Subgroups — #15783116

Order 3328.

298 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718091	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#88034	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#88034	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
6	C₁₃×#504	—	2	2	no	—
7	C₁₃×#504	—	2	2	no	—
8	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₄	1	2	yes	—
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	#11718091	—	2	2	no	—
13	C₂×#88034	—	2	2	no	—
14	C₂×#88034	C₂²	1	2	yes	—
15	C₂×#88034	C₂²	1	2	yes	—
16	C₂×C₁₃×#41	C₈	1	1	yes	characteristic
17	C₁₃×#131	—	2	2	no	—
18	C₁₃×#131	—	2	2	no	—
19	C₂×C₁₃×M₁₆	C₈	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	C₈	1	2	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₁₃×M₃₂	—	2	2	no	—
24	C₁₃×M₃₂	—	2	4	no	—
25	C₂×C₁₃×C₁₆	—	2	2	no	—
26	C₂×C₁₃×C₁₆	—	2	2	no	—
27	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
30	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
31	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
32	#88034	—	2	2	no	—
33	#88034	—	2	2	no	—
34	#13554	—	13	1	no	maximal, Sylow
35	C₁₃×#41	—	2	2	no	—
36	C₁₃×#41	—	2	2	no	—
37	C₂×Q₈×C₁₃	C₁₆	1	1	yes	characteristic
38	C₂×D₄×C₁₃	C₁₆	1	2	yes	—
39	C₂×D₄×C₁₃	C₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₁₃×M₁₆	—	2	2	no	—
43	C₁₃×M₁₆	—	2	4	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₁₃×C₁₆	—	2	2	no	—
52	C₁₃×C₁₆	C₂²⋊C₄	1	2	yes	—
53	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
54	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
55	C₂×#504	D₁₃	1	1	yes	characteristic
56	C₂×M₆₄	—	13	1	no	—
57	C₂²×C₃₂	—	13	1	no	—
58	C₂²⋊C₃₂	—	13	4	no	—
59	C₂³×C₁₃	C₂×C₁₆	1	2	yes	—
60	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
61	Q₈×C₁₃	—	2	2	no	—
62	D₄×C₁₃	—	2	4	no	—
63	D₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
68	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
70	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
71	C₈×C₁₃	—	2	2	no	—
72	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
73	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×#131	Dic₁₃	1	1	yes	characteristic
76	#504	—	2	2	no	—
77	#504	—	2	2	no	—
78	C₂×M₃₂	Dic₁₃	1	1	yes	characteristic
79	C₂×M₃₂	Dic₁₃	1	2	yes	—
80	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
81	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
82	M₆₄	—	26	2	no	—
83	C₂×C₃₂	—	13	2	no	—
84	C₂×C₃₂	—	13	2	no	—
85	C₂×C₃₂	—	26	2	no	—
86	C₂²×C₁₃	—	2	2	no	—
87	C₂²×C₁₃	—	2	2	no	—
88	C₂²×C₁₃	—	2	4	no	—
89	C₂²×C₁₃	C₂²⋊C₁₆	1	2	yes	—
90	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	C₂²⋊C₁₆	1	2	yes	—
93	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
94	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
95	C₂×#41	#1055	1	1	yes	characteristic
96	#131	—	2	2	no	—
97	#131	—	2	2	no	—
98	C₂×M₁₆	#1055	1	1	yes	characteristic
99	C₂×M₁₆	#1055	1	2	yes	—
100	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
101	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
102	M₃₂	—	2	2	no	—
103	M₃₂	—	2	4	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
107	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
108	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
109	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
110	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
111	C₃₂	—	26	2	no	—
112	C₃₂	—	26	2	no	—
113	C₂×C₁₃	—	2	2	no	—
114	C₂×C₁₃	—	4	2	no	—
115	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
116	C₂×C₁₃	#2258	1	2	yes	—
117	#41	—	2	2	no	—
118	#41	—	2	2	no	—
119	C₂×Q₈	#6090	1	1	yes	characteristic
120	C₂×D₄	#6090	1	2	yes	—
121	C₂×D₄	#6090	1	1	yes	characteristic
122	C₂²×C₄	C₂×#1055	1	2	yes	—
123	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
124	M₁₆	—	2	2	no	—
125	M₁₆	—	2	4	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	C₂×#1055	1	2	yes	—
129	C₂×C₈	C₂×#1055	1	1	yes	characteristic
130	C₂×C₈	#6099	1	2	yes	—
131	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
132	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
133	C₁₆	—	2	2	no	—
134	C₁₆	C₂²⋊Dic₁₃	1	2	yes	—
135	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
136	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
137	C₁₃	#13554	1	1	yes	characteristic, complemented, Sylow
138	C₂³	C₂×#6090	1	2	yes	—
139	C₂³	C₂×#6090	1	1	yes	characteristic
140	Q₈	—	2	2	no	—
141	D₄	—	2	4	no	—
142	D₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	C₂×#6090	1	2	yes	—
146	C₂×C₄	C₂×#6090	1	1	yes	characteristic
147	C₂×C₄	#88052	1	2	yes	—
148	C₂×C₄	C₂²⋊#1055	1	2	yes	—
149	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	C₂²⋊#1055	1	2	yes	—
152	C₈	C₂×#6099	1	1	yes	characteristic
153	C₈	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
154	C₂²	—	2	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	4	no	—
157	C₂²	C₂²⋊#6090	1	2	yes	—
158	C₂²	C₂²×#6090	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	C₂²⋊#6090	1	2	yes	—
161	C₄	C₂×#88052	1	1	yes	characteristic
162	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	4	2	no	—
165	C₂	C₂×C₂²⋊#6090	1	1	yes	characteristic
166	C₂	#13117130	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂²×C₁₆
Fitting	C₂×C₁₃×#504	C₂
prime radical	C₂×C₁₃×#41	C₈
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	C₂×#504	D₁₃
hypocenter	C₁₃	#13554
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#11718091	C₂	semidirect
2. C₂²×#88034	C₂	semidirect
3. C₂²⋊#88034	C₂	semidirect
137. C₁₃	#13554	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718091	2
C₂²×#88034	2
C₂²⋊#88034	2
C₂×C₁₃×#504	2
#13554	13