Subgroups — C₁₃×#33768 #15720107

Order 3328.

552 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3059	C₂	1	2	yes	maximal
2	C₁₃×#2237	C₂	1	4	yes	maximal, complemented
3	C₁₃×#2219	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2216	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2188	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2113	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
8	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
9	C₁₃×#536	—	2	2	no	—
10	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
11	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₁₃×#470	—	2	2	no	—
15	C₁₃×#470	—	2	2	no	—
16	C₁₃×#467	C₂²	1	4	yes	—
17	C₁₃×#467	C₂²	1	4	yes	—
18	C₁₃×#427	C₂²	1	2	yes	—
19	C₁₃×#419	—	2	2	no	—
20	C₁₃×#367	C₂²	1	2	yes	—
21	C₁₃×#366	C₂²	1	2	yes	—
22	C₁₃×#358	C₂²	1	4	yes	—
23	C₁₃×#358	C₂²	1	4	yes	—
24	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
25	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
26	C₂²×Q₈×C₁₃	—	2	2	no	—
27	C₂×C₁₃×Q₁₆	—	2	2	no	—
28	C₂×C₁₃×Q₁₆	—	4	2	no	—
29	C₂×C₁₃×Q₁₆	—	4	2	no	—
30	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
31	C₁₃×#131	—	4	1	no	—
32	C₁₃×#131	—	4	1	no	—
33	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
34	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×C₄⋊Q₈	—	4	2	no	—
38	C₁₃×#124	—	4	2	no	—
39	C₁₃×#123	—	4	2	no	—
40	C₁₃×#123	—	4	2	no	—
41	C₁₃×C₂²⋊Q₈	—	2	4	no	—
42	C₁₃×C₂²⋊Q₈	—	2	4	no	—
43	C₁₃×#117	—	2	2	no	—
44	C₁₃×Q₃₂	—	4	2	no	—
45	C₁₃×Q₃₂	—	4	2	no	—
46	C₁₃×M₃₂	—	4	1	no	—
47	C₂×C₁₃×C₁₆	—	4	1	no	—
48	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₁₃×#107	C₂³	1	4	yes	—
51	C₁₃×#106	—	2	2	no	—
52	C₁₃×#103	—	2	4	no	—
53	C₁₃×#103	—	2	4	no	—
54	#33768	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₁₃×#41	—	4	1	no	—
56	C₁₃×#41	—	4	1	no	—
57	C₂×Q₈×C₁₃	—	2	4	no	—
58	C₂×Q₈×C₁₃	—	4	4	no	—
59	C₂×Q₈×C₁₃	—	4	4	no	—
60	C₂×Q₈×C₁₃	—	4	2	no	—
61	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
62	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
63	C₂×D₄×C₁₃	D₈	1	2	yes	—
64	C₂²×C₄×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₁₃×Q₁₆	—	2	4	no	—
68	C₁₃×Q₁₆	—	2	4	no	—
69	C₁₃×Q₁₆	—	4	4	no	—
70	C₁₃×M₁₆	—	4	2	no	—
71	C₁₃×M₁₆	—	4	1	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	4	no	—
81	C₁₃×C₄⋊C₄	—	4	4	no	—
82	C₁₃×C₂²⋊C₄	—	4	2	no	—
83	C₁₃×C₂²⋊C₄	—	4	4	no	—
84	C₁₃×C₂²⋊C₄	—	4	4	no	—
85	C₄²×C₁₃	—	4	2	no	—
86	C₁₃×C₁₆	—	2	2	no	—
87	C₁₃×C₁₆	—	2	2	no	—
88	#3059	C₂×C₁₃	1	2	yes	—
89	#2237	C₂×C₁₃	1	4	yes	complemented
90	#2219	C₂×C₁₃	1	2	yes	complemented
91	#2216	C₂×C₁₃	1	2	yes	complemented
92	#2188	C₂×C₁₃	1	4	yes	complemented
93	#2113	C₂×C₁₃	1	1	yes	characteristic
94	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
95	C₂³×C₁₃	C₂×D₈	1	2	yes	—
96	Q₈×C₁₃	—	4	1	no	—
97	Q₈×C₁₃	—	4	4	no	—
98	Q₈×C₁₃	—	4	4	no	—
99	Q₈×C₁₃	—	8	4	no	—
100	D₄×C₁₃	—	4	1	no	—
101	D₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	8	4	no	—
109	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₁₃	—	2	1	no	—
113	C₈×C₁₃	—	2	2	no	—
114	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
117	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
118	#536	—	2	2	no	—
119	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
120	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
121	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
122	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
123	#470	—	2	2	no	—
124	#470	—	2	2	no	—
125	#467	C₂²×C₁₃	1	4	yes	—
