Subgroups — #46825750

Order 34432.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771626	C₂	1	2	yes	maximal
2	C₂×#22575382	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14524137	—	2	2	no	—
4	C₂²⋊#14524137	—	2	2	no	—
5	#22575508	C₄	1	4	yes	—
6	C₂²×#14524130	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
8	#22575382	—	2	2	no	—
9	C₂²×C₈×C₂₆₉	C₄	1	1	yes	characteristic
10	C₂×#14524137	—	2	4	no	—
11	C₂²×Dic₂₆₉	D₄	1	1	yes	characteristic
12	C₂²×Dic₂₆₉	Q₈	1	1	yes	characteristic
13	C₂×#14524130	—	2	1	no	—
14	C₂×#14524130	—	4	1	no	—
15	C₄×Dic₂₆₉	C₂×C₄	1	2	yes	—
16	C₄×Dic₂₆₉	D₄	1	2	yes	—
17	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₂₆₉	—	2	1	no	—
19	C₂×C₈×C₂₆₉	—	4	1	no	—
20	#14524137	—	4	4	no	—
21	C₂×Dic₂₆₉	—	2	1	no	—
22	C₂×Dic₂₆₉	—	2	1	no	—
23	C₂×Dic₂₆₉	—	4	1	no	—
24	C₂×Dic₂₆₉	C₂²⋊C₄	1	2	yes	—
25	C₂×Dic₂₆₉	C₄⋊C₄	1	2	yes	—
26	#14524130	—	4	1	no	—
27	C₂³×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₆₉	—	2	1	no	—
29	C₂×C₄×C₂₆₉	—	2	1	no	—
30	C₂×C₄×C₂₆₉	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₈×C₂₆₉	—	4	1	no	—
33	C₂²×C₂₆₉	—	2	1	no	—
34	C₂²×C₂₆₉	—	2	1	no	—
35	C₂²×C₂₆₉	#95	1	1	yes	characteristic
36	C₂²×C₂₆₉	#108	1	1	yes	characteristic
37	C₂²×C₂₆₉	#108	1	1	yes	characteristic
38	C₄×C₂₆₉	—	2	1	no	—
39	C₄×C₂₆₉	#100	1	1	yes	characteristic
40	C₄×C₂₆₉	#101	1	1	yes	characteristic
41	Dic₂₆₉	—	2	2	no	—
42	Dic₂₆₉	—	2	2	no	—
43	C₂×C₂₆₉	—	2	1	no	—
44	C₂×C₂₆₉	—	2	1	no	—
45	C₂×C₂₆₉	H(3,ℤ/4)	1	1	yes	characteristic
46	C₂×C₂₆₉	#341	1	1	yes	characteristic
47	C₂×C₂₆₉	#343	1	1	yes	characteristic
48	C₂₆₉	#1290	1	1	yes	characteristic, complemented, Sylow
49	#1290	—	269	1	no	maximal, Sylow
50	#432	—	269	2	no	—
51	C₂×C₄⋊C₈	—	269	1	no	—
52	C₂²×C₈	—	269	1	no	—
53	C₂²×C₈	#11761640	1	1	yes	characteristic
54	C₂×C₄²	—	269	1	no	—
55	C₄⋊C₈	—	538	2	no	—
56	C₄⋊C₈	—	538	2	no	—
57	C₂²⋊C₈	—	538	2	no	—
58	#97	—	269	4	no	—
59	C₂²×C₄	—	269	1	no	—
60	C₂²×C₄	—	269	1	no	—
61	C₂²×C₄	C₂×#11761640	1	1	yes	characteristic
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	—	538	4	no	—
65	C₂×C₈	—	538	1	no	—
66	C₂×C₈	—	1076	1	no	—
67	C₄²	—	269	2	no	—
68	C₄²	—	269	2	no	—
69	C₂³	C₄⋊#11761640	1	1	yes	characteristic
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	269	2	no	—
73	C₂×C₄	—	269	2	no	—
74	C₂×C₄	—	538	1	no	—
75	C₂×C₄	—	538	1	no	—
76	C₂×C₄	—	1076	1	no	—
77	C₂×C₄	C₄×#11761640	1	1	yes	characteristic
78	C₂×C₄	C₂²⋊#11761640	1	1	yes	characteristic
79	C₈	—	4	1	no	—
80	C₈	—	1076	4	no	—
81	C₈	—	1076	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	1	no	—
84	C₂²	(C₂×#11761640)⋊C₄	1	1	yes	characteristic
85	C₂²	#22575523	1	1	yes	characteristic
86	C₂²	#22575524	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	538	2	no	—
89	C₄	—	538	2	no	—
90	C₄	#22575513	1	1	yes	characteristic
91	C₄	#22575514	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	2	1	no	—
94	C₂	#31771473	1	1	yes	characteristic
95	C₂	#31771481	1	1	yes	characteristic
96	C₂	#31771485	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#11761640)⋊C₄
commutator	C₂×C₄×C₂₆₉	C₄²
Frattini	C₂²×C₄	C₂×#11761640
Fitting	C₂²×C₈×C₂₆₉	C₄
socle	C₂²×C₂₆₉	#95
prime radical	C₂³×C₂₆₉	C₄⋊C₄
cosocle	C₂×C₄×Dic₂₆₉	C₂²
hypercenter	C₂²×C₈	#11761640
hypocenter	C₂₆₉	#1290
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
48. C₂₆₉	#1290	semidirect

Maximal subgroups

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

subgroup	index
#31771626	2
C₂×#22575382	2
#1290	269