Subgroups — C₁₃×#33908 #15720247

Order 3328.

492 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3341	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2234	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2226	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2215	C₂	1	2	yes	maximal, complemented
6	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2158	C₂	1	2	yes	maximal, complemented
9	C₁₃×#555	—	2	2	no	—
10	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
12	C₁₃×#500	C₂²	1	2	yes	—
13	C₁₃×#482	—	2	4	no	—
14	C₁₃×#476	—	2	2	no	—
15	C₁₃×#466	C₂²	1	4	yes	—
16	C₄×C₁₃×D₈	C₂²	1	2	yes	—
17	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#427	C₂²	1	4	yes	—
19	C₁₃×#367	C₂²	1	4	yes	—
20	C₁₃×#366	C₂²	1	2	yes	—
21	C₁₃×#366	C₂²	1	2	yes	—
22	C₁₃×#366	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₁₃×#357	C₂²	1	2	yes	—
25	C₁₃×#357	C₂²	1	4	yes	—
26	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×D₈	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₁₃×C₄⋊Q₈	—	2	2	no	—
31	C₁₃×#124	—	4	4	no	—
32	C₁₃×#121	—	2	4	no	—
33	C₁₃×#121	—	4	2	no	—
34	C₄×D₄×C₁₃	—	2	2	no	—
35	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
36	C₁₃×#117	—	2	4	no	—
37	C₁₃×#117	—	4	2	no	—
38	C₁₃×D₁₆	—	2	2	no	—
39	C₂×C₁₃×C₁₆	—	2	4	no	—
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	C₂³	1	2	yes	—
45	C₁₃×#107	C₂³	1	4	yes	—
46	C₁₃×#106	—	2	4	no	—
47	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
48	C₁₃×#102	—	2	2	no	—
49	C₁₃×#102	—	2	4	no	—
50	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	#33908	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×Q₈×C₁₃	—	4	2	no	—
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	—	4	4	no	—
56	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₁₃×D₈	—	2	2	no	—
60	C₁₃×D₈	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₁₃×C₄⋊C₄	—	2	4	no	—
68	C₁₃×C₄⋊C₄	—	4	2	no	—
69	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
70	C₁₃×C₂²⋊C₄	—	4	4	no	—
71	C₁₃×C₂²⋊C₄	—	4	4	no	—
72	C₁₃×C₂²⋊C₄	—	4	2	no	—
73	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
74	C₄²×C₁₃	—	4	4	no	—
75	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₁₃×C₁₆	—	2	2	no	—
77	C₁₃×C₁₆	—	2	1	no	—
78	C₁₃×C₁₆	#41	1	2	yes	—
79	#3341	C₂×C₁₃	1	2	yes	complemented
80	#2252	C₂×C₁₃	1	1	yes	characteristic, complemented
81	#2234	C₂×C₁₃	1	4	yes	complemented
82	#2226	C₂×C₁₃	1	2	yes	complemented
83	#2215	C₂×C₁₃	1	2	yes	complemented
84	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
85	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
86	#2158	C₂×C₁₃	1	2	yes	complemented
87	C₂³×C₁₃	—	4	2	no	—
88	C₂³×C₁₃	C₂×D₈	1	2	yes	—
89	Q₈×C₁₃	—	16	2	no	—
90	D₄×C₁₃	—	2	2	no	—
91	D₄×C₁₃	—	4	2	no	—
92	D₄×C₁₃	—	4	2	no	—
93	D₄×C₁₃	—	16	4	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	—	8	4	no	—
99	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
103	C₈×C₁₃	—	2	2	no	—
104	C₈×C₁₃	—	2	1	no	—
105	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
106	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
107	#555	—	2	2	no	—
108	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
109	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
110	#500	C₂²×C₁₃	1	2	yes	—
111	#482	—	2	4	no	—
112	#476	—	2	2	no	—
113	#466	C₂²×C₁₃	1	4	yes	—
114	C₄×D₈	C₂²×C₁₃	1	2	yes	—
115	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
116	#427	C₂²×C₁₃	1	4	yes	—
117	#367	C₂²×C₁₃	1	4	yes	—
118	#366	C₂²×C₁₃	1	1	yes	characteristic
119	#366	C₂²×C₁₃	1	2	yes	—
120	#366	C₂²×C₁₃	1	2	yes	—
121	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
122	#357	C₂²×C₁₃	1	2	yes	—
123	#357	C₂²×C₁₃	1	4	yes	—
124	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
