Subgroups — #15756727

Order 3328.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114974	C₂	1	2	yes	maximal
2	#13113595	C₂	1	2	yes	maximal
3	#13113592	C₂	1	2	yes	maximal
4	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	#11718585	C₂²	1	4	yes	—
9	C₈²×C₁₃	C₄	1	1	yes	characteristic
10	C₈²×C₁₃	C₄	1	1	yes	characteristic
11	C₈²×C₁₃	C₄	1	1	yes	characteristic
12	C₈²×C₁₃	C₄	1	1	yes	characteristic
13	#11718144	—	2	4	no	—
14	#11718050	—	2	4	no	—
15	#11718049	—	2	4	no	—
16	C₂²×C₈×C₁₃	D₄	1	2	yes	—
17	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
18	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×#6099	—	2	4	no	—
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	D₄	1	2	yes	—
28	C₄×C₈×C₁₃	D₄	1	2	yes	—
29	C₄×C₈×C₁₃	Q₈	1	2	yes	—
30	C₄⋊#1055	—	2	4	no	—
31	C₄⋊#1055	—	2	4	no	—
32	#8824	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃	—	2	4	no	—
37	C₂×C₈×C₁₃	—	2	4	no	—
38	C₂×C₈×C₁₃	—	2	4	no	—
39	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
49	#6099	—	4	8	no	—
50	C₂×#1055	—	4	4	no	—
51	#1837	—	13	2	no	—
52	#1567	—	13	2	no	—
53	#1566	—	13	2	no	—
54	C₂×C₈²	D₁₃	1	1	yes	characteristic
55	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#135	1	4	yes	—
65	C₈×C₁₃	—	2	2	no	—
66	C₈×C₁₃	—	2	2	no	—
67	C₈×C₁₃	—	2	2	no	—
68	C₈×C₁₃	—	2	2	no	—
69	C₈×C₁₃	#108	1	4	yes	—
70	C₈×C₁₃	#108	1	4	yes	—
71	#1055	—	8	4	no	—
72	#423	—	13	4	no	—
73	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
76	#341	—	26	4	no	—
77	#335	—	26	4	no	—
78	#334	—	26	4	no	—
79	C₈²	Dic₁₃	1	1	yes	characteristic
80	C₈²	Dic₁₃	1	1	yes	characteristic
81	C₈²	Dic₁₃	1	1	yes	characteristic
82	C₈²	Dic₁₃	1	1	yes	characteristic
83	C₂²×C₁₃	—	2	1	no	—
84	C₂²×C₁₃	—	2	1	no	—
85	C₂²×C₁₃	#384	1	1	yes	characteristic
86	C₂²×C₁₃	#499	1	2	yes	—
87	C₄×C₁₃	—	2	1	no	—
88	C₄×C₁₃	—	2	1	no	—
89	C₄×C₁₃	#427	1	2	yes	—
90	C₄×C₁₃	C₂×#108	1	2	yes	—
91	C₄×C₁₃	C₂×#108	1	2	yes	—
92	C₄×C₁₃	#495	1	2	yes	—
93	C₂×M₁₆	—	26	4	no	—
94	C₂²×C₈	Dic₂₆	1	2	yes	—
95	C₂²×C₈	Dic₂₆	1	2	yes	—
96	C₂²×C₈	D₅₂	1	2	yes	—
97	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
98	C₄⋊C₈	—	26	4	no	—
99	C₄⋊C₈	—	26	4	no	—
100	C₄×C₈	Dic₂₆	1	2	yes	—
101	C₄×C₈	D₅₂	1	2	yes	—
102	C₄×C₈	D₅₂	1	2	yes	—
103	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
105	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
106	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
108	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
109	C₂×C₁₃	—	2	2	no	—
110	C₂×C₁₃	#1846	1	1	yes	characteristic
111	C₂×C₁₃	#1848	1	1	yes	characteristic
112	C₂×C₁₃	#1848	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
114	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
115	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
116	M₁₆	—	52	8	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	52	4	no	—
121	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
122	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
123	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
124	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
125	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
126	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
127	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
128	C₄²	C₂×D₅₂	1	1	yes	characteristic
129	C₄²	C₂×D₅₂	1	1	yes	characteristic
130	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
131	C₁₃	#8824	1	1	yes	characteristic, complemented, Sylow
132	C₂³	#88123	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	#88123	1	1	yes	characteristic
136	C₂×C₄	#88123	1	1	yes	characteristic
137	C₂×C₄	#88128	1	1	yes	characteristic
138	C₂×C₄	#88158	1	4	yes	—
139	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	104	4	no	—
147	C₈	#88059	1	4	yes	—
148	C₈	#88059	1	4	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#11718344	1	2	yes	—
152	C₂²	#11718596	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#11718357	1	2	yes	—
156	C₄	#11718767	1	2	yes	—
157	C₄	C₂×#88059	1	2	yes	—
158	C₄	C₂×#88059	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	#13115000	1	1	yes	characteristic
161	C₂	#13115008	1	1	yes	characteristic
162	C₂	#13115008	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#8824
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
131. C₁₃	#8824	semidirect

Maximal subgroups

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

subgroup	index
#13114974	2
#13113595	2
#13113592	2
C₂×C₈²×C₁₃	2
#8824	13