Subgroups — #15774537

Order 3328.

238 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116639	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic
3	#13112906	C₂	1	2	yes	maximal, complemented
4	#13112900	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#431	C₄	1	1	yes	characteristic
7	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6090	—	2	1	no	—
10	C₁₃×#322	C₄	1	2	yes	—
11	C₈²×C₁₃	C₂²	1	2	yes	—
12	C₂²⋊#6090	—	2	1	no	—
13	C₄⋊#6090	—	2	1	no	—
14	C₄⋊#6090	C₂²	1	1	yes	characteristic
15	C₄⋊#6090	C₂²	1	2	yes	—
16	C₄⋊#6090	C₂²	1	1	yes	characteristic
17	#11718053	—	2	1	no	—
18	C₂×C₁₃×M₁₆	—	2	1	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₁₃×C₄⋊C₈	C₈	1	2	yes	—
23	C₁₃×C₂²⋊C₈	C₈	1	2	yes	—
24	C₁₃×#97	—	2	1	no	—
25	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
30	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×#6090	—	2	2	no	—
32	C₂×#6090	—	2	1	no	—
33	C₂×#6090	—	2	1	no	—
34	C₂×#6090	—	4	2	no	—
35	#12016	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
39	C₁₃×M₁₆	—	2	4	no	—
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	—	4	1	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	4	yes	—
47	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
52	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	#6090	—	4	1	no	—
54	#6090	—	4	2	no	—
55	#6090	—	4	1	no	—
56	#2163	—	13	1	no	—
57	C₂×C₄⋊C₁₆	—	13	1	no	—
58	C₈×M₁₆	D₁₃	1	1	yes	characteristic
59	#1370	—	13	2	no	—
60	#1367	—	13	2	no	—
61	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
72	C₈×C₁₃	—	2	4	no	—
73	C₈×C₁₃	—	2	1	no	—
74	C₈×C₁₃	—	2	1	no	—
75	C₈×C₁₃	—	2	1	no	—
76	C₈×C₁₃	—	2	4	no	—
77	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₈×C₁₃	#137	1	1	yes	characteristic
79	C₂²×C₁₆	—	26	1	no	—
80	#431	Dic₁₃	1	1	yes	characteristic
81	C₄×M₁₆	Dic₁₃	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
83	C₄⋊C₁₆	—	13	1	no	—
84	C₄⋊C₁₆	—	13	2	no	—
85	C₄⋊C₁₆	—	13	1	no	—
86	C₄⋊C₁₆	—	26	1	no	—
87	C₂²⋊C₁₆	—	26	1	no	—
88	#346	—	26	1	no	—
89	#322	Dic₁₃	1	2	yes	—
90	C₈²	C₂×D₁₃	1	2	yes	—
91	C₂²×C₁₃	—	2	1	no	—
92	C₂²×C₁₃	—	2	1	no	—
93	C₂²×C₁₃	#364	1	2	yes	—
94	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₁₃	—	2	1	no	—
96	C₄×C₁₃	—	2	1	no	—
97	C₄×C₁₃	—	2	1	no	—
98	C₄×C₁₃	—	2	1	no	—
99	C₄×C₁₃	#428	1	1	yes	characteristic
100	C₄×C₁₃	#430	1	1	yes	characteristic
101	C₄×C₁₃	#504	1	2	yes	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
104	C₂²×C₈	D₅₂	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
106	C₂×C₁₆	—	26	1	no	—
107	C₂×C₁₆	—	26	1	no	—
108	C₂×C₁₆	—	26	2	no	—
109	C₂×C₁₆	—	52	2	no	—
110	C₄⋊C₈	#1055	1	2	yes	—
111	C₂²⋊C₈	#1055	1	2	yes	—
112	#97	—	2	1	no	—
113	C₄×C₈	Dic₂₆	1	1	yes	characteristic
114	C₄×C₈	D₅₂	1	1	yes	characteristic
115	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
116	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
117	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
118	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
119	C₂×C₁₃	—	2	2	no	—
120	C₂×C₁₃	#1565	1	1	yes	characteristic
121	C₂×C₁₃	#2151	1	1	yes	characteristic
122	C₂×C₁₃	C₂×#364	1	1	yes	characteristic
123	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
124	C₂²×C₄	#6099	1	1	yes	characteristic
125	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	C₂×#1055	1	4	yes	—
132	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
134	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
135	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
136	C₄²	C₂×#1055	1	1	yes	characteristic
137	C₄²	#6099	1	1	yes	characteristic
138	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
139	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
140	C₁₆	—	52	2	no	—
141	C₁₆	—	52	1	no	—
142	C₁₆	—	52	1	no	—
143	C₁₃	#12016	1	1	yes	characteristic, complemented, Sylow
144	C₂³	C₄⋊#1055	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
150	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
151	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
152	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
153	C₂×C₄	C₂×#6099	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	4	no	—
160	C₈	#88162	1	1	yes	characteristic
161	C₈	#88163	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#11718055	1	2	yes	—
165	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#11718782	1	1	yes	characteristic
171	C₄	#11718786	1	1	yes	characteristic
172	C₄	#11718801	1	2	yes	—
173	C₂	—	2	2	no	—
174	C₂	#13113588	1	1	yes	characteristic
175	C₂	#13116595	1	1	yes	characteristic
176	C₂	C₂×#11718055	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#12016
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13112906	C₂	semidirect
4. #13112900	C₂	semidirect
143. C₁₃	#12016	semidirect

Maximal subgroups

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

subgroup	index
#13116639	2
C₂×C₄⋊#6090	2
#13112906	2
#13112900	2
C₈×C₁₃×M₁₆	2
#12016	13