Subgroups — #15774494

Order 3328.

242 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×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic
2	#13113119	C₂	1	1	yes	maximal, characteristic
3	#13112905	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×M₃₂	—	2	1	no	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×#6090	—	2	1	no	—
8	#11718586	C₄	1	1	yes	characteristic
9	#11718586	C₄	1	1	yes	characteristic
10	C₁₃×C₄⋊C₁₆	—	2	1	no	—
11	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
12	C₄⋊#6090	—	2	1	no	—
13	C₄⋊#6090	C₂²	1	2	yes	—
14	#11718047	C₄	1	2	yes	—
15	C₈×#1055	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₁₃×M₃₂	—	4	2	no	—
20	C₂×C₁₃×C₁₆	—	2	2	no	—
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²⋊#1055	—	2	1	no	—
26	C₂²⋊#1055	C₈	1	2	yes	—
27	C₂×#6090	—	2	2	no	—
28	C₂×#6090	—	4	2	no	—
29	C₄⋊#1055	—	2	1	no	—
30	C₄⋊#1055	C₈	1	2	yes	—
31	C₄×#1055	C₂×C₄	1	2	yes	—
32	C₄×#1055	C₂×C₄	1	2	yes	—
33	#12005	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	D₈	1	2	yes	—
41	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₁₃	C₂×D₄	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	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
47	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×#1055	—	2	2	no	—
49	C₂×#1055	C₂×C₈	1	4	yes	—
50	C₁₃×C₁₆	—	4	2	no	—
51	#6090	—	4	2	no	—
52	#2150	D₁₃	1	1	yes	characteristic
53	C₂×C₄⋊C₁₆	—	13	1	no	—
54	#1450	—	13	1	no	—
55	#1370	—	13	4	no	—
56	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	4	1	no	—
59	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#107	1	2	yes	—
63	C₂×C₄×C₁₃	#108	1	2	yes	—
64	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
67	C₈×C₁₃	—	2	1	no	—
68	C₈×C₁₃	—	2	1	no	—
69	C₈×C₁₃	#107	1	2	yes	—
70	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
71	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
72	#1055	—	2	4	no	—
73	#1055	—	4	2	no	—
74	C₂×M₃₂	—	2	1	no	—
75	C₂²×C₁₆	—	26	1	no	—
76	#431	—	13	1	no	—
77	#431	—	13	1	no	—
78	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
79	C₄⋊C₁₆	—	2	1	no	—
80	C₄⋊C₁₆	—	13	2	no	—
81	C₄⋊C₁₆	—	26	1	no	—
82	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
83	#322	—	13	2	no	—
84	C₈²	—	13	2	no	—
85	C₂²×C₁₃	—	2	1	no	—
86	C₂²×C₁₃	—	2	1	no	—
87	C₂²×C₁₃	#335	1	2	yes	—
88	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₄×C₁₃	—	2	1	no	—
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	—	2	1	no	—
92	C₄×C₁₃	#335	1	2	yes	—
93	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
94	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
95	C₄×C₁₃	#504	1	2	yes	—
96	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
97	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
98	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
99	M₃₂	—	4	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	—	26	2	no	—
102	C₂×C₁₆	—	52	2	no	—
103	C₄⋊C₈	—	13	2	no	—
104	C₄⋊C₈	—	26	1	no	—
105	C₂²⋊C₈	—	13	2	no	—
106	C₂²⋊C₈	—	26	1	no	—
107	C₄×C₈	—	13	2	no	—
108	C₄×C₈	—	13	2	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	C₂×#335	1	1	yes	characteristic
115	C₂×C₁₃	#2151	1	1	yes	characteristic
116	C₂×C₁₃	#2153	1	1	yes	characteristic
117	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
118	C₂²×C₄	#6094	1	1	yes	characteristic
119	C₂²×C₄	#6101	1	1	yes	characteristic
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	13	4	no	—
122	C₂×C₈	—	26	2	no	—
123	C₂×C₈	#6101	1	1	yes	characteristic
124	C₂×C₈	#6101	1	1	yes	characteristic
125	C₂×C₈	#6104	1	2	yes	—
126	C₂×C₈	#6107	1	2	yes	—
127	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
128	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
129	C₄²	C₈×D₁₃	1	1	yes	characteristic
130	C₄²	#6094	1	1	yes	characteristic
131	C₄²	#6101	1	1	yes	characteristic
132	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
133	C₁₆	—	4	2	no	—
134	C₁₆	—	52	2	no	—
135	C₁₃	#12005	1	1	yes	characteristic, complemented, Sylow
136	C₂³	#88054	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	#88047	1	2	yes	—
140	C₂×C₄	#88054	1	1	yes	characteristic
141	C₂×C₄	#88054	1	1	yes	characteristic
142	C₂×C₄	#88054	1	1	yes	characteristic
143	C₂×C₄	#88062	1	2	yes	—
144	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
145	C₂×C₄	C₂×#6094	1	1	yes	characteristic
146	C₂×C₄	C₂×#6101	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	26	4	no	—
150	C₈	—	52	2	no	—
151	C₈	#88047	1	2	yes	—
152	C₈	C₂×#6104	1	1	yes	characteristic
153	C₈	C₂×#6107	1	1	yes	characteristic
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#11718071	1	2	yes	—
157	C₂²	C₂×#88054	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#11718071	1	2	yes	—
162	C₄	#11718568	1	2	yes	—
163	C₄	C₂×#88047	1	1	yes	characteristic
164	C₄	C₂×#88062	1	1	yes	characteristic
165	C₂	—	2	2	no	—
166	C₂	C₂×#11718071	1	1	yes	characteristic
167	C₂	#13116596	1	1	yes	characteristic
168	C₂	#13116601	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2150	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#2150	D₁₃
hypocenter	C₁₃	#12005
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #13112905	C₂	semidirect
135. C₁₃	#12005	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
#13113119	2
#13112905	2
C₁₃×#2150	2
#12005	13