Finite Group Databasebeta

Subgroups — #46813064

Order 34432.

150 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31770838	C₂	1	1	yes	maximal, characteristic
2	#107×D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#31770431	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770430	C₂	1	1	yes	maximal, characteristic, complemented
5	#31770406	C₂	1	1	yes	maximal, characteristic, complemented
6	#31770384	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₂₆₉	C₂²	1	1	yes	characteristic
9	#22575424	—	2	1	no	—
10	C₄⋊C₄×D₂₆₉	—	2	1	no	—
11	C₂×#16864377	C₂²	1	1	yes	characteristic
12	#22575387	C₂²	1	1	yes	characteristic
13	#22575387	C₂²	1	1	yes	characteristic
14	#22575386	—	2	1	no	—
15	#22575374	—	2	1	no	—
16	C₂×D₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
18	#107×C₂₆₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊D₂₆₉	—	4	1	no	—
20	C₂×C₄×D₂₆₉	—	4	1	no	—
21	C₂×C₄×D₂₆₉	D₄	1	1	yes	characteristic
22	C₂×#14524130	D₄	1	1	yes	characteristic
23	C₈×D₂₆₉	—	2	1	no	—
24	C₈×D₂₆₉	—	2	1	no	—
25	C₄⋊Dic₂₆₉	—	2	1	no	—
26	C₄⋊Dic₂₆₉	—	2	1	no	—
27	#16864381	—	4	1	no	—
28	C₂²⋊Dic₂₆₉	—	4	1	no	—
29	#16864377	—	2	1	no	—
30	C₂×D₄×C₂₆₉	—	2	1	no	—
31	D₈×C₂₆₉	—	2	1	no	—
32	D₈×C₂₆₉	—	2	1	no	—
33	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₂₆₉	—	2	1	no	—
35	C₁₆×C₂₆₉	—	2	1	no	—
36	C₂²×D₂₆₉	D₈	1	1	yes	characteristic
37	C₂²⋊D₂₆₉	—	8	2	no	—
38	C₂×Dic₂₆₉	—	4	1	no	—
39	C₂×Dic₂₆₉	—	4	1	no	—
40	C₂×Dic₂₆₉	D₈	1	1	yes	characteristic
41	C₄×D₂₆₉	—	2	1	no	—
42	C₄×D₂₆₉	—	2	1	no	—
43	C₄×D₂₆₉	—	8	2	no	—
44	#14524130	—	2	1	no	—
45	C₂³×C₂₆₉	—	4	1	no	—
46	D₄×C₂₆₉	—	4	1	no	—
47	D₄×C₂₆₉	—	4	1	no	—
48	C₂×C₄×C₂₆₉	—	4	1	no	—
49	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
50	C₈×C₂₆₉	#41	1	1	yes	characteristic
51	C₈×C₂₆₉	#41	1	1	yes	characteristic
52	C₂²×C₂₆₉	—	8	1	no	—
53	C₂²×C₂₆₉	—	8	1	no	—
54	C₂²×C₂₆₉	—	8	1	no	—
55	C₂²×C₂₆₉	C₂×D₈	1	1	yes	characteristic
56	C₂×D₂₆₉	—	2	1	no	—
57	C₂×D₂₆₉	—	2	1	no	—
58	C₂×D₂₆₉	D₁₆	1	2	yes	—
59	C₄×C₂₆₉	—	8	1	no	—
60	C₄×C₂₆₉	#123	1	1	yes	characteristic
61	C₄×C₂₆₉	#137	1	1	yes	characteristic
62	Dic₂₆₉	—	2	1	no	—
63	Dic₂₆₉	—	8	1	no	—
64	Dic₂₆₉	—	8	1	no	—
65	C₂×C₂₆₉	—	16	1	no	—
66	C₂×C₂₆₉	#480	1	1	yes	characteristic
67	C₂×C₂₆₉	C₂×D₁₆	1	1	yes	characteristic
68	C₂×C₂₆₉	#510	1	1	yes	characteristic
69	D₂₆₉	—	2	2	no	—
70	C₂₆₉	#2232	1	1	yes	characteristic, complemented, Sylow
71	#2232	—	269	1	no	maximal, Sylow
72	#466	—	269	1	no	—
73	C₂×#107	—	269	1	no	—
74	#366	—	269	1	no	—
75	#366	—	269	1	no	—
76	#357	—	269	1	no	—
