Subgroups — C₁₃×#33729 #15720068

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3164	C₂	1	2	yes	maximal
2	C₁₃×#2255	C₂	1	4	yes	maximal, complemented
3	C₁₃×#2253	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2252	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2159	C₂	1	2	yes	maximal
6	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#544	—	2	2	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×#501	—	2	4	no	—
13	C₁₃×#500	C₂²	1	4	yes	—
14	C₁₃×#499	C₂²	1	4	yes	—
15	C₁₃×#428	—	2	2	no	—
16	C₁₃×#427	C₂²	1	2	yes	—
17	C₂×C₁₃×#107	C₂²	1	2	yes	—
18	C₁₃×#405	C₂²	1	1	yes	characteristic
19	C₁₃×#367	C₂²	1	4	yes	—
20	C₁₃×#367	C₂²	1	2	yes	—
21	C₁₃×#366	C₂²	1	1	yes	characteristic
22	C₁₃×#366	C₂²	1	2	yes	—
23	C₁₃×#366	C₂²	1	4	yes	—
24	C₁₃×#366	C₂²	1	1	yes	characteristic
25	C₁₃×#366	C₂²	1	2	yes	—
26	C₁₃×#346	C₂²	1	2	yes	—
27	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
28	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₁₃×C₄⋊Q₈	—	2	4	no	—
31	C₁₃×C₄⋊Q₈	—	4	2	no	—
32	C₁₃×#125	—	2	4	no	—
33	C₁₃×#125	—	4	2	no	—
34	C₁₃×C₂²⋊Q₈	—	4	4	no	—
35	C₁₃×#117	—	2	2	no	—
36	C₁₃×#117	D₄	1	1	yes	characteristic
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₁₃×M₃₂	Q₈	1	4	yes	complemented
39	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
40	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
43	C₁₃×#107	—	2	2	no	—
44	C₁₃×#107	—	2	4	no	—
45	C₁₃×#107	C₂³	1	4	yes	—
46	C₁₃×#107	C₂³	1	2	yes	—
47	C₁₃×#107	C₂³	1	2	yes	—
48	C₁₃×#106	—	2	2	no	—
49	C₁₃×#106	—	2	4	no	—
50	C₁₃×#97	D₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	2	yes	—
52	#33729	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×Q₈×C₁₃	—	4	4	no	—
54	C₂²×C₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₁₃×M₁₆	Q₁₆	1	4	yes	—
57	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	—	2	4	no	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₄⋊C₄	—	4	4	no	—
67	C₁₃×C₄⋊C₄	—	4	4	no	—
68	C₁₃×C₄⋊C₄	—	4	4	no	—
69	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
71	C₁₃×C₂²⋊C₄	—	4	2	no	—
72	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
73	C₄²×C₁₃	—	4	2	no	—
74	C₄²×C₁₃	C₂×D₄	1	2	yes	—
75	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
76	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
77	C₁₃×C₁₆	C₂×Q₈	1	4	yes	—
78	#3164	C₂×C₁₃	1	2	yes	—
79	#2255	C₂×C₁₃	1	4	yes	complemented
80	#2253	C₂×C₁₃	1	2	yes	complemented
81	#2252	C₂×C₁₃	1	2	yes	complemented
82	#2159	C₂×C₁₃	1	2	yes	—
83	#2158	C₂×C₁₃	1	1	yes	characteristic
84	C₂×#366	C₂×C₁₃	1	1	yes	characteristic
85	#2108	C₂×C₁₃	1	1	yes	characteristic
86	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
87	Q₈×C₁₃	—	16	4	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	4	4	no	—
91	C₂×C₄×C₁₃	—	16	2	no	—
92	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
98	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
99	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
100	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
101	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
102	#544	—	2	2	no	—
103	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
104	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
105	#501	—	2	4	no	—
106	#500	C₂²×C₁₃	1	4	yes	—
107	#499	C₂²×C₁₃	1	4	yes	—
108	#428	—	2	2	no	—
109	#427	C₂²×C₁₃	1	2	yes	—
110	C₂×#107	C₂²×C₁₃	1	2	yes	—
111	#405	C₂²×C₁₃	1	1	yes	characteristic
112	#367	C₂²×C₁₃	1	2	yes	—
113	#367	C₂²×C₁₃	1	4	yes	—
114	#366	C₂²×C₁₃	1	2	yes	—
115	#366	C₂²×C₁₃	1	2	yes	—
116	#366	C₂²×C₁₃	1	1	yes	characteristic
117	#366	C₂²×C₁₃	1	1	yes	characteristic
118	#366	C₂²×C₁₃	1	4	yes	—
119	#346	C₂²×C₁₃	1	2	yes	—