126	#467	C₂²×C₁₃	1	4	yes	—
127	#427	C₂²×C₁₃	1	2	yes	—
128	#419	—	2	2	no	—
129	#367	C₂²×C₁₃	1	2	yes	—
130	#366	C₂²×C₁₃	1	2	yes	—
131	#358	C₂²×C₁₃	1	4	yes	—
132	#358	C₂²×C₁₃	1	4	yes	—
133	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	4	no	—
136	C₂²×C₁₃	—	2	2	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	8	4	no	—
142	C₄×C₁₃	—	8	4	no	—
143	C₄×C₁₃	#534	1	1	yes	characteristic
144	C₄×C₁₃	#575	1	1	yes	characteristic
145	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
146	C₂²×Q₈	—	2	2	no	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	—	4	2	no	—
149	C₂×Q₁₆	—	4	2	no	—
150	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
151	#131	—	4	1	no	—
152	#131	—	4	1	no	—
153	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
154	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
156	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
157	C₄⋊Q₈	—	4	2	no	—
158	#124	—	4	2	no	—
159	#123	—	4	2	no	—
160	#123	—	4	2	no	—
161	C₂²⋊Q₈	—	2	4	no	—
162	C₂²⋊Q₈	—	2	4	no	—
163	#117	—	2	2	no	—
164	Q₃₂	—	4	2	no	—
165	Q₃₂	—	4	2	no	—
166	M₃₂	—	4	1	no	—
167	C₂×C₁₆	—	4	1	no	—
168	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
169	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
170	#107	C₂³×C₁₃	1	4	yes	—
171	#106	—	2	2	no	—
172	#103	—	2	4	no	—
173	#103	—	2	4	no	—
174	C₂×C₁₃	—	4	2	no	—
175	C₂×C₁₃	—	4	1	no	—
176	C₂×C₁₃	#3186	1	1	yes	characteristic
177	C₂×C₁₃	#3417	1	2	yes	—
178	#41	—	4	1	no	—
179	#41	—	4	1	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	4	4	no	—
182	C₂×Q₈	—	4	4	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
185	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
186	C₂×D₄	C₁₃×D₈	1	2	yes	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
190	Q₁₆	—	2	4	no	—
191	Q₁₆	—	2	4	no	—
192	Q₁₆	—	4	4	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
201	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	4	4	no	—
207	C₂²⋊C₄	—	4	4	no	—
208	C₄²	—	4	2	no	—
209	C₁₆	—	2	2	no	—
210	C₁₆	—	2	2	no	—
211	C₁₃	#33768	1	1	yes	characteristic, direct product factor, Sylow
212	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
213	C₂³	C₂×C₁₃×D₈	1	2	yes	—
214	Q₈	—	4	1	no	—
215	Q₈	—	4	4	no	—
216	Q₈	—	4	4	no	—
217	Q₈	—	8	4	no	—
218	D₄	—	4	1	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	8	4	no	—
227	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
228	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
229	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	2	no	—
232	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
233	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	1	no	—
238	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	8	4	no	—
242	C₄	—	8	4	no	—
243	C₄	C₁₃×#534	1	1	yes	characteristic
244	C₄	C₁₃×#575	1	1	yes	characteristic
245	C₂	—	4	2	no	—
246	C₂	—	4	1	no	—
247	C₂	C₁₃×#3186	1	1	yes	characteristic
248	C₂	C₁₃×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₁₃×#41 ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2237	C₂	semidirect
3. C₁₃×#2219	C₂	semidirect
4. C₁₃×#2216	C₂	semidirect
5. C₁₃×#2188	C₂	semidirect
54. #33768	C₁₃	direct (N × H)
89. #2237	C₂×C₁₃	semidirect
90. #2219	C₂×C₁₃	semidirect
91. #2216	C₂×C₁₃	semidirect
92. #2188	C₂×C₁₃	semidirect
211. C₁₃	#33768	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×#3059	2
C₁₃×#2237	2
C₁₃×#2219	2
C₁₃×#2216	2
C₁₃×#2188	2
C₁₃×#2113	2
#33768	13

Subgroups — C13×#33768 #15720107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33768 #15720107