Subgroups — C₁₃×#13172 #15699511

Order 3328.

274 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1952	C₂	1	2	yes	maximal
5	C₁₃×#1387	C₂	1	2	yes	maximal
6	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#108	C₂²	1	2	yes	—
9	C₁₃×#428	—	2	2	no	—
10	C₂×C₁₃×#107	C₂²	1	2	yes	—
11	C₁₃×#405	C₂²	1	1	yes	characteristic
12	C₁₃×#368	C₄	1	4	yes	complemented
13	C₁₃×#367	—	2	1	no	—
14	C₁₃×#367	—	2	1	no	—
15	C₁₃×#346	—	2	1	no	—
16	C₄×C₁₃×C₁₆	—	2	1	no	—
17	C₁₃×#343	—	2	2	no	—
18	C₂×C₁₃×M₁₆	—	2	2	no	—
19	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₁₃×#117	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₁₃×M₃₂	—	2	1	no	—
24	C₁₃×M₃₂	Q₈	1	2	yes	—
25	C₂×C₁₃×C₁₆	—	2	1	no	—
26	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
28	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₁₃×C₁₆	Q₈	1	2	yes	—
30	C₁₃×#108	—	2	2	no	—
31	C₁₃×#108	C₂×C₄	1	4	yes	—
32	C₁₃×#107	—	2	2	no	—
33	C₁₃×#107	—	2	2	no	—
34	C₁₃×#107	—	4	2	no	—
35	C₁₃×#106	—	4	2	no	—
36	C₁₃×#97	—	2	1	no	—
37	C₄×C₈×C₁₃	—	2	1	no	—
38	#13172	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₄×C₁₃	—	4	2	no	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₁₃×M₁₆	—	2	1	no	—
42	C₁₃×M₁₆	—	2	4	no	—
43	C₁₃×M₁₆	—	4	2	no	—
44	C₁₃×M₁₆	Q₁₆	1	2	yes	—
45	C₂×C₈×C₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	—	4	2	no	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	—	4	2	no	—
54	C₁₃×C₄⋊C₄	—	4	2	no	—
55	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
56	C₁₃×C₂²⋊C₄	—	2	1	no	—
57	C₄²×C₁₃	—	2	1	no	—
58	C₁₃×C₁₆	—	2	2	no	—
59	C₁₃×C₁₆	C₄⋊C₄	1	2	yes	—
60	C₁₃×C₁₆	C₄⋊C₄	1	2	yes	—
61	C₂×#368	C₂×C₁₃	1	1	yes	characteristic
62	C₂×#367	C₂×C₁₃	1	1	yes	characteristic
63	#2108	C₂×C₁₃	1	1	yes	characteristic
64	#1952	C₂×C₁₃	1	2	yes	—
65	#1387	C₂×C₁₃	1	2	yes	—
66	C₂³×C₁₃	#135	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	8	2	no	—
69	C₂×C₄×C₁₃	—	16	2	no	—
70	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
76	C₈×C₁₃	—	2	2	no	—
77	C₈×C₁₃	—	4	4	no	—
78	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₈×C₁₃	#125	1	1	yes	characteristic
82	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
83	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
84	C₂×#108	C₂²×C₁₃	1	2	yes	—
85	#428	—	2	2	no	—
86	C₂×#107	C₂²×C₁₃	1	2	yes	—
87	#405	C₂²×C₁₃	1	1	yes	characteristic
88	#368	C₄×C₁₃	1	4	yes	complemented
89	#367	—	2	1	no	—
90	#367	—	2	1	no	—
91	#346	—	2	1	no	—
92	C₄×C₁₆	—	2	1	no	—
93	#343	—	2	2	no	—
94	C₂²×C₁₃	—	2	2	no	—
95	C₂²×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
96	C₂²×C₁₃	#466	1	1	yes	characteristic
97	C₂²×C₁₃	#499	1	1	yes	characteristic
98	C₄×C₁₃	—	4	2	no	—
99	C₄×C₁₃	—	16	2	no	—
100	C₄×C₁₃	#389	1	1	yes	characteristic
101	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
102	C₄×C₁₃	#467	1	1	yes	characteristic
103	C₄×C₁₃	#500	1	1	yes	characteristic
104	C₂×M₁₆	—	2	2	no	—
105	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
107	#117	D₄×C₁₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	2	2	no	—
109	M₃₂	—	2	1	no	—
110	M₃₂	Q₈×C₁₃	1	2	yes	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
113	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
114	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
115	C₂×C₁₆	Q₈×C₁₃	1	2	yes	—
116	#108	—	2	2	no	—
117	#108	C₂×C₄×C₁₃	1	4	yes	—
118	#107	—	2	2	no	—
119	#107	—	2	2	no	—
120	#107	—	4	2	no	—
121	#106	—	4	2	no	—
122	#97	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₂×C₁₃	—	2	1	no	—
125	C₂×C₁₃	—	2	1	no	—
126	C₂×C₁₃	#1942	1	1	yes	characteristic
127	C₂×C₁₃	#2223	1	2	yes	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
130	M₁₆	—	2	1	no	—
131	M₁₆	—	2	4	no	—
132	M₁₆	—	4	2	no	—
133	M₁₆	C₁₃×Q₁₆	1	2	yes	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
137	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
138	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
139	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
140	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
141	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
145	C₂²⋊C₄	—	2	1	no	—
146	C₄²	—	2	1	no	—
147	C₁₆	—	2	2	no	—
148	C₁₆	C₁₃×C₄⋊C₄	1	2	yes	—
149	C₁₆	C₁₃×C₄⋊C₄	1	2	yes	—
150	C₁₃	#13172	1	1	yes	characteristic, direct product factor, Sylow
151	C₂³	C₁₃×#135	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	16	2	no	—
155	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
156	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
157	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
158	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
159	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
160	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	4	4	no	—
163	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
164	C₈	C₄×Q₈×C₁₃	1	1	yes	characteristic
165	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
166	C₈	C₁₃×#125	1	1	yes	characteristic
167	C₂²	—	2	2	no	—
168	C₂²	C₄×C₁₃×Q₁₆	1	1	yes	characteristic
169	C₂²	C₁₃×#466	1	1	yes	characteristic
170	C₂²	C₁₃×#499	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	16	2	no	—
173	C₄	C₁₃×#389	1	1	yes	characteristic
174	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
175	C₄	C₁₃×#467	1	1	yes	characteristic
176	C₄	C₁₃×#500	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₁₃×#1942	1	1	yes	characteristic
180	C₂	C₁₃×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#466
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂³×C₁₃	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₃×#368	C₄	semidirect
38. #13172	C₁₃	direct (N × H)
88. #368	C₄×C₁₃	semidirect
150. C₁₃	#13172	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#368	2
C₂×C₁₃×#367	2
C₁₃×#2108	2
C₁₃×#1952	2
C₁₃×#1387	2
#13172	13

Subgroups — C13×#13172 #15699511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13172 #15699511