125	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	8	2	no	—
130	C₂²×C₁₃	—	8	2	no	—
131	C₂²×C₁₃	—	8	2	no	—
132	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	1	no	—
135	C₄×C₁₃	—	8	2	no	—
136	C₄×C₁₃	—	8	4	no	—
137	C₄×C₁₃	#508	1	2	yes	—
138	C₄×C₁₃	#547	1	1	yes	characteristic
139	C₄×C₁₃	#578	1	1	yes	characteristic
140	C₂×D₈	C₂³×C₁₃	1	2	yes	—
141	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
142	C₄⋊Q₈	—	2	2	no	—
143	#124	—	4	4	no	—
144	#121	—	2	4	no	—
145	#121	—	4	2	no	—
146	C₄×D₄	—	2	2	no	—
147	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
148	#117	—	2	4	no	—
149	#117	—	4	2	no	—
150	D₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	4	no	—
152	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
153	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
154	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
155	#107	—	2	2	no	—
156	#107	C₂³×C₁₃	1	4	yes	—
157	#107	C₂³×C₁₃	1	2	yes	—
158	#106	—	2	4	no	—
159	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
160	#102	—	2	2	no	—
161	#102	—	2	4	no	—
162	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
163	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
164	C₂×C₁₃	—	4	2	no	—
165	C₂×C₁₃	—	16	2	no	—
166	C₂×C₁₃	#3334	1	1	yes	characteristic
167	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
168	C₂×C₁₃	#3415	1	1	yes	characteristic
169	C₂×Q₈	—	4	2	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
175	D₈	—	2	2	no	—
176	D₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
180	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
181	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
186	C₂²⋊C₄	—	4	4	no	—
187	C₂²⋊C₄	—	4	4	no	—
188	C₂²⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
190	C₄²	—	4	4	no	—
191	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
192	C₁₆	—	2	2	no	—
193	C₁₆	—	2	1	no	—
194	C₁₆	C₁₃×#41	1	2	yes	—
195	C₁₃	#33908	1	1	yes	characteristic, direct product factor, Sylow
196	C₂³	—	4	2	no	—
197	C₂³	C₂×C₁₃×D₈	1	2	yes	—
198	Q₈	—	16	2	no	—
199	D₄	—	2	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	16	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	4	no	—
208	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
209	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
210	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
211	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	1	no	—
214	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
215	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	—	8	2	no	—
221	C₂²	—	8	2	no	—
222	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	1	no	—
225	C₄	—	8	2	no	—
226	C₄	—	8	4	no	—
227	C₄	C₁₃×#508	1	2	yes	—
228	C₄	C₁₃×#547	1	1	yes	characteristic
229	C₄	C₁₃×#578	1	1	yes	characteristic
230	C₂	—	4	2	no	—
231	C₂	—	16	2	no	—
232	C₂	C₁₃×#3334	1	1	yes	characteristic
233	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
234	C₂	C₁₃×#3415	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3341	C₂	semidirect
2. C₁₃×#2252	C₂	semidirect
3. C₁₃×#2234	C₂	semidirect
4. C₁₃×#2226	C₂	semidirect
5. C₁₃×#2215	C₂	semidirect
6. C₄×C₁₃×D₁₆	C₂	semidirect
7. D₄×C₁₃×C₁₆	C₂	semidirect
8. C₁₃×#2158	C₂	semidirect
52. #33908	C₁₃	direct (N × H)
79. #3341	C₂×C₁₃	semidirect
80. #2252	C₂×C₁₃	semidirect
81. #2234	C₂×C₁₃	semidirect
82. #2226	C₂×C₁₃	semidirect
83. #2215	C₂×C₁₃	semidirect
84. C₄×D₁₆	C₂×C₁₃	semidirect
85. D₄×C₁₆	C₂×C₁₃	semidirect
86. #2158	C₂×C₁₃	semidirect
195. C₁₃	#33908	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×#3341	2
C₁₃×#2252	2
C₁₃×#2234	2
C₁₃×#2226	2
C₁₃×#2215	2
C₄×C₁₃×D₁₆	2
D₄×C₁₃×C₁₆	2
C₁₃×#2158	2
#33908	13

Subgroups — C13×#33908 #15720247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33908 #15720247