120	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
121	C₂²×C₁₃	—	2	1	no	—
122	C₂²×C₁₃	—	2	1	no	—
123	C₂²×C₁₃	#500	1	2	yes	—
124	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	8	4	no	—
128	C₄×C₁₃	—	8	2	no	—
129	C₄×C₁₃	—	8	2	no	—
130	C₄×C₁₃	#500	1	2	yes	—
131	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
133	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	4	no	—
136	C₄⋊Q₈	—	4	2	no	—
137	#125	—	2	4	no	—
138	#125	—	4	2	no	—
139	C₂²⋊Q₈	—	4	4	no	—
140	#117	—	2	2	no	—
141	#117	D₄×C₁₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	2	2	no	—
143	M₃₂	Q₈×C₁₃	1	4	yes	complemented
144	C₂×C₁₆	Q₈×C₁₃	1	4	yes	complemented
145	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
146	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
147	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
148	#107	—	2	2	no	—
149	#107	—	2	4	no	—
150	#107	C₂³×C₁₃	1	4	yes	—
151	#107	C₂³×C₁₃	1	2	yes	—
152	#107	C₂³×C₁₃	1	2	yes	—
153	#106	—	2	2	no	—
154	#106	—	2	4	no	—
155	#97	D₄×C₁₃	1	2	yes	—
156	C₄×C₈	C₂³×C₁₃	1	2	yes	—
157	C₂×C₁₃	—	2	2	no	—
158	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
159	C₂×C₁₃	#3415	1	1	yes	characteristic
160	C₂×C₁₃	#3417	1	1	yes	characteristic
161	C₂×Q₈	—	4	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
164	M₁₆	C₁₃×Q₁₆	1	4	yes	—
165	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
166	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
167	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
168	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
169	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
178	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
179	C₂²⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
181	C₄²	—	4	2	no	—
182	C₄²	C₂×D₄×C₁₃	1	2	yes	—
183	C₁₆	C₂×Q₈×C₁₃	1	2	yes	—
184	C₁₆	C₂×Q₈×C₁₃	1	2	yes	—
185	C₁₆	C₂×Q₈×C₁₃	1	4	yes	—
186	C₁₃	#33729	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
188	Q₈	—	16	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	16	2	no	—
193	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
194	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
195	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
196	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
197	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
198	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
199	C₈	C₁₃×C₄⋊Q₈	1	2	yes	—
200	C₈	C₂×C₁₃×Q₁₆	1	4	yes	—
201	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
202	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	C₁₃×#500	1	2	yes	—
206	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	4	no	—
210	C₄	—	8	2	no	—
211	C₄	—	8	2	no	—
212	C₄	C₁₃×#500	1	2	yes	—
213	C₄	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
214	C₄	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×C₁₃×#500	1	1	yes	characteristic
217	C₂	C₁₃×#3415	1	1	yes	characteristic
218	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	C₂³×C₁₃	C₂×D₈
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
2. C₁₃×#2255	C₂	semidirect
3. C₁₃×#2253	C₂	semidirect
4. C₁₃×#2252	C₂	semidirect
38. C₁₃×M₃₂	Q₈	semidirect
39. C₂×C₁₃×C₁₆	Q₈	semidirect
52. #33729	C₁₃	direct (N × H)
79. #2255	C₂×C₁₃	semidirect
80. #2253	C₂×C₁₃	semidirect
81. #2252	C₂×C₁₃	semidirect
143. M₃₂	Q₈×C₁₃	semidirect
144. C₂×C₁₆	Q₈×C₁₃	semidirect
186. C₁₃	#33729	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×#3164	2
C₁₃×#2255	2
C₁₃×#2253	2
C₁₃×#2252	2
C₁₃×#2159	2
C₁₃×#2158	2
C₂×C₁₃×#366	2
C₁₃×#2108	2
#33729	13

Subgroups — C13×#33729 #15720068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33729 #15720068