₂²	1	2	yes	—
30	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
31	C₁₃×2₋¹⁺⁴	—	2	4	no	—
32	C₁₃×#137	—	2	4	no	—
33	C₁₃×#137	—	2	4	no	—
34	C₁₃×#137	—	4	4	no	—
35	C₁₃×#137	—	4	4	no	—
36	C₁₃×#135	—	2	4	no	—
37	C₁₃×#135	—	4	2	no	—
38	C₁₃×#135	D₄	1	1	yes	characteristic
39	C₁₃×#135	C₂³	1	2	yes	—
40	C₂×C₁₃×Q₁₆	—	2	4	no	—
41	C₂×C₁₃×Q₁₆	—	2	2	no	—
42	C₂×C₁₃×Q₁₆	—	4	2	no	—
43	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
44	C₁₃×#131	—	2	2	no	—
45	C₁₃×#131	C₂³	1	2	yes	—
46	C₁₃×C₄⋊Q₈	—	2	2	no	—
47	C₁₃×Q₃₂	—	2	4	no	—
48	C₁₃×Q₃₂	—	2	2	no	—
49	C₁₃×Q₃₂	—	2	2	no	—
50	C₁₃×Q₃₂	—	4	2	no	—
51	C₁₃×Q₃₂	D₄	1	1	yes	characteristic
52	C₁₃×SD₃₂	—	2	4	no	—
53	C₁₃×SD₃₂	D₄	1	2	yes	—
54	C₁₃×D₁₆	D₄	1	1	yes	characteristic
55	C₁₃×M₃₂	—	2	2	no	—
56	C₁₃×M₃₂	C₂³	1	2	yes	—
57	C₂×C₁₃×C₁₆	—	2	2	no	—
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₁₃×#108	D₄	1	1	yes	characteristic
61	C₁₃×#108	C₂³	1	2	yes	—
62	C₁₃×#104	—	2	2	no	—
63	C₁₃×#104	D₄	1	2	yes	—
64	C₁₃×#101	—	2	4	no	—
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#33860	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₁₃×#41	—	2	2	no	—
68	C₁₃×#41	—	4	4	no	—
69	C₁₃×#41	—	4	4	no	—
70	C₁₃×#41	—	8	4	no	—
71	C₁₃×#41	C₂×D₄	1	2	yes	—
72	C₂×Q₈×C₁₃	—	2	4	no	—
73	C₂×Q₈×C₁₃	—	4	4	no	—
74	C₂×Q₈×C₁₃	—	4	4	no	—
75	C₁₃×Q₁₆	—	2	4	no	—
76	C₁₃×Q₁₆	—	2	4	no	—
77	C₁₃×Q₁₆	—	4	4	no	—
78	C₁₃×Q₁₆	—	4	4	no	—
79	C₁₃×Q₁₆	—	4	2	no	—
80	C₁₃×Q₁₆	—	4	2	no	—
81	C₁₃×Q₁₆	D₈	1	1	yes	characteristic
82	C₁₃×Q₁₆	C₂×D₄	1	2	yes	—
83	C₁₃×SD₁₆	—	2	2	no	—
84	C₁₃×SD₁₆	—	4	4	no	—
85	C₁₃×SD₁₆	—	4	4	no	—
86	C₁₃×SD₁₆	D₈	1	2	yes	—
87	C₁₃×D₈	D₈	1	1	yes	characteristic
88	C₁₃×D₈	C₂×D₄	1	2	yes	—
89	C₁₃×M₁₆	—	2	2	no	—
90	C₁₃×M₁₆	—	2	2	no	—
91	C₁₃×M₁₆	—	4	2	no	—
92	C₁₃×M₁₆	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	4	2	no	—
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	8	2	no	—
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₁₃×C₁₆	—	2	2	no	—
100	C₁₃×C₁₆	—	2	1	no	—
101	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
102	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
103	#3417	C₂×C₁₃	1	2	yes	complemented
104	#3293	C₂×C₁₃	1	2	yes	complemented
105	#2247	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#2222	C₂×C₁₃	1	2	yes	complemented
107	#2214	C₂×C₁₃	1	2	yes	complemented
108	#2195	C₂×C₁₃	1	4	yes	complemented
109	#2178	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#2170	C₂×C₁₃	1	1	yes	characteristic, complemented
111	Q₈×C₁₃	—	2	2	no	—
112	Q₈×C₁₃	—	4	4	no	—
113	Q₈×C₁₃	—	4	4	no	—
114	Q₈×C₁₃	—	8	4	no	—
115	Q₈×C₁₃	C₂×D₈	1	2	yes	—
116	D₄×C₁₃	—	2	2	no	—
117	D₄×C₁₃	—	2	2	no	—
118	D₄×C₁₃	—	4	2	no	—
119	D₄×C₁₃	—	8	4	no	—
120	D₄×C₁₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	8	4	no	—
126	C₂×C₄×C₁₃	—	8	4	no	—
127	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	4	2	no	—
130	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
134	#578	—	2	2	no	—
135	#578	C₂²×C₁₃	1	4	yes	—
136	#510	—	2	4	no	—
137	#510	C₂²×C₁₃	1	4	yes	—
138	#508	—	2	2	no	—
139	#508	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×Q₃₂	—	2	2	no	—
141	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
142	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
143	#504	C₂²×C₁₃	1	2	yes	—
144	#494	C₂²×C₁₃	1	2	yes	—
145	#473	—	2	2	no	—
146	#473	C₂²×C₁₃	1	4	yes	—
147	#456	—	2	4	no	—
148	#443	C₂²×C₁₃	1	2	yes	—
149	#443	C₂²×C₁₃	1	1	yes	characteristic
150	#368	C₂²×C₁₃	1	1	yes	characteristic
151	#364	C₂²×C₁₃	1	1	yes	characteristic
152	#362	C₂²×C₁₃	1	4	yes	—
153	#359	C₂²×C₁₃	1	2	yes	—
154	#350	C₂²×C₁₃	1	2	yes	—
155	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	—	4	2	no	—
158	C₂²×C₁₃	#576	1	1	yes	characteristic
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	4	2	no	—
162	C₄×C₁₃	—	8	4	no	—
