Subgroups — #15756601

Order 3328.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115000	C₂	1	1	yes	maximal, characteristic
2	#13113119	C₂	1	1	yes	maximal, characteristic
3	#13112696	C₂	1	4	yes	maximal
4	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#108	—	2	1	no	—
6	C₂×C₁₃×#108	—	2	1	no	—
7	C₁₃×#423	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88058	—	2	1	no	—
10	C₂×#88057	—	2	1	no	—
11	#11718597	C₂²	1	2	yes	—
12	#11718586	C₂²	1	2	yes	—
13	#11718047	C₄	1	2	yes	complemented
14	C₈×#1055	C₄	1	2	yes	complemented
15	C₂×C₁₃×M₁₆	—	2	2	no	—
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	—	2	2	no	—
20	C₁₃×#108	—	4	2	no	—
21	C₁₃×#108	—	4	2	no	—
22	C₁₃×C₄⋊C₈	—	2	4	no	—
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
27	#88071	—	2	4	no	—
28	C₂²⋊#1055	—	2	2	no	—
29	#88058	—	4	2	no	—
30	#88057	—	4	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄×#1055	C₂×C₄	1	4	yes	—
33	#8790	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	#41	1	2	yes	—
36	C₂²×Dic₁₃	—	4	2	no	—
37	C₁₃×M₁₆	—	4	2	no	—
38	C₁₃×M₁₆	—	4	2	no	—
39	C₂×C₈×C₁₃	—	4	2	no	—
40	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	D₈	1	2	yes	—
45	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
46	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
47	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
48	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₁₃	#41	1	2	yes	—
51	C₄⋊Dic₁₃	—	4	2	no	—
52	C₄⋊Dic₁₃	—	4	2	no	—
53	C₂×#1055	—	2	4	no	—
54	#1848	D₁₃	1	1	yes	characteristic
55	#1846	—	13	1	no	—
56	#1450	—	13	1	no	—
57	#1286	—	13	4	no	—
58	C₂³×C₁₃	#124	1	1	yes	characteristic
59	C₂×C₄×C₁₃	—	4	2	no	—
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#117	1	2	yes	—
62	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#125	1	2	yes	—
64	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×Dic₁₃	—	8	4	no	—
69	C₂×Dic₁₃	—	8	2	no	—
70	C₈×C₁₃	—	8	2	no	—
71	C₈×C₁₃	#106	1	2	yes	—
72	C₈×C₁₃	#106	1	2	yes	—
73	C₈×C₁₃	#107	1	2	yes	—
74	C₈×C₁₃	#107	1	2	yes	—
75	#1055	—	4	4	no	—
76	#431	—	13	2	no	—
77	C₂×#108	—	2	1	no	—
78	C₂×#108	—	2	1	no	—
79	C₂×#107	—	26	1	no	—
80	C₂×#106	—	26	1	no	—
81	#423	C₂×D₁₃	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
83	#382	—	13	2	no	—
84	#322	—	13	2	no	—
85	C₈²	—	13	2	no	—
86	C₂²×C₁₃	—	2	2	no	—
87	C₂²×C₁₃	#382	1	1	yes	characteristic
88	C₂²×C₁₃	#486	1	1	yes	characteristic
89	C₂²×C₁₃	#487	1	1	yes	characteristic
90	C₄×C₁₃	—	2	2	no	—
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
93	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
94	C₄×C₁₃	#486	1	1	yes	characteristic
95	C₄×C₁₃	#487	1	1	yes	characteristic
96	Dic₁₃	—	16	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	26	2	no	—
101	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
102	#108	—	4	2	no	—
103	#108	—	4	2	no	—
104	#107	—	52	2	no	—
105	#106	—	52	2	no	—
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	26	2	no	—
108	C₂²⋊C₈	—	26	2	no	—
109	C₄×C₈	—	13	4	no	—
110	C₄×C₈	Dic₂₆	1	1	yes	characteristic
111	C₄×C₈	Dic₂₆	1	1	yes	characteristic
112	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
113	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
114	#95	—	26	4	no	—
115	C₂×C₁₃	—	2	1	no	—
116	C₂×C₁₃	—	2	1	no	—
117	C₂×C₁₃	#1833	1	2	yes	—
118	C₂×C₁₃	#1834	1	1	yes	characteristic
119	C₂²×C₄	—	52	2	no	—
120	C₂²×C₄	#6131	1	2	yes	—
121	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
122	M₁₆	—	4	2	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	26	4	no	—
126	C₂×C₈	#6101	1	1	yes	characteristic
127	C₂×C₈	#6101	1	1	yes	characteristic
128	C₂×C₈	#6101	1	1	yes	characteristic
129	C₂×C₈	#6101	1	1	yes	characteristic
130	C₂×C₈	#6104	1	2	yes	—
131	C₂×C₈	#6105	1	2	yes	—
132	C₂×C₈	#6106	1	2	yes	—
133	C₂×C₈	#6107	1	2	yes	—
134	C₄⋊C₄	—	52	2	no	—
135	C₄⋊C₄	—	52	2	no	—
136	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
138	C₄²	#6129	1	2	yes	—
139	C₁₃	#8790	1	1	yes	characteristic, complemented, Sylow
140	C₂³	#88201	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	104	4	no	—
143	C₂×C₄	—	104	2	no	—
144	C₂×C₄	#88144	1	2	yes	—
145	C₂×C₄	#88146	1	2	yes	—
146	C₂×C₄	C₂×#6101	1	1	yes	characteristic
147	C₂×C₄	C₂×#6104	1	1	yes	characteristic
148	C₂×C₄	C₂×#6105	1	1	yes	characteristic
149	C₂×C₄	#88190	1	1	yes	characteristic
150	C₂×C₄	C₂×#6106	1	1	yes	characteristic
151	C₂×C₄	C₂×#6107	1	1	yes	characteristic
152	C₈	—	8	2	no	—
153	C₈	—	52	4	no	—
154	C₈	#88047	1	2	yes	—
155	C₈	#88047	1	2	yes	—
156	C₈	#88048	1	2	yes	—
157	C₈	#88048	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#11718638	1	1	yes	characteristic
160	C₂²	#11718749	1	1	yes	characteristic
161	C₂²	#11718754	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	208	2	no	—
165	C₄	C₂×#88047	1	1	yes	characteristic
166	C₄	C₂×#88048	1	1	yes	characteristic
167	C₄	#11718731	1	1	yes	characteristic
168	C₄	#11718742	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#13114961	1	2	yes	—
172	C₂	#13114963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718638
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂²×C₁₃	#382
prime radical	C₂³×C₁₃	#124
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#8790
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
13. #11718047	C₄	semidirect
14. C₈×#1055	C₄	semidirect
139. C₁₃	#8790	semidirect

Maximal subgroups

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

subgroup	index
#13115000	2
#13113119	2
#13112696	2
C₁₃×#1848	2
#8790	13