Subgroups — #15774502

Order 3328.

242 conjugacy classes of subgroups.

The subgroup lattice has 184 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	#13113584	C₂	1	1	yes	maximal, characteristic
3	#13112909	C₂	1	2	yes	maximal, complemented
4	#13112907	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#431	—	2	1	no	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×#6090	C₄	1	1	yes	characteristic
9	C₂²×#6090	C₄	1	1	yes	characteristic
10	#11718586	—	2	1	no	—
11	C₁₃×#335	C₂²	1	2	yes	—
12	C₁₃×#334	—	2	1	no	—
13	C₄×#6090	C₄	1	1	yes	characteristic
14	C₄×#6090	C₄	1	1	yes	characteristic
15	C₄×#6090	C₂²	1	1	yes	characteristic
16	C₄×#6090	C₂²	1	1	yes	characteristic
17	#11718050	C₂²	1	2	yes	—
18	#11718049	—	2	1	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	C₁₃×C₄⋊C₈	—	2	2	no	—
23	C₁₃×C₂²⋊C₈	—	4	1	no	—
24	C₄×C₈×C₁₃	—	4	1	no	—
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²⋊#1055	—	4	1	no	—
30	C₂×#6090	—	2	2	no	—
31	C₂×#6090	C₈	1	4	yes	complemented
32	C₂×#6090	C₂×C₄	1	1	yes	characteristic
33	C₂×#6090	C₂×C₄	1	1	yes	characteristic
34	C₂×#6090	C₂×C₄	1	1	yes	characteristic
35	C₂×#6090	C₂×C₄	1	1	yes	characteristic
36	C₄⋊#1055	—	2	2	no	—
37	C₄×#1055	—	4	1	no	—
38	#12008	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	4	2	no	—
44	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₁₃	D₈	1	1	yes	characteristic
52	C₄²×C₁₃	Q₁₆	1	1	yes	characteristic
53	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×#1055	—	4	2	no	—
55	#6090	—	2	1	no	—
56	#6090	—	2	1	no	—
57	#6090	C₂×C₈	1	2	yes	—
58	#6090	C₂×C₈	1	2	yes	—
59	C₂×C₄×C₁₆	—	13	1	no	—
60	#1564	—	13	1	no	—
61	#1564	D₁₃	1	1	yes	characteristic
62	#1372	—	13	2	no	—
63	#1371	—	13	2	no	—
64	C₂³×C₁₃	#107	1	1	yes	characteristic
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
68	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
69	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#108	1	2	yes	—
72	C₂×C₄×C₁₃	C₂×C₄⋊C₄	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	C₈×C₁₃	—	2	1	no	—
76	C₈×C₁₃	—	2	1	no	—
77	C₈×C₁₃	—	8	2	no	—
78	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
79	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
81	#1055	—	8	2	no	—
82	C₂²×C₁₆	—	13	1	no	—
83	C₂²×C₁₆	—	13	1	no	—
84	#431	—	2	1	no	—
85	#431	—	26	1	no	—
86	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
87	C₄×C₁₆	—	13	1	no	—
88	C₄×C₁₆	—	13	1	no	—
89	C₄×C₁₆	—	13	1	no	—
90	C₄×C₁₆	—	13	1	no	—
91	#335	—	13	2	no	—
92	#335	C₂×D₁₃	1	2	yes	—
93	#334	—	2	1	no	—
94	#334	—	26	1	no	—
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	#335	1	2	yes	—
98	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
99	C₄×C₁₃	—	2	1	no	—
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	#335	1	2	yes	—
102	C₄×C₁₃	#368	1	2	yes	—
103	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
105	C₄×C₁₃	#508	1	2	yes	—
106	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
107	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
108	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
109	C₂×C₁₆	—	13	4	no	—
110	C₂×C₁₆	—	13	1	no	—
111	C₂×C₁₆	—	13	1	no	—
112	C₂×C₁₆	—	13	1	no	—
113	C₂×C₁₆	—	13	1	no	—
114	C₂×C₁₆	—	26	2	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	26	2	no	—
117	C₂²⋊C₈	—	4	1	no	—
118	C₂²⋊C₈	—	52	1	no	—
119	C₄×C₈	—	4	1	no	—
120	C₄×C₈	—	52	1	no	—
121	C₄×C₈	Dic₂₆	1	1	yes	characteristic
122	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
123	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
125	C₂×C₁₃	—	2	2	no	—
126	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
127	C₂×C₁₃	#2158	1	1	yes	characteristic
128	C₂×C₁₃	C₂×#368	1	1	yes	characteristic
129	C₂²×C₄	#6101	1	1	yes	characteristic
130	C₂²×C₄	#6104	1	1	yes	characteristic
131	C₂²×C₄	#6107	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	52	2	no	—
135	C₂×C₈	C₈×D₁₃	1	2	yes	—
136	C₂×C₈	#6094	1	2	yes	—
137	C₂×C₈	#6101	1	1	yes	characteristic
138	C₂×C₈	#6101	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
140	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
141	C₄²	#6101	1	1	yes	characteristic
142	C₄²	#6104	1	1	yes	characteristic
143	C₄²	#6107	1	1	yes	characteristic
144	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
145	C₁₆	—	13	2	no	—
146	C₁₆	—	13	2	no	—
147	C₁₆	—	26	1	no	—
148	C₁₆	—	26	1	no	—
149	C₁₃	#12008	1	1	yes	characteristic, complemented, Sylow
150	C₂³	#88047	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#88047	1	1	yes	characteristic
154	C₂×C₄	#88047	1	1	yes	characteristic
155	C₂×C₄	#88047	1	1	yes	characteristic
156	C₂×C₄	#88054	1	2	yes	—
157	C₂×C₄	#88062	1	2	yes	—
158	C₂×C₄	C₂×#6101	1	1	yes	characteristic
159	C₂×C₄	C₂×#6104	1	1	yes	characteristic
160	C₂×C₄	C₂×#6107	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	8	2	no	—
164	C₈	—	104	2	no	—
165	C₈	#88054	1	2	yes	—
166	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
167	C₈	C₂×#6094	1	1	yes	characteristic
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#11718071	1	2	yes	—
171	C₂²	C₂×#88047	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	#11718071	1	2	yes	—
175	C₄	#11718085	1	2	yes	—
176	C₄	C₂×#88054	1	1	yes	characteristic
177	C₄	C₂×#88062	1	1	yes	characteristic
178	C₄	#11718855	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	C₂×#11718071	1	1	yes	characteristic
181	C₂	#13116618	1	1	yes	characteristic
182	C₂	C₂×#11718085	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#6104
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	#107
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#12008
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #13112909	C₂	semidirect
4. #13112907	C₂	semidirect
31. C₂×#6090	C₈	semidirect
149. C₁₃	#12008	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
#13113584	2
#13112909	2
#13112907	2
C₁₃×#1564	2
#12008	13