Subgroups — #41×#6090 #16016438

Order 3328.

602 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	Q₈×#6090	C₂	1	2	yes	maximal, complemented
2	D₄×#6090	C₂	1	6	yes	maximal, complemented
3	#13116635	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×#6090	C₂	1	3	yes	maximal, complemented
5	C₈×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
7	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
8	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₈×D₄×C₁₃	C₄	1	3	yes	—
10	C₈×D₄×C₁₃	C₂²	1	3	yes	—
11	C₁₃×#431	C₄	1	3	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
13	C₂²×#6090	C₂²	1	6	yes	—
14	C₂²⋊#6090	C₂²	1	6	yes	—
15	C₄⋊#6090	C₂²	1	6	yes	—
16	C₄×#6090	C₂²	1	1	yes	characteristic
17	C₄×#6090	C₂²	1	6	yes	—
18	C₄×#6090	C₂²	1	3	yes	—
19	C₂×C₁₃×#41	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
22	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
23	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×D₄×C₁₃	C₈	1	3	yes	—
25	C₄×D₄×C₁₃	C₂×C₄	1	3	yes	—
26	C₁₃×#117	C₈	1	3	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
28	C₁₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
29	C₁₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
30	C₁₃×C₂²⋊C₈	C₂×C₄	1	6	yes	—
31	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₁₃	C₂³	1	3	yes	—
34	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×#6090	—	2	12	no	—
36	C₂×#6090	C₂³	1	6	yes	—
37	C₂×#6090	C₂³	1	2	yes	—
38	C₁₆×#41	—	13	1	no	maximal, Sylow
39	C₁₃×#41	C₁₆	1	8	yes	complemented
40	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂×D₄×C₁₃	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
45	C₂×C₈×C₁₃	—	2	6	no	—
46	C₂×C₈×C₁₃	—	2	6	no	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
51	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
52	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
53	C₁₃×C₂²⋊C₄	C₂×C₈	1	6	yes	—
54	C₄²×C₁₃	C₂×C₈	1	3	yes	—
55	C₄²×C₁₃	C₂×C₈	1	3	yes	—
56	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
58	#6090	—	2	6	no	—
59	#6090	#41	1	4	yes	direct product factor
60	C₈×#41	D₁₃	1	1	yes	characteristic
61	Q₈×C₁₆	—	13	2	no	—
62	D₄×C₁₆	—	13	6	no	—
63	#2162	—	13	3	no	—
64	C₂×C₄×C₁₆	—	13	3	no	—
65	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
66	Q₈×C₁₃	C₂×C₁₆	1	4	yes	complemented
67	D₄×C₁₃	C₂×C₁₆	1	12	yes	complemented
68	C₂×C₄×C₁₃	—	2	3	no	—
69	C₂×C₄×C₁₃	—	2	3	no	—
70	C₂×C₄×C₁₃	—	2	6	no	—
71	C₂×C₄×C₁₃	C₂×C₁₆	1	12	yes	complemented
72	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
78	C₈×C₁₃	—	2	3	no	—
79	C₈×C₁₃	—	2	3	no	—
80	C₈×C₁₃	#131	1	2	yes	—
81	C₈×C₁₃	C₂×#41	1	2	yes	—
82	C₄×#41	Dic₁₃	1	1	yes	characteristic
83	C₂²×C₁₆	—	13	6	no	—
84	C₈×Q₈	Dic₁₃	1	1	yes	characteristic
85	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
86	C₈×D₄	Dic₁₃	1	3	yes	—
87	C₈×D₄	C₂×D₁₃	1	3	yes	—
88	#431	Dic₁₃	1	3	yes	—
89	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
90	C₄⋊C₁₆	—	13	6	no	—
91	C₂²⋊C₁₆	—	13	6	no	—
92	C₄×C₁₆	—	13	6	no	—
93	C₄×C₁₆	—	13	1	no	—
94	C₄×C₁₆	—	13	3	no	—
95	C₂²×C₁₃	—	2	3	no	—
96	C₂²×C₁₃	—	2	3	no	—
97	C₂²×C₁₃	C₂²×C₁₆	1	6	yes	—
98	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
99	C₄×C₁₃	—	2	3	no	—
100	C₄×C₁₃	—	2	3	no	—
101	C₄×C₁₃	C₂²×C₁₆	1	2	yes	—
102	C₄×C₁₃	C₂²×C₁₆	1	6	yes	—
103	C₄×C₁₃	#504	1	2	yes	—
104	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
105	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
106	C₂×#41	#1055	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
