Subgroups — #15749752

Order 3328.

262 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718047	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×M₃₂	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×#88052	C₄	1	2	yes	—
7	C₂×C₄×#1055	C₄	1	2	yes	—
8	C₁₃×#346	—	2	1	no	—
9	C₄×C₁₃×C₁₆	—	2	1	no	—
10	#11718053	—	2	1	no	—
11	C₄×#6090	—	2	1	no	—
12	#11718047	—	2	1	no	—
13	#11718047	—	2	1	no	—
14	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×#1055	C₈	1	4	yes	—
18	C₁₃×M₃₂	—	2	2	no	—
19	C₁₃×M₃₂	C₈	1	4	yes	complemented
20	C₂×C₁₃×C₁₆	—	2	2	no	—
21	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
25	#88052	—	2	2	no	—
26	#88052	C₈	1	4	yes	complemented
27	C₂×#6090	—	2	2	no	—
28	C₄×#1055	—	2	2	no	—
29	C₄×#1055	—	2	2	no	—
30	#7139	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
40	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
41	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₃	M₁₆	1	2	yes	—
44	C₂×#1055	—	2	4	no	—
45	C₂×#1055	—	4	4	no	—
46	C₁₃×C₁₆	—	2	4	no	—
47	#6090	—	2	4	no	—
48	C₄×M₃₂	—	13	1	no	—
49	C₄×M₃₂	D₁₃	1	1	yes	characteristic
50	C₂×#322	—	13	1	no	—
51	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃	—	2	4	no	—
53	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
61	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
62	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
63	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
64	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
65	#1055	—	4	4	no	—
66	C₂×M₃₂	—	13	2	no	—
67	C₂×M₃₂	Dic₁₃	1	2	yes	—
68	C₂×C₄×C₈	—	13	2	no	—
69	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
70	#346	—	2	1	no	—
71	#346	—	26	1	no	—
72	C₄×C₁₆	—	2	1	no	—
73	C₄×C₁₆	—	26	1	no	—
74	#322	—	26	1	no	—
75	#322	—	26	1	no	—
76	C₂²×C₁₃	—	2	2	no	—
77	C₂²×C₁₃	C₈²	1	1	yes	characteristic
78	C₂²×C₁₃	#322	1	1	yes	characteristic
79	C₂²×C₁₃	#336	1	1	yes	characteristic
80	C₄×C₁₃	—	2	2	no	—
81	C₄×C₁₃	#322	1	1	yes	characteristic
82	C₄×C₁₃	#322	1	1	yes	characteristic
83	C₄×C₁₃	#336	1	1	yes	characteristic
84	C₄×C₁₃	#336	1	1	yes	characteristic
85	C₄×C₁₃	#349	1	4	yes	—
86	C₂²×C₈	—	13	4	no	—
87	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
88	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
89	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
90	M₃₂	—	2	2	no	—
91	M₃₂	—	13	4	no	—
92	M₃₂	—	26	2	no	—
93	M₃₂	#1055	1	4	yes	complemented
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	26	2	no	—
96	C₄×C₈	—	26	2	no	—
97	C₄×C₈	—	26	2	no	—
98	C₄×C₈	Dic₂₆	1	1	yes	characteristic
99	C₄×C₈	D₅₂	1	1	yes	characteristic
100	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
101	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
102	C₂×C₁₃	—	2	1	no	—
103	C₂×C₁₃	—	2	1	no	—
104	C₂×C₁₃	#1273	1	1	yes	characteristic
105	C₂×C₁₃	#1377	1	2	yes	—
106	C₂²×C₄	C₈×D₁₃	1	2	yes	—
107	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
108	C₂×C₈	—	26	4	no	—
109	C₂×C₈	—	52	4	no	—
110	C₂×C₈	C₈×D₁₃	1	2	yes	—
111	C₂×C₈	#6094	1	2	yes	—
112	C₂×C₈	C₂×#1055	1	2	yes	—
113	C₂×C₈	#6099	1	2	yes	—
114	C₂×C₈	#6101	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
116	C₂×C₈	#6103	1	1	yes	characteristic
117	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
118	C₄²	#6094	1	2	yes	—
119	C₄²	#6101	1	1	yes	characteristic
120	C₄²	#6103	1	1	yes	characteristic
121	C₁₆	—	2	4	no	—
122	C₁₆	—	26	4	no	—
123	C₁₃	#7139	1	1	yes	characteristic, complemented, Sylow
124	C₂³	C₈×Dic₁₃	1	1	yes	characteristic
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	C₄×#1055	1	1	yes	characteristic
127	C₂×C₄	#88043	1	1	yes	characteristic
128	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
129	C₂×C₄	#88054	1	2	yes	—
130	C₂×C₄	#88055	1	1	yes	characteristic
131	C₂×C₄	#88055	1	1	yes	characteristic
132	C₂×C₄	#88060	1	2	yes	—
133	C₂×C₄	#88071	1	1	yes	characteristic
134	C₈	—	52	4	no	—
135	C₈	C₄⋊#1055	1	2	yes	—
136	C₈	#88054	1	2	yes	—
137	C₈	#88060	1	2	yes	—
138	C₈	C₂²⋊#1055	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₈×#1055	1	1	yes	characteristic
141	C₂²	#11718046	1	1	yes	characteristic
142	C₂²	#11718142	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	#11718046	1	1	yes	characteristic
145	C₄	#11718047	1	1	yes	characteristic
146	C₄	#11718106	1	4	yes	—
147	C₄	#11718116	1	1	yes	characteristic
148	C₄	#11718142	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#13112663	1	1	yes	characteristic
152	C₂	#13112919	1	2	yes	—

	subgroup	G/H
center	C₄²	#6103
commutator	C₂²×C₁₃	C₈²
Frattini	C₂×C₄×C₈	C₂×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	#336
prime radical	C₂³×C₁₃	C₄×C₈
cosocle	C₂×C₄×C₈×C₁₃	C₂²
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#7139
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
19. C₁₃×M₃₂	C₈	semidirect
26. #88052	C₈	semidirect
93. M₃₂	#1055	semidirect
123. C₁₃	#7139	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88052	2
C₂×#11718047	2
C₄×C₁₃×M₃₂	2
#7139	13