77	#357	D₂₆₉	1	1	yes	characteristic, complemented
78	C₂²⋊C₁₆	—	269	1	no	—
79	C₂×D₈	C₂×D₂₆₉	1	1	yes	characteristic
80	C₂²×C₈	—	269	1	no	—
81	#121	—	538	1	no	—
82	C₂×C₄⋊C₄	—	538	1	no	—
83	C₂×C₁₆	—	269	1	no	—
84	C₂×C₁₆	C₂×D₂₆₉	1	1	yes	characteristic
85	#107	—	269	1	no	—
86	#107	—	269	1	no	—
87	#107	—	538	1	no	—
88	#107	C₂×D₂₆₉	1	1	yes	characteristic
89	#102	—	538	1	no	—
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	—	1076	1	no	—
92	C₂²×C₄	—	269	1	no	—
93	C₂²×C₄	—	1076	1	no	—
94	D₈	—	2	1	no	—
95	D₈	—	2	1	no	—
96	C₂×C₈	—	269	1	no	—
97	C₂×C₈	—	538	1	no	—
98	C₂×C₈	—	538	1	no	—
99	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	538	1	no	—
102	C₄⋊C₄	—	538	1	no	—
103	C₄⋊C₄	—	1076	1	no	—
104	C₂²⋊C₄	—	1076	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	538	1	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	269	1	no	—
109	D₄	—	4	1	no	—
110	D₄	—	4	1	no	—
111	D₄	—	2152	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	269	1	no	—
114	C₂×C₄	—	538	1	no	—
115	C₂×C₄	—	538	1	no	—
116	C₂×C₄	—	1076	1	no	—
117	C₂×C₄	—	1076	1	no	—
118	C₂×C₄	—	2152	2	no	—
119	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
120	C₈	—	538	1	no	—
121	C₈	#16864401	1	1	yes	characteristic
122	C₈	#16864402	1	1	yes	characteristic
123	C₂²	—	8	1	no	—
124	C₂²	—	8	1	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	269	2	no	—
127	C₂²	—	538	1	no	—
128	C₂²	—	538	1	no	—
129	C₂²	D₈×D₂₆₉	1	1	yes	characteristic
130	C₄	—	8	1	no	—
131	C₄	—	538	1	no	—
132	C₄	—	2152	1	no	—
133	C₄	—	2152	1	no	—
134	C₄	#22575435	1	1	yes	characteristic
135	C₄	#22575482	1	1	yes	characteristic
136	C₂	—	16	1	no	—
137	C₂	—	538	2	no	—
138	C₂	#31770917	1	1	yes	characteristic
139	C₂	D₁₆×D₂₆₉	1	1	yes	characteristic
140	C₂	#31771044	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₆₉
commutator, cosocle	C₂×C₈×C₂₆₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₆₉
Fitting	#357×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×D₈
prime radical	#31770838	C₂
hypercenter	#357	D₂₆₉
hypocenter	C₂₆₉	#2232
LCS	G ▷ C₂×C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

Complements

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

N	H ≅ G/N	type
2. #107×D₂₆₉	C₂	semidirect
3. #31770431	C₂	semidirect
4. #31770430	C₂	semidirect
5. #31770406	C₂	semidirect
6. #31770384	C₂	semidirect
7. #357×C₂₆₉	C₂	semidirect
70. C₂₆₉	#2232	semidirect
77. #357	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31770838	2
#107×D₂₆₉	2
#31770431	2
#31770430	2
#31770406	2
#31770384	2
#357×C₂₆₉	2
#2232	269