163	C₄×C₁₃	D₄²	1	1	yes	characteristic
164	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
165	2₋¹⁺⁴	—	2	4	no	—
166	#137	—	2	4	no	—
167	#137	—	2	4	no	—
168	#137	—	4	4	no	—
169	#137	—	4	4	no	—
170	#135	—	2	4	no	—
171	#135	—	4	2	no	—
172	#135	D₄×C₁₃	1	1	yes	characteristic
173	#135	C₂³×C₁₃	1	2	yes	—
174	C₂×Q₁₆	—	2	4	no	—
175	C₂×Q₁₆	—	2	2	no	—
176	C₂×Q₁₆	—	4	2	no	—
177	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
178	#131	—	2	2	no	—
179	#131	C₂³×C₁₃	1	2	yes	—
180	C₄⋊Q₈	—	2	2	no	—
181	Q₃₂	—	2	4	no	—
182	Q₃₂	—	2	2	no	—
183	Q₃₂	—	2	2	no	—
184	Q₃₂	—	4	2	no	—
185	Q₃₂	D₄×C₁₃	1	1	yes	characteristic
186	SD₃₂	—	2	4	no	—
187	SD₃₂	D₄×C₁₃	1	2	yes	—
188	D₁₆	D₄×C₁₃	1	1	yes	characteristic
189	M₃₂	—	2	2	no	—
190	M₃₂	C₂³×C₁₃	1	2	yes	—
191	C₂×C₁₆	—	2	2	no	—
192	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
193	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
194	#108	D₄×C₁₃	1	1	yes	characteristic
195	#108	C₂³×C₁₃	1	2	yes	—
196	#104	—	2	2	no	—
197	#104	D₄×C₁₃	1	2	yes	—
198	#101	—	2	4	no	—
199	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	—	4	2	no	—
202	C₂×C₁₃	—	8	2	no	—
203	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
204	#41	—	2	2	no	—
205	#41	—	4	4	no	—
206	#41	—	4	4	no	—
207	#41	—	8	4	no	—
208	#41	C₂×D₄×C₁₃	1	2	yes	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	—	4	4	no	—
212	Q₁₆	—	2	4	no	—
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	4	4	no	—
215	Q₁₆	—	4	4	no	—
216	Q₁₆	—	4	2	no	—
217	Q₁₆	—	4	2	no	—
218	Q₁₆	C₁₃×D₈	1	1	yes	characteristic
219	Q₁₆	C₂×D₄×C₁₃	1	2	yes	—
220	SD₁₆	—	2	2	no	—
221	SD₁₆	—	4	4	no	—
222	SD₁₆	—	4	4	no	—
223	SD₁₆	C₁₃×D₈	1	2	yes	—
224	D₈	C₁₃×D₈	1	1	yes	characteristic
225	D₈	C₂×D₄×C₁₃	1	2	yes	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	C₂×D₄×C₁₃	1	2	yes	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
233	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
234	C₄⋊C₄	—	8	2	no	—
235	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₁₆	—	2	2	no	—
237	C₁₆	—	2	1	no	—
238	C₁₆	C₂×D₄×C₁₃	1	1	yes	characteristic
239	C₁₆	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₁₃	#33860	1	1	yes	characteristic, direct product factor, Sylow
241	Q₈	—	2	2	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	8	4	no	—
245	Q₈	C₂×C₁₃×D₈	1	2	yes	—
246	D₄	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	8	4	no	—
250	D₄	C₂×C₁₃×D₈	1	2	yes	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	4	2	no	—
260	C₈	C₂×C₁₃×D₈	1	1	yes	characteristic
261	C₈	C₂×C₁₃×D₈	1	1	yes	characteristic
262	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
263	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₁₃×#576	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	8	4	no	—
271	C₄	D₄²×C₁₃	1	1	yes	characteristic
272	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	4	2	no	—
275	C₂	—	8	2	no	—
276	C₂	D₄×C₁₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₁₃×#443 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3417	C₂	semidirect
2. C₁₃×#3293	C₂	semidirect
3. C₁₃×#2247	C₂	semidirect
4. C₁₃×#2222	C₂	semidirect
5. C₁₃×#2214	C₂	semidirect
6. C₁₃×#2195	C₂	semidirect
7. C₁₃×#2178	C₂	semidirect
8. C₁₃×#2170	C₂	semidirect
66. #33860	C₁₃	direct (N × H)
103. #3417	C₂×C₁₃	semidirect
104. #3293	C₂×C₁₃	semidirect
105. #2247	C₂×C₁₃	semidirect
106. #2222	C₂×C₁₃	semidirect
107. #2214	C₂×C₁₃	semidirect
108. #2195	C₂×C₁₃	semidirect
109. #2178	C₂×C₁₃	semidirect
110. #2170	C₂×C₁₃	semidirect
240. C₁₃	#33860	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×#3417	2
C₁₃×#3293	2
C₁₃×#2247	2
C₁₃×#2222	2
C₁₃×#2214	2
C₁₃×#2195	2
C₁₃×#2178	2
C₁₃×#2170	2
#33860	13

Subgroups — C13×#33860 #15720199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33860 #15720199