Subgroups — #15779367

Order 3328.

314 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718085	C₂	1	4	yes	maximal
2	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
3	#13115008	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#108	—	2	1	no	—
6	C₂×C₁₃×#108	C₂²	1	2	yes	—
7	C₁₃×#423	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88059	—	2	1	no	—
10	C₂×#88059	C₂²	1	2	yes	—
11	C₂²×#6090	C₂²	1	2	yes	—
12	#11718585	—	2	1	no	—
13	#11718085	—	2	8	no	—
14	C₄×#6090	C₄	1	2	yes	—
15	C₄×#6090	C₄	1	2	yes	—
16	C₂×C₁₃×M₁₆	—	2	2	no	—
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×#6099	—	2	2	no	—
21	C₁₃×#108	—	2	4	no	—
22	C₁₃×#108	—	4	2	no	—
23	C₁₃×C₄⋊C₈	—	4	2	no	—
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
28	#88059	—	2	4	no	—
29	#88059	—	4	2	no	—
30	C₂×#6090	C₂×C₄	1	4	yes	—
31	C₂×#6090	D₄	1	4	yes	—
32	C₂×#6090	Q₈	1	4	yes	—
33	C₄⋊#1055	—	4	2	no	—
34	#12909	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	Q₁₆	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
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	D₈	1	2	yes	—
43	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
44	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₃	D₈	1	2	yes	—
51	#6099	—	4	2	no	—
52	#6099	—	4	2	no	—
53	C₂×#1055	—	4	2	no	—
54	#6090	C₄⋊C₄	1	8	yes	—
55	C₂×#368	—	13	4	no	—
56	C₂×C₄×C₁₆	—	13	1	no	—
57	#1848	—	13	1	no	—
58	#1848	D₁₃	1	1	yes	characteristic
59	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×C₄×C₁₃	—	2	4	no	—
61	C₂×C₄×C₁₃	#107	1	2	yes	—
62	C₂×C₄×C₁₃	#107	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₈×C₁₃	—	8	2	no	—
70	C₈×C₁₃	#107	1	2	yes	—
71	C₈×C₁₃	#107	1	2	yes	—
72	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
73	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
74	#1055	—	8	2	no	—
75	C₂²×C₁₆	—	13	2	no	—
76	C₂×#108	—	2	1	no	—
77	C₂×#108	—	13	2	no	—
78	C₂×#108	—	26	1	no	—
79	C₂×#108	C₂×D₁₃	1	2	yes	—
80	#423	—	2	1	no	—
81	#423	—	26	1	no	—
82	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
83	#368	—	26	8	no	—
84	C₄×C₁₆	—	13	2	no	—
85	C₄×C₁₆	—	13	2	no	—
86	C₂²×C₁₃	—	2	2	no	—
87	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
88	C₂²×C₁₃	#494	1	1	yes	characteristic
89	C₂²×C₁₃	#500	1	1	yes	characteristic
90	C₄×C₁₃	—	2	2	no	—
91	C₄×C₁₃	#368	1	4	yes	complemented
92	C₄×C₁₃	#384	1	1	yes	characteristic
93	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
94	C₄×C₁₃	#493	1	1	yes	characteristic
95	C₄×C₁₃	#500	1	1	yes	characteristic
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	26	2	no	—
98	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
100	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
101	C₂×C₁₆	—	13	4	no	—
102	C₂×C₁₆	—	13	4	no	—
103	C₂×C₁₆	—	13	4	no	—
104	#108	—	2	4	no	—
105	#108	—	4	2	no	—
106	#108	—	26	4	no	—
107	#108	—	52	2	no	—
108	C₄⋊C₈	—	4	2	no	—
109	C₄⋊C₈	—	52	2	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	C₂×C₁₃	—	2	1	no	—
115	C₂×C₁₃	—	2	1	no	—
116	C₂×C₁₃	#1845	1	1	yes	characteristic
117	C₂×C₁₃	C₂×#368	1	2	yes	—
118	C₂²×C₄	#6107	1	2	yes	—
119	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
120	M₁₆	—	4	2	no	—
121	M₁₆	—	4	2	no	—
122	M₁₆	—	52	2	no	—
123	M₁₆	—	52	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	52	2	no	—
126	C₂×C₈	#6101	1	1	yes	characteristic
127	C₂×C₈	#6101	1	1	yes	characteristic
128	C₂×C₈	#6104	1	2	yes	—
129	C₂×C₈	#6107	1	2	yes	—
130	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
132	C₂×C₈	D₄×D₁₃	1	2	yes	—
133	C₂×C₈	Q₈×D₁₃	1	2	yes	—
134	C₄²	#6101	1	1	yes	characteristic
135	C₄²	#6101	1	1	yes	characteristic
136	C₄²	#6104	1	2	yes	—
137	C₁₆	—	13	8	no	—
138	C₁₃	#12909	1	1	yes	characteristic, complemented, Sylow
139	C₂³	C₂×#6107	1	1	yes	characteristic
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	#88047	1	2	yes	—
142	C₂×C₄	#88047	1	2	yes	—
143	C₂×C₄	C₂×#6101	1	1	yes	characteristic
144	C₂×C₄	C₂×#6104	1	1	yes	characteristic
145	C₂×C₄	C₂×#6104	1	1	yes	characteristic
146	C₂×C₄	#88194	1	1	yes	characteristic
147	C₂×C₄	C₂×#6107	1	1	yes	characteristic
148	C₂×C₄	#88198	1	1	yes	characteristic
149	C₈	—	8	2	no	—
150	C₈	—	104	2	no	—
151	C₈	#88047	1	2	yes	—
152	C₈	#88047	1	2	yes	—
153	C₈	#88142	1	2	yes	—
154	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#88047	1	1	yes	characteristic
157	C₂²	#11718739	1	1	yes	characteristic
158	C₂²	#11718753	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	#11718085	1	4	yes	complemented
161	C₄	C₂×#88047	1	1	yes	characteristic
162	C₄	#11718637	1	1	yes	characteristic
163	C₄	#11718734	1	1	yes	characteristic
164	C₄	#11718739	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#13114995	1	1	yes	characteristic
168	C₂	C₂×#11718085	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#12909
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
91. C₄×C₁₃	#368	semidirect
138. C₁₃	#12909	semidirect
160. C₄	#11718085	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718085	2
C₂×C₄×#6090	2
#13115008	2
C₁₃×#1848	2
#12909	13