Subgroups — #15774560

Order 3328.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
2	#13113595	C₂	1	1	yes	maximal, characteristic
3	#13112907	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1567	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#423	—	2	1	no	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×#6090	C₄	1	2	yes	—
8	#11718585	—	2	1	no	—
9	C₁₃×#335	—	2	1	no	—
10	C₁₃×#335	C₂²	1	2	yes	—
11	C₄×#6090	C₄	1	1	yes	characteristic
12	C₄×#6090	C₄	1	1	yes	characteristic
13	C₄×#6090	C₂²	1	2	yes	—
14	#11718050	—	2	1	no	—
15	#11718050	C₂²	1	2	yes	—
16	C₂×C₁₃×M₁₆	—	4	1	no	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×#6099	—	4	1	no	—
21	C₁₃×C₄⋊C₈	—	2	2	no	—
22	C₁₃×C₄⋊C₈	—	4	1	no	—
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×#6090	—	2	4	no	—
28	C₂×#6090	C₂×C₄	1	2	yes	—
29	C₂×#6090	C₂×C₄	1	2	yes	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	4	1	no	—
32	#12022	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
36	C₁₃×M₁₆	—	8	2	no	—
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	—	2	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	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₃	D₈	1	1	yes	characteristic
46	C₄²×C₁₃	Q₁₆	1	1	yes	characteristic
47	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
48	#6099	—	8	2	no	—
49	C₂×#1055	—	4	2	no	—
50	#6090	—	2	2	no	—
51	#6090	M₁₆	1	4	yes	complemented
52	C₂×C₄×C₁₆	—	13	1	no	—
53	#1567	—	13	1	no	—
54	#1567	D₁₃	1	1	yes	characteristic
55	#1371	—	13	4	no	—
56	C₂³×C₁₃	#107	1	1	yes	characteristic
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	—	2	1	no	—
60	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
63	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
64	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₈×C₁₃	—	2	1	no	—
69	C₈×C₁₃	—	2	1	no	—
70	C₈×C₁₃	—	8	2	no	—
71	C₈×C₁₃	#108	1	2	yes	—
72	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
73	#1055	—	8	2	no	—
74	C₂²×C₁₆	—	13	2	no	—
75	#423	—	2	1	no	—
76	#423	—	26	1	no	—
77	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
78	C₄×C₁₆	—	13	1	no	—
79	C₄×C₁₆	—	13	1	no	—
80	C₄×C₁₆	—	13	2	no	—
81	#335	—	2	1	no	—
82	#335	—	13	2	no	—
83	#335	—	26	1	no	—
84	#335	C₂×D₁₃	1	2	yes	—
85	C₂²×C₁₃	—	2	1	no	—
86	C₂²×C₁₃	—	2	1	no	—
87	C₂²×C₁₃	#366	1	2	yes	—
88	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
89	C₄×C₁₃	—	2	1	no	—
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	#366	1	2	yes	—
92	C₄×C₁₃	#368	1	2	yes	—
93	C₄×C₁₃	#423	1	1	yes	characteristic
94	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
95	C₄×C₁₃	C₂×D₁₆	1	1	yes	characteristic
96	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
97	C₂×M₁₆	—	4	1	no	—
98	C₂×M₁₆	—	52	1	no	—
99	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
100	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
102	C₂×C₁₆	—	13	2	no	—
103	C₂×C₁₆	—	13	2	no	—
104	C₂×C₁₆	—	26	4	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₄⋊C₈	—	26	2	no	—
108	C₄⋊C₈	—	52	1	no	—
109	C₄×C₈	Dic₂₆	1	1	yes	characteristic
110	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
113	C₂×C₁₃	—	2	2	no	—
114	C₂×C₁₃	#1567	1	1	yes	characteristic
115	C₂×C₁₃	C₂×#366	1	1	yes	characteristic
116	C₂×C₁₃	C₂×#368	1	1	yes	characteristic
117	C₂²×C₄	#6101	1	1	yes	characteristic
118	C₂²×C₄	#6104	1	1	yes	characteristic
119	C₂²×C₄	#6107	1	1	yes	characteristic
120	M₁₆	—	8	2	no	—
121	M₁₆	—	104	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	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	C₂×Dic₂₆	1	1	yes	characteristic
129	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
130	C₄²	#6101	1	1	yes	characteristic
131	C₄²	#6104	1	1	yes	characteristic
132	C₄²	#6107	1	1	yes	characteristic
133	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
134	C₁₆	—	13	4	no	—
135	C₁₆	—	26	2	no	—
136	C₁₃	#12022	1	1	yes	characteristic, complemented, Sylow
137	C₂³	#88047	1	1	yes	characteristic
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	#88047	1	1	yes	characteristic
142	C₂×C₄	#88047	1	1	yes	characteristic
143	C₂×C₄	#88047	1	1	yes	characteristic
144	C₂×C₄	#88066	1	2	yes	—
145	C₂×C₄	#88069	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×#6107	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	104	2	no	—
153	C₈	#88062	1	2	yes	—
154	C₈	M₁₆×D₁₃	1	2	yes	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#11718081	1	2	yes	—
158	C₂²	C₂×#88047	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#11718081	1	2	yes	—
162	C₄	#11718085	1	2	yes	—
163	C₄	#11718779	1	1	yes	characteristic
164	C₄	C₂×#88062	1	1	yes	characteristic
165	C₄	C₂×#88066	1	1	yes	characteristic
166	C₄	C₂×#88069	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	#13113594	1	1	yes	characteristic
169	C₂	C₂×#11718081	1	1	yes	characteristic
170	C₂	C₂×#11718085	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1567	C₂
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	#107
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1567	D₁₃
hypocenter	C₁₃	#12022
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #13112907	C₂	semidirect
51. #6090	M₁₆	semidirect
136. C₁₃	#12022	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#6090	2
#13113595	2
#13112907	2
C₁₃×#1567	2
#12022	13