108	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
109	C₄×Q₈	#1055	1	1	yes	characteristic
110	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
111	C₄×D₄	#1055	1	3	yes	—
112	C₄×D₄	C₂×Dic₁₃	1	3	yes	—
113	#117	#1055	1	3	yes	—
114	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
115	C₂×C₁₆	—	13	2	no	—
116	C₂×C₁₆	—	13	6	no	—
117	C₂×C₁₆	—	26	12	no	—
118	C₄⋊C₈	C₂×Dic₁₃	1	3	yes	—
119	C₄⋊C₈	C₂×Dic₁₃	1	3	yes	—
120	C₂²⋊C₈	C₂×Dic₁₃	1	6	yes	—
121	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
122	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
123	C₄×C₈	C₂²×D₁₃	1	3	yes	—
124	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
125	C₂×C₁₃	—	2	6	no	—
126	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
127	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
128	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
129	#41	#6090	1	8	yes	direct product factor
130	C₂×Q₈	C₂×#1055	1	1	yes	characteristic
131	C₂×D₄	C₂×#1055	1	3	yes	—
132	C₂²×C₄	C₂×#1055	1	3	yes	—
133	C₂²×C₄	C₂×#1055	1	3	yes	—
134	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
138	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
139	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
140	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
141	C₄⋊C₄	C₂×#1055	1	3	yes	—
142	C₄⋊C₄	C₂×#1055	1	3	yes	—
143	C₂²⋊C₄	C₂×#1055	1	6	yes	—
144	C₄²	C₂×#1055	1	3	yes	—
145	C₄²	C₂×#1055	1	3	yes	—
146	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
147	C₄²	C₂²×Dic₁₃	1	3	yes	—
148	C₁₆	—	13	4	no	—
149	C₁₆	—	26	6	no	—
150	C₁₃	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₂²×#1055	1	3	yes	—
152	Q₈	C₂×#6090	1	4	yes	complemented
153	D₄	C₂×#6090	1	12	yes	complemented
154	C₂×C₄	—	2	3	no	—
155	C₂×C₄	—	2	3	no	—
156	C₂×C₄	—	2	6	no	—
157	C₂×C₄	C₂×#6090	1	12	yes	complemented
158	C₂×C₄	C₂²×#1055	1	3	yes	—
159	C₂×C₄	C₂²×#1055	1	3	yes	—
160	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
161	C₂×C₄	C₂²×#1055	1	3	yes	—
162	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
163	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	2	3	no	—
166	C₈	#88202	1	2	yes	—
167	C₈	#41×D₁₃	1	2	yes	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	C₂²×#6090	1	6	yes	—
171	C₂²	C₂³×#1055	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	2	3	no	—
174	C₄	C₂²×#6090	1	2	yes	—
175	C₄	C₂²×#6090	1	6	yes	—
176	C₄	#11718801	1	2	yes	—
177	C₄	C₂×#88202	1	1	yes	characteristic
178	C₄	#41×Dic₁₃	1	1	yes	characteristic
179	C₂	—	2	6	no	—
180	C₂	#41×#1055	1	1	yes	characteristic
181	C₂	C₂³×#6090	1	1	yes	characteristic
182	C₂	C₂×#11718801	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₈×C₁₃×#41	C₂
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂×C₁₃×#41	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₈×#41	D₁₃
hypocenter	C₁₃	C₁₆×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×#6090	C₂	semidirect
2. D₄×#6090	C₂	semidirect
3. #13116635	C₂	semidirect
4. C₂×C₄×#6090	C₂	semidirect
39. C₁₃×#41	C₁₆	semidirect
59. #6090	#41	direct (N × H)
66. Q₈×C₁₃	C₂×C₁₆	semidirect
67. D₄×C₁₃	C₂×C₁₆	semidirect
71. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
129. #41	#6090	direct (N × H)
150. C₁₃	C₁₆×#41	semidirect
152. Q₈	C₂×#6090	semidirect
153. D₄	C₂×#6090	semidirect
157. C₂×C₄	C₂×#6090	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6090	2
D₄×#6090	2
#13116635	2
C₂×C₄×#6090	2
C₈×C₁₃×#41	2
C₁₆